Research ArticleFree Vibration and Hardening Behavior ofTruss Core Sandwich Beam
J E Chen1 W Zhang2 M Sun3 and M H Yao2
1Tianjin Key Laboratory of the Design and Intelligent Control of the Advanced Mechatronical SystemTianjin University of Technology Tianjin 300384 China2College of Mechanical Engineering Beijing University of Technology Beijing 100124 China3School of Science Tianjin Chengjian University Tianjin 300384 China
Correspondence should be addressed to M Sun sunmin0537163com
Received 1 October 2015 Revised 19 February 2016 Accepted 6 March 2016
Academic Editor Emiliano Mucchi
Copyright copy 2016 J E Chen et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The dynamic characteristics of simply supported pyramidal truss core sandwich beam are investigated The nonlinear governingequation of motion for the beam is obtained by using a Zig-Zag theory The averaged equations of the beam with primarysubharmonic and superharmonic resonances are derived by using the method of multiple scales and then the correspondingfrequency response equations are obtained The influences of strut radius and core height on the linear natural frequencies andhardening behaviors of the beam are studied It is illustrated that the first-order natural frequency decreases continuously and thesecond-order and third-order natural frequencies initially increase and then decrease with the increase of strut radius and the firstthree natural frequencies all increase with the rise of the core height Furthermore the results indicate that the hardening behaviorsof the beam become weaker with the increase of the rise of strut radius and core heightThemechanisms of variations in hardeningbehavior of the sandwich beam with the three types of resonances are detailed and discussed
1 Introduction
Truss core sandwich structures are drawing more and moreattention from researchers due to their excellent mechanicalproperties andmany functional applications Cellularmateri-als have always played a significant role in many engineeringfields As a new member of the family of cellular materialstruss-like material has broad application prospects Evanset al [1] compared the integrated performances of the differ-ent types of cellular materials the advantages of mechanicalcharacteristics andmultifunction of the lattice materials werehighlighted Deshpande and Fleck [2] investigated the col-lapse responses of the sandwich beams with truss coresTheyrevealed that truss core sandwich beams were significantlylighter than the sandwich beams with metallic foam coreagainst the design constraint of collapse load Wallach andGibson [3] analyzed the elastic moduli uniaxial compressivestrengths and shear strength of three-dimensional trussstructures It was concluded that the truss material hadimproved properties over the closed-cell aluminum foam
Next the manufacturing techniques mechanical behaviorsand functionalities of the sandwich structures with differenttruss cores were paid close attention to by researchers fromdifferent research fields [4ndash9]
Unfortunately the investigations on vibration responsesof the truss core sandwich structures are extremely limitedLou et al [10] studied the free vibration of the simplysupported truss core sandwich beams the natural frequenciesof the sandwich beams were obtained and the theoreticalresults agree well with the numerical results Xu and Qiu[11] investigated the free vibration of the truss core sandwichbeams with interval parameters The effects of geometric andmaterial parameters on the natural frequencies were ana-lyzed Lou et al [12] analyzed the effects of local damage onvibration characteristics of the truss core sandwich structuresby numerical and experimental methods Li and Lyu [13]investigated the active vibration control of lattice sandwichbeams with pyramidal lattice core using the piezoelectricactuatorsensor pairs Song and Li [14] studied the nonlinearaeroelastic characteristics and the active flutter control of
Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 7348518 13 pageshttpdxdoiorg10115520167348518
2 Shock and Vibration
sandwich beams The face sheets and piezoelectric materialwere modeled by the Euler-Bernoulli beam theory and thecore was modeled by the Reddy third-order shear deforma-tion theory
Moreover the vibration behaviors of the truss coresandwich structures can be deduced with the help of a largenumber of dynamical studies on the sandwich structureswith honeycomb core foam core and other cores FrostigandThomsen [15] used high-order sandwich panel theory toanalyze the free vibration of sandwich panels with a flexiblecore Biglari and Jafari [16] studied the free vibrations ofdoubly curved sandwich panels based on a refined three-layered theory Jam et al [17] investigated the free vibrationof sandwich panels by using an improved high-order theoryIn several papers like the aforementioned three referencesthe three-dimensional elasticity theory was used for the corewhich has a better accuracy to predict the response of thethick sandwich structures
For thin and some moderately thick sandwich struc-tures the equivalent single layer theory and the layerwisetheory were widely used E Nilsson and A C Nilsson [18]investigated the dynamic properties of sandwich structureswith honeycomb and foam cores Li and Zhu [19] studiedthe free vibration of honeycomb sandwich plates by usingthe improved Reddy third-order shear deformation platetheory J H Zhang andW Zhang [20] investigated the globalbifurcations and multipulse chaotic dynamics of a simplysupported honeycomb sandwich rectangular plate by usingthe extended Melnikov method Yang et al [21] investigatedthe transverse vibrations and stability of an axially movingsandwich beam Sahoo and Singh [22] analyzed the bucklingand free vibration analysis of the laminated composite andsandwich plates by using a new trigonometric Zig-Zag theoryKhare et al [23] presented the free vibration analysis ofcomposite and sandwich laminates by using a high-ordershear deformation theory Sakiyama et al [24] studied the freevibration of a three-layer sandwich beam with an elastic orviscoelastic core and arbitrary boundary conditions Banerjee[25] analyzed the free vibration of three-layered symmetricsandwich beams by using the dynamic stiffness methodCetkovic and Vuksanovic [26] studied the bending freevibrations and buckling of the laminated composite andsandwich plates by using the generalized layerwise theoryof Reddy Ferreira [27] analyzed the mechanical behaviorsof the laminated composite and sandwich plates by using alayerwise shear deformation theory and the multiquadricsdiscretization method
Although the studies on the vibrations of the sandwichstructures are relatively detailed the unique dynamic charac-teristics especially the nonlinear behaviors of the sandwichstructures with truss core are still required to be exploredThe main aim of this paper is to particularly study the linearand nonlinear dynamic behaviors of the truss core sandwichbeam In this paper the nonlinear governing equation ofmotion for the simply supported truss core sandwich beamwith immovable edges is obtained by using the von Karmantype equation Hamiltonrsquos principle and a Zig-Zag theoryThe influences of strut radius and core height on the linear
r
a
120579
Figure 1 Sketches of a unit cell of the sandwich beam withpyramidal truss core
natural frequencies and hardening behaviors of the sandwichbeam are investigated The nonlinear frequency responsesof the sandwich beam with primary subharmonic andsuperharmonic resonances are obtained and the numericalresults are discussed
2 Formulation
The simply supported sandwich beam with pyramidal trusscore subjected to a transverse uniform load is considered inthe paper A unit cell of the pyramidal truss core is shown inFigure 1The length radius and inclination angle of the trussare represented by 119886 119903 and 120579 respectively Thus the relativedensity 120588 and the equivalent shear modulus 119866 are expressedas follows [2]
120588 =
120588119888
120588
=
2120587
cos2120579 sin 120579(
119903
119886
)
2
(1a)
119866 = 120587 sin 120579 ( 119903119886
)
2
119864 (1b)
where120588119888is the equivalent density of the truss core and120588 and119864
are the density and elasticity modulus of the mother materialof the truss
The truss core sandwich beam can be modeled as alaminated beam which is composed of two solid layers andan equivalent continuum core layer Generally for a sandwichstructure the following assumptions [28] can be given
(1) The thickness of the truss core sandwich beamremains constant during deformation
(2) Only bending deformation is considered for both thinface sheets and only shear deformation is consideredfor the core
(3) The face sheets and truss core are combined closelyThus the deflections between each layer are continu-ous
These assumptions are widely used to study the sandwichstructures which have thin face sheets and thick core Based
Shock and Vibration 3
z
hhc x
w0
u0
120601x
120597w0
120597x
Figure 2 Undeformed and deformed geometries of the truss coresandwich beam
on undeformed and deformed geometries of the truss coresandwich beam as shown in Figure 2 the displacement fieldcan be expressed in terms of the midplane displacements 119906
0
and1199080and rotations 120601
119909 the face sheets of the sandwich beam
adopt the Kirchhoff hypothesis and the truss core adopts thefirst-order shear deformation theory Hence
minus
ℎ
2
le 119911 le minus
ℎ119888
2
119906119905= 1199060minus 119911
1205971199080
120597119909
+
ℎ119888
2
(120601119909minus
1205971199080
120597119909
)
119908119905= 1199080
(2a)
minus
ℎ119888
2
le 119911 le
ℎ119888
2
119906119888= 1199060minus 119911120601119909
119908119888= 1199080
(2b)
ℎ119888
2
le 119911 le
ℎ
2
119906119887= 1199060minus 119911
1205971199080
120597119909
minus
ℎ119888
2
(120601119909minus
1205971199080
120597119909
)
119908119887= 1199080
(2c)
where the subscripts 119905 119888 and 119887 represent the top face sheetthe truss core and the bottom face sheet respectively ℎrepresents the total thickness of the truss core sandwich beamand ℎ119888represents the height of the truss core
Using von Karmanrsquos theory for the geometric nonlinear-ity the strain-displacement relations have the form
1205761199091199091=
120597119906119905
120597119909
+
1
2
(
120597119908119905
120597119909
)
2
120574119909119911=
120597119906119888
120597119911
+
120597119908119888
120597119909
1205761199091199093=
120597119906119887
120597119909
+
1
2
(
120597119908119887
120597119909
)
2
(3)
where 1205761199091199091
and 1205761199091199093
represent the strains of the face sheets and120574119909119911
represents the strain of the truss coreSubstituting (2a) (2b) and (2c) into (3) we have
1205761199091199091
1205761199091199093
120574119909119911
=
120576(0)
1199091199091
120576(0)
1199091199093
120574(0)
119909119911
+ 119911
120576(1)
1199091199091
120576(1)
1199091199093
120574(1)
119909119911
120576(0)
1199091199091
120576(0)
1199091199093
120574(0)
119909119911
=
1205971199060
120597119909
+
ℎ119888
2
(
120597120601119909
120597119909
minus
1205972
1199080
1205971199092) +
1
2
(
1205971199080
120597119909
)
2
1205971199060
120597119909
minus
ℎ119888
2
(
120597120601119909
120597119909
minus
1205972
1199080
1205971199092) +
1
2
(
1205971199080
120597119909
)
2
minus120601119909+
1205971199080
120597119909
120576(1)
1199091199091
120576(1)
1199091199093
120574(1)
119909119911
=
minus
1205972
1199080
1205971199092
minus
1205972
1199080
1205971199092
0
(4)
The mathematical statement of Hamiltonrsquos principle isgiven as
int
1199052
1199051
(120575119880 + 120575119882 minus 120575119870) 119889119905 = 0 (5)
The virtual kinetic energy 120575119870 virtual strain energy 120575119880and the virtual work 120575119882 are respectively given by
120575119880 = 1205751198801+ 1205751198802+ 1205751198803
120575119870 = 1205751198701+ 1205751198702+ 1205751198703
120575119882 = minusint
119860
1198651205751199080minus 12058301205751199080119889119909 119889119910
(6a)
120575119880 = int
119897
0
int
minusℎ1198882
minusℎ2
12059011990911990911205751205761199091199091119889119909 119889119911
+ int
119897
0
int
ℎ1198882
minusℎ1198882
120590119909119911120575120574119909119911119889119909 119889119911 + int
119897
0
int
ℎ2
ℎ1198882
12059011990911990931205751205761199091199093119889119909 119889119911
= int
119897
0
int
minusℎ1198882
minusℎ2
1205901199091199091(120575120576(0)
1199091199091+ 119911120575120576(1)
1199091199091) 119889119909 119889119911
+ int
119897
0
int
ℎ1198882
minusℎ1198882
120590119909119911120575120574119909119911119889119909 119889119911 + int
119897
0
int
ℎ2
ℎ1198882
1205901199091199093(120575120576(0)
1199091199093
+ 119911120575120576(1)
1199091199093) 119889119909 119889119911 = int
119897
0
1198731199091199091120575120576(0)
1199091199091
+1198721199091199091120575120576(1)
1199091199091119889119909 + int
119897
0
int
ℎ1198882
minusℎ1198882
119876119909120575120574119909119911119889119909 + int
119897
0
1198731199091199093
sdot 120575120576(0)
1199091199093+1198721199091199093120575120576(1)
1199091199093119889119909 119889119911
(6b)
4 Shock and Vibration
120575119870 = int
119897
0
int
minusℎ1198882
minusℎ2
120588 [(0minus 119911
1205970
120597119909
+
ℎ119888
2
120601119909minus
ℎ119888
2
1205970
120597119909
)
sdot (1205750minus 119911
1205971205750
120597119909
+
ℎ119888
2
120575120601119909minus
ℎ119888
2
1205971205750
120597119909
)
+ 01205750] 119889119909 119889119911
+ int
119897
0
int
ℎ1198882
minusℎ1198882
120588119888[(0minus 119911
120601119909) (1205750minus 119911120575
120601119909)
+ 01205750] 119889119909 119889119911
+ int
119897
0
int
ℎ2
ℎ1198882
120588 [(0minus 119911
1205970
120597119909
minus
ℎ119888
2
120601119909+
ℎ119888
2
1205970
120597119909
)
sdot (1205750minus 119911
1205971205750
120597119909
minus
ℎ119888
2
120575120601119909+
ℎ119888
2
1205971205750
120597119909
)
+ 01205750] 119889119909 119889119911
(6c)
where the superposed dot on a variable indicates its firstderivative with respect to time
0= 1205971199060120597119905 and so on
Substituting (6a) (6b) and (6c) into (5) the nonlineargoverning equations of motion for the truss core sandwichbeam are obtained
(1198731199091199091+ 1198731199091199093)119909= (211986801+ 11986802) 0minus 11986812
120601119909 (7a)
ℎ119888
2
(1198731199091199091minus 1198731199091199093)119909+ 119876119909= (
ℎ2
119888
2
11986801+ 11986822)120601119909
minus (
ℎ2
119888
2
11986801+ ℎ11988811986811)
1205970
120597119909
minus 119868120
(7b)
(1198731199091199091
1205971199080
120597119909
+ 1198731199091199093
1205971199080
120597119909
)
119909
+
ℎ119888
2
(1198731199091199091minus 1198731199091199093)119909119909
+ (1198721199091199091+1198721199091199093)119909119909+ 119876119909119909+ 119865
minus 1205830= (
ℎ2
119888
2
11986801+ ℎ11988811986811)
120597120601119909
120597119909
minus (
ℎ2
119888
2
11986801+ 2ℎ11988811986811+ 211986821)
1205972
0
1205971199092
+ (211986801+ 11986802) 0
(7c)
where
11986801
11986811
11986821
= int
minusℎ1198882
minusℎ2
1
119911
1199112
120588119889119911
11986802
11986812
11986822
= int
ℎ1198882
minusℎ1198882
1
119911
1199112
120588119888119889119911
(8a)
1198731199091199091= int
minusℎ1198882
minusℎ2
1205901199091199091119889119911
1198731199091199093= int
ℎ2
ℎ1198882
1205901199091199093119889119911
119876119909= int
ℎ1198882
minusℎ1198882
120590119909119911119889119911
1198721199091199091= int
minusℎ1198882
minusℎ2
1199111205901199091199091119889119911
1198721199091199093= int
ℎ2
ℎ1198882
1199111205901199091199093119889119911
(8b)
We consider the face sheets and the truss core aremade ofthe same isotropy material The stress-strain relations of thetruss core sandwich beam can be represented by
1205901199091199091= 1198641205761199091199091
1205901199091199093= 1198641205761199091199093
120590119909119911= 119866120574119909119911
(9)
The stress-displacement relations can be expressed as
1198731199091199091= 1198601120576(0)
1199091199091+ 1198611120576(1)
1199091199091
1198731199091199093= 1198603120576(0)
1199091199093+ 1198613120576(1)
1199091199093
1198721199091199091= 1198611120576(0)
1199091199091+ 1198631120576(1)
1199091199091
1198721199091199093= 1198613120576(0)
1199091199093+ 1198633120576(1)
1199091199093
119876119909= 119862120574(0)
119909119911
(10a)
where
(1198601 1198611 1198631) = int
minusℎ1198882
minusℎ2
119864 (1 119911 1199112
) 119889119911
(1198603 1198613 1198633) = int
ℎ2
ℎ1198882
119864 (1 119911 1199112
) 119889119911
119862 = int
ℎ1198882
minusℎ1198882
119866119889119911
(10b)
1198601and 119860
3are called extensional stiffness 119861
1and 119861
3are
called bending-extensional stiffness 1198631and 119863
3are called
bending stiffness 119862 is called shear stiffness and 119868119894119895are the
mass moments of inertiaConsidering the symmetry of the two face sheets of the
truss core sandwich beam about the 119909-axis we have
1198601= 1198603
1198611= minus1198613
1198631= 1198633
(11)
Shock and Vibration 5
Using the stress-displacement relations (7a)ndash(7c) can beexpressed in terms of generalized displacements
21198601
1205972
1199060
1205971199092+ 21198601
1205971199080
120597119909
1205972
1199080
1205971199092= (211986801+ 11986802) 0minus 11986812
120601119909 (12a)
ℎ2
119888
2
1198601
1205972
120601119909
1205971199092minus
ℎ2
119888
2
1198601
1205973
1199080
1205971199093minus ℎ1198881198611
1205973
1199080
1205971199093minus 119862120601119909
+ 119862
1205971199080
120597119909
= (
ℎ2
119888
2
11986801+ 11986822)120601119909
minus (
ℎ2
119888
2
11986801+ ℎ11988811986811)
1205970
120597119909
minus 119868120
(12b)
(
ℎ2
119888
2
1198601+ 1198611ℎ119888)
1205973
120601119909
1205971199093
+ (minus
ℎ2
119888
2
1198601minus 21198611ℎ119888minus 21198631)
1205974
1199080
1205971199094minus 119862
120597120601119909
120597119909
+ 119862
1205972
1199080
1205971199092+ 31198601(
1205971199080
120597119909
)
2
1205972
1199080
1205971199092+ 21198601
1205972
1199060
1205971199092
1205971199080
120597119909
+ 21198601
1205971199060
120597119909
1205972
1199080
1205971199092+ 119865
minus 1205830= (
ℎ2
119888
2
11986801+ ℎ11988811986811)
120597120601119909
120597119909
minus (
ℎ2
119888
2
11986801+ 2ℎ11988811986811+ 211986821)
1205972
0
1205971199092
+ (211986801+ 11986802) 0
(12c)
3 Validation
In order to check the accuracy of the presented model thenatural frequencies are calculated and compared with theresults presented by [10] For linear vibration the in-planemotion of the structure can be neglected Therefore thesimply supported boundary conditions of the sandwich beamcan be expressed as
1199080=
120597120601119909
120597119909
= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (13)
The displacements 120601119909and 119908
0which satisfy (13) can be
represented as
120601119909=
119873
sum
119899=1
120601119899(119905) cos119899120587119909
119897
(14a)
1199080=
119873
sum
119899=1
119908119899(119905) sin119899120587119909
119897
(14b)
We only consider the transverse vibration of the trusscore sandwich beam thus we can neglect the rotatory inertiaterms Substituting (14a) and (14b) into the linear form of
Table 1 Comparison of natural frequencies of the sandwich beamwith the first group of parameters
Mode Ahmed Hwu Lou Present1 56 58 57 57142 mdash 221 219 219583 451 459 465 465174 mdash 742 767 768185 1073 1048 1105 110669
Table 2 Comparison of natural frequencies of the sandwich beamwith the second group of parameters
Mode Lou Present1 13774 137812 52309 523953 109138 109411
(12a) (12b) and (12c) and applying the Galerkin method thenatural frequencies are obtained as shown in Tables 1 and2 The good agreement between the present and publishedresults validates the accuracy of the present method
The dimensions and material properties in Table 1 are
119864119891= 689GPa
ℎ119891= 04572mm
120588119891= 2680 kgm3
119866119888= 008268GPa
ℎ119888= 127mm
120588119888= 328 kgm3
119897 = 09144m
(15)
The dimensions and material properties in Table 2 are
119864119891= 210GPa
ℎ119891= 1mm
120588119891= 7930 kgm3
119866119888= 103667GPa
ℎ119888= 15mm
120588119888= 3131739 kgm3
119897 = 06364m
(16)
4 Nonlinear Frequency Response Equation
In the following analysis the nonlinear vibrations of the trusscore sandwich beam with simply supported boundary con-ditions are considered For nonlinear vibration the simply
6 Shock and Vibration
supported boundary conditions with immoveable edges ofthe beam can be expressed as
1199060= 1199080=
120597120601119909
120597119909
= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (17)
The transformations of variables and parameters areintroduced to obtain the dimensionless equation
1199060=
1199060
119897
1199080=
1199080
ℎ
120601119909= 120601119909
119909 =
119909
119897
119865 =
1198974
119864ℎ4119865
120583 =
119897
ℎ
(
1
120588119864
)
12
120583
119905 =
ℎ
1198972(
119864
120588
)
12
119905
Ω =
1198972
ℎ
(
120588
119864
)
12
Ω
ℎ119888=
ℎ119888
ℎ
119860119894=
119897
119864ℎ2119860119894
119861119894=
119897
119864ℎ3119861119894
119862 =
119897
119864ℎ2119862
119863119894=
119897
119864ℎ4119863119894
119868119905119894=
1
119897(119905+1)120588
119868119905119894 119905 = 0 1 2
(18)
Therefore the dimensionless equations of motion for thetruss core sandwich beam are obtained For convenience ofour study we drop the overbar in the following analysisThe displacements 119906
0 120601119909 and 119908
0which satisfy (17) can be
represented as
1199060=
119873
sum
119899=1
119906 (119905) sin (2119899120587119909) (19a)
120601119909=
119873
sum
119899=1
120601 (119905) cos (119899120587119909) (19b)
1199080=
119873
sum
119899=1
119908 (119905) sin (119899120587119909) (19c)
In addition the excitation has the form
119865 =
119873
sum
119899=1
119891 (119905) sin (119899120587119909) (19d)
Next theGalerkinmethod is used to obtain the nonlinearordinary differential equations of motion for the sandwichbeam The transverse vibrations of the sandwich beam aremainly considered In the same way we can neglect thelongitude and rotatory inertia terms Substituting the dis-placements 119906
0 120601119909 and 119908
0into the resulting equations the
ordinary differential equation is derived
119899+ 120583119899119899+ 1205962
119899119908119899+ 1205721198991199083
119899+ 119892119899(1199081 1199082 119908
119873)
= 120573119899119891119899cos (Ω
119899119905) 119899 = 1 2 119873
(20)
where 119892119899(1199081 1199082 119908
119873) represents the coupled relation
between the different modes and the coupled term does notappear when the single mode response is considered Thecoefficients in (20) are presented in the Appendix
The following scales transformations must be introducedfor obtaining a system which is suitable for the application ofthe method of multiple scales and studying the resonances ofthe sandwich beam
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
120573119899997888rarr 120576120573
119899
(21)
where 120576 is a small dimensionless parameterBecause the coupled term 119892
119899(1199081 1199082 119908
119873) has no
contribution to the primary resonance of each mode intheoretical analysis therefore we can neglect this term andobtain the following equation
119899+ 120576120583119899119899+ 1205962
119899119908119899+ 1205761205721198991199083
119899= 120576120573119899119891119899cos (Ω
119899119905) (22)
The method of multiple scales [29] is used to obtain theuniform solution of (22) in the following form
119908119899(119905 120576) = 119909
1198990(1198790 1198791) + 120576119909
1198991(1198790 1198791) + sdot sdot sdot (23)
where 1198790= 119905 and 119879
1= 120576119905
Then we have the differential operators
119889
119889119905
=
120597
1205971198790
1205971198790
120597119905
+
120597
1205971198791
1205971198791
120597119905
+ sdot sdot sdot = 1198630+ 1205761198631+ sdot sdot sdot (24a)
1198892
1198891199052= (1198630+ 1205761198631+ sdot sdot sdot)
2
= 1198632
0+ 2120576119863
01198631+ sdot sdot sdot (24b)
where1198630= 120597120597119879
0and119863
1= 120597120597119879
1
We have the following relation
Ω119899= 120596119899+ 120576120590119899 (25)
120590119899
is the detuning parameter which quantitativelydescribes the nearness of excitation frequency to the linearnatural frequency
Substituting (23)ndash(25) into (22) and balancing the coef-ficients of the like power of 120576 yield the following differentialequations
Shock and Vibration 7
1205760 order
1198632
01199091198990+ 1205962
1198991199091198990= 0 (26a)
1205761 order
1198632
01199091198991+ 1205962
1198991199091198991= minus2119863
011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093
1198990
+ 120573119899119891119899cos (120596
1198991198790+ 1205901198991198791)
(26b)
The solution of (26a) in the complex form can beexpressed as
1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)
where 119860119899is the complex conjugate of 119860
119899
Substituting (27) into (26b) yields
1198632
01199091198991+ 1205962
1198991199091198991= (minus2119894120596
1198991198631119860119899minus 120583119899119894120596119899119860119899
minus 31205721198991198602
119899119860119899+
120573119899119891119899
2
1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST
(28)
where symbol cc denotes the parts of the complex conjugateof function on the right-hand side of (28) andNST representsthe terms that do not produce secular terms
Function 119860119899may be denoted in the complex form
119860119899=
1
2
119886119899119890119894120593119899 (29)
Substituting (29) into simplified (28) and separating thereal and imaginary parts from the resulting equations thetwo-dimensional averaged equation in the polar form isobtained as
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin (120590
1198991198791minus 120593119899)
2120596119899
(30a)
119886119899119899=
31205721198991198863
119899
8120596119899
minus
120573119899119891119899cos (120590
1198991198791minus 120593119899)
2120596119899
(30b)
Letting
120574119899= 1205901198991198791minus 120593119899 (31)
we have
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin 120574119899
2120596119899
(32a)
119886119899119899= 119886119899120590119899minus
31205721198991198863
119899
8120596119899
+
120573119899119891119899cos 120574119899
2120596119899
(32b)
Steady-state motions occur when 119899and
119899equal zero
which correspond to the singular points of averaged (32a) and(32b) Thus 119886
119899and 120574119899are constants Eliminating 120574
119899by using
the relations between trigonometric functions we obtain thefrequency response equation of the truss core sandwich beamwith primary resonance as follows
(
120583119899119886119899
2
)
2
+ (120590119899119886119899minus
31205721198991198863
119899
8120596119899
)
2
= (
120573119899119891119899
2120596119899
)
2
(33)
Subsequently the subharmonic resonance and the super-harmonic resonance of the sandwich beams are consideredIn the two cases the following two transformations arerequired
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
(34)
For subharmonic resonance the detuning parameter 120590119899
is introduced according toΩ119899= 3120596119899+ 120576120590119899 (35)
Similarly the averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
31205721198991205731198991198862
119899Λ sin 120574
119899
4120596119899
(36a)
119886119899119899= 119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
minus
9120572119899120573119899Λ1198862
119899cos 120574119899
4120596119899
(36b)
where Λ = (12)119891119899(1205962
119899minus Ω2
119899)minus1
The frequency response equation of the truss core sand-wich beam with subharmonic resonance is
(minus
120583119899119886119899
2
)
2
+
1
9
[119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
]
2
=
91205722
1198991205732
1198991198864
119899Λ2
161205962
119899
(37)
For superharmonic resonance the detuning parameter120590119899
is introduced according to3Ω119899= 120596119899+ 120576120590119899 (38)
The averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
1205721198991205733
119899Λ3 sin 120574
119899
120596119899
(39a)
119886119899119899= 119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
minus
1205721198991205733
119899Λ3 cos 120574
119899
120596119899
(39b)
The frequency response equation of the truss core sand-wich beam with superharmonic resonance is
(minus
120583119899119886119899
2
)
2
+ [119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
]
2
=
1205722
1198991205736
119899Λ6
1205962
119899
(40)
The plot of 119886119899as a function of 120590
119899for given 119891 is called
a nonlinear frequency response curve which can be usedto describe the hardening behavior of a structure In thefollowing analysis the parameters in (33) (37) and (40)should be calculated by using the Appendix
8 Shock and Vibration
120596
32
28
24
20
16
12
8
4
120588c120588
000 002 004 006 008 010
Figure 3 Variations of dimensionless linear natural frequencieswith respect to the strut radius Solid line first-order frequencydash line second-order frequency dot-dash line third-order fre-quency
5 Numerical Results
In the study the influences of the structural parameterson the dimensionless linear natural frequencies of the trusscore sandwich beam are studied at first by neglecting thenonlinear terms in (20) The face sheets and the truss of thesandwich beam are made of the same material The materialproperties are taken as follows 119864 = 689GPa and 120588 =
2680 kgm3 Figure 3 indicates that the first-order naturalfrequency decreases continuously and the second-order andthird-order natural frequencies initially increase and thendecrease with the increase of strut radiusThe relative densitycorresponding to the turning point in the changing curveof the second-order natural frequency is smaller than thatof the third-order natural frequency When the strut radiusincreases the mass moment of inertia and shear stiffness ofthe sandwich beam both increase the changes of the naturalfrequencies depend upon the combined effect of the twofactors The structural parameters in the figure are given asℎ = 0012m ℎ
119888= 001m 120579 = 1205874 and 119897 = 07m
Figure 4 shows that the first three natural frequenciesall increase with the rise of the core height and the higher-order natural frequency increases faster than the lower-ordernatural frequency When the core height increases the massmoment of inertia extensional stiffness bending stiffnessand shear stiffness of the sandwich beam all decrease thechanges of the natural frequencies depend upon the com-bined effect of those factors The structural parameters in thefigure are given as ℎ = 0012m 119903 = 06mm 120579 = 1205874 and119897 = 07m
The steady-state responses of the first three modes ofthe sandwich beam with primary resonance are investigatedThe hardening-type nonlinearity of the first three modes allbecomes weaker with increasing strut radius and core heightas shown in Figures 5 and 6 The bending of the frequencyresponse curve mainly depends on the nonlinear term andthe frequency parameter according to (33) It can be known
120596
32
28
24
20
16
12
8
4
hch
072 076 080 084 088 092
Figure 4 Variations of dimensionless linear natural frequencieswith respect to the core height Solid line first-order frequency dashline second-order frequency dot-dash line third-order frequency
from 120572119899in the Appendix that the nonlinear term is related to
the extensional stiffness of the face sheets as well as the massmoment of inertia thickness and length of the sandwichbeam The theoretical analyses indicate that the reasons forthe weakening of the hardening-type nonlinearity with therise of strut radius and core height are different
At first we discuss the influence of strut radius on thehardening behavior of the sandwich beam With the increaseof the strut radius the hardening behavior of the beam isweakened by the rise of themassmoment of inertia and shearstiffness It is concluded from (33) that the increases of themass moment of inertia and shear stiffness both weaken thehardening behavior The structural parameters in Figure 5are the same as in Figure 3 With the increase of the coreheight the hardening behavior of the beam is weakened It isconcluded that the decrease of extensional stiffness tends toweaken the hardening behavior and the decrease of the massmoment of inertia shear stiffness and bending stiffness tendsto strengthen the hardening behavior of the sandwich beamObviously the effect of the extensional stiffness is dominantaccording to Figure 6The structural parameters in this figureare the same as in Figure 4
Furthermore the hardening-type nonlinearity of thehigher mode is stronger than that of the lower mode andthe influences of the structural parameters on the hardeningbehavior for the lower mode are slightly larger than for thehigher mode In the analysis of the primary resonance of thesandwich beam 120583 = 0001 and 119891 = 1 are always chosen
Subharmonic and superharmonic resonances are rela-tively easy to be generated at the lower modes especially atthe first mode The nonlinear frequency response curves forthe first mode of the sandwich beam with subharmonic andsuperharmonic resonances are investigated in the paper thenumerical results are shown in Figures 7 and 8 respectivelyFigure 7 demonstrates that the subharmonic responses onlyoccur in the frequency ranges which are higher than thelinear natural frequency no matter which and how structuralparameters change Increasing strut radius and core height
Shock and Vibration 9
a1
25
2
15
1
05
1205901
minus10 minus5 0 5 10
(a)a2
10
12
06
04
08
02
1205902
minus10 minus5 0 5 10
(b)
a3
07
08
05
04
06
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 5 Influences of the strut radius on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line 120588 = 002 dash line 120588 = 004 dot-dash line 120588 = 006
will weaken the hardening-type nonlinearity in the resonancecase just as the influences of the structural parameters onthe responses of the beam with primary resonance Figure 8demonstrates that the shapes of the frequency responsecurves for the first mode with superharmonic responses aresimilar to that of the sandwich beamwith primary resonanceThe variations in the structural parameters can not only affectthe hardening behaviors of the beam but also significantlychange the frequency range of the resonance responses
Decreasing strut radius and core height can broaden thefrequency ranges of the resonance responses
Based on the analysis of the frequency response equa-tions we conclude that in the cases of subharmonic andsuperharmonic resonances the mechanism of the variationof the hardening behavior via the structural parameters is thesame as that in the case of primary resonance 120583 = 0001and 119891 = 2 are chosen in Figures 7 and 8 Moreover thestructural parameters in Figures 7(a) and 8(a) are the same
10 Shock and Vibration
a1
14
12
10
08
06
04
02
1205901
minus10 minus5 0 5 10
(a)a2
06
05
04
03
02
01
1205902
minus10 minus5 0 5 10
(b)
a3
04
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 6 Influences of the core height on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash line ℎ
119888ℎ = 09
as in Figure 3 and the structural parameters in Figures 7(b)and 8(b) are the same as in Figure 4
6 Conclusions
The natural frequency and hardening behavior of sandwichbeam with pyramidal truss core are investigated The trusscore is equivalent to a continuous homogeneousmaterial anda Zig-Zag theory is used to derive the nonlinear governingequation of motion for the sandwich beam At first thenatural frequencies are calculated to validate the presentmodel and the changing rules of the natural frequency with
the strut radius and core height are obtained It is illustratedthat the first-order natural frequency decreases continuouslyand the second-order and third-order natural frequenciesinitially increase and then decrease with the increase of strutradius and the first three natural frequencies all increase withthe rise of the core height Then the hardening behaviorsof the sandwich beam are analyzed Numerical simulationsindicate that the effects of the two structural parameters onthe hardening behavior in the cases of primary resonancesubharmonic resonance and superharmonic resonance aresimilar However the mechanisms of the variations of thehardening behavior are very different The influences of
Shock and Vibration 11
a1
15
1
05
0
1205901
minus10 minus5 0 5 10
(a)a1
1205901
minus10 minus5 0 5 10
08
07
06
05
04
03
02
01
0
(b)
Figure 7 The hardening behavior of the first mode with subharmonic resonance (a) The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
1205901
minus4 minus2 0 2 4
a1
16
14
12
10
08
06
04
02
(a)
1205901
minus5 0 5 10 15
a1
14
12
10
08
06
04
02
(b)
Figure 8The hardening behavior of the first mode with superharmonic resonance (a)The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
the structural parameters on the frequency parameter and thenonlinear term of the sandwich beam are studied in the paperfor analyzing the variations of the hardening behavior
Thehardening nonlinearity bends the frequency responsecurve to the right Generally under the same excitationthe stronger the hardening-type nonlinearity the smaller
the response amplitude We investigate the influences ofstrut radius and core height on the dynamical responses ofthe sandwich beam Using the equations obtained in thispaper the influences of the other structural parameters ofthe sandwich beam on the vibrations can be convenientlyanalyzed
12 Shock and Vibration
It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics
Appendix
The relations between the in-plane displacements and trans-verse displacement are
119906119899= 1198781198991199082
119899
119878119899= minus
1198991198861120587
8
(A1)
The relations between the rotation angles and transversedisplacement are
120601119899= 119870119899119908119899
119870119899=
1198991198873120587 minus 1198993
11988721205873
119899211988711205872minus 1
(A2)
The coefficients in (20) are given as follows
120574119899=
1198792119899
1198791119899
1205962
119899= minus
1198793119899
1198791119899
120572119899= minus
1198794119899
1198791119899
120573119899=
1198795119899
1198791119899
1198791119899=
1198889
2
1198792119899=
1198888
2
1198793119899=
1198994
1205874
1198882
2
minus
1198992
1205872
1198884
2
minus
1198991205871198701198991198883
2
+
1198993
1205873
1198701198991198881
2
1198794119899= minus
1198994
1205874
8
+
1198993
1205873
1198781198991198886
2
minus 1198993
1205873
1198781198991198885
1198795119899=
1198887
2
(A3)
where
1198861=
ℎ2
1198972
1198871= minus
ℎ2
ℎ
2
119888
21198972
1198601
119862
1198872=
ℎ3
ℎ
2
119888
21198973
1198601
119862
+
ℎ3
ℎ119888
1198973
1198611
119862
1198873= minus
ℎ
119897
1198881=
119897
3ℎ
(
ℎ119888
2
+ ℎ
1198611
1198601
)
1198882= minus
1
3
(
ℎ
2
119888
2
+ 2ℎ
1198611
1198601
+ 2
1198631
1198601
)
1198883= minus
1198973
3ℎ3
119862
1198601
1198884=
1198972
3ℎ2
119862
1198601
1198885=
21198972
3ℎ2
1198886=
21198972
3ℎ2
1198887=
119897
3ℎ1198601
1198888=
1198972
3ℎ21198601
1198889=
1198972
3ℎ21198601
(211986801+ 11986802)
(A4)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)
References
[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
2 Shock and Vibration
sandwich beams The face sheets and piezoelectric materialwere modeled by the Euler-Bernoulli beam theory and thecore was modeled by the Reddy third-order shear deforma-tion theory
Moreover the vibration behaviors of the truss coresandwich structures can be deduced with the help of a largenumber of dynamical studies on the sandwich structureswith honeycomb core foam core and other cores FrostigandThomsen [15] used high-order sandwich panel theory toanalyze the free vibration of sandwich panels with a flexiblecore Biglari and Jafari [16] studied the free vibrations ofdoubly curved sandwich panels based on a refined three-layered theory Jam et al [17] investigated the free vibrationof sandwich panels by using an improved high-order theoryIn several papers like the aforementioned three referencesthe three-dimensional elasticity theory was used for the corewhich has a better accuracy to predict the response of thethick sandwich structures
For thin and some moderately thick sandwich struc-tures the equivalent single layer theory and the layerwisetheory were widely used E Nilsson and A C Nilsson [18]investigated the dynamic properties of sandwich structureswith honeycomb and foam cores Li and Zhu [19] studiedthe free vibration of honeycomb sandwich plates by usingthe improved Reddy third-order shear deformation platetheory J H Zhang andW Zhang [20] investigated the globalbifurcations and multipulse chaotic dynamics of a simplysupported honeycomb sandwich rectangular plate by usingthe extended Melnikov method Yang et al [21] investigatedthe transverse vibrations and stability of an axially movingsandwich beam Sahoo and Singh [22] analyzed the bucklingand free vibration analysis of the laminated composite andsandwich plates by using a new trigonometric Zig-Zag theoryKhare et al [23] presented the free vibration analysis ofcomposite and sandwich laminates by using a high-ordershear deformation theory Sakiyama et al [24] studied the freevibration of a three-layer sandwich beam with an elastic orviscoelastic core and arbitrary boundary conditions Banerjee[25] analyzed the free vibration of three-layered symmetricsandwich beams by using the dynamic stiffness methodCetkovic and Vuksanovic [26] studied the bending freevibrations and buckling of the laminated composite andsandwich plates by using the generalized layerwise theoryof Reddy Ferreira [27] analyzed the mechanical behaviorsof the laminated composite and sandwich plates by using alayerwise shear deformation theory and the multiquadricsdiscretization method
Although the studies on the vibrations of the sandwichstructures are relatively detailed the unique dynamic charac-teristics especially the nonlinear behaviors of the sandwichstructures with truss core are still required to be exploredThe main aim of this paper is to particularly study the linearand nonlinear dynamic behaviors of the truss core sandwichbeam In this paper the nonlinear governing equation ofmotion for the simply supported truss core sandwich beamwith immovable edges is obtained by using the von Karmantype equation Hamiltonrsquos principle and a Zig-Zag theoryThe influences of strut radius and core height on the linear
r
a
120579
Figure 1 Sketches of a unit cell of the sandwich beam withpyramidal truss core
natural frequencies and hardening behaviors of the sandwichbeam are investigated The nonlinear frequency responsesof the sandwich beam with primary subharmonic andsuperharmonic resonances are obtained and the numericalresults are discussed
2 Formulation
The simply supported sandwich beam with pyramidal trusscore subjected to a transverse uniform load is considered inthe paper A unit cell of the pyramidal truss core is shown inFigure 1The length radius and inclination angle of the trussare represented by 119886 119903 and 120579 respectively Thus the relativedensity 120588 and the equivalent shear modulus 119866 are expressedas follows [2]
120588 =
120588119888
120588
=
2120587
cos2120579 sin 120579(
119903
119886
)
2
(1a)
119866 = 120587 sin 120579 ( 119903119886
)
2
119864 (1b)
where120588119888is the equivalent density of the truss core and120588 and119864
are the density and elasticity modulus of the mother materialof the truss
The truss core sandwich beam can be modeled as alaminated beam which is composed of two solid layers andan equivalent continuum core layer Generally for a sandwichstructure the following assumptions [28] can be given
(1) The thickness of the truss core sandwich beamremains constant during deformation
(2) Only bending deformation is considered for both thinface sheets and only shear deformation is consideredfor the core
(3) The face sheets and truss core are combined closelyThus the deflections between each layer are continu-ous
These assumptions are widely used to study the sandwichstructures which have thin face sheets and thick core Based
Shock and Vibration 3
z
hhc x
w0
u0
120601x
120597w0
120597x
Figure 2 Undeformed and deformed geometries of the truss coresandwich beam
on undeformed and deformed geometries of the truss coresandwich beam as shown in Figure 2 the displacement fieldcan be expressed in terms of the midplane displacements 119906
0
and1199080and rotations 120601
119909 the face sheets of the sandwich beam
adopt the Kirchhoff hypothesis and the truss core adopts thefirst-order shear deformation theory Hence
minus
ℎ
2
le 119911 le minus
ℎ119888
2
119906119905= 1199060minus 119911
1205971199080
120597119909
+
ℎ119888
2
(120601119909minus
1205971199080
120597119909
)
119908119905= 1199080
(2a)
minus
ℎ119888
2
le 119911 le
ℎ119888
2
119906119888= 1199060minus 119911120601119909
119908119888= 1199080
(2b)
ℎ119888
2
le 119911 le
ℎ
2
119906119887= 1199060minus 119911
1205971199080
120597119909
minus
ℎ119888
2
(120601119909minus
1205971199080
120597119909
)
119908119887= 1199080
(2c)
where the subscripts 119905 119888 and 119887 represent the top face sheetthe truss core and the bottom face sheet respectively ℎrepresents the total thickness of the truss core sandwich beamand ℎ119888represents the height of the truss core
Using von Karmanrsquos theory for the geometric nonlinear-ity the strain-displacement relations have the form
1205761199091199091=
120597119906119905
120597119909
+
1
2
(
120597119908119905
120597119909
)
2
120574119909119911=
120597119906119888
120597119911
+
120597119908119888
120597119909
1205761199091199093=
120597119906119887
120597119909
+
1
2
(
120597119908119887
120597119909
)
2
(3)
where 1205761199091199091
and 1205761199091199093
represent the strains of the face sheets and120574119909119911
represents the strain of the truss coreSubstituting (2a) (2b) and (2c) into (3) we have
1205761199091199091
1205761199091199093
120574119909119911
=
120576(0)
1199091199091
120576(0)
1199091199093
120574(0)
119909119911
+ 119911
120576(1)
1199091199091
120576(1)
1199091199093
120574(1)
119909119911
120576(0)
1199091199091
120576(0)
1199091199093
120574(0)
119909119911
=
1205971199060
120597119909
+
ℎ119888
2
(
120597120601119909
120597119909
minus
1205972
1199080
1205971199092) +
1
2
(
1205971199080
120597119909
)
2
1205971199060
120597119909
minus
ℎ119888
2
(
120597120601119909
120597119909
minus
1205972
1199080
1205971199092) +
1
2
(
1205971199080
120597119909
)
2
minus120601119909+
1205971199080
120597119909
120576(1)
1199091199091
120576(1)
1199091199093
120574(1)
119909119911
=
minus
1205972
1199080
1205971199092
minus
1205972
1199080
1205971199092
0
(4)
The mathematical statement of Hamiltonrsquos principle isgiven as
int
1199052
1199051
(120575119880 + 120575119882 minus 120575119870) 119889119905 = 0 (5)
The virtual kinetic energy 120575119870 virtual strain energy 120575119880and the virtual work 120575119882 are respectively given by
120575119880 = 1205751198801+ 1205751198802+ 1205751198803
120575119870 = 1205751198701+ 1205751198702+ 1205751198703
120575119882 = minusint
119860
1198651205751199080minus 12058301205751199080119889119909 119889119910
(6a)
120575119880 = int
119897
0
int
minusℎ1198882
minusℎ2
12059011990911990911205751205761199091199091119889119909 119889119911
+ int
119897
0
int
ℎ1198882
minusℎ1198882
120590119909119911120575120574119909119911119889119909 119889119911 + int
119897
0
int
ℎ2
ℎ1198882
12059011990911990931205751205761199091199093119889119909 119889119911
= int
119897
0
int
minusℎ1198882
minusℎ2
1205901199091199091(120575120576(0)
1199091199091+ 119911120575120576(1)
1199091199091) 119889119909 119889119911
+ int
119897
0
int
ℎ1198882
minusℎ1198882
120590119909119911120575120574119909119911119889119909 119889119911 + int
119897
0
int
ℎ2
ℎ1198882
1205901199091199093(120575120576(0)
1199091199093
+ 119911120575120576(1)
1199091199093) 119889119909 119889119911 = int
119897
0
1198731199091199091120575120576(0)
1199091199091
+1198721199091199091120575120576(1)
1199091199091119889119909 + int
119897
0
int
ℎ1198882
minusℎ1198882
119876119909120575120574119909119911119889119909 + int
119897
0
1198731199091199093
sdot 120575120576(0)
1199091199093+1198721199091199093120575120576(1)
1199091199093119889119909 119889119911
(6b)
4 Shock and Vibration
120575119870 = int
119897
0
int
minusℎ1198882
minusℎ2
120588 [(0minus 119911
1205970
120597119909
+
ℎ119888
2
120601119909minus
ℎ119888
2
1205970
120597119909
)
sdot (1205750minus 119911
1205971205750
120597119909
+
ℎ119888
2
120575120601119909minus
ℎ119888
2
1205971205750
120597119909
)
+ 01205750] 119889119909 119889119911
+ int
119897
0
int
ℎ1198882
minusℎ1198882
120588119888[(0minus 119911
120601119909) (1205750minus 119911120575
120601119909)
+ 01205750] 119889119909 119889119911
+ int
119897
0
int
ℎ2
ℎ1198882
120588 [(0minus 119911
1205970
120597119909
minus
ℎ119888
2
120601119909+
ℎ119888
2
1205970
120597119909
)
sdot (1205750minus 119911
1205971205750
120597119909
minus
ℎ119888
2
120575120601119909+
ℎ119888
2
1205971205750
120597119909
)
+ 01205750] 119889119909 119889119911
(6c)
where the superposed dot on a variable indicates its firstderivative with respect to time
0= 1205971199060120597119905 and so on
Substituting (6a) (6b) and (6c) into (5) the nonlineargoverning equations of motion for the truss core sandwichbeam are obtained
(1198731199091199091+ 1198731199091199093)119909= (211986801+ 11986802) 0minus 11986812
120601119909 (7a)
ℎ119888
2
(1198731199091199091minus 1198731199091199093)119909+ 119876119909= (
ℎ2
119888
2
11986801+ 11986822)120601119909
minus (
ℎ2
119888
2
11986801+ ℎ11988811986811)
1205970
120597119909
minus 119868120
(7b)
(1198731199091199091
1205971199080
120597119909
+ 1198731199091199093
1205971199080
120597119909
)
119909
+
ℎ119888
2
(1198731199091199091minus 1198731199091199093)119909119909
+ (1198721199091199091+1198721199091199093)119909119909+ 119876119909119909+ 119865
minus 1205830= (
ℎ2
119888
2
11986801+ ℎ11988811986811)
120597120601119909
120597119909
minus (
ℎ2
119888
2
11986801+ 2ℎ11988811986811+ 211986821)
1205972
0
1205971199092
+ (211986801+ 11986802) 0
(7c)
where
11986801
11986811
11986821
= int
minusℎ1198882
minusℎ2
1
119911
1199112
120588119889119911
11986802
11986812
11986822
= int
ℎ1198882
minusℎ1198882
1
119911
1199112
120588119888119889119911
(8a)
1198731199091199091= int
minusℎ1198882
minusℎ2
1205901199091199091119889119911
1198731199091199093= int
ℎ2
ℎ1198882
1205901199091199093119889119911
119876119909= int
ℎ1198882
minusℎ1198882
120590119909119911119889119911
1198721199091199091= int
minusℎ1198882
minusℎ2
1199111205901199091199091119889119911
1198721199091199093= int
ℎ2
ℎ1198882
1199111205901199091199093119889119911
(8b)
We consider the face sheets and the truss core aremade ofthe same isotropy material The stress-strain relations of thetruss core sandwich beam can be represented by
1205901199091199091= 1198641205761199091199091
1205901199091199093= 1198641205761199091199093
120590119909119911= 119866120574119909119911
(9)
The stress-displacement relations can be expressed as
1198731199091199091= 1198601120576(0)
1199091199091+ 1198611120576(1)
1199091199091
1198731199091199093= 1198603120576(0)
1199091199093+ 1198613120576(1)
1199091199093
1198721199091199091= 1198611120576(0)
1199091199091+ 1198631120576(1)
1199091199091
1198721199091199093= 1198613120576(0)
1199091199093+ 1198633120576(1)
1199091199093
119876119909= 119862120574(0)
119909119911
(10a)
where
(1198601 1198611 1198631) = int
minusℎ1198882
minusℎ2
119864 (1 119911 1199112
) 119889119911
(1198603 1198613 1198633) = int
ℎ2
ℎ1198882
119864 (1 119911 1199112
) 119889119911
119862 = int
ℎ1198882
minusℎ1198882
119866119889119911
(10b)
1198601and 119860
3are called extensional stiffness 119861
1and 119861
3are
called bending-extensional stiffness 1198631and 119863
3are called
bending stiffness 119862 is called shear stiffness and 119868119894119895are the
mass moments of inertiaConsidering the symmetry of the two face sheets of the
truss core sandwich beam about the 119909-axis we have
1198601= 1198603
1198611= minus1198613
1198631= 1198633
(11)
Shock and Vibration 5
Using the stress-displacement relations (7a)ndash(7c) can beexpressed in terms of generalized displacements
21198601
1205972
1199060
1205971199092+ 21198601
1205971199080
120597119909
1205972
1199080
1205971199092= (211986801+ 11986802) 0minus 11986812
120601119909 (12a)
ℎ2
119888
2
1198601
1205972
120601119909
1205971199092minus
ℎ2
119888
2
1198601
1205973
1199080
1205971199093minus ℎ1198881198611
1205973
1199080
1205971199093minus 119862120601119909
+ 119862
1205971199080
120597119909
= (
ℎ2
119888
2
11986801+ 11986822)120601119909
minus (
ℎ2
119888
2
11986801+ ℎ11988811986811)
1205970
120597119909
minus 119868120
(12b)
(
ℎ2
119888
2
1198601+ 1198611ℎ119888)
1205973
120601119909
1205971199093
+ (minus
ℎ2
119888
2
1198601minus 21198611ℎ119888minus 21198631)
1205974
1199080
1205971199094minus 119862
120597120601119909
120597119909
+ 119862
1205972
1199080
1205971199092+ 31198601(
1205971199080
120597119909
)
2
1205972
1199080
1205971199092+ 21198601
1205972
1199060
1205971199092
1205971199080
120597119909
+ 21198601
1205971199060
120597119909
1205972
1199080
1205971199092+ 119865
minus 1205830= (
ℎ2
119888
2
11986801+ ℎ11988811986811)
120597120601119909
120597119909
minus (
ℎ2
119888
2
11986801+ 2ℎ11988811986811+ 211986821)
1205972
0
1205971199092
+ (211986801+ 11986802) 0
(12c)
3 Validation
In order to check the accuracy of the presented model thenatural frequencies are calculated and compared with theresults presented by [10] For linear vibration the in-planemotion of the structure can be neglected Therefore thesimply supported boundary conditions of the sandwich beamcan be expressed as
1199080=
120597120601119909
120597119909
= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (13)
The displacements 120601119909and 119908
0which satisfy (13) can be
represented as
120601119909=
119873
sum
119899=1
120601119899(119905) cos119899120587119909
119897
(14a)
1199080=
119873
sum
119899=1
119908119899(119905) sin119899120587119909
119897
(14b)
We only consider the transverse vibration of the trusscore sandwich beam thus we can neglect the rotatory inertiaterms Substituting (14a) and (14b) into the linear form of
Table 1 Comparison of natural frequencies of the sandwich beamwith the first group of parameters
Mode Ahmed Hwu Lou Present1 56 58 57 57142 mdash 221 219 219583 451 459 465 465174 mdash 742 767 768185 1073 1048 1105 110669
Table 2 Comparison of natural frequencies of the sandwich beamwith the second group of parameters
Mode Lou Present1 13774 137812 52309 523953 109138 109411
(12a) (12b) and (12c) and applying the Galerkin method thenatural frequencies are obtained as shown in Tables 1 and2 The good agreement between the present and publishedresults validates the accuracy of the present method
The dimensions and material properties in Table 1 are
119864119891= 689GPa
ℎ119891= 04572mm
120588119891= 2680 kgm3
119866119888= 008268GPa
ℎ119888= 127mm
120588119888= 328 kgm3
119897 = 09144m
(15)
The dimensions and material properties in Table 2 are
119864119891= 210GPa
ℎ119891= 1mm
120588119891= 7930 kgm3
119866119888= 103667GPa
ℎ119888= 15mm
120588119888= 3131739 kgm3
119897 = 06364m
(16)
4 Nonlinear Frequency Response Equation
In the following analysis the nonlinear vibrations of the trusscore sandwich beam with simply supported boundary con-ditions are considered For nonlinear vibration the simply
6 Shock and Vibration
supported boundary conditions with immoveable edges ofthe beam can be expressed as
1199060= 1199080=
120597120601119909
120597119909
= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (17)
The transformations of variables and parameters areintroduced to obtain the dimensionless equation
1199060=
1199060
119897
1199080=
1199080
ℎ
120601119909= 120601119909
119909 =
119909
119897
119865 =
1198974
119864ℎ4119865
120583 =
119897
ℎ
(
1
120588119864
)
12
120583
119905 =
ℎ
1198972(
119864
120588
)
12
119905
Ω =
1198972
ℎ
(
120588
119864
)
12
Ω
ℎ119888=
ℎ119888
ℎ
119860119894=
119897
119864ℎ2119860119894
119861119894=
119897
119864ℎ3119861119894
119862 =
119897
119864ℎ2119862
119863119894=
119897
119864ℎ4119863119894
119868119905119894=
1
119897(119905+1)120588
119868119905119894 119905 = 0 1 2
(18)
Therefore the dimensionless equations of motion for thetruss core sandwich beam are obtained For convenience ofour study we drop the overbar in the following analysisThe displacements 119906
0 120601119909 and 119908
0which satisfy (17) can be
represented as
1199060=
119873
sum
119899=1
119906 (119905) sin (2119899120587119909) (19a)
120601119909=
119873
sum
119899=1
120601 (119905) cos (119899120587119909) (19b)
1199080=
119873
sum
119899=1
119908 (119905) sin (119899120587119909) (19c)
In addition the excitation has the form
119865 =
119873
sum
119899=1
119891 (119905) sin (119899120587119909) (19d)
Next theGalerkinmethod is used to obtain the nonlinearordinary differential equations of motion for the sandwichbeam The transverse vibrations of the sandwich beam aremainly considered In the same way we can neglect thelongitude and rotatory inertia terms Substituting the dis-placements 119906
0 120601119909 and 119908
0into the resulting equations the
ordinary differential equation is derived
119899+ 120583119899119899+ 1205962
119899119908119899+ 1205721198991199083
119899+ 119892119899(1199081 1199082 119908
119873)
= 120573119899119891119899cos (Ω
119899119905) 119899 = 1 2 119873
(20)
where 119892119899(1199081 1199082 119908
119873) represents the coupled relation
between the different modes and the coupled term does notappear when the single mode response is considered Thecoefficients in (20) are presented in the Appendix
The following scales transformations must be introducedfor obtaining a system which is suitable for the application ofthe method of multiple scales and studying the resonances ofthe sandwich beam
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
120573119899997888rarr 120576120573
119899
(21)
where 120576 is a small dimensionless parameterBecause the coupled term 119892
119899(1199081 1199082 119908
119873) has no
contribution to the primary resonance of each mode intheoretical analysis therefore we can neglect this term andobtain the following equation
119899+ 120576120583119899119899+ 1205962
119899119908119899+ 1205761205721198991199083
119899= 120576120573119899119891119899cos (Ω
119899119905) (22)
The method of multiple scales [29] is used to obtain theuniform solution of (22) in the following form
119908119899(119905 120576) = 119909
1198990(1198790 1198791) + 120576119909
1198991(1198790 1198791) + sdot sdot sdot (23)
where 1198790= 119905 and 119879
1= 120576119905
Then we have the differential operators
119889
119889119905
=
120597
1205971198790
1205971198790
120597119905
+
120597
1205971198791
1205971198791
120597119905
+ sdot sdot sdot = 1198630+ 1205761198631+ sdot sdot sdot (24a)
1198892
1198891199052= (1198630+ 1205761198631+ sdot sdot sdot)
2
= 1198632
0+ 2120576119863
01198631+ sdot sdot sdot (24b)
where1198630= 120597120597119879
0and119863
1= 120597120597119879
1
We have the following relation
Ω119899= 120596119899+ 120576120590119899 (25)
120590119899
is the detuning parameter which quantitativelydescribes the nearness of excitation frequency to the linearnatural frequency
Substituting (23)ndash(25) into (22) and balancing the coef-ficients of the like power of 120576 yield the following differentialequations
Shock and Vibration 7
1205760 order
1198632
01199091198990+ 1205962
1198991199091198990= 0 (26a)
1205761 order
1198632
01199091198991+ 1205962
1198991199091198991= minus2119863
011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093
1198990
+ 120573119899119891119899cos (120596
1198991198790+ 1205901198991198791)
(26b)
The solution of (26a) in the complex form can beexpressed as
1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)
where 119860119899is the complex conjugate of 119860
119899
Substituting (27) into (26b) yields
1198632
01199091198991+ 1205962
1198991199091198991= (minus2119894120596
1198991198631119860119899minus 120583119899119894120596119899119860119899
minus 31205721198991198602
119899119860119899+
120573119899119891119899
2
1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST
(28)
where symbol cc denotes the parts of the complex conjugateof function on the right-hand side of (28) andNST representsthe terms that do not produce secular terms
Function 119860119899may be denoted in the complex form
119860119899=
1
2
119886119899119890119894120593119899 (29)
Substituting (29) into simplified (28) and separating thereal and imaginary parts from the resulting equations thetwo-dimensional averaged equation in the polar form isobtained as
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin (120590
1198991198791minus 120593119899)
2120596119899
(30a)
119886119899119899=
31205721198991198863
119899
8120596119899
minus
120573119899119891119899cos (120590
1198991198791minus 120593119899)
2120596119899
(30b)
Letting
120574119899= 1205901198991198791minus 120593119899 (31)
we have
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin 120574119899
2120596119899
(32a)
119886119899119899= 119886119899120590119899minus
31205721198991198863
119899
8120596119899
+
120573119899119891119899cos 120574119899
2120596119899
(32b)
Steady-state motions occur when 119899and
119899equal zero
which correspond to the singular points of averaged (32a) and(32b) Thus 119886
119899and 120574119899are constants Eliminating 120574
119899by using
the relations between trigonometric functions we obtain thefrequency response equation of the truss core sandwich beamwith primary resonance as follows
(
120583119899119886119899
2
)
2
+ (120590119899119886119899minus
31205721198991198863
119899
8120596119899
)
2
= (
120573119899119891119899
2120596119899
)
2
(33)
Subsequently the subharmonic resonance and the super-harmonic resonance of the sandwich beams are consideredIn the two cases the following two transformations arerequired
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
(34)
For subharmonic resonance the detuning parameter 120590119899
is introduced according toΩ119899= 3120596119899+ 120576120590119899 (35)
Similarly the averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
31205721198991205731198991198862
119899Λ sin 120574
119899
4120596119899
(36a)
119886119899119899= 119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
minus
9120572119899120573119899Λ1198862
119899cos 120574119899
4120596119899
(36b)
where Λ = (12)119891119899(1205962
119899minus Ω2
119899)minus1
The frequency response equation of the truss core sand-wich beam with subharmonic resonance is
(minus
120583119899119886119899
2
)
2
+
1
9
[119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
]
2
=
91205722
1198991205732
1198991198864
119899Λ2
161205962
119899
(37)
For superharmonic resonance the detuning parameter120590119899
is introduced according to3Ω119899= 120596119899+ 120576120590119899 (38)
The averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
1205721198991205733
119899Λ3 sin 120574
119899
120596119899
(39a)
119886119899119899= 119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
minus
1205721198991205733
119899Λ3 cos 120574
119899
120596119899
(39b)
The frequency response equation of the truss core sand-wich beam with superharmonic resonance is
(minus
120583119899119886119899
2
)
2
+ [119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
]
2
=
1205722
1198991205736
119899Λ6
1205962
119899
(40)
The plot of 119886119899as a function of 120590
119899for given 119891 is called
a nonlinear frequency response curve which can be usedto describe the hardening behavior of a structure In thefollowing analysis the parameters in (33) (37) and (40)should be calculated by using the Appendix
8 Shock and Vibration
120596
32
28
24
20
16
12
8
4
120588c120588
000 002 004 006 008 010
Figure 3 Variations of dimensionless linear natural frequencieswith respect to the strut radius Solid line first-order frequencydash line second-order frequency dot-dash line third-order fre-quency
5 Numerical Results
In the study the influences of the structural parameterson the dimensionless linear natural frequencies of the trusscore sandwich beam are studied at first by neglecting thenonlinear terms in (20) The face sheets and the truss of thesandwich beam are made of the same material The materialproperties are taken as follows 119864 = 689GPa and 120588 =
2680 kgm3 Figure 3 indicates that the first-order naturalfrequency decreases continuously and the second-order andthird-order natural frequencies initially increase and thendecrease with the increase of strut radiusThe relative densitycorresponding to the turning point in the changing curveof the second-order natural frequency is smaller than thatof the third-order natural frequency When the strut radiusincreases the mass moment of inertia and shear stiffness ofthe sandwich beam both increase the changes of the naturalfrequencies depend upon the combined effect of the twofactors The structural parameters in the figure are given asℎ = 0012m ℎ
119888= 001m 120579 = 1205874 and 119897 = 07m
Figure 4 shows that the first three natural frequenciesall increase with the rise of the core height and the higher-order natural frequency increases faster than the lower-ordernatural frequency When the core height increases the massmoment of inertia extensional stiffness bending stiffnessand shear stiffness of the sandwich beam all decrease thechanges of the natural frequencies depend upon the com-bined effect of those factors The structural parameters in thefigure are given as ℎ = 0012m 119903 = 06mm 120579 = 1205874 and119897 = 07m
The steady-state responses of the first three modes ofthe sandwich beam with primary resonance are investigatedThe hardening-type nonlinearity of the first three modes allbecomes weaker with increasing strut radius and core heightas shown in Figures 5 and 6 The bending of the frequencyresponse curve mainly depends on the nonlinear term andthe frequency parameter according to (33) It can be known
120596
32
28
24
20
16
12
8
4
hch
072 076 080 084 088 092
Figure 4 Variations of dimensionless linear natural frequencieswith respect to the core height Solid line first-order frequency dashline second-order frequency dot-dash line third-order frequency
from 120572119899in the Appendix that the nonlinear term is related to
the extensional stiffness of the face sheets as well as the massmoment of inertia thickness and length of the sandwichbeam The theoretical analyses indicate that the reasons forthe weakening of the hardening-type nonlinearity with therise of strut radius and core height are different
At first we discuss the influence of strut radius on thehardening behavior of the sandwich beam With the increaseof the strut radius the hardening behavior of the beam isweakened by the rise of themassmoment of inertia and shearstiffness It is concluded from (33) that the increases of themass moment of inertia and shear stiffness both weaken thehardening behavior The structural parameters in Figure 5are the same as in Figure 3 With the increase of the coreheight the hardening behavior of the beam is weakened It isconcluded that the decrease of extensional stiffness tends toweaken the hardening behavior and the decrease of the massmoment of inertia shear stiffness and bending stiffness tendsto strengthen the hardening behavior of the sandwich beamObviously the effect of the extensional stiffness is dominantaccording to Figure 6The structural parameters in this figureare the same as in Figure 4
Furthermore the hardening-type nonlinearity of thehigher mode is stronger than that of the lower mode andthe influences of the structural parameters on the hardeningbehavior for the lower mode are slightly larger than for thehigher mode In the analysis of the primary resonance of thesandwich beam 120583 = 0001 and 119891 = 1 are always chosen
Subharmonic and superharmonic resonances are rela-tively easy to be generated at the lower modes especially atthe first mode The nonlinear frequency response curves forthe first mode of the sandwich beam with subharmonic andsuperharmonic resonances are investigated in the paper thenumerical results are shown in Figures 7 and 8 respectivelyFigure 7 demonstrates that the subharmonic responses onlyoccur in the frequency ranges which are higher than thelinear natural frequency no matter which and how structuralparameters change Increasing strut radius and core height
Shock and Vibration 9
a1
25
2
15
1
05
1205901
minus10 minus5 0 5 10
(a)a2
10
12
06
04
08
02
1205902
minus10 minus5 0 5 10
(b)
a3
07
08
05
04
06
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 5 Influences of the strut radius on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line 120588 = 002 dash line 120588 = 004 dot-dash line 120588 = 006
will weaken the hardening-type nonlinearity in the resonancecase just as the influences of the structural parameters onthe responses of the beam with primary resonance Figure 8demonstrates that the shapes of the frequency responsecurves for the first mode with superharmonic responses aresimilar to that of the sandwich beamwith primary resonanceThe variations in the structural parameters can not only affectthe hardening behaviors of the beam but also significantlychange the frequency range of the resonance responses
Decreasing strut radius and core height can broaden thefrequency ranges of the resonance responses
Based on the analysis of the frequency response equa-tions we conclude that in the cases of subharmonic andsuperharmonic resonances the mechanism of the variationof the hardening behavior via the structural parameters is thesame as that in the case of primary resonance 120583 = 0001and 119891 = 2 are chosen in Figures 7 and 8 Moreover thestructural parameters in Figures 7(a) and 8(a) are the same
10 Shock and Vibration
a1
14
12
10
08
06
04
02
1205901
minus10 minus5 0 5 10
(a)a2
06
05
04
03
02
01
1205902
minus10 minus5 0 5 10
(b)
a3
04
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 6 Influences of the core height on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash line ℎ
119888ℎ = 09
as in Figure 3 and the structural parameters in Figures 7(b)and 8(b) are the same as in Figure 4
6 Conclusions
The natural frequency and hardening behavior of sandwichbeam with pyramidal truss core are investigated The trusscore is equivalent to a continuous homogeneousmaterial anda Zig-Zag theory is used to derive the nonlinear governingequation of motion for the sandwich beam At first thenatural frequencies are calculated to validate the presentmodel and the changing rules of the natural frequency with
the strut radius and core height are obtained It is illustratedthat the first-order natural frequency decreases continuouslyand the second-order and third-order natural frequenciesinitially increase and then decrease with the increase of strutradius and the first three natural frequencies all increase withthe rise of the core height Then the hardening behaviorsof the sandwich beam are analyzed Numerical simulationsindicate that the effects of the two structural parameters onthe hardening behavior in the cases of primary resonancesubharmonic resonance and superharmonic resonance aresimilar However the mechanisms of the variations of thehardening behavior are very different The influences of
Shock and Vibration 11
a1
15
1
05
0
1205901
minus10 minus5 0 5 10
(a)a1
1205901
minus10 minus5 0 5 10
08
07
06
05
04
03
02
01
0
(b)
Figure 7 The hardening behavior of the first mode with subharmonic resonance (a) The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
1205901
minus4 minus2 0 2 4
a1
16
14
12
10
08
06
04
02
(a)
1205901
minus5 0 5 10 15
a1
14
12
10
08
06
04
02
(b)
Figure 8The hardening behavior of the first mode with superharmonic resonance (a)The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
the structural parameters on the frequency parameter and thenonlinear term of the sandwich beam are studied in the paperfor analyzing the variations of the hardening behavior
Thehardening nonlinearity bends the frequency responsecurve to the right Generally under the same excitationthe stronger the hardening-type nonlinearity the smaller
the response amplitude We investigate the influences ofstrut radius and core height on the dynamical responses ofthe sandwich beam Using the equations obtained in thispaper the influences of the other structural parameters ofthe sandwich beam on the vibrations can be convenientlyanalyzed
12 Shock and Vibration
It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics
Appendix
The relations between the in-plane displacements and trans-verse displacement are
119906119899= 1198781198991199082
119899
119878119899= minus
1198991198861120587
8
(A1)
The relations between the rotation angles and transversedisplacement are
120601119899= 119870119899119908119899
119870119899=
1198991198873120587 minus 1198993
11988721205873
119899211988711205872minus 1
(A2)
The coefficients in (20) are given as follows
120574119899=
1198792119899
1198791119899
1205962
119899= minus
1198793119899
1198791119899
120572119899= minus
1198794119899
1198791119899
120573119899=
1198795119899
1198791119899
1198791119899=
1198889
2
1198792119899=
1198888
2
1198793119899=
1198994
1205874
1198882
2
minus
1198992
1205872
1198884
2
minus
1198991205871198701198991198883
2
+
1198993
1205873
1198701198991198881
2
1198794119899= minus
1198994
1205874
8
+
1198993
1205873
1198781198991198886
2
minus 1198993
1205873
1198781198991198885
1198795119899=
1198887
2
(A3)
where
1198861=
ℎ2
1198972
1198871= minus
ℎ2
ℎ
2
119888
21198972
1198601
119862
1198872=
ℎ3
ℎ
2
119888
21198973
1198601
119862
+
ℎ3
ℎ119888
1198973
1198611
119862
1198873= minus
ℎ
119897
1198881=
119897
3ℎ
(
ℎ119888
2
+ ℎ
1198611
1198601
)
1198882= minus
1
3
(
ℎ
2
119888
2
+ 2ℎ
1198611
1198601
+ 2
1198631
1198601
)
1198883= minus
1198973
3ℎ3
119862
1198601
1198884=
1198972
3ℎ2
119862
1198601
1198885=
21198972
3ℎ2
1198886=
21198972
3ℎ2
1198887=
119897
3ℎ1198601
1198888=
1198972
3ℎ21198601
1198889=
1198972
3ℎ21198601
(211986801+ 11986802)
(A4)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)
References
[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 3
z
hhc x
w0
u0
120601x
120597w0
120597x
Figure 2 Undeformed and deformed geometries of the truss coresandwich beam
on undeformed and deformed geometries of the truss coresandwich beam as shown in Figure 2 the displacement fieldcan be expressed in terms of the midplane displacements 119906
0
and1199080and rotations 120601
119909 the face sheets of the sandwich beam
adopt the Kirchhoff hypothesis and the truss core adopts thefirst-order shear deformation theory Hence
minus
ℎ
2
le 119911 le minus
ℎ119888
2
119906119905= 1199060minus 119911
1205971199080
120597119909
+
ℎ119888
2
(120601119909minus
1205971199080
120597119909
)
119908119905= 1199080
(2a)
minus
ℎ119888
2
le 119911 le
ℎ119888
2
119906119888= 1199060minus 119911120601119909
119908119888= 1199080
(2b)
ℎ119888
2
le 119911 le
ℎ
2
119906119887= 1199060minus 119911
1205971199080
120597119909
minus
ℎ119888
2
(120601119909minus
1205971199080
120597119909
)
119908119887= 1199080
(2c)
where the subscripts 119905 119888 and 119887 represent the top face sheetthe truss core and the bottom face sheet respectively ℎrepresents the total thickness of the truss core sandwich beamand ℎ119888represents the height of the truss core
Using von Karmanrsquos theory for the geometric nonlinear-ity the strain-displacement relations have the form
1205761199091199091=
120597119906119905
120597119909
+
1
2
(
120597119908119905
120597119909
)
2
120574119909119911=
120597119906119888
120597119911
+
120597119908119888
120597119909
1205761199091199093=
120597119906119887
120597119909
+
1
2
(
120597119908119887
120597119909
)
2
(3)
where 1205761199091199091
and 1205761199091199093
represent the strains of the face sheets and120574119909119911
represents the strain of the truss coreSubstituting (2a) (2b) and (2c) into (3) we have
1205761199091199091
1205761199091199093
120574119909119911
=
120576(0)
1199091199091
120576(0)
1199091199093
120574(0)
119909119911
+ 119911
120576(1)
1199091199091
120576(1)
1199091199093
120574(1)
119909119911
120576(0)
1199091199091
120576(0)
1199091199093
120574(0)
119909119911
=
1205971199060
120597119909
+
ℎ119888
2
(
120597120601119909
120597119909
minus
1205972
1199080
1205971199092) +
1
2
(
1205971199080
120597119909
)
2
1205971199060
120597119909
minus
ℎ119888
2
(
120597120601119909
120597119909
minus
1205972
1199080
1205971199092) +
1
2
(
1205971199080
120597119909
)
2
minus120601119909+
1205971199080
120597119909
120576(1)
1199091199091
120576(1)
1199091199093
120574(1)
119909119911
=
minus
1205972
1199080
1205971199092
minus
1205972
1199080
1205971199092
0
(4)
The mathematical statement of Hamiltonrsquos principle isgiven as
int
1199052
1199051
(120575119880 + 120575119882 minus 120575119870) 119889119905 = 0 (5)
The virtual kinetic energy 120575119870 virtual strain energy 120575119880and the virtual work 120575119882 are respectively given by
120575119880 = 1205751198801+ 1205751198802+ 1205751198803
120575119870 = 1205751198701+ 1205751198702+ 1205751198703
120575119882 = minusint
119860
1198651205751199080minus 12058301205751199080119889119909 119889119910
(6a)
120575119880 = int
119897
0
int
minusℎ1198882
minusℎ2
12059011990911990911205751205761199091199091119889119909 119889119911
+ int
119897
0
int
ℎ1198882
minusℎ1198882
120590119909119911120575120574119909119911119889119909 119889119911 + int
119897
0
int
ℎ2
ℎ1198882
12059011990911990931205751205761199091199093119889119909 119889119911
= int
119897
0
int
minusℎ1198882
minusℎ2
1205901199091199091(120575120576(0)
1199091199091+ 119911120575120576(1)
1199091199091) 119889119909 119889119911
+ int
119897
0
int
ℎ1198882
minusℎ1198882
120590119909119911120575120574119909119911119889119909 119889119911 + int
119897
0
int
ℎ2
ℎ1198882
1205901199091199093(120575120576(0)
1199091199093
+ 119911120575120576(1)
1199091199093) 119889119909 119889119911 = int
119897
0
1198731199091199091120575120576(0)
1199091199091
+1198721199091199091120575120576(1)
1199091199091119889119909 + int
119897
0
int
ℎ1198882
minusℎ1198882
119876119909120575120574119909119911119889119909 + int
119897
0
1198731199091199093
sdot 120575120576(0)
1199091199093+1198721199091199093120575120576(1)
1199091199093119889119909 119889119911
(6b)
4 Shock and Vibration
120575119870 = int
119897
0
int
minusℎ1198882
minusℎ2
120588 [(0minus 119911
1205970
120597119909
+
ℎ119888
2
120601119909minus
ℎ119888
2
1205970
120597119909
)
sdot (1205750minus 119911
1205971205750
120597119909
+
ℎ119888
2
120575120601119909minus
ℎ119888
2
1205971205750
120597119909
)
+ 01205750] 119889119909 119889119911
+ int
119897
0
int
ℎ1198882
minusℎ1198882
120588119888[(0minus 119911
120601119909) (1205750minus 119911120575
120601119909)
+ 01205750] 119889119909 119889119911
+ int
119897
0
int
ℎ2
ℎ1198882
120588 [(0minus 119911
1205970
120597119909
minus
ℎ119888
2
120601119909+
ℎ119888
2
1205970
120597119909
)
sdot (1205750minus 119911
1205971205750
120597119909
minus
ℎ119888
2
120575120601119909+
ℎ119888
2
1205971205750
120597119909
)
+ 01205750] 119889119909 119889119911
(6c)
where the superposed dot on a variable indicates its firstderivative with respect to time
0= 1205971199060120597119905 and so on
Substituting (6a) (6b) and (6c) into (5) the nonlineargoverning equations of motion for the truss core sandwichbeam are obtained
(1198731199091199091+ 1198731199091199093)119909= (211986801+ 11986802) 0minus 11986812
120601119909 (7a)
ℎ119888
2
(1198731199091199091minus 1198731199091199093)119909+ 119876119909= (
ℎ2
119888
2
11986801+ 11986822)120601119909
minus (
ℎ2
119888
2
11986801+ ℎ11988811986811)
1205970
120597119909
minus 119868120
(7b)
(1198731199091199091
1205971199080
120597119909
+ 1198731199091199093
1205971199080
120597119909
)
119909
+
ℎ119888
2
(1198731199091199091minus 1198731199091199093)119909119909
+ (1198721199091199091+1198721199091199093)119909119909+ 119876119909119909+ 119865
minus 1205830= (
ℎ2
119888
2
11986801+ ℎ11988811986811)
120597120601119909
120597119909
minus (
ℎ2
119888
2
11986801+ 2ℎ11988811986811+ 211986821)
1205972
0
1205971199092
+ (211986801+ 11986802) 0
(7c)
where
11986801
11986811
11986821
= int
minusℎ1198882
minusℎ2
1
119911
1199112
120588119889119911
11986802
11986812
11986822
= int
ℎ1198882
minusℎ1198882
1
119911
1199112
120588119888119889119911
(8a)
1198731199091199091= int
minusℎ1198882
minusℎ2
1205901199091199091119889119911
1198731199091199093= int
ℎ2
ℎ1198882
1205901199091199093119889119911
119876119909= int
ℎ1198882
minusℎ1198882
120590119909119911119889119911
1198721199091199091= int
minusℎ1198882
minusℎ2
1199111205901199091199091119889119911
1198721199091199093= int
ℎ2
ℎ1198882
1199111205901199091199093119889119911
(8b)
We consider the face sheets and the truss core aremade ofthe same isotropy material The stress-strain relations of thetruss core sandwich beam can be represented by
1205901199091199091= 1198641205761199091199091
1205901199091199093= 1198641205761199091199093
120590119909119911= 119866120574119909119911
(9)
The stress-displacement relations can be expressed as
1198731199091199091= 1198601120576(0)
1199091199091+ 1198611120576(1)
1199091199091
1198731199091199093= 1198603120576(0)
1199091199093+ 1198613120576(1)
1199091199093
1198721199091199091= 1198611120576(0)
1199091199091+ 1198631120576(1)
1199091199091
1198721199091199093= 1198613120576(0)
1199091199093+ 1198633120576(1)
1199091199093
119876119909= 119862120574(0)
119909119911
(10a)
where
(1198601 1198611 1198631) = int
minusℎ1198882
minusℎ2
119864 (1 119911 1199112
) 119889119911
(1198603 1198613 1198633) = int
ℎ2
ℎ1198882
119864 (1 119911 1199112
) 119889119911
119862 = int
ℎ1198882
minusℎ1198882
119866119889119911
(10b)
1198601and 119860
3are called extensional stiffness 119861
1and 119861
3are
called bending-extensional stiffness 1198631and 119863
3are called
bending stiffness 119862 is called shear stiffness and 119868119894119895are the
mass moments of inertiaConsidering the symmetry of the two face sheets of the
truss core sandwich beam about the 119909-axis we have
1198601= 1198603
1198611= minus1198613
1198631= 1198633
(11)
Shock and Vibration 5
Using the stress-displacement relations (7a)ndash(7c) can beexpressed in terms of generalized displacements
21198601
1205972
1199060
1205971199092+ 21198601
1205971199080
120597119909
1205972
1199080
1205971199092= (211986801+ 11986802) 0minus 11986812
120601119909 (12a)
ℎ2
119888
2
1198601
1205972
120601119909
1205971199092minus
ℎ2
119888
2
1198601
1205973
1199080
1205971199093minus ℎ1198881198611
1205973
1199080
1205971199093minus 119862120601119909
+ 119862
1205971199080
120597119909
= (
ℎ2
119888
2
11986801+ 11986822)120601119909
minus (
ℎ2
119888
2
11986801+ ℎ11988811986811)
1205970
120597119909
minus 119868120
(12b)
(
ℎ2
119888
2
1198601+ 1198611ℎ119888)
1205973
120601119909
1205971199093
+ (minus
ℎ2
119888
2
1198601minus 21198611ℎ119888minus 21198631)
1205974
1199080
1205971199094minus 119862
120597120601119909
120597119909
+ 119862
1205972
1199080
1205971199092+ 31198601(
1205971199080
120597119909
)
2
1205972
1199080
1205971199092+ 21198601
1205972
1199060
1205971199092
1205971199080
120597119909
+ 21198601
1205971199060
120597119909
1205972
1199080
1205971199092+ 119865
minus 1205830= (
ℎ2
119888
2
11986801+ ℎ11988811986811)
120597120601119909
120597119909
minus (
ℎ2
119888
2
11986801+ 2ℎ11988811986811+ 211986821)
1205972
0
1205971199092
+ (211986801+ 11986802) 0
(12c)
3 Validation
In order to check the accuracy of the presented model thenatural frequencies are calculated and compared with theresults presented by [10] For linear vibration the in-planemotion of the structure can be neglected Therefore thesimply supported boundary conditions of the sandwich beamcan be expressed as
1199080=
120597120601119909
120597119909
= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (13)
The displacements 120601119909and 119908
0which satisfy (13) can be
represented as
120601119909=
119873
sum
119899=1
120601119899(119905) cos119899120587119909
119897
(14a)
1199080=
119873
sum
119899=1
119908119899(119905) sin119899120587119909
119897
(14b)
We only consider the transverse vibration of the trusscore sandwich beam thus we can neglect the rotatory inertiaterms Substituting (14a) and (14b) into the linear form of
Table 1 Comparison of natural frequencies of the sandwich beamwith the first group of parameters
Mode Ahmed Hwu Lou Present1 56 58 57 57142 mdash 221 219 219583 451 459 465 465174 mdash 742 767 768185 1073 1048 1105 110669
Table 2 Comparison of natural frequencies of the sandwich beamwith the second group of parameters
Mode Lou Present1 13774 137812 52309 523953 109138 109411
(12a) (12b) and (12c) and applying the Galerkin method thenatural frequencies are obtained as shown in Tables 1 and2 The good agreement between the present and publishedresults validates the accuracy of the present method
The dimensions and material properties in Table 1 are
119864119891= 689GPa
ℎ119891= 04572mm
120588119891= 2680 kgm3
119866119888= 008268GPa
ℎ119888= 127mm
120588119888= 328 kgm3
119897 = 09144m
(15)
The dimensions and material properties in Table 2 are
119864119891= 210GPa
ℎ119891= 1mm
120588119891= 7930 kgm3
119866119888= 103667GPa
ℎ119888= 15mm
120588119888= 3131739 kgm3
119897 = 06364m
(16)
4 Nonlinear Frequency Response Equation
In the following analysis the nonlinear vibrations of the trusscore sandwich beam with simply supported boundary con-ditions are considered For nonlinear vibration the simply
6 Shock and Vibration
supported boundary conditions with immoveable edges ofthe beam can be expressed as
1199060= 1199080=
120597120601119909
120597119909
= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (17)
The transformations of variables and parameters areintroduced to obtain the dimensionless equation
1199060=
1199060
119897
1199080=
1199080
ℎ
120601119909= 120601119909
119909 =
119909
119897
119865 =
1198974
119864ℎ4119865
120583 =
119897
ℎ
(
1
120588119864
)
12
120583
119905 =
ℎ
1198972(
119864
120588
)
12
119905
Ω =
1198972
ℎ
(
120588
119864
)
12
Ω
ℎ119888=
ℎ119888
ℎ
119860119894=
119897
119864ℎ2119860119894
119861119894=
119897
119864ℎ3119861119894
119862 =
119897
119864ℎ2119862
119863119894=
119897
119864ℎ4119863119894
119868119905119894=
1
119897(119905+1)120588
119868119905119894 119905 = 0 1 2
(18)
Therefore the dimensionless equations of motion for thetruss core sandwich beam are obtained For convenience ofour study we drop the overbar in the following analysisThe displacements 119906
0 120601119909 and 119908
0which satisfy (17) can be
represented as
1199060=
119873
sum
119899=1
119906 (119905) sin (2119899120587119909) (19a)
120601119909=
119873
sum
119899=1
120601 (119905) cos (119899120587119909) (19b)
1199080=
119873
sum
119899=1
119908 (119905) sin (119899120587119909) (19c)
In addition the excitation has the form
119865 =
119873
sum
119899=1
119891 (119905) sin (119899120587119909) (19d)
Next theGalerkinmethod is used to obtain the nonlinearordinary differential equations of motion for the sandwichbeam The transverse vibrations of the sandwich beam aremainly considered In the same way we can neglect thelongitude and rotatory inertia terms Substituting the dis-placements 119906
0 120601119909 and 119908
0into the resulting equations the
ordinary differential equation is derived
119899+ 120583119899119899+ 1205962
119899119908119899+ 1205721198991199083
119899+ 119892119899(1199081 1199082 119908
119873)
= 120573119899119891119899cos (Ω
119899119905) 119899 = 1 2 119873
(20)
where 119892119899(1199081 1199082 119908
119873) represents the coupled relation
between the different modes and the coupled term does notappear when the single mode response is considered Thecoefficients in (20) are presented in the Appendix
The following scales transformations must be introducedfor obtaining a system which is suitable for the application ofthe method of multiple scales and studying the resonances ofthe sandwich beam
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
120573119899997888rarr 120576120573
119899
(21)
where 120576 is a small dimensionless parameterBecause the coupled term 119892
119899(1199081 1199082 119908
119873) has no
contribution to the primary resonance of each mode intheoretical analysis therefore we can neglect this term andobtain the following equation
119899+ 120576120583119899119899+ 1205962
119899119908119899+ 1205761205721198991199083
119899= 120576120573119899119891119899cos (Ω
119899119905) (22)
The method of multiple scales [29] is used to obtain theuniform solution of (22) in the following form
119908119899(119905 120576) = 119909
1198990(1198790 1198791) + 120576119909
1198991(1198790 1198791) + sdot sdot sdot (23)
where 1198790= 119905 and 119879
1= 120576119905
Then we have the differential operators
119889
119889119905
=
120597
1205971198790
1205971198790
120597119905
+
120597
1205971198791
1205971198791
120597119905
+ sdot sdot sdot = 1198630+ 1205761198631+ sdot sdot sdot (24a)
1198892
1198891199052= (1198630+ 1205761198631+ sdot sdot sdot)
2
= 1198632
0+ 2120576119863
01198631+ sdot sdot sdot (24b)
where1198630= 120597120597119879
0and119863
1= 120597120597119879
1
We have the following relation
Ω119899= 120596119899+ 120576120590119899 (25)
120590119899
is the detuning parameter which quantitativelydescribes the nearness of excitation frequency to the linearnatural frequency
Substituting (23)ndash(25) into (22) and balancing the coef-ficients of the like power of 120576 yield the following differentialequations
Shock and Vibration 7
1205760 order
1198632
01199091198990+ 1205962
1198991199091198990= 0 (26a)
1205761 order
1198632
01199091198991+ 1205962
1198991199091198991= minus2119863
011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093
1198990
+ 120573119899119891119899cos (120596
1198991198790+ 1205901198991198791)
(26b)
The solution of (26a) in the complex form can beexpressed as
1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)
where 119860119899is the complex conjugate of 119860
119899
Substituting (27) into (26b) yields
1198632
01199091198991+ 1205962
1198991199091198991= (minus2119894120596
1198991198631119860119899minus 120583119899119894120596119899119860119899
minus 31205721198991198602
119899119860119899+
120573119899119891119899
2
1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST
(28)
where symbol cc denotes the parts of the complex conjugateof function on the right-hand side of (28) andNST representsthe terms that do not produce secular terms
Function 119860119899may be denoted in the complex form
119860119899=
1
2
119886119899119890119894120593119899 (29)
Substituting (29) into simplified (28) and separating thereal and imaginary parts from the resulting equations thetwo-dimensional averaged equation in the polar form isobtained as
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin (120590
1198991198791minus 120593119899)
2120596119899
(30a)
119886119899119899=
31205721198991198863
119899
8120596119899
minus
120573119899119891119899cos (120590
1198991198791minus 120593119899)
2120596119899
(30b)
Letting
120574119899= 1205901198991198791minus 120593119899 (31)
we have
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin 120574119899
2120596119899
(32a)
119886119899119899= 119886119899120590119899minus
31205721198991198863
119899
8120596119899
+
120573119899119891119899cos 120574119899
2120596119899
(32b)
Steady-state motions occur when 119899and
119899equal zero
which correspond to the singular points of averaged (32a) and(32b) Thus 119886
119899and 120574119899are constants Eliminating 120574
119899by using
the relations between trigonometric functions we obtain thefrequency response equation of the truss core sandwich beamwith primary resonance as follows
(
120583119899119886119899
2
)
2
+ (120590119899119886119899minus
31205721198991198863
119899
8120596119899
)
2
= (
120573119899119891119899
2120596119899
)
2
(33)
Subsequently the subharmonic resonance and the super-harmonic resonance of the sandwich beams are consideredIn the two cases the following two transformations arerequired
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
(34)
For subharmonic resonance the detuning parameter 120590119899
is introduced according toΩ119899= 3120596119899+ 120576120590119899 (35)
Similarly the averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
31205721198991205731198991198862
119899Λ sin 120574
119899
4120596119899
(36a)
119886119899119899= 119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
minus
9120572119899120573119899Λ1198862
119899cos 120574119899
4120596119899
(36b)
where Λ = (12)119891119899(1205962
119899minus Ω2
119899)minus1
The frequency response equation of the truss core sand-wich beam with subharmonic resonance is
(minus
120583119899119886119899
2
)
2
+
1
9
[119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
]
2
=
91205722
1198991205732
1198991198864
119899Λ2
161205962
119899
(37)
For superharmonic resonance the detuning parameter120590119899
is introduced according to3Ω119899= 120596119899+ 120576120590119899 (38)
The averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
1205721198991205733
119899Λ3 sin 120574
119899
120596119899
(39a)
119886119899119899= 119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
minus
1205721198991205733
119899Λ3 cos 120574
119899
120596119899
(39b)
The frequency response equation of the truss core sand-wich beam with superharmonic resonance is
(minus
120583119899119886119899
2
)
2
+ [119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
]
2
=
1205722
1198991205736
119899Λ6
1205962
119899
(40)
The plot of 119886119899as a function of 120590
119899for given 119891 is called
a nonlinear frequency response curve which can be usedto describe the hardening behavior of a structure In thefollowing analysis the parameters in (33) (37) and (40)should be calculated by using the Appendix
8 Shock and Vibration
120596
32
28
24
20
16
12
8
4
120588c120588
000 002 004 006 008 010
Figure 3 Variations of dimensionless linear natural frequencieswith respect to the strut radius Solid line first-order frequencydash line second-order frequency dot-dash line third-order fre-quency
5 Numerical Results
In the study the influences of the structural parameterson the dimensionless linear natural frequencies of the trusscore sandwich beam are studied at first by neglecting thenonlinear terms in (20) The face sheets and the truss of thesandwich beam are made of the same material The materialproperties are taken as follows 119864 = 689GPa and 120588 =
2680 kgm3 Figure 3 indicates that the first-order naturalfrequency decreases continuously and the second-order andthird-order natural frequencies initially increase and thendecrease with the increase of strut radiusThe relative densitycorresponding to the turning point in the changing curveof the second-order natural frequency is smaller than thatof the third-order natural frequency When the strut radiusincreases the mass moment of inertia and shear stiffness ofthe sandwich beam both increase the changes of the naturalfrequencies depend upon the combined effect of the twofactors The structural parameters in the figure are given asℎ = 0012m ℎ
119888= 001m 120579 = 1205874 and 119897 = 07m
Figure 4 shows that the first three natural frequenciesall increase with the rise of the core height and the higher-order natural frequency increases faster than the lower-ordernatural frequency When the core height increases the massmoment of inertia extensional stiffness bending stiffnessand shear stiffness of the sandwich beam all decrease thechanges of the natural frequencies depend upon the com-bined effect of those factors The structural parameters in thefigure are given as ℎ = 0012m 119903 = 06mm 120579 = 1205874 and119897 = 07m
The steady-state responses of the first three modes ofthe sandwich beam with primary resonance are investigatedThe hardening-type nonlinearity of the first three modes allbecomes weaker with increasing strut radius and core heightas shown in Figures 5 and 6 The bending of the frequencyresponse curve mainly depends on the nonlinear term andthe frequency parameter according to (33) It can be known
120596
32
28
24
20
16
12
8
4
hch
072 076 080 084 088 092
Figure 4 Variations of dimensionless linear natural frequencieswith respect to the core height Solid line first-order frequency dashline second-order frequency dot-dash line third-order frequency
from 120572119899in the Appendix that the nonlinear term is related to
the extensional stiffness of the face sheets as well as the massmoment of inertia thickness and length of the sandwichbeam The theoretical analyses indicate that the reasons forthe weakening of the hardening-type nonlinearity with therise of strut radius and core height are different
At first we discuss the influence of strut radius on thehardening behavior of the sandwich beam With the increaseof the strut radius the hardening behavior of the beam isweakened by the rise of themassmoment of inertia and shearstiffness It is concluded from (33) that the increases of themass moment of inertia and shear stiffness both weaken thehardening behavior The structural parameters in Figure 5are the same as in Figure 3 With the increase of the coreheight the hardening behavior of the beam is weakened It isconcluded that the decrease of extensional stiffness tends toweaken the hardening behavior and the decrease of the massmoment of inertia shear stiffness and bending stiffness tendsto strengthen the hardening behavior of the sandwich beamObviously the effect of the extensional stiffness is dominantaccording to Figure 6The structural parameters in this figureare the same as in Figure 4
Furthermore the hardening-type nonlinearity of thehigher mode is stronger than that of the lower mode andthe influences of the structural parameters on the hardeningbehavior for the lower mode are slightly larger than for thehigher mode In the analysis of the primary resonance of thesandwich beam 120583 = 0001 and 119891 = 1 are always chosen
Subharmonic and superharmonic resonances are rela-tively easy to be generated at the lower modes especially atthe first mode The nonlinear frequency response curves forthe first mode of the sandwich beam with subharmonic andsuperharmonic resonances are investigated in the paper thenumerical results are shown in Figures 7 and 8 respectivelyFigure 7 demonstrates that the subharmonic responses onlyoccur in the frequency ranges which are higher than thelinear natural frequency no matter which and how structuralparameters change Increasing strut radius and core height
Shock and Vibration 9
a1
25
2
15
1
05
1205901
minus10 minus5 0 5 10
(a)a2
10
12
06
04
08
02
1205902
minus10 minus5 0 5 10
(b)
a3
07
08
05
04
06
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 5 Influences of the strut radius on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line 120588 = 002 dash line 120588 = 004 dot-dash line 120588 = 006
will weaken the hardening-type nonlinearity in the resonancecase just as the influences of the structural parameters onthe responses of the beam with primary resonance Figure 8demonstrates that the shapes of the frequency responsecurves for the first mode with superharmonic responses aresimilar to that of the sandwich beamwith primary resonanceThe variations in the structural parameters can not only affectthe hardening behaviors of the beam but also significantlychange the frequency range of the resonance responses
Decreasing strut radius and core height can broaden thefrequency ranges of the resonance responses
Based on the analysis of the frequency response equa-tions we conclude that in the cases of subharmonic andsuperharmonic resonances the mechanism of the variationof the hardening behavior via the structural parameters is thesame as that in the case of primary resonance 120583 = 0001and 119891 = 2 are chosen in Figures 7 and 8 Moreover thestructural parameters in Figures 7(a) and 8(a) are the same
10 Shock and Vibration
a1
14
12
10
08
06
04
02
1205901
minus10 minus5 0 5 10
(a)a2
06
05
04
03
02
01
1205902
minus10 minus5 0 5 10
(b)
a3
04
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 6 Influences of the core height on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash line ℎ
119888ℎ = 09
as in Figure 3 and the structural parameters in Figures 7(b)and 8(b) are the same as in Figure 4
6 Conclusions
The natural frequency and hardening behavior of sandwichbeam with pyramidal truss core are investigated The trusscore is equivalent to a continuous homogeneousmaterial anda Zig-Zag theory is used to derive the nonlinear governingequation of motion for the sandwich beam At first thenatural frequencies are calculated to validate the presentmodel and the changing rules of the natural frequency with
the strut radius and core height are obtained It is illustratedthat the first-order natural frequency decreases continuouslyand the second-order and third-order natural frequenciesinitially increase and then decrease with the increase of strutradius and the first three natural frequencies all increase withthe rise of the core height Then the hardening behaviorsof the sandwich beam are analyzed Numerical simulationsindicate that the effects of the two structural parameters onthe hardening behavior in the cases of primary resonancesubharmonic resonance and superharmonic resonance aresimilar However the mechanisms of the variations of thehardening behavior are very different The influences of
Shock and Vibration 11
a1
15
1
05
0
1205901
minus10 minus5 0 5 10
(a)a1
1205901
minus10 minus5 0 5 10
08
07
06
05
04
03
02
01
0
(b)
Figure 7 The hardening behavior of the first mode with subharmonic resonance (a) The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
1205901
minus4 minus2 0 2 4
a1
16
14
12
10
08
06
04
02
(a)
1205901
minus5 0 5 10 15
a1
14
12
10
08
06
04
02
(b)
Figure 8The hardening behavior of the first mode with superharmonic resonance (a)The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
the structural parameters on the frequency parameter and thenonlinear term of the sandwich beam are studied in the paperfor analyzing the variations of the hardening behavior
Thehardening nonlinearity bends the frequency responsecurve to the right Generally under the same excitationthe stronger the hardening-type nonlinearity the smaller
the response amplitude We investigate the influences ofstrut radius and core height on the dynamical responses ofthe sandwich beam Using the equations obtained in thispaper the influences of the other structural parameters ofthe sandwich beam on the vibrations can be convenientlyanalyzed
12 Shock and Vibration
It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics
Appendix
The relations between the in-plane displacements and trans-verse displacement are
119906119899= 1198781198991199082
119899
119878119899= minus
1198991198861120587
8
(A1)
The relations between the rotation angles and transversedisplacement are
120601119899= 119870119899119908119899
119870119899=
1198991198873120587 minus 1198993
11988721205873
119899211988711205872minus 1
(A2)
The coefficients in (20) are given as follows
120574119899=
1198792119899
1198791119899
1205962
119899= minus
1198793119899
1198791119899
120572119899= minus
1198794119899
1198791119899
120573119899=
1198795119899
1198791119899
1198791119899=
1198889
2
1198792119899=
1198888
2
1198793119899=
1198994
1205874
1198882
2
minus
1198992
1205872
1198884
2
minus
1198991205871198701198991198883
2
+
1198993
1205873
1198701198991198881
2
1198794119899= minus
1198994
1205874
8
+
1198993
1205873
1198781198991198886
2
minus 1198993
1205873
1198781198991198885
1198795119899=
1198887
2
(A3)
where
1198861=
ℎ2
1198972
1198871= minus
ℎ2
ℎ
2
119888
21198972
1198601
119862
1198872=
ℎ3
ℎ
2
119888
21198973
1198601
119862
+
ℎ3
ℎ119888
1198973
1198611
119862
1198873= minus
ℎ
119897
1198881=
119897
3ℎ
(
ℎ119888
2
+ ℎ
1198611
1198601
)
1198882= minus
1
3
(
ℎ
2
119888
2
+ 2ℎ
1198611
1198601
+ 2
1198631
1198601
)
1198883= minus
1198973
3ℎ3
119862
1198601
1198884=
1198972
3ℎ2
119862
1198601
1198885=
21198972
3ℎ2
1198886=
21198972
3ℎ2
1198887=
119897
3ℎ1198601
1198888=
1198972
3ℎ21198601
1198889=
1198972
3ℎ21198601
(211986801+ 11986802)
(A4)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)
References
[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 Shock and Vibration
120575119870 = int
119897
0
int
minusℎ1198882
minusℎ2
120588 [(0minus 119911
1205970
120597119909
+
ℎ119888
2
120601119909minus
ℎ119888
2
1205970
120597119909
)
sdot (1205750minus 119911
1205971205750
120597119909
+
ℎ119888
2
120575120601119909minus
ℎ119888
2
1205971205750
120597119909
)
+ 01205750] 119889119909 119889119911
+ int
119897
0
int
ℎ1198882
minusℎ1198882
120588119888[(0minus 119911
120601119909) (1205750minus 119911120575
120601119909)
+ 01205750] 119889119909 119889119911
+ int
119897
0
int
ℎ2
ℎ1198882
120588 [(0minus 119911
1205970
120597119909
minus
ℎ119888
2
120601119909+
ℎ119888
2
1205970
120597119909
)
sdot (1205750minus 119911
1205971205750
120597119909
minus
ℎ119888
2
120575120601119909+
ℎ119888
2
1205971205750
120597119909
)
+ 01205750] 119889119909 119889119911
(6c)
where the superposed dot on a variable indicates its firstderivative with respect to time
0= 1205971199060120597119905 and so on
Substituting (6a) (6b) and (6c) into (5) the nonlineargoverning equations of motion for the truss core sandwichbeam are obtained
(1198731199091199091+ 1198731199091199093)119909= (211986801+ 11986802) 0minus 11986812
120601119909 (7a)
ℎ119888
2
(1198731199091199091minus 1198731199091199093)119909+ 119876119909= (
ℎ2
119888
2
11986801+ 11986822)120601119909
minus (
ℎ2
119888
2
11986801+ ℎ11988811986811)
1205970
120597119909
minus 119868120
(7b)
(1198731199091199091
1205971199080
120597119909
+ 1198731199091199093
1205971199080
120597119909
)
119909
+
ℎ119888
2
(1198731199091199091minus 1198731199091199093)119909119909
+ (1198721199091199091+1198721199091199093)119909119909+ 119876119909119909+ 119865
minus 1205830= (
ℎ2
119888
2
11986801+ ℎ11988811986811)
120597120601119909
120597119909
minus (
ℎ2
119888
2
11986801+ 2ℎ11988811986811+ 211986821)
1205972
0
1205971199092
+ (211986801+ 11986802) 0
(7c)
where
11986801
11986811
11986821
= int
minusℎ1198882
minusℎ2
1
119911
1199112
120588119889119911
11986802
11986812
11986822
= int
ℎ1198882
minusℎ1198882
1
119911
1199112
120588119888119889119911
(8a)
1198731199091199091= int
minusℎ1198882
minusℎ2
1205901199091199091119889119911
1198731199091199093= int
ℎ2
ℎ1198882
1205901199091199093119889119911
119876119909= int
ℎ1198882
minusℎ1198882
120590119909119911119889119911
1198721199091199091= int
minusℎ1198882
minusℎ2
1199111205901199091199091119889119911
1198721199091199093= int
ℎ2
ℎ1198882
1199111205901199091199093119889119911
(8b)
We consider the face sheets and the truss core aremade ofthe same isotropy material The stress-strain relations of thetruss core sandwich beam can be represented by
1205901199091199091= 1198641205761199091199091
1205901199091199093= 1198641205761199091199093
120590119909119911= 119866120574119909119911
(9)
The stress-displacement relations can be expressed as
1198731199091199091= 1198601120576(0)
1199091199091+ 1198611120576(1)
1199091199091
1198731199091199093= 1198603120576(0)
1199091199093+ 1198613120576(1)
1199091199093
1198721199091199091= 1198611120576(0)
1199091199091+ 1198631120576(1)
1199091199091
1198721199091199093= 1198613120576(0)
1199091199093+ 1198633120576(1)
1199091199093
119876119909= 119862120574(0)
119909119911
(10a)
where
(1198601 1198611 1198631) = int
minusℎ1198882
minusℎ2
119864 (1 119911 1199112
) 119889119911
(1198603 1198613 1198633) = int
ℎ2
ℎ1198882
119864 (1 119911 1199112
) 119889119911
119862 = int
ℎ1198882
minusℎ1198882
119866119889119911
(10b)
1198601and 119860
3are called extensional stiffness 119861
1and 119861
3are
called bending-extensional stiffness 1198631and 119863
3are called
bending stiffness 119862 is called shear stiffness and 119868119894119895are the
mass moments of inertiaConsidering the symmetry of the two face sheets of the
truss core sandwich beam about the 119909-axis we have
1198601= 1198603
1198611= minus1198613
1198631= 1198633
(11)
Shock and Vibration 5
Using the stress-displacement relations (7a)ndash(7c) can beexpressed in terms of generalized displacements
21198601
1205972
1199060
1205971199092+ 21198601
1205971199080
120597119909
1205972
1199080
1205971199092= (211986801+ 11986802) 0minus 11986812
120601119909 (12a)
ℎ2
119888
2
1198601
1205972
120601119909
1205971199092minus
ℎ2
119888
2
1198601
1205973
1199080
1205971199093minus ℎ1198881198611
1205973
1199080
1205971199093minus 119862120601119909
+ 119862
1205971199080
120597119909
= (
ℎ2
119888
2
11986801+ 11986822)120601119909
minus (
ℎ2
119888
2
11986801+ ℎ11988811986811)
1205970
120597119909
minus 119868120
(12b)
(
ℎ2
119888
2
1198601+ 1198611ℎ119888)
1205973
120601119909
1205971199093
+ (minus
ℎ2
119888
2
1198601minus 21198611ℎ119888minus 21198631)
1205974
1199080
1205971199094minus 119862
120597120601119909
120597119909
+ 119862
1205972
1199080
1205971199092+ 31198601(
1205971199080
120597119909
)
2
1205972
1199080
1205971199092+ 21198601
1205972
1199060
1205971199092
1205971199080
120597119909
+ 21198601
1205971199060
120597119909
1205972
1199080
1205971199092+ 119865
minus 1205830= (
ℎ2
119888
2
11986801+ ℎ11988811986811)
120597120601119909
120597119909
minus (
ℎ2
119888
2
11986801+ 2ℎ11988811986811+ 211986821)
1205972
0
1205971199092
+ (211986801+ 11986802) 0
(12c)
3 Validation
In order to check the accuracy of the presented model thenatural frequencies are calculated and compared with theresults presented by [10] For linear vibration the in-planemotion of the structure can be neglected Therefore thesimply supported boundary conditions of the sandwich beamcan be expressed as
1199080=
120597120601119909
120597119909
= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (13)
The displacements 120601119909and 119908
0which satisfy (13) can be
represented as
120601119909=
119873
sum
119899=1
120601119899(119905) cos119899120587119909
119897
(14a)
1199080=
119873
sum
119899=1
119908119899(119905) sin119899120587119909
119897
(14b)
We only consider the transverse vibration of the trusscore sandwich beam thus we can neglect the rotatory inertiaterms Substituting (14a) and (14b) into the linear form of
Table 1 Comparison of natural frequencies of the sandwich beamwith the first group of parameters
Mode Ahmed Hwu Lou Present1 56 58 57 57142 mdash 221 219 219583 451 459 465 465174 mdash 742 767 768185 1073 1048 1105 110669
Table 2 Comparison of natural frequencies of the sandwich beamwith the second group of parameters
Mode Lou Present1 13774 137812 52309 523953 109138 109411
(12a) (12b) and (12c) and applying the Galerkin method thenatural frequencies are obtained as shown in Tables 1 and2 The good agreement between the present and publishedresults validates the accuracy of the present method
The dimensions and material properties in Table 1 are
119864119891= 689GPa
ℎ119891= 04572mm
120588119891= 2680 kgm3
119866119888= 008268GPa
ℎ119888= 127mm
120588119888= 328 kgm3
119897 = 09144m
(15)
The dimensions and material properties in Table 2 are
119864119891= 210GPa
ℎ119891= 1mm
120588119891= 7930 kgm3
119866119888= 103667GPa
ℎ119888= 15mm
120588119888= 3131739 kgm3
119897 = 06364m
(16)
4 Nonlinear Frequency Response Equation
In the following analysis the nonlinear vibrations of the trusscore sandwich beam with simply supported boundary con-ditions are considered For nonlinear vibration the simply
6 Shock and Vibration
supported boundary conditions with immoveable edges ofthe beam can be expressed as
1199060= 1199080=
120597120601119909
120597119909
= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (17)
The transformations of variables and parameters areintroduced to obtain the dimensionless equation
1199060=
1199060
119897
1199080=
1199080
ℎ
120601119909= 120601119909
119909 =
119909
119897
119865 =
1198974
119864ℎ4119865
120583 =
119897
ℎ
(
1
120588119864
)
12
120583
119905 =
ℎ
1198972(
119864
120588
)
12
119905
Ω =
1198972
ℎ
(
120588
119864
)
12
Ω
ℎ119888=
ℎ119888
ℎ
119860119894=
119897
119864ℎ2119860119894
119861119894=
119897
119864ℎ3119861119894
119862 =
119897
119864ℎ2119862
119863119894=
119897
119864ℎ4119863119894
119868119905119894=
1
119897(119905+1)120588
119868119905119894 119905 = 0 1 2
(18)
Therefore the dimensionless equations of motion for thetruss core sandwich beam are obtained For convenience ofour study we drop the overbar in the following analysisThe displacements 119906
0 120601119909 and 119908
0which satisfy (17) can be
represented as
1199060=
119873
sum
119899=1
119906 (119905) sin (2119899120587119909) (19a)
120601119909=
119873
sum
119899=1
120601 (119905) cos (119899120587119909) (19b)
1199080=
119873
sum
119899=1
119908 (119905) sin (119899120587119909) (19c)
In addition the excitation has the form
119865 =
119873
sum
119899=1
119891 (119905) sin (119899120587119909) (19d)
Next theGalerkinmethod is used to obtain the nonlinearordinary differential equations of motion for the sandwichbeam The transverse vibrations of the sandwich beam aremainly considered In the same way we can neglect thelongitude and rotatory inertia terms Substituting the dis-placements 119906
0 120601119909 and 119908
0into the resulting equations the
ordinary differential equation is derived
119899+ 120583119899119899+ 1205962
119899119908119899+ 1205721198991199083
119899+ 119892119899(1199081 1199082 119908
119873)
= 120573119899119891119899cos (Ω
119899119905) 119899 = 1 2 119873
(20)
where 119892119899(1199081 1199082 119908
119873) represents the coupled relation
between the different modes and the coupled term does notappear when the single mode response is considered Thecoefficients in (20) are presented in the Appendix
The following scales transformations must be introducedfor obtaining a system which is suitable for the application ofthe method of multiple scales and studying the resonances ofthe sandwich beam
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
120573119899997888rarr 120576120573
119899
(21)
where 120576 is a small dimensionless parameterBecause the coupled term 119892
119899(1199081 1199082 119908
119873) has no
contribution to the primary resonance of each mode intheoretical analysis therefore we can neglect this term andobtain the following equation
119899+ 120576120583119899119899+ 1205962
119899119908119899+ 1205761205721198991199083
119899= 120576120573119899119891119899cos (Ω
119899119905) (22)
The method of multiple scales [29] is used to obtain theuniform solution of (22) in the following form
119908119899(119905 120576) = 119909
1198990(1198790 1198791) + 120576119909
1198991(1198790 1198791) + sdot sdot sdot (23)
where 1198790= 119905 and 119879
1= 120576119905
Then we have the differential operators
119889
119889119905
=
120597
1205971198790
1205971198790
120597119905
+
120597
1205971198791
1205971198791
120597119905
+ sdot sdot sdot = 1198630+ 1205761198631+ sdot sdot sdot (24a)
1198892
1198891199052= (1198630+ 1205761198631+ sdot sdot sdot)
2
= 1198632
0+ 2120576119863
01198631+ sdot sdot sdot (24b)
where1198630= 120597120597119879
0and119863
1= 120597120597119879
1
We have the following relation
Ω119899= 120596119899+ 120576120590119899 (25)
120590119899
is the detuning parameter which quantitativelydescribes the nearness of excitation frequency to the linearnatural frequency
Substituting (23)ndash(25) into (22) and balancing the coef-ficients of the like power of 120576 yield the following differentialequations
Shock and Vibration 7
1205760 order
1198632
01199091198990+ 1205962
1198991199091198990= 0 (26a)
1205761 order
1198632
01199091198991+ 1205962
1198991199091198991= minus2119863
011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093
1198990
+ 120573119899119891119899cos (120596
1198991198790+ 1205901198991198791)
(26b)
The solution of (26a) in the complex form can beexpressed as
1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)
where 119860119899is the complex conjugate of 119860
119899
Substituting (27) into (26b) yields
1198632
01199091198991+ 1205962
1198991199091198991= (minus2119894120596
1198991198631119860119899minus 120583119899119894120596119899119860119899
minus 31205721198991198602
119899119860119899+
120573119899119891119899
2
1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST
(28)
where symbol cc denotes the parts of the complex conjugateof function on the right-hand side of (28) andNST representsthe terms that do not produce secular terms
Function 119860119899may be denoted in the complex form
119860119899=
1
2
119886119899119890119894120593119899 (29)
Substituting (29) into simplified (28) and separating thereal and imaginary parts from the resulting equations thetwo-dimensional averaged equation in the polar form isobtained as
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin (120590
1198991198791minus 120593119899)
2120596119899
(30a)
119886119899119899=
31205721198991198863
119899
8120596119899
minus
120573119899119891119899cos (120590
1198991198791minus 120593119899)
2120596119899
(30b)
Letting
120574119899= 1205901198991198791minus 120593119899 (31)
we have
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin 120574119899
2120596119899
(32a)
119886119899119899= 119886119899120590119899minus
31205721198991198863
119899
8120596119899
+
120573119899119891119899cos 120574119899
2120596119899
(32b)
Steady-state motions occur when 119899and
119899equal zero
which correspond to the singular points of averaged (32a) and(32b) Thus 119886
119899and 120574119899are constants Eliminating 120574
119899by using
the relations between trigonometric functions we obtain thefrequency response equation of the truss core sandwich beamwith primary resonance as follows
(
120583119899119886119899
2
)
2
+ (120590119899119886119899minus
31205721198991198863
119899
8120596119899
)
2
= (
120573119899119891119899
2120596119899
)
2
(33)
Subsequently the subharmonic resonance and the super-harmonic resonance of the sandwich beams are consideredIn the two cases the following two transformations arerequired
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
(34)
For subharmonic resonance the detuning parameter 120590119899
is introduced according toΩ119899= 3120596119899+ 120576120590119899 (35)
Similarly the averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
31205721198991205731198991198862
119899Λ sin 120574
119899
4120596119899
(36a)
119886119899119899= 119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
minus
9120572119899120573119899Λ1198862
119899cos 120574119899
4120596119899
(36b)
where Λ = (12)119891119899(1205962
119899minus Ω2
119899)minus1
The frequency response equation of the truss core sand-wich beam with subharmonic resonance is
(minus
120583119899119886119899
2
)
2
+
1
9
[119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
]
2
=
91205722
1198991205732
1198991198864
119899Λ2
161205962
119899
(37)
For superharmonic resonance the detuning parameter120590119899
is introduced according to3Ω119899= 120596119899+ 120576120590119899 (38)
The averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
1205721198991205733
119899Λ3 sin 120574
119899
120596119899
(39a)
119886119899119899= 119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
minus
1205721198991205733
119899Λ3 cos 120574
119899
120596119899
(39b)
The frequency response equation of the truss core sand-wich beam with superharmonic resonance is
(minus
120583119899119886119899
2
)
2
+ [119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
]
2
=
1205722
1198991205736
119899Λ6
1205962
119899
(40)
The plot of 119886119899as a function of 120590
119899for given 119891 is called
a nonlinear frequency response curve which can be usedto describe the hardening behavior of a structure In thefollowing analysis the parameters in (33) (37) and (40)should be calculated by using the Appendix
8 Shock and Vibration
120596
32
28
24
20
16
12
8
4
120588c120588
000 002 004 006 008 010
Figure 3 Variations of dimensionless linear natural frequencieswith respect to the strut radius Solid line first-order frequencydash line second-order frequency dot-dash line third-order fre-quency
5 Numerical Results
In the study the influences of the structural parameterson the dimensionless linear natural frequencies of the trusscore sandwich beam are studied at first by neglecting thenonlinear terms in (20) The face sheets and the truss of thesandwich beam are made of the same material The materialproperties are taken as follows 119864 = 689GPa and 120588 =
2680 kgm3 Figure 3 indicates that the first-order naturalfrequency decreases continuously and the second-order andthird-order natural frequencies initially increase and thendecrease with the increase of strut radiusThe relative densitycorresponding to the turning point in the changing curveof the second-order natural frequency is smaller than thatof the third-order natural frequency When the strut radiusincreases the mass moment of inertia and shear stiffness ofthe sandwich beam both increase the changes of the naturalfrequencies depend upon the combined effect of the twofactors The structural parameters in the figure are given asℎ = 0012m ℎ
119888= 001m 120579 = 1205874 and 119897 = 07m
Figure 4 shows that the first three natural frequenciesall increase with the rise of the core height and the higher-order natural frequency increases faster than the lower-ordernatural frequency When the core height increases the massmoment of inertia extensional stiffness bending stiffnessand shear stiffness of the sandwich beam all decrease thechanges of the natural frequencies depend upon the com-bined effect of those factors The structural parameters in thefigure are given as ℎ = 0012m 119903 = 06mm 120579 = 1205874 and119897 = 07m
The steady-state responses of the first three modes ofthe sandwich beam with primary resonance are investigatedThe hardening-type nonlinearity of the first three modes allbecomes weaker with increasing strut radius and core heightas shown in Figures 5 and 6 The bending of the frequencyresponse curve mainly depends on the nonlinear term andthe frequency parameter according to (33) It can be known
120596
32
28
24
20
16
12
8
4
hch
072 076 080 084 088 092
Figure 4 Variations of dimensionless linear natural frequencieswith respect to the core height Solid line first-order frequency dashline second-order frequency dot-dash line third-order frequency
from 120572119899in the Appendix that the nonlinear term is related to
the extensional stiffness of the face sheets as well as the massmoment of inertia thickness and length of the sandwichbeam The theoretical analyses indicate that the reasons forthe weakening of the hardening-type nonlinearity with therise of strut radius and core height are different
At first we discuss the influence of strut radius on thehardening behavior of the sandwich beam With the increaseof the strut radius the hardening behavior of the beam isweakened by the rise of themassmoment of inertia and shearstiffness It is concluded from (33) that the increases of themass moment of inertia and shear stiffness both weaken thehardening behavior The structural parameters in Figure 5are the same as in Figure 3 With the increase of the coreheight the hardening behavior of the beam is weakened It isconcluded that the decrease of extensional stiffness tends toweaken the hardening behavior and the decrease of the massmoment of inertia shear stiffness and bending stiffness tendsto strengthen the hardening behavior of the sandwich beamObviously the effect of the extensional stiffness is dominantaccording to Figure 6The structural parameters in this figureare the same as in Figure 4
Furthermore the hardening-type nonlinearity of thehigher mode is stronger than that of the lower mode andthe influences of the structural parameters on the hardeningbehavior for the lower mode are slightly larger than for thehigher mode In the analysis of the primary resonance of thesandwich beam 120583 = 0001 and 119891 = 1 are always chosen
Subharmonic and superharmonic resonances are rela-tively easy to be generated at the lower modes especially atthe first mode The nonlinear frequency response curves forthe first mode of the sandwich beam with subharmonic andsuperharmonic resonances are investigated in the paper thenumerical results are shown in Figures 7 and 8 respectivelyFigure 7 demonstrates that the subharmonic responses onlyoccur in the frequency ranges which are higher than thelinear natural frequency no matter which and how structuralparameters change Increasing strut radius and core height
Shock and Vibration 9
a1
25
2
15
1
05
1205901
minus10 minus5 0 5 10
(a)a2
10
12
06
04
08
02
1205902
minus10 minus5 0 5 10
(b)
a3
07
08
05
04
06
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 5 Influences of the strut radius on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line 120588 = 002 dash line 120588 = 004 dot-dash line 120588 = 006
will weaken the hardening-type nonlinearity in the resonancecase just as the influences of the structural parameters onthe responses of the beam with primary resonance Figure 8demonstrates that the shapes of the frequency responsecurves for the first mode with superharmonic responses aresimilar to that of the sandwich beamwith primary resonanceThe variations in the structural parameters can not only affectthe hardening behaviors of the beam but also significantlychange the frequency range of the resonance responses
Decreasing strut radius and core height can broaden thefrequency ranges of the resonance responses
Based on the analysis of the frequency response equa-tions we conclude that in the cases of subharmonic andsuperharmonic resonances the mechanism of the variationof the hardening behavior via the structural parameters is thesame as that in the case of primary resonance 120583 = 0001and 119891 = 2 are chosen in Figures 7 and 8 Moreover thestructural parameters in Figures 7(a) and 8(a) are the same
10 Shock and Vibration
a1
14
12
10
08
06
04
02
1205901
minus10 minus5 0 5 10
(a)a2
06
05
04
03
02
01
1205902
minus10 minus5 0 5 10
(b)
a3
04
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 6 Influences of the core height on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash line ℎ
119888ℎ = 09
as in Figure 3 and the structural parameters in Figures 7(b)and 8(b) are the same as in Figure 4
6 Conclusions
The natural frequency and hardening behavior of sandwichbeam with pyramidal truss core are investigated The trusscore is equivalent to a continuous homogeneousmaterial anda Zig-Zag theory is used to derive the nonlinear governingequation of motion for the sandwich beam At first thenatural frequencies are calculated to validate the presentmodel and the changing rules of the natural frequency with
the strut radius and core height are obtained It is illustratedthat the first-order natural frequency decreases continuouslyand the second-order and third-order natural frequenciesinitially increase and then decrease with the increase of strutradius and the first three natural frequencies all increase withthe rise of the core height Then the hardening behaviorsof the sandwich beam are analyzed Numerical simulationsindicate that the effects of the two structural parameters onthe hardening behavior in the cases of primary resonancesubharmonic resonance and superharmonic resonance aresimilar However the mechanisms of the variations of thehardening behavior are very different The influences of
Shock and Vibration 11
a1
15
1
05
0
1205901
minus10 minus5 0 5 10
(a)a1
1205901
minus10 minus5 0 5 10
08
07
06
05
04
03
02
01
0
(b)
Figure 7 The hardening behavior of the first mode with subharmonic resonance (a) The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
1205901
minus4 minus2 0 2 4
a1
16
14
12
10
08
06
04
02
(a)
1205901
minus5 0 5 10 15
a1
14
12
10
08
06
04
02
(b)
Figure 8The hardening behavior of the first mode with superharmonic resonance (a)The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
the structural parameters on the frequency parameter and thenonlinear term of the sandwich beam are studied in the paperfor analyzing the variations of the hardening behavior
Thehardening nonlinearity bends the frequency responsecurve to the right Generally under the same excitationthe stronger the hardening-type nonlinearity the smaller
the response amplitude We investigate the influences ofstrut radius and core height on the dynamical responses ofthe sandwich beam Using the equations obtained in thispaper the influences of the other structural parameters ofthe sandwich beam on the vibrations can be convenientlyanalyzed
12 Shock and Vibration
It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics
Appendix
The relations between the in-plane displacements and trans-verse displacement are
119906119899= 1198781198991199082
119899
119878119899= minus
1198991198861120587
8
(A1)
The relations between the rotation angles and transversedisplacement are
120601119899= 119870119899119908119899
119870119899=
1198991198873120587 minus 1198993
11988721205873
119899211988711205872minus 1
(A2)
The coefficients in (20) are given as follows
120574119899=
1198792119899
1198791119899
1205962
119899= minus
1198793119899
1198791119899
120572119899= minus
1198794119899
1198791119899
120573119899=
1198795119899
1198791119899
1198791119899=
1198889
2
1198792119899=
1198888
2
1198793119899=
1198994
1205874
1198882
2
minus
1198992
1205872
1198884
2
minus
1198991205871198701198991198883
2
+
1198993
1205873
1198701198991198881
2
1198794119899= minus
1198994
1205874
8
+
1198993
1205873
1198781198991198886
2
minus 1198993
1205873
1198781198991198885
1198795119899=
1198887
2
(A3)
where
1198861=
ℎ2
1198972
1198871= minus
ℎ2
ℎ
2
119888
21198972
1198601
119862
1198872=
ℎ3
ℎ
2
119888
21198973
1198601
119862
+
ℎ3
ℎ119888
1198973
1198611
119862
1198873= minus
ℎ
119897
1198881=
119897
3ℎ
(
ℎ119888
2
+ ℎ
1198611
1198601
)
1198882= minus
1
3
(
ℎ
2
119888
2
+ 2ℎ
1198611
1198601
+ 2
1198631
1198601
)
1198883= minus
1198973
3ℎ3
119862
1198601
1198884=
1198972
3ℎ2
119862
1198601
1198885=
21198972
3ℎ2
1198886=
21198972
3ℎ2
1198887=
119897
3ℎ1198601
1198888=
1198972
3ℎ21198601
1198889=
1198972
3ℎ21198601
(211986801+ 11986802)
(A4)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)
References
[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 5
Using the stress-displacement relations (7a)ndash(7c) can beexpressed in terms of generalized displacements
21198601
1205972
1199060
1205971199092+ 21198601
1205971199080
120597119909
1205972
1199080
1205971199092= (211986801+ 11986802) 0minus 11986812
120601119909 (12a)
ℎ2
119888
2
1198601
1205972
120601119909
1205971199092minus
ℎ2
119888
2
1198601
1205973
1199080
1205971199093minus ℎ1198881198611
1205973
1199080
1205971199093minus 119862120601119909
+ 119862
1205971199080
120597119909
= (
ℎ2
119888
2
11986801+ 11986822)120601119909
minus (
ℎ2
119888
2
11986801+ ℎ11988811986811)
1205970
120597119909
minus 119868120
(12b)
(
ℎ2
119888
2
1198601+ 1198611ℎ119888)
1205973
120601119909
1205971199093
+ (minus
ℎ2
119888
2
1198601minus 21198611ℎ119888minus 21198631)
1205974
1199080
1205971199094minus 119862
120597120601119909
120597119909
+ 119862
1205972
1199080
1205971199092+ 31198601(
1205971199080
120597119909
)
2
1205972
1199080
1205971199092+ 21198601
1205972
1199060
1205971199092
1205971199080
120597119909
+ 21198601
1205971199060
120597119909
1205972
1199080
1205971199092+ 119865
minus 1205830= (
ℎ2
119888
2
11986801+ ℎ11988811986811)
120597120601119909
120597119909
minus (
ℎ2
119888
2
11986801+ 2ℎ11988811986811+ 211986821)
1205972
0
1205971199092
+ (211986801+ 11986802) 0
(12c)
3 Validation
In order to check the accuracy of the presented model thenatural frequencies are calculated and compared with theresults presented by [10] For linear vibration the in-planemotion of the structure can be neglected Therefore thesimply supported boundary conditions of the sandwich beamcan be expressed as
1199080=
120597120601119909
120597119909
= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (13)
The displacements 120601119909and 119908
0which satisfy (13) can be
represented as
120601119909=
119873
sum
119899=1
120601119899(119905) cos119899120587119909
119897
(14a)
1199080=
119873
sum
119899=1
119908119899(119905) sin119899120587119909
119897
(14b)
We only consider the transverse vibration of the trusscore sandwich beam thus we can neglect the rotatory inertiaterms Substituting (14a) and (14b) into the linear form of
Table 1 Comparison of natural frequencies of the sandwich beamwith the first group of parameters
Mode Ahmed Hwu Lou Present1 56 58 57 57142 mdash 221 219 219583 451 459 465 465174 mdash 742 767 768185 1073 1048 1105 110669
Table 2 Comparison of natural frequencies of the sandwich beamwith the second group of parameters
Mode Lou Present1 13774 137812 52309 523953 109138 109411
(12a) (12b) and (12c) and applying the Galerkin method thenatural frequencies are obtained as shown in Tables 1 and2 The good agreement between the present and publishedresults validates the accuracy of the present method
The dimensions and material properties in Table 1 are
119864119891= 689GPa
ℎ119891= 04572mm
120588119891= 2680 kgm3
119866119888= 008268GPa
ℎ119888= 127mm
120588119888= 328 kgm3
119897 = 09144m
(15)
The dimensions and material properties in Table 2 are
119864119891= 210GPa
ℎ119891= 1mm
120588119891= 7930 kgm3
119866119888= 103667GPa
ℎ119888= 15mm
120588119888= 3131739 kgm3
119897 = 06364m
(16)
4 Nonlinear Frequency Response Equation
In the following analysis the nonlinear vibrations of the trusscore sandwich beam with simply supported boundary con-ditions are considered For nonlinear vibration the simply
6 Shock and Vibration
supported boundary conditions with immoveable edges ofthe beam can be expressed as
1199060= 1199080=
120597120601119909
120597119909
= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (17)
The transformations of variables and parameters areintroduced to obtain the dimensionless equation
1199060=
1199060
119897
1199080=
1199080
ℎ
120601119909= 120601119909
119909 =
119909
119897
119865 =
1198974
119864ℎ4119865
120583 =
119897
ℎ
(
1
120588119864
)
12
120583
119905 =
ℎ
1198972(
119864
120588
)
12
119905
Ω =
1198972
ℎ
(
120588
119864
)
12
Ω
ℎ119888=
ℎ119888
ℎ
119860119894=
119897
119864ℎ2119860119894
119861119894=
119897
119864ℎ3119861119894
119862 =
119897
119864ℎ2119862
119863119894=
119897
119864ℎ4119863119894
119868119905119894=
1
119897(119905+1)120588
119868119905119894 119905 = 0 1 2
(18)
Therefore the dimensionless equations of motion for thetruss core sandwich beam are obtained For convenience ofour study we drop the overbar in the following analysisThe displacements 119906
0 120601119909 and 119908
0which satisfy (17) can be
represented as
1199060=
119873
sum
119899=1
119906 (119905) sin (2119899120587119909) (19a)
120601119909=
119873
sum
119899=1
120601 (119905) cos (119899120587119909) (19b)
1199080=
119873
sum
119899=1
119908 (119905) sin (119899120587119909) (19c)
In addition the excitation has the form
119865 =
119873
sum
119899=1
119891 (119905) sin (119899120587119909) (19d)
Next theGalerkinmethod is used to obtain the nonlinearordinary differential equations of motion for the sandwichbeam The transverse vibrations of the sandwich beam aremainly considered In the same way we can neglect thelongitude and rotatory inertia terms Substituting the dis-placements 119906
0 120601119909 and 119908
0into the resulting equations the
ordinary differential equation is derived
119899+ 120583119899119899+ 1205962
119899119908119899+ 1205721198991199083
119899+ 119892119899(1199081 1199082 119908
119873)
= 120573119899119891119899cos (Ω
119899119905) 119899 = 1 2 119873
(20)
where 119892119899(1199081 1199082 119908
119873) represents the coupled relation
between the different modes and the coupled term does notappear when the single mode response is considered Thecoefficients in (20) are presented in the Appendix
The following scales transformations must be introducedfor obtaining a system which is suitable for the application ofthe method of multiple scales and studying the resonances ofthe sandwich beam
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
120573119899997888rarr 120576120573
119899
(21)
where 120576 is a small dimensionless parameterBecause the coupled term 119892
119899(1199081 1199082 119908
119873) has no
contribution to the primary resonance of each mode intheoretical analysis therefore we can neglect this term andobtain the following equation
119899+ 120576120583119899119899+ 1205962
119899119908119899+ 1205761205721198991199083
119899= 120576120573119899119891119899cos (Ω
119899119905) (22)
The method of multiple scales [29] is used to obtain theuniform solution of (22) in the following form
119908119899(119905 120576) = 119909
1198990(1198790 1198791) + 120576119909
1198991(1198790 1198791) + sdot sdot sdot (23)
where 1198790= 119905 and 119879
1= 120576119905
Then we have the differential operators
119889
119889119905
=
120597
1205971198790
1205971198790
120597119905
+
120597
1205971198791
1205971198791
120597119905
+ sdot sdot sdot = 1198630+ 1205761198631+ sdot sdot sdot (24a)
1198892
1198891199052= (1198630+ 1205761198631+ sdot sdot sdot)
2
= 1198632
0+ 2120576119863
01198631+ sdot sdot sdot (24b)
where1198630= 120597120597119879
0and119863
1= 120597120597119879
1
We have the following relation
Ω119899= 120596119899+ 120576120590119899 (25)
120590119899
is the detuning parameter which quantitativelydescribes the nearness of excitation frequency to the linearnatural frequency
Substituting (23)ndash(25) into (22) and balancing the coef-ficients of the like power of 120576 yield the following differentialequations
Shock and Vibration 7
1205760 order
1198632
01199091198990+ 1205962
1198991199091198990= 0 (26a)
1205761 order
1198632
01199091198991+ 1205962
1198991199091198991= minus2119863
011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093
1198990
+ 120573119899119891119899cos (120596
1198991198790+ 1205901198991198791)
(26b)
The solution of (26a) in the complex form can beexpressed as
1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)
where 119860119899is the complex conjugate of 119860
119899
Substituting (27) into (26b) yields
1198632
01199091198991+ 1205962
1198991199091198991= (minus2119894120596
1198991198631119860119899minus 120583119899119894120596119899119860119899
minus 31205721198991198602
119899119860119899+
120573119899119891119899
2
1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST
(28)
where symbol cc denotes the parts of the complex conjugateof function on the right-hand side of (28) andNST representsthe terms that do not produce secular terms
Function 119860119899may be denoted in the complex form
119860119899=
1
2
119886119899119890119894120593119899 (29)
Substituting (29) into simplified (28) and separating thereal and imaginary parts from the resulting equations thetwo-dimensional averaged equation in the polar form isobtained as
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin (120590
1198991198791minus 120593119899)
2120596119899
(30a)
119886119899119899=
31205721198991198863
119899
8120596119899
minus
120573119899119891119899cos (120590
1198991198791minus 120593119899)
2120596119899
(30b)
Letting
120574119899= 1205901198991198791minus 120593119899 (31)
we have
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin 120574119899
2120596119899
(32a)
119886119899119899= 119886119899120590119899minus
31205721198991198863
119899
8120596119899
+
120573119899119891119899cos 120574119899
2120596119899
(32b)
Steady-state motions occur when 119899and
119899equal zero
which correspond to the singular points of averaged (32a) and(32b) Thus 119886
119899and 120574119899are constants Eliminating 120574
119899by using
the relations between trigonometric functions we obtain thefrequency response equation of the truss core sandwich beamwith primary resonance as follows
(
120583119899119886119899
2
)
2
+ (120590119899119886119899minus
31205721198991198863
119899
8120596119899
)
2
= (
120573119899119891119899
2120596119899
)
2
(33)
Subsequently the subharmonic resonance and the super-harmonic resonance of the sandwich beams are consideredIn the two cases the following two transformations arerequired
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
(34)
For subharmonic resonance the detuning parameter 120590119899
is introduced according toΩ119899= 3120596119899+ 120576120590119899 (35)
Similarly the averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
31205721198991205731198991198862
119899Λ sin 120574
119899
4120596119899
(36a)
119886119899119899= 119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
minus
9120572119899120573119899Λ1198862
119899cos 120574119899
4120596119899
(36b)
where Λ = (12)119891119899(1205962
119899minus Ω2
119899)minus1
The frequency response equation of the truss core sand-wich beam with subharmonic resonance is
(minus
120583119899119886119899
2
)
2
+
1
9
[119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
]
2
=
91205722
1198991205732
1198991198864
119899Λ2
161205962
119899
(37)
For superharmonic resonance the detuning parameter120590119899
is introduced according to3Ω119899= 120596119899+ 120576120590119899 (38)
The averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
1205721198991205733
119899Λ3 sin 120574
119899
120596119899
(39a)
119886119899119899= 119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
minus
1205721198991205733
119899Λ3 cos 120574
119899
120596119899
(39b)
The frequency response equation of the truss core sand-wich beam with superharmonic resonance is
(minus
120583119899119886119899
2
)
2
+ [119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
]
2
=
1205722
1198991205736
119899Λ6
1205962
119899
(40)
The plot of 119886119899as a function of 120590
119899for given 119891 is called
a nonlinear frequency response curve which can be usedto describe the hardening behavior of a structure In thefollowing analysis the parameters in (33) (37) and (40)should be calculated by using the Appendix
8 Shock and Vibration
120596
32
28
24
20
16
12
8
4
120588c120588
000 002 004 006 008 010
Figure 3 Variations of dimensionless linear natural frequencieswith respect to the strut radius Solid line first-order frequencydash line second-order frequency dot-dash line third-order fre-quency
5 Numerical Results
In the study the influences of the structural parameterson the dimensionless linear natural frequencies of the trusscore sandwich beam are studied at first by neglecting thenonlinear terms in (20) The face sheets and the truss of thesandwich beam are made of the same material The materialproperties are taken as follows 119864 = 689GPa and 120588 =
2680 kgm3 Figure 3 indicates that the first-order naturalfrequency decreases continuously and the second-order andthird-order natural frequencies initially increase and thendecrease with the increase of strut radiusThe relative densitycorresponding to the turning point in the changing curveof the second-order natural frequency is smaller than thatof the third-order natural frequency When the strut radiusincreases the mass moment of inertia and shear stiffness ofthe sandwich beam both increase the changes of the naturalfrequencies depend upon the combined effect of the twofactors The structural parameters in the figure are given asℎ = 0012m ℎ
119888= 001m 120579 = 1205874 and 119897 = 07m
Figure 4 shows that the first three natural frequenciesall increase with the rise of the core height and the higher-order natural frequency increases faster than the lower-ordernatural frequency When the core height increases the massmoment of inertia extensional stiffness bending stiffnessand shear stiffness of the sandwich beam all decrease thechanges of the natural frequencies depend upon the com-bined effect of those factors The structural parameters in thefigure are given as ℎ = 0012m 119903 = 06mm 120579 = 1205874 and119897 = 07m
The steady-state responses of the first three modes ofthe sandwich beam with primary resonance are investigatedThe hardening-type nonlinearity of the first three modes allbecomes weaker with increasing strut radius and core heightas shown in Figures 5 and 6 The bending of the frequencyresponse curve mainly depends on the nonlinear term andthe frequency parameter according to (33) It can be known
120596
32
28
24
20
16
12
8
4
hch
072 076 080 084 088 092
Figure 4 Variations of dimensionless linear natural frequencieswith respect to the core height Solid line first-order frequency dashline second-order frequency dot-dash line third-order frequency
from 120572119899in the Appendix that the nonlinear term is related to
the extensional stiffness of the face sheets as well as the massmoment of inertia thickness and length of the sandwichbeam The theoretical analyses indicate that the reasons forthe weakening of the hardening-type nonlinearity with therise of strut radius and core height are different
At first we discuss the influence of strut radius on thehardening behavior of the sandwich beam With the increaseof the strut radius the hardening behavior of the beam isweakened by the rise of themassmoment of inertia and shearstiffness It is concluded from (33) that the increases of themass moment of inertia and shear stiffness both weaken thehardening behavior The structural parameters in Figure 5are the same as in Figure 3 With the increase of the coreheight the hardening behavior of the beam is weakened It isconcluded that the decrease of extensional stiffness tends toweaken the hardening behavior and the decrease of the massmoment of inertia shear stiffness and bending stiffness tendsto strengthen the hardening behavior of the sandwich beamObviously the effect of the extensional stiffness is dominantaccording to Figure 6The structural parameters in this figureare the same as in Figure 4
Furthermore the hardening-type nonlinearity of thehigher mode is stronger than that of the lower mode andthe influences of the structural parameters on the hardeningbehavior for the lower mode are slightly larger than for thehigher mode In the analysis of the primary resonance of thesandwich beam 120583 = 0001 and 119891 = 1 are always chosen
Subharmonic and superharmonic resonances are rela-tively easy to be generated at the lower modes especially atthe first mode The nonlinear frequency response curves forthe first mode of the sandwich beam with subharmonic andsuperharmonic resonances are investigated in the paper thenumerical results are shown in Figures 7 and 8 respectivelyFigure 7 demonstrates that the subharmonic responses onlyoccur in the frequency ranges which are higher than thelinear natural frequency no matter which and how structuralparameters change Increasing strut radius and core height
Shock and Vibration 9
a1
25
2
15
1
05
1205901
minus10 minus5 0 5 10
(a)a2
10
12
06
04
08
02
1205902
minus10 minus5 0 5 10
(b)
a3
07
08
05
04
06
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 5 Influences of the strut radius on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line 120588 = 002 dash line 120588 = 004 dot-dash line 120588 = 006
will weaken the hardening-type nonlinearity in the resonancecase just as the influences of the structural parameters onthe responses of the beam with primary resonance Figure 8demonstrates that the shapes of the frequency responsecurves for the first mode with superharmonic responses aresimilar to that of the sandwich beamwith primary resonanceThe variations in the structural parameters can not only affectthe hardening behaviors of the beam but also significantlychange the frequency range of the resonance responses
Decreasing strut radius and core height can broaden thefrequency ranges of the resonance responses
Based on the analysis of the frequency response equa-tions we conclude that in the cases of subharmonic andsuperharmonic resonances the mechanism of the variationof the hardening behavior via the structural parameters is thesame as that in the case of primary resonance 120583 = 0001and 119891 = 2 are chosen in Figures 7 and 8 Moreover thestructural parameters in Figures 7(a) and 8(a) are the same
10 Shock and Vibration
a1
14
12
10
08
06
04
02
1205901
minus10 minus5 0 5 10
(a)a2
06
05
04
03
02
01
1205902
minus10 minus5 0 5 10
(b)
a3
04
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 6 Influences of the core height on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash line ℎ
119888ℎ = 09
as in Figure 3 and the structural parameters in Figures 7(b)and 8(b) are the same as in Figure 4
6 Conclusions
The natural frequency and hardening behavior of sandwichbeam with pyramidal truss core are investigated The trusscore is equivalent to a continuous homogeneousmaterial anda Zig-Zag theory is used to derive the nonlinear governingequation of motion for the sandwich beam At first thenatural frequencies are calculated to validate the presentmodel and the changing rules of the natural frequency with
the strut radius and core height are obtained It is illustratedthat the first-order natural frequency decreases continuouslyand the second-order and third-order natural frequenciesinitially increase and then decrease with the increase of strutradius and the first three natural frequencies all increase withthe rise of the core height Then the hardening behaviorsof the sandwich beam are analyzed Numerical simulationsindicate that the effects of the two structural parameters onthe hardening behavior in the cases of primary resonancesubharmonic resonance and superharmonic resonance aresimilar However the mechanisms of the variations of thehardening behavior are very different The influences of
Shock and Vibration 11
a1
15
1
05
0
1205901
minus10 minus5 0 5 10
(a)a1
1205901
minus10 minus5 0 5 10
08
07
06
05
04
03
02
01
0
(b)
Figure 7 The hardening behavior of the first mode with subharmonic resonance (a) The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
1205901
minus4 minus2 0 2 4
a1
16
14
12
10
08
06
04
02
(a)
1205901
minus5 0 5 10 15
a1
14
12
10
08
06
04
02
(b)
Figure 8The hardening behavior of the first mode with superharmonic resonance (a)The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
the structural parameters on the frequency parameter and thenonlinear term of the sandwich beam are studied in the paperfor analyzing the variations of the hardening behavior
Thehardening nonlinearity bends the frequency responsecurve to the right Generally under the same excitationthe stronger the hardening-type nonlinearity the smaller
the response amplitude We investigate the influences ofstrut radius and core height on the dynamical responses ofthe sandwich beam Using the equations obtained in thispaper the influences of the other structural parameters ofthe sandwich beam on the vibrations can be convenientlyanalyzed
12 Shock and Vibration
It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics
Appendix
The relations between the in-plane displacements and trans-verse displacement are
119906119899= 1198781198991199082
119899
119878119899= minus
1198991198861120587
8
(A1)
The relations between the rotation angles and transversedisplacement are
120601119899= 119870119899119908119899
119870119899=
1198991198873120587 minus 1198993
11988721205873
119899211988711205872minus 1
(A2)
The coefficients in (20) are given as follows
120574119899=
1198792119899
1198791119899
1205962
119899= minus
1198793119899
1198791119899
120572119899= minus
1198794119899
1198791119899
120573119899=
1198795119899
1198791119899
1198791119899=
1198889
2
1198792119899=
1198888
2
1198793119899=
1198994
1205874
1198882
2
minus
1198992
1205872
1198884
2
minus
1198991205871198701198991198883
2
+
1198993
1205873
1198701198991198881
2
1198794119899= minus
1198994
1205874
8
+
1198993
1205873
1198781198991198886
2
minus 1198993
1205873
1198781198991198885
1198795119899=
1198887
2
(A3)
where
1198861=
ℎ2
1198972
1198871= minus
ℎ2
ℎ
2
119888
21198972
1198601
119862
1198872=
ℎ3
ℎ
2
119888
21198973
1198601
119862
+
ℎ3
ℎ119888
1198973
1198611
119862
1198873= minus
ℎ
119897
1198881=
119897
3ℎ
(
ℎ119888
2
+ ℎ
1198611
1198601
)
1198882= minus
1
3
(
ℎ
2
119888
2
+ 2ℎ
1198611
1198601
+ 2
1198631
1198601
)
1198883= minus
1198973
3ℎ3
119862
1198601
1198884=
1198972
3ℎ2
119862
1198601
1198885=
21198972
3ℎ2
1198886=
21198972
3ℎ2
1198887=
119897
3ℎ1198601
1198888=
1198972
3ℎ21198601
1198889=
1198972
3ℎ21198601
(211986801+ 11986802)
(A4)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)
References
[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Shock and Vibration
supported boundary conditions with immoveable edges ofthe beam can be expressed as
1199060= 1199080=
120597120601119909
120597119909
= 1198721199091199091= 1198721199091199093= 0 at 119909 = 0 119897 (17)
The transformations of variables and parameters areintroduced to obtain the dimensionless equation
1199060=
1199060
119897
1199080=
1199080
ℎ
120601119909= 120601119909
119909 =
119909
119897
119865 =
1198974
119864ℎ4119865
120583 =
119897
ℎ
(
1
120588119864
)
12
120583
119905 =
ℎ
1198972(
119864
120588
)
12
119905
Ω =
1198972
ℎ
(
120588
119864
)
12
Ω
ℎ119888=
ℎ119888
ℎ
119860119894=
119897
119864ℎ2119860119894
119861119894=
119897
119864ℎ3119861119894
119862 =
119897
119864ℎ2119862
119863119894=
119897
119864ℎ4119863119894
119868119905119894=
1
119897(119905+1)120588
119868119905119894 119905 = 0 1 2
(18)
Therefore the dimensionless equations of motion for thetruss core sandwich beam are obtained For convenience ofour study we drop the overbar in the following analysisThe displacements 119906
0 120601119909 and 119908
0which satisfy (17) can be
represented as
1199060=
119873
sum
119899=1
119906 (119905) sin (2119899120587119909) (19a)
120601119909=
119873
sum
119899=1
120601 (119905) cos (119899120587119909) (19b)
1199080=
119873
sum
119899=1
119908 (119905) sin (119899120587119909) (19c)
In addition the excitation has the form
119865 =
119873
sum
119899=1
119891 (119905) sin (119899120587119909) (19d)
Next theGalerkinmethod is used to obtain the nonlinearordinary differential equations of motion for the sandwichbeam The transverse vibrations of the sandwich beam aremainly considered In the same way we can neglect thelongitude and rotatory inertia terms Substituting the dis-placements 119906
0 120601119909 and 119908
0into the resulting equations the
ordinary differential equation is derived
119899+ 120583119899119899+ 1205962
119899119908119899+ 1205721198991199083
119899+ 119892119899(1199081 1199082 119908
119873)
= 120573119899119891119899cos (Ω
119899119905) 119899 = 1 2 119873
(20)
where 119892119899(1199081 1199082 119908
119873) represents the coupled relation
between the different modes and the coupled term does notappear when the single mode response is considered Thecoefficients in (20) are presented in the Appendix
The following scales transformations must be introducedfor obtaining a system which is suitable for the application ofthe method of multiple scales and studying the resonances ofthe sandwich beam
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
120573119899997888rarr 120576120573
119899
(21)
where 120576 is a small dimensionless parameterBecause the coupled term 119892
119899(1199081 1199082 119908
119873) has no
contribution to the primary resonance of each mode intheoretical analysis therefore we can neglect this term andobtain the following equation
119899+ 120576120583119899119899+ 1205962
119899119908119899+ 1205761205721198991199083
119899= 120576120573119899119891119899cos (Ω
119899119905) (22)
The method of multiple scales [29] is used to obtain theuniform solution of (22) in the following form
119908119899(119905 120576) = 119909
1198990(1198790 1198791) + 120576119909
1198991(1198790 1198791) + sdot sdot sdot (23)
where 1198790= 119905 and 119879
1= 120576119905
Then we have the differential operators
119889
119889119905
=
120597
1205971198790
1205971198790
120597119905
+
120597
1205971198791
1205971198791
120597119905
+ sdot sdot sdot = 1198630+ 1205761198631+ sdot sdot sdot (24a)
1198892
1198891199052= (1198630+ 1205761198631+ sdot sdot sdot)
2
= 1198632
0+ 2120576119863
01198631+ sdot sdot sdot (24b)
where1198630= 120597120597119879
0and119863
1= 120597120597119879
1
We have the following relation
Ω119899= 120596119899+ 120576120590119899 (25)
120590119899
is the detuning parameter which quantitativelydescribes the nearness of excitation frequency to the linearnatural frequency
Substituting (23)ndash(25) into (22) and balancing the coef-ficients of the like power of 120576 yield the following differentialequations
Shock and Vibration 7
1205760 order
1198632
01199091198990+ 1205962
1198991199091198990= 0 (26a)
1205761 order
1198632
01199091198991+ 1205962
1198991199091198991= minus2119863
011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093
1198990
+ 120573119899119891119899cos (120596
1198991198790+ 1205901198991198791)
(26b)
The solution of (26a) in the complex form can beexpressed as
1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)
where 119860119899is the complex conjugate of 119860
119899
Substituting (27) into (26b) yields
1198632
01199091198991+ 1205962
1198991199091198991= (minus2119894120596
1198991198631119860119899minus 120583119899119894120596119899119860119899
minus 31205721198991198602
119899119860119899+
120573119899119891119899
2
1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST
(28)
where symbol cc denotes the parts of the complex conjugateof function on the right-hand side of (28) andNST representsthe terms that do not produce secular terms
Function 119860119899may be denoted in the complex form
119860119899=
1
2
119886119899119890119894120593119899 (29)
Substituting (29) into simplified (28) and separating thereal and imaginary parts from the resulting equations thetwo-dimensional averaged equation in the polar form isobtained as
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin (120590
1198991198791minus 120593119899)
2120596119899
(30a)
119886119899119899=
31205721198991198863
119899
8120596119899
minus
120573119899119891119899cos (120590
1198991198791minus 120593119899)
2120596119899
(30b)
Letting
120574119899= 1205901198991198791minus 120593119899 (31)
we have
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin 120574119899
2120596119899
(32a)
119886119899119899= 119886119899120590119899minus
31205721198991198863
119899
8120596119899
+
120573119899119891119899cos 120574119899
2120596119899
(32b)
Steady-state motions occur when 119899and
119899equal zero
which correspond to the singular points of averaged (32a) and(32b) Thus 119886
119899and 120574119899are constants Eliminating 120574
119899by using
the relations between trigonometric functions we obtain thefrequency response equation of the truss core sandwich beamwith primary resonance as follows
(
120583119899119886119899
2
)
2
+ (120590119899119886119899minus
31205721198991198863
119899
8120596119899
)
2
= (
120573119899119891119899
2120596119899
)
2
(33)
Subsequently the subharmonic resonance and the super-harmonic resonance of the sandwich beams are consideredIn the two cases the following two transformations arerequired
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
(34)
For subharmonic resonance the detuning parameter 120590119899
is introduced according toΩ119899= 3120596119899+ 120576120590119899 (35)
Similarly the averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
31205721198991205731198991198862
119899Λ sin 120574
119899
4120596119899
(36a)
119886119899119899= 119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
minus
9120572119899120573119899Λ1198862
119899cos 120574119899
4120596119899
(36b)
where Λ = (12)119891119899(1205962
119899minus Ω2
119899)minus1
The frequency response equation of the truss core sand-wich beam with subharmonic resonance is
(minus
120583119899119886119899
2
)
2
+
1
9
[119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
]
2
=
91205722
1198991205732
1198991198864
119899Λ2
161205962
119899
(37)
For superharmonic resonance the detuning parameter120590119899
is introduced according to3Ω119899= 120596119899+ 120576120590119899 (38)
The averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
1205721198991205733
119899Λ3 sin 120574
119899
120596119899
(39a)
119886119899119899= 119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
minus
1205721198991205733
119899Λ3 cos 120574
119899
120596119899
(39b)
The frequency response equation of the truss core sand-wich beam with superharmonic resonance is
(minus
120583119899119886119899
2
)
2
+ [119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
]
2
=
1205722
1198991205736
119899Λ6
1205962
119899
(40)
The plot of 119886119899as a function of 120590
119899for given 119891 is called
a nonlinear frequency response curve which can be usedto describe the hardening behavior of a structure In thefollowing analysis the parameters in (33) (37) and (40)should be calculated by using the Appendix
8 Shock and Vibration
120596
32
28
24
20
16
12
8
4
120588c120588
000 002 004 006 008 010
Figure 3 Variations of dimensionless linear natural frequencieswith respect to the strut radius Solid line first-order frequencydash line second-order frequency dot-dash line third-order fre-quency
5 Numerical Results
In the study the influences of the structural parameterson the dimensionless linear natural frequencies of the trusscore sandwich beam are studied at first by neglecting thenonlinear terms in (20) The face sheets and the truss of thesandwich beam are made of the same material The materialproperties are taken as follows 119864 = 689GPa and 120588 =
2680 kgm3 Figure 3 indicates that the first-order naturalfrequency decreases continuously and the second-order andthird-order natural frequencies initially increase and thendecrease with the increase of strut radiusThe relative densitycorresponding to the turning point in the changing curveof the second-order natural frequency is smaller than thatof the third-order natural frequency When the strut radiusincreases the mass moment of inertia and shear stiffness ofthe sandwich beam both increase the changes of the naturalfrequencies depend upon the combined effect of the twofactors The structural parameters in the figure are given asℎ = 0012m ℎ
119888= 001m 120579 = 1205874 and 119897 = 07m
Figure 4 shows that the first three natural frequenciesall increase with the rise of the core height and the higher-order natural frequency increases faster than the lower-ordernatural frequency When the core height increases the massmoment of inertia extensional stiffness bending stiffnessand shear stiffness of the sandwich beam all decrease thechanges of the natural frequencies depend upon the com-bined effect of those factors The structural parameters in thefigure are given as ℎ = 0012m 119903 = 06mm 120579 = 1205874 and119897 = 07m
The steady-state responses of the first three modes ofthe sandwich beam with primary resonance are investigatedThe hardening-type nonlinearity of the first three modes allbecomes weaker with increasing strut radius and core heightas shown in Figures 5 and 6 The bending of the frequencyresponse curve mainly depends on the nonlinear term andthe frequency parameter according to (33) It can be known
120596
32
28
24
20
16
12
8
4
hch
072 076 080 084 088 092
Figure 4 Variations of dimensionless linear natural frequencieswith respect to the core height Solid line first-order frequency dashline second-order frequency dot-dash line third-order frequency
from 120572119899in the Appendix that the nonlinear term is related to
the extensional stiffness of the face sheets as well as the massmoment of inertia thickness and length of the sandwichbeam The theoretical analyses indicate that the reasons forthe weakening of the hardening-type nonlinearity with therise of strut radius and core height are different
At first we discuss the influence of strut radius on thehardening behavior of the sandwich beam With the increaseof the strut radius the hardening behavior of the beam isweakened by the rise of themassmoment of inertia and shearstiffness It is concluded from (33) that the increases of themass moment of inertia and shear stiffness both weaken thehardening behavior The structural parameters in Figure 5are the same as in Figure 3 With the increase of the coreheight the hardening behavior of the beam is weakened It isconcluded that the decrease of extensional stiffness tends toweaken the hardening behavior and the decrease of the massmoment of inertia shear stiffness and bending stiffness tendsto strengthen the hardening behavior of the sandwich beamObviously the effect of the extensional stiffness is dominantaccording to Figure 6The structural parameters in this figureare the same as in Figure 4
Furthermore the hardening-type nonlinearity of thehigher mode is stronger than that of the lower mode andthe influences of the structural parameters on the hardeningbehavior for the lower mode are slightly larger than for thehigher mode In the analysis of the primary resonance of thesandwich beam 120583 = 0001 and 119891 = 1 are always chosen
Subharmonic and superharmonic resonances are rela-tively easy to be generated at the lower modes especially atthe first mode The nonlinear frequency response curves forthe first mode of the sandwich beam with subharmonic andsuperharmonic resonances are investigated in the paper thenumerical results are shown in Figures 7 and 8 respectivelyFigure 7 demonstrates that the subharmonic responses onlyoccur in the frequency ranges which are higher than thelinear natural frequency no matter which and how structuralparameters change Increasing strut radius and core height
Shock and Vibration 9
a1
25
2
15
1
05
1205901
minus10 minus5 0 5 10
(a)a2
10
12
06
04
08
02
1205902
minus10 minus5 0 5 10
(b)
a3
07
08
05
04
06
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 5 Influences of the strut radius on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line 120588 = 002 dash line 120588 = 004 dot-dash line 120588 = 006
will weaken the hardening-type nonlinearity in the resonancecase just as the influences of the structural parameters onthe responses of the beam with primary resonance Figure 8demonstrates that the shapes of the frequency responsecurves for the first mode with superharmonic responses aresimilar to that of the sandwich beamwith primary resonanceThe variations in the structural parameters can not only affectthe hardening behaviors of the beam but also significantlychange the frequency range of the resonance responses
Decreasing strut radius and core height can broaden thefrequency ranges of the resonance responses
Based on the analysis of the frequency response equa-tions we conclude that in the cases of subharmonic andsuperharmonic resonances the mechanism of the variationof the hardening behavior via the structural parameters is thesame as that in the case of primary resonance 120583 = 0001and 119891 = 2 are chosen in Figures 7 and 8 Moreover thestructural parameters in Figures 7(a) and 8(a) are the same
10 Shock and Vibration
a1
14
12
10
08
06
04
02
1205901
minus10 minus5 0 5 10
(a)a2
06
05
04
03
02
01
1205902
minus10 minus5 0 5 10
(b)
a3
04
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 6 Influences of the core height on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash line ℎ
119888ℎ = 09
as in Figure 3 and the structural parameters in Figures 7(b)and 8(b) are the same as in Figure 4
6 Conclusions
The natural frequency and hardening behavior of sandwichbeam with pyramidal truss core are investigated The trusscore is equivalent to a continuous homogeneousmaterial anda Zig-Zag theory is used to derive the nonlinear governingequation of motion for the sandwich beam At first thenatural frequencies are calculated to validate the presentmodel and the changing rules of the natural frequency with
the strut radius and core height are obtained It is illustratedthat the first-order natural frequency decreases continuouslyand the second-order and third-order natural frequenciesinitially increase and then decrease with the increase of strutradius and the first three natural frequencies all increase withthe rise of the core height Then the hardening behaviorsof the sandwich beam are analyzed Numerical simulationsindicate that the effects of the two structural parameters onthe hardening behavior in the cases of primary resonancesubharmonic resonance and superharmonic resonance aresimilar However the mechanisms of the variations of thehardening behavior are very different The influences of
Shock and Vibration 11
a1
15
1
05
0
1205901
minus10 minus5 0 5 10
(a)a1
1205901
minus10 minus5 0 5 10
08
07
06
05
04
03
02
01
0
(b)
Figure 7 The hardening behavior of the first mode with subharmonic resonance (a) The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
1205901
minus4 minus2 0 2 4
a1
16
14
12
10
08
06
04
02
(a)
1205901
minus5 0 5 10 15
a1
14
12
10
08
06
04
02
(b)
Figure 8The hardening behavior of the first mode with superharmonic resonance (a)The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
the structural parameters on the frequency parameter and thenonlinear term of the sandwich beam are studied in the paperfor analyzing the variations of the hardening behavior
Thehardening nonlinearity bends the frequency responsecurve to the right Generally under the same excitationthe stronger the hardening-type nonlinearity the smaller
the response amplitude We investigate the influences ofstrut radius and core height on the dynamical responses ofthe sandwich beam Using the equations obtained in thispaper the influences of the other structural parameters ofthe sandwich beam on the vibrations can be convenientlyanalyzed
12 Shock and Vibration
It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics
Appendix
The relations between the in-plane displacements and trans-verse displacement are
119906119899= 1198781198991199082
119899
119878119899= minus
1198991198861120587
8
(A1)
The relations between the rotation angles and transversedisplacement are
120601119899= 119870119899119908119899
119870119899=
1198991198873120587 minus 1198993
11988721205873
119899211988711205872minus 1
(A2)
The coefficients in (20) are given as follows
120574119899=
1198792119899
1198791119899
1205962
119899= minus
1198793119899
1198791119899
120572119899= minus
1198794119899
1198791119899
120573119899=
1198795119899
1198791119899
1198791119899=
1198889
2
1198792119899=
1198888
2
1198793119899=
1198994
1205874
1198882
2
minus
1198992
1205872
1198884
2
minus
1198991205871198701198991198883
2
+
1198993
1205873
1198701198991198881
2
1198794119899= minus
1198994
1205874
8
+
1198993
1205873
1198781198991198886
2
minus 1198993
1205873
1198781198991198885
1198795119899=
1198887
2
(A3)
where
1198861=
ℎ2
1198972
1198871= minus
ℎ2
ℎ
2
119888
21198972
1198601
119862
1198872=
ℎ3
ℎ
2
119888
21198973
1198601
119862
+
ℎ3
ℎ119888
1198973
1198611
119862
1198873= minus
ℎ
119897
1198881=
119897
3ℎ
(
ℎ119888
2
+ ℎ
1198611
1198601
)
1198882= minus
1
3
(
ℎ
2
119888
2
+ 2ℎ
1198611
1198601
+ 2
1198631
1198601
)
1198883= minus
1198973
3ℎ3
119862
1198601
1198884=
1198972
3ℎ2
119862
1198601
1198885=
21198972
3ℎ2
1198886=
21198972
3ℎ2
1198887=
119897
3ℎ1198601
1198888=
1198972
3ℎ21198601
1198889=
1198972
3ℎ21198601
(211986801+ 11986802)
(A4)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)
References
[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 7
1205760 order
1198632
01199091198990+ 1205962
1198991199091198990= 0 (26a)
1205761 order
1198632
01199091198991+ 1205962
1198991199091198991= minus2119863
011986311199091198990minus 12058311989911986301199091198990minus 1205721198991199093
1198990
+ 120573119899119891119899cos (120596
1198991198790+ 1205901198991198791)
(26b)
The solution of (26a) in the complex form can beexpressed as
1199091198990= 119860119899(1198791) 1198901198941205961198991198790+ 119860119899(1198791) 119890minus1198941205961198991198790 (27)
where 119860119899is the complex conjugate of 119860
119899
Substituting (27) into (26b) yields
1198632
01199091198991+ 1205962
1198991199091198991= (minus2119894120596
1198991198631119860119899minus 120583119899119894120596119899119860119899
minus 31205721198991198602
119899119860119899+
120573119899119891119899
2
1198901198941205901198991198791) 1198901198941205961198991198790+ cc +NST
(28)
where symbol cc denotes the parts of the complex conjugateof function on the right-hand side of (28) andNST representsthe terms that do not produce secular terms
Function 119860119899may be denoted in the complex form
119860119899=
1
2
119886119899119890119894120593119899 (29)
Substituting (29) into simplified (28) and separating thereal and imaginary parts from the resulting equations thetwo-dimensional averaged equation in the polar form isobtained as
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin (120590
1198991198791minus 120593119899)
2120596119899
(30a)
119886119899119899=
31205721198991198863
119899
8120596119899
minus
120573119899119891119899cos (120590
1198991198791minus 120593119899)
2120596119899
(30b)
Letting
120574119899= 1205901198991198791minus 120593119899 (31)
we have
119899= minus
120583119899119886119899
2
+
120573119899119891119899sin 120574119899
2120596119899
(32a)
119886119899119899= 119886119899120590119899minus
31205721198991198863
119899
8120596119899
+
120573119899119891119899cos 120574119899
2120596119899
(32b)
Steady-state motions occur when 119899and
119899equal zero
which correspond to the singular points of averaged (32a) and(32b) Thus 119886
119899and 120574119899are constants Eliminating 120574
119899by using
the relations between trigonometric functions we obtain thefrequency response equation of the truss core sandwich beamwith primary resonance as follows
(
120583119899119886119899
2
)
2
+ (120590119899119886119899minus
31205721198991198863
119899
8120596119899
)
2
= (
120573119899119891119899
2120596119899
)
2
(33)
Subsequently the subharmonic resonance and the super-harmonic resonance of the sandwich beams are consideredIn the two cases the following two transformations arerequired
120583119899997888rarr 120576120583
119899
120572119899997888rarr 120576120572
119899
(34)
For subharmonic resonance the detuning parameter 120590119899
is introduced according toΩ119899= 3120596119899+ 120576120590119899 (35)
Similarly the averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
31205721198991205731198991198862
119899Λ sin 120574
119899
4120596119899
(36a)
119886119899119899= 119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
minus
9120572119899120573119899Λ1198862
119899cos 120574119899
4120596119899
(36b)
where Λ = (12)119891119899(1205962
119899minus Ω2
119899)minus1
The frequency response equation of the truss core sand-wich beam with subharmonic resonance is
(minus
120583119899119886119899
2
)
2
+
1
9
[119886119899(120590119899minus
91205721198991205732
119899Λ2
120596119899
) minus
91205721198991198863
119899
8120596119899
]
2
=
91205722
1198991205732
1198991198864
119899Λ2
161205962
119899
(37)
For superharmonic resonance the detuning parameter120590119899
is introduced according to3Ω119899= 120596119899+ 120576120590119899 (38)
The averaged equations in the case are obtained
119899= minus
120583119899119886119899
2
minus
1205721198991205733
119899Λ3 sin 120574
119899
120596119899
(39a)
119886119899119899= 119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
minus
1205721198991205733
119899Λ3 cos 120574
119899
120596119899
(39b)
The frequency response equation of the truss core sand-wich beam with superharmonic resonance is
(minus
120583119899119886119899
2
)
2
+ [119886119899(120590119899minus
31205721198991205732
119899Λ2
120596119899
) minus
31205721198991198863
119899
8120596119899
]
2
=
1205722
1198991205736
119899Λ6
1205962
119899
(40)
The plot of 119886119899as a function of 120590
119899for given 119891 is called
a nonlinear frequency response curve which can be usedto describe the hardening behavior of a structure In thefollowing analysis the parameters in (33) (37) and (40)should be calculated by using the Appendix
8 Shock and Vibration
120596
32
28
24
20
16
12
8
4
120588c120588
000 002 004 006 008 010
Figure 3 Variations of dimensionless linear natural frequencieswith respect to the strut radius Solid line first-order frequencydash line second-order frequency dot-dash line third-order fre-quency
5 Numerical Results
In the study the influences of the structural parameterson the dimensionless linear natural frequencies of the trusscore sandwich beam are studied at first by neglecting thenonlinear terms in (20) The face sheets and the truss of thesandwich beam are made of the same material The materialproperties are taken as follows 119864 = 689GPa and 120588 =
2680 kgm3 Figure 3 indicates that the first-order naturalfrequency decreases continuously and the second-order andthird-order natural frequencies initially increase and thendecrease with the increase of strut radiusThe relative densitycorresponding to the turning point in the changing curveof the second-order natural frequency is smaller than thatof the third-order natural frequency When the strut radiusincreases the mass moment of inertia and shear stiffness ofthe sandwich beam both increase the changes of the naturalfrequencies depend upon the combined effect of the twofactors The structural parameters in the figure are given asℎ = 0012m ℎ
119888= 001m 120579 = 1205874 and 119897 = 07m
Figure 4 shows that the first three natural frequenciesall increase with the rise of the core height and the higher-order natural frequency increases faster than the lower-ordernatural frequency When the core height increases the massmoment of inertia extensional stiffness bending stiffnessand shear stiffness of the sandwich beam all decrease thechanges of the natural frequencies depend upon the com-bined effect of those factors The structural parameters in thefigure are given as ℎ = 0012m 119903 = 06mm 120579 = 1205874 and119897 = 07m
The steady-state responses of the first three modes ofthe sandwich beam with primary resonance are investigatedThe hardening-type nonlinearity of the first three modes allbecomes weaker with increasing strut radius and core heightas shown in Figures 5 and 6 The bending of the frequencyresponse curve mainly depends on the nonlinear term andthe frequency parameter according to (33) It can be known
120596
32
28
24
20
16
12
8
4
hch
072 076 080 084 088 092
Figure 4 Variations of dimensionless linear natural frequencieswith respect to the core height Solid line first-order frequency dashline second-order frequency dot-dash line third-order frequency
from 120572119899in the Appendix that the nonlinear term is related to
the extensional stiffness of the face sheets as well as the massmoment of inertia thickness and length of the sandwichbeam The theoretical analyses indicate that the reasons forthe weakening of the hardening-type nonlinearity with therise of strut radius and core height are different
At first we discuss the influence of strut radius on thehardening behavior of the sandwich beam With the increaseof the strut radius the hardening behavior of the beam isweakened by the rise of themassmoment of inertia and shearstiffness It is concluded from (33) that the increases of themass moment of inertia and shear stiffness both weaken thehardening behavior The structural parameters in Figure 5are the same as in Figure 3 With the increase of the coreheight the hardening behavior of the beam is weakened It isconcluded that the decrease of extensional stiffness tends toweaken the hardening behavior and the decrease of the massmoment of inertia shear stiffness and bending stiffness tendsto strengthen the hardening behavior of the sandwich beamObviously the effect of the extensional stiffness is dominantaccording to Figure 6The structural parameters in this figureare the same as in Figure 4
Furthermore the hardening-type nonlinearity of thehigher mode is stronger than that of the lower mode andthe influences of the structural parameters on the hardeningbehavior for the lower mode are slightly larger than for thehigher mode In the analysis of the primary resonance of thesandwich beam 120583 = 0001 and 119891 = 1 are always chosen
Subharmonic and superharmonic resonances are rela-tively easy to be generated at the lower modes especially atthe first mode The nonlinear frequency response curves forthe first mode of the sandwich beam with subharmonic andsuperharmonic resonances are investigated in the paper thenumerical results are shown in Figures 7 and 8 respectivelyFigure 7 demonstrates that the subharmonic responses onlyoccur in the frequency ranges which are higher than thelinear natural frequency no matter which and how structuralparameters change Increasing strut radius and core height
Shock and Vibration 9
a1
25
2
15
1
05
1205901
minus10 minus5 0 5 10
(a)a2
10
12
06
04
08
02
1205902
minus10 minus5 0 5 10
(b)
a3
07
08
05
04
06
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 5 Influences of the strut radius on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line 120588 = 002 dash line 120588 = 004 dot-dash line 120588 = 006
will weaken the hardening-type nonlinearity in the resonancecase just as the influences of the structural parameters onthe responses of the beam with primary resonance Figure 8demonstrates that the shapes of the frequency responsecurves for the first mode with superharmonic responses aresimilar to that of the sandwich beamwith primary resonanceThe variations in the structural parameters can not only affectthe hardening behaviors of the beam but also significantlychange the frequency range of the resonance responses
Decreasing strut radius and core height can broaden thefrequency ranges of the resonance responses
Based on the analysis of the frequency response equa-tions we conclude that in the cases of subharmonic andsuperharmonic resonances the mechanism of the variationof the hardening behavior via the structural parameters is thesame as that in the case of primary resonance 120583 = 0001and 119891 = 2 are chosen in Figures 7 and 8 Moreover thestructural parameters in Figures 7(a) and 8(a) are the same
10 Shock and Vibration
a1
14
12
10
08
06
04
02
1205901
minus10 minus5 0 5 10
(a)a2
06
05
04
03
02
01
1205902
minus10 minus5 0 5 10
(b)
a3
04
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 6 Influences of the core height on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash line ℎ
119888ℎ = 09
as in Figure 3 and the structural parameters in Figures 7(b)and 8(b) are the same as in Figure 4
6 Conclusions
The natural frequency and hardening behavior of sandwichbeam with pyramidal truss core are investigated The trusscore is equivalent to a continuous homogeneousmaterial anda Zig-Zag theory is used to derive the nonlinear governingequation of motion for the sandwich beam At first thenatural frequencies are calculated to validate the presentmodel and the changing rules of the natural frequency with
the strut radius and core height are obtained It is illustratedthat the first-order natural frequency decreases continuouslyand the second-order and third-order natural frequenciesinitially increase and then decrease with the increase of strutradius and the first three natural frequencies all increase withthe rise of the core height Then the hardening behaviorsof the sandwich beam are analyzed Numerical simulationsindicate that the effects of the two structural parameters onthe hardening behavior in the cases of primary resonancesubharmonic resonance and superharmonic resonance aresimilar However the mechanisms of the variations of thehardening behavior are very different The influences of
Shock and Vibration 11
a1
15
1
05
0
1205901
minus10 minus5 0 5 10
(a)a1
1205901
minus10 minus5 0 5 10
08
07
06
05
04
03
02
01
0
(b)
Figure 7 The hardening behavior of the first mode with subharmonic resonance (a) The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
1205901
minus4 minus2 0 2 4
a1
16
14
12
10
08
06
04
02
(a)
1205901
minus5 0 5 10 15
a1
14
12
10
08
06
04
02
(b)
Figure 8The hardening behavior of the first mode with superharmonic resonance (a)The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
the structural parameters on the frequency parameter and thenonlinear term of the sandwich beam are studied in the paperfor analyzing the variations of the hardening behavior
Thehardening nonlinearity bends the frequency responsecurve to the right Generally under the same excitationthe stronger the hardening-type nonlinearity the smaller
the response amplitude We investigate the influences ofstrut radius and core height on the dynamical responses ofthe sandwich beam Using the equations obtained in thispaper the influences of the other structural parameters ofthe sandwich beam on the vibrations can be convenientlyanalyzed
12 Shock and Vibration
It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics
Appendix
The relations between the in-plane displacements and trans-verse displacement are
119906119899= 1198781198991199082
119899
119878119899= minus
1198991198861120587
8
(A1)
The relations between the rotation angles and transversedisplacement are
120601119899= 119870119899119908119899
119870119899=
1198991198873120587 minus 1198993
11988721205873
119899211988711205872minus 1
(A2)
The coefficients in (20) are given as follows
120574119899=
1198792119899
1198791119899
1205962
119899= minus
1198793119899
1198791119899
120572119899= minus
1198794119899
1198791119899
120573119899=
1198795119899
1198791119899
1198791119899=
1198889
2
1198792119899=
1198888
2
1198793119899=
1198994
1205874
1198882
2
minus
1198992
1205872
1198884
2
minus
1198991205871198701198991198883
2
+
1198993
1205873
1198701198991198881
2
1198794119899= minus
1198994
1205874
8
+
1198993
1205873
1198781198991198886
2
minus 1198993
1205873
1198781198991198885
1198795119899=
1198887
2
(A3)
where
1198861=
ℎ2
1198972
1198871= minus
ℎ2
ℎ
2
119888
21198972
1198601
119862
1198872=
ℎ3
ℎ
2
119888
21198973
1198601
119862
+
ℎ3
ℎ119888
1198973
1198611
119862
1198873= minus
ℎ
119897
1198881=
119897
3ℎ
(
ℎ119888
2
+ ℎ
1198611
1198601
)
1198882= minus
1
3
(
ℎ
2
119888
2
+ 2ℎ
1198611
1198601
+ 2
1198631
1198601
)
1198883= minus
1198973
3ℎ3
119862
1198601
1198884=
1198972
3ℎ2
119862
1198601
1198885=
21198972
3ℎ2
1198886=
21198972
3ℎ2
1198887=
119897
3ℎ1198601
1198888=
1198972
3ℎ21198601
1198889=
1198972
3ℎ21198601
(211986801+ 11986802)
(A4)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)
References
[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 Shock and Vibration
120596
32
28
24
20
16
12
8
4
120588c120588
000 002 004 006 008 010
Figure 3 Variations of dimensionless linear natural frequencieswith respect to the strut radius Solid line first-order frequencydash line second-order frequency dot-dash line third-order fre-quency
5 Numerical Results
In the study the influences of the structural parameterson the dimensionless linear natural frequencies of the trusscore sandwich beam are studied at first by neglecting thenonlinear terms in (20) The face sheets and the truss of thesandwich beam are made of the same material The materialproperties are taken as follows 119864 = 689GPa and 120588 =
2680 kgm3 Figure 3 indicates that the first-order naturalfrequency decreases continuously and the second-order andthird-order natural frequencies initially increase and thendecrease with the increase of strut radiusThe relative densitycorresponding to the turning point in the changing curveof the second-order natural frequency is smaller than thatof the third-order natural frequency When the strut radiusincreases the mass moment of inertia and shear stiffness ofthe sandwich beam both increase the changes of the naturalfrequencies depend upon the combined effect of the twofactors The structural parameters in the figure are given asℎ = 0012m ℎ
119888= 001m 120579 = 1205874 and 119897 = 07m
Figure 4 shows that the first three natural frequenciesall increase with the rise of the core height and the higher-order natural frequency increases faster than the lower-ordernatural frequency When the core height increases the massmoment of inertia extensional stiffness bending stiffnessand shear stiffness of the sandwich beam all decrease thechanges of the natural frequencies depend upon the com-bined effect of those factors The structural parameters in thefigure are given as ℎ = 0012m 119903 = 06mm 120579 = 1205874 and119897 = 07m
The steady-state responses of the first three modes ofthe sandwich beam with primary resonance are investigatedThe hardening-type nonlinearity of the first three modes allbecomes weaker with increasing strut radius and core heightas shown in Figures 5 and 6 The bending of the frequencyresponse curve mainly depends on the nonlinear term andthe frequency parameter according to (33) It can be known
120596
32
28
24
20
16
12
8
4
hch
072 076 080 084 088 092
Figure 4 Variations of dimensionless linear natural frequencieswith respect to the core height Solid line first-order frequency dashline second-order frequency dot-dash line third-order frequency
from 120572119899in the Appendix that the nonlinear term is related to
the extensional stiffness of the face sheets as well as the massmoment of inertia thickness and length of the sandwichbeam The theoretical analyses indicate that the reasons forthe weakening of the hardening-type nonlinearity with therise of strut radius and core height are different
At first we discuss the influence of strut radius on thehardening behavior of the sandwich beam With the increaseof the strut radius the hardening behavior of the beam isweakened by the rise of themassmoment of inertia and shearstiffness It is concluded from (33) that the increases of themass moment of inertia and shear stiffness both weaken thehardening behavior The structural parameters in Figure 5are the same as in Figure 3 With the increase of the coreheight the hardening behavior of the beam is weakened It isconcluded that the decrease of extensional stiffness tends toweaken the hardening behavior and the decrease of the massmoment of inertia shear stiffness and bending stiffness tendsto strengthen the hardening behavior of the sandwich beamObviously the effect of the extensional stiffness is dominantaccording to Figure 6The structural parameters in this figureare the same as in Figure 4
Furthermore the hardening-type nonlinearity of thehigher mode is stronger than that of the lower mode andthe influences of the structural parameters on the hardeningbehavior for the lower mode are slightly larger than for thehigher mode In the analysis of the primary resonance of thesandwich beam 120583 = 0001 and 119891 = 1 are always chosen
Subharmonic and superharmonic resonances are rela-tively easy to be generated at the lower modes especially atthe first mode The nonlinear frequency response curves forthe first mode of the sandwich beam with subharmonic andsuperharmonic resonances are investigated in the paper thenumerical results are shown in Figures 7 and 8 respectivelyFigure 7 demonstrates that the subharmonic responses onlyoccur in the frequency ranges which are higher than thelinear natural frequency no matter which and how structuralparameters change Increasing strut radius and core height
Shock and Vibration 9
a1
25
2
15
1
05
1205901
minus10 minus5 0 5 10
(a)a2
10
12
06
04
08
02
1205902
minus10 minus5 0 5 10
(b)
a3
07
08
05
04
06
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 5 Influences of the strut radius on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line 120588 = 002 dash line 120588 = 004 dot-dash line 120588 = 006
will weaken the hardening-type nonlinearity in the resonancecase just as the influences of the structural parameters onthe responses of the beam with primary resonance Figure 8demonstrates that the shapes of the frequency responsecurves for the first mode with superharmonic responses aresimilar to that of the sandwich beamwith primary resonanceThe variations in the structural parameters can not only affectthe hardening behaviors of the beam but also significantlychange the frequency range of the resonance responses
Decreasing strut radius and core height can broaden thefrequency ranges of the resonance responses
Based on the analysis of the frequency response equa-tions we conclude that in the cases of subharmonic andsuperharmonic resonances the mechanism of the variationof the hardening behavior via the structural parameters is thesame as that in the case of primary resonance 120583 = 0001and 119891 = 2 are chosen in Figures 7 and 8 Moreover thestructural parameters in Figures 7(a) and 8(a) are the same
10 Shock and Vibration
a1
14
12
10
08
06
04
02
1205901
minus10 minus5 0 5 10
(a)a2
06
05
04
03
02
01
1205902
minus10 minus5 0 5 10
(b)
a3
04
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 6 Influences of the core height on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash line ℎ
119888ℎ = 09
as in Figure 3 and the structural parameters in Figures 7(b)and 8(b) are the same as in Figure 4
6 Conclusions
The natural frequency and hardening behavior of sandwichbeam with pyramidal truss core are investigated The trusscore is equivalent to a continuous homogeneousmaterial anda Zig-Zag theory is used to derive the nonlinear governingequation of motion for the sandwich beam At first thenatural frequencies are calculated to validate the presentmodel and the changing rules of the natural frequency with
the strut radius and core height are obtained It is illustratedthat the first-order natural frequency decreases continuouslyand the second-order and third-order natural frequenciesinitially increase and then decrease with the increase of strutradius and the first three natural frequencies all increase withthe rise of the core height Then the hardening behaviorsof the sandwich beam are analyzed Numerical simulationsindicate that the effects of the two structural parameters onthe hardening behavior in the cases of primary resonancesubharmonic resonance and superharmonic resonance aresimilar However the mechanisms of the variations of thehardening behavior are very different The influences of
Shock and Vibration 11
a1
15
1
05
0
1205901
minus10 minus5 0 5 10
(a)a1
1205901
minus10 minus5 0 5 10
08
07
06
05
04
03
02
01
0
(b)
Figure 7 The hardening behavior of the first mode with subharmonic resonance (a) The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
1205901
minus4 minus2 0 2 4
a1
16
14
12
10
08
06
04
02
(a)
1205901
minus5 0 5 10 15
a1
14
12
10
08
06
04
02
(b)
Figure 8The hardening behavior of the first mode with superharmonic resonance (a)The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
the structural parameters on the frequency parameter and thenonlinear term of the sandwich beam are studied in the paperfor analyzing the variations of the hardening behavior
Thehardening nonlinearity bends the frequency responsecurve to the right Generally under the same excitationthe stronger the hardening-type nonlinearity the smaller
the response amplitude We investigate the influences ofstrut radius and core height on the dynamical responses ofthe sandwich beam Using the equations obtained in thispaper the influences of the other structural parameters ofthe sandwich beam on the vibrations can be convenientlyanalyzed
12 Shock and Vibration
It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics
Appendix
The relations between the in-plane displacements and trans-verse displacement are
119906119899= 1198781198991199082
119899
119878119899= minus
1198991198861120587
8
(A1)
The relations between the rotation angles and transversedisplacement are
120601119899= 119870119899119908119899
119870119899=
1198991198873120587 minus 1198993
11988721205873
119899211988711205872minus 1
(A2)
The coefficients in (20) are given as follows
120574119899=
1198792119899
1198791119899
1205962
119899= minus
1198793119899
1198791119899
120572119899= minus
1198794119899
1198791119899
120573119899=
1198795119899
1198791119899
1198791119899=
1198889
2
1198792119899=
1198888
2
1198793119899=
1198994
1205874
1198882
2
minus
1198992
1205872
1198884
2
minus
1198991205871198701198991198883
2
+
1198993
1205873
1198701198991198881
2
1198794119899= minus
1198994
1205874
8
+
1198993
1205873
1198781198991198886
2
minus 1198993
1205873
1198781198991198885
1198795119899=
1198887
2
(A3)
where
1198861=
ℎ2
1198972
1198871= minus
ℎ2
ℎ
2
119888
21198972
1198601
119862
1198872=
ℎ3
ℎ
2
119888
21198973
1198601
119862
+
ℎ3
ℎ119888
1198973
1198611
119862
1198873= minus
ℎ
119897
1198881=
119897
3ℎ
(
ℎ119888
2
+ ℎ
1198611
1198601
)
1198882= minus
1
3
(
ℎ
2
119888
2
+ 2ℎ
1198611
1198601
+ 2
1198631
1198601
)
1198883= minus
1198973
3ℎ3
119862
1198601
1198884=
1198972
3ℎ2
119862
1198601
1198885=
21198972
3ℎ2
1198886=
21198972
3ℎ2
1198887=
119897
3ℎ1198601
1198888=
1198972
3ℎ21198601
1198889=
1198972
3ℎ21198601
(211986801+ 11986802)
(A4)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)
References
[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 9
a1
25
2
15
1
05
1205901
minus10 minus5 0 5 10
(a)a2
10
12
06
04
08
02
1205902
minus10 minus5 0 5 10
(b)
a3
07
08
05
04
06
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 5 Influences of the strut radius on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line 120588 = 002 dash line 120588 = 004 dot-dash line 120588 = 006
will weaken the hardening-type nonlinearity in the resonancecase just as the influences of the structural parameters onthe responses of the beam with primary resonance Figure 8demonstrates that the shapes of the frequency responsecurves for the first mode with superharmonic responses aresimilar to that of the sandwich beamwith primary resonanceThe variations in the structural parameters can not only affectthe hardening behaviors of the beam but also significantlychange the frequency range of the resonance responses
Decreasing strut radius and core height can broaden thefrequency ranges of the resonance responses
Based on the analysis of the frequency response equa-tions we conclude that in the cases of subharmonic andsuperharmonic resonances the mechanism of the variationof the hardening behavior via the structural parameters is thesame as that in the case of primary resonance 120583 = 0001and 119891 = 2 are chosen in Figures 7 and 8 Moreover thestructural parameters in Figures 7(a) and 8(a) are the same
10 Shock and Vibration
a1
14
12
10
08
06
04
02
1205901
minus10 minus5 0 5 10
(a)a2
06
05
04
03
02
01
1205902
minus10 minus5 0 5 10
(b)
a3
04
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 6 Influences of the core height on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash line ℎ
119888ℎ = 09
as in Figure 3 and the structural parameters in Figures 7(b)and 8(b) are the same as in Figure 4
6 Conclusions
The natural frequency and hardening behavior of sandwichbeam with pyramidal truss core are investigated The trusscore is equivalent to a continuous homogeneousmaterial anda Zig-Zag theory is used to derive the nonlinear governingequation of motion for the sandwich beam At first thenatural frequencies are calculated to validate the presentmodel and the changing rules of the natural frequency with
the strut radius and core height are obtained It is illustratedthat the first-order natural frequency decreases continuouslyand the second-order and third-order natural frequenciesinitially increase and then decrease with the increase of strutradius and the first three natural frequencies all increase withthe rise of the core height Then the hardening behaviorsof the sandwich beam are analyzed Numerical simulationsindicate that the effects of the two structural parameters onthe hardening behavior in the cases of primary resonancesubharmonic resonance and superharmonic resonance aresimilar However the mechanisms of the variations of thehardening behavior are very different The influences of
Shock and Vibration 11
a1
15
1
05
0
1205901
minus10 minus5 0 5 10
(a)a1
1205901
minus10 minus5 0 5 10
08
07
06
05
04
03
02
01
0
(b)
Figure 7 The hardening behavior of the first mode with subharmonic resonance (a) The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
1205901
minus4 minus2 0 2 4
a1
16
14
12
10
08
06
04
02
(a)
1205901
minus5 0 5 10 15
a1
14
12
10
08
06
04
02
(b)
Figure 8The hardening behavior of the first mode with superharmonic resonance (a)The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
the structural parameters on the frequency parameter and thenonlinear term of the sandwich beam are studied in the paperfor analyzing the variations of the hardening behavior
Thehardening nonlinearity bends the frequency responsecurve to the right Generally under the same excitationthe stronger the hardening-type nonlinearity the smaller
the response amplitude We investigate the influences ofstrut radius and core height on the dynamical responses ofthe sandwich beam Using the equations obtained in thispaper the influences of the other structural parameters ofthe sandwich beam on the vibrations can be convenientlyanalyzed
12 Shock and Vibration
It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics
Appendix
The relations between the in-plane displacements and trans-verse displacement are
119906119899= 1198781198991199082
119899
119878119899= minus
1198991198861120587
8
(A1)
The relations between the rotation angles and transversedisplacement are
120601119899= 119870119899119908119899
119870119899=
1198991198873120587 minus 1198993
11988721205873
119899211988711205872minus 1
(A2)
The coefficients in (20) are given as follows
120574119899=
1198792119899
1198791119899
1205962
119899= minus
1198793119899
1198791119899
120572119899= minus
1198794119899
1198791119899
120573119899=
1198795119899
1198791119899
1198791119899=
1198889
2
1198792119899=
1198888
2
1198793119899=
1198994
1205874
1198882
2
minus
1198992
1205872
1198884
2
minus
1198991205871198701198991198883
2
+
1198993
1205873
1198701198991198881
2
1198794119899= minus
1198994
1205874
8
+
1198993
1205873
1198781198991198886
2
minus 1198993
1205873
1198781198991198885
1198795119899=
1198887
2
(A3)
where
1198861=
ℎ2
1198972
1198871= minus
ℎ2
ℎ
2
119888
21198972
1198601
119862
1198872=
ℎ3
ℎ
2
119888
21198973
1198601
119862
+
ℎ3
ℎ119888
1198973
1198611
119862
1198873= minus
ℎ
119897
1198881=
119897
3ℎ
(
ℎ119888
2
+ ℎ
1198611
1198601
)
1198882= minus
1
3
(
ℎ
2
119888
2
+ 2ℎ
1198611
1198601
+ 2
1198631
1198601
)
1198883= minus
1198973
3ℎ3
119862
1198601
1198884=
1198972
3ℎ2
119862
1198601
1198885=
21198972
3ℎ2
1198886=
21198972
3ℎ2
1198887=
119897
3ℎ1198601
1198888=
1198972
3ℎ21198601
1198889=
1198972
3ℎ21198601
(211986801+ 11986802)
(A4)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)
References
[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 Shock and Vibration
a1
14
12
10
08
06
04
02
1205901
minus10 minus5 0 5 10
(a)a2
06
05
04
03
02
01
1205902
minus10 minus5 0 5 10
(b)
a3
04
03
02
01
1205903
minus10 minus5 0 5 10
(c)
Figure 6 Influences of the core height on the hardening behavior of the first three modes (a) the first mode (b) the second mode (c) thethird mode Solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash line ℎ
119888ℎ = 09
as in Figure 3 and the structural parameters in Figures 7(b)and 8(b) are the same as in Figure 4
6 Conclusions
The natural frequency and hardening behavior of sandwichbeam with pyramidal truss core are investigated The trusscore is equivalent to a continuous homogeneousmaterial anda Zig-Zag theory is used to derive the nonlinear governingequation of motion for the sandwich beam At first thenatural frequencies are calculated to validate the presentmodel and the changing rules of the natural frequency with
the strut radius and core height are obtained It is illustratedthat the first-order natural frequency decreases continuouslyand the second-order and third-order natural frequenciesinitially increase and then decrease with the increase of strutradius and the first three natural frequencies all increase withthe rise of the core height Then the hardening behaviorsof the sandwich beam are analyzed Numerical simulationsindicate that the effects of the two structural parameters onthe hardening behavior in the cases of primary resonancesubharmonic resonance and superharmonic resonance aresimilar However the mechanisms of the variations of thehardening behavior are very different The influences of
Shock and Vibration 11
a1
15
1
05
0
1205901
minus10 minus5 0 5 10
(a)a1
1205901
minus10 minus5 0 5 10
08
07
06
05
04
03
02
01
0
(b)
Figure 7 The hardening behavior of the first mode with subharmonic resonance (a) The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
1205901
minus4 minus2 0 2 4
a1
16
14
12
10
08
06
04
02
(a)
1205901
minus5 0 5 10 15
a1
14
12
10
08
06
04
02
(b)
Figure 8The hardening behavior of the first mode with superharmonic resonance (a)The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
the structural parameters on the frequency parameter and thenonlinear term of the sandwich beam are studied in the paperfor analyzing the variations of the hardening behavior
Thehardening nonlinearity bends the frequency responsecurve to the right Generally under the same excitationthe stronger the hardening-type nonlinearity the smaller
the response amplitude We investigate the influences ofstrut radius and core height on the dynamical responses ofthe sandwich beam Using the equations obtained in thispaper the influences of the other structural parameters ofthe sandwich beam on the vibrations can be convenientlyanalyzed
12 Shock and Vibration
It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics
Appendix
The relations between the in-plane displacements and trans-verse displacement are
119906119899= 1198781198991199082
119899
119878119899= minus
1198991198861120587
8
(A1)
The relations between the rotation angles and transversedisplacement are
120601119899= 119870119899119908119899
119870119899=
1198991198873120587 minus 1198993
11988721205873
119899211988711205872minus 1
(A2)
The coefficients in (20) are given as follows
120574119899=
1198792119899
1198791119899
1205962
119899= minus
1198793119899
1198791119899
120572119899= minus
1198794119899
1198791119899
120573119899=
1198795119899
1198791119899
1198791119899=
1198889
2
1198792119899=
1198888
2
1198793119899=
1198994
1205874
1198882
2
minus
1198992
1205872
1198884
2
minus
1198991205871198701198991198883
2
+
1198993
1205873
1198701198991198881
2
1198794119899= minus
1198994
1205874
8
+
1198993
1205873
1198781198991198886
2
minus 1198993
1205873
1198781198991198885
1198795119899=
1198887
2
(A3)
where
1198861=
ℎ2
1198972
1198871= minus
ℎ2
ℎ
2
119888
21198972
1198601
119862
1198872=
ℎ3
ℎ
2
119888
21198973
1198601
119862
+
ℎ3
ℎ119888
1198973
1198611
119862
1198873= minus
ℎ
119897
1198881=
119897
3ℎ
(
ℎ119888
2
+ ℎ
1198611
1198601
)
1198882= minus
1
3
(
ℎ
2
119888
2
+ 2ℎ
1198611
1198601
+ 2
1198631
1198601
)
1198883= minus
1198973
3ℎ3
119862
1198601
1198884=
1198972
3ℎ2
119862
1198601
1198885=
21198972
3ℎ2
1198886=
21198972
3ℎ2
1198887=
119897
3ℎ1198601
1198888=
1198972
3ℎ21198601
1198889=
1198972
3ℎ21198601
(211986801+ 11986802)
(A4)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)
References
[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 11
a1
15
1
05
0
1205901
minus10 minus5 0 5 10
(a)a1
1205901
minus10 minus5 0 5 10
08
07
06
05
04
03
02
01
0
(b)
Figure 7 The hardening behavior of the first mode with subharmonic resonance (a) The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
1205901
minus4 minus2 0 2 4
a1
16
14
12
10
08
06
04
02
(a)
1205901
minus5 0 5 10 15
a1
14
12
10
08
06
04
02
(b)
Figure 8The hardening behavior of the first mode with superharmonic resonance (a)The influences of the strut radius (solid line 120588 = 002dash line 120588 = 004 dot-dash line 120588 = 006) (b) The influences of the core height (solid line ℎ
119888ℎ = 074 dash line ℎ
119888ℎ = 082 dot-dash
line ℎ119888ℎ = 09)
the structural parameters on the frequency parameter and thenonlinear term of the sandwich beam are studied in the paperfor analyzing the variations of the hardening behavior
Thehardening nonlinearity bends the frequency responsecurve to the right Generally under the same excitationthe stronger the hardening-type nonlinearity the smaller
the response amplitude We investigate the influences ofstrut radius and core height on the dynamical responses ofthe sandwich beam Using the equations obtained in thispaper the influences of the other structural parameters ofthe sandwich beam on the vibrations can be convenientlyanalyzed
12 Shock and Vibration
It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics
Appendix
The relations between the in-plane displacements and trans-verse displacement are
119906119899= 1198781198991199082
119899
119878119899= minus
1198991198861120587
8
(A1)
The relations between the rotation angles and transversedisplacement are
120601119899= 119870119899119908119899
119870119899=
1198991198873120587 minus 1198993
11988721205873
119899211988711205872minus 1
(A2)
The coefficients in (20) are given as follows
120574119899=
1198792119899
1198791119899
1205962
119899= minus
1198793119899
1198791119899
120572119899= minus
1198794119899
1198791119899
120573119899=
1198795119899
1198791119899
1198791119899=
1198889
2
1198792119899=
1198888
2
1198793119899=
1198994
1205874
1198882
2
minus
1198992
1205872
1198884
2
minus
1198991205871198701198991198883
2
+
1198993
1205873
1198701198991198881
2
1198794119899= minus
1198994
1205874
8
+
1198993
1205873
1198781198991198886
2
minus 1198993
1205873
1198781198991198885
1198795119899=
1198887
2
(A3)
where
1198861=
ℎ2
1198972
1198871= minus
ℎ2
ℎ
2
119888
21198972
1198601
119862
1198872=
ℎ3
ℎ
2
119888
21198973
1198601
119862
+
ℎ3
ℎ119888
1198973
1198611
119862
1198873= minus
ℎ
119897
1198881=
119897
3ℎ
(
ℎ119888
2
+ ℎ
1198611
1198601
)
1198882= minus
1
3
(
ℎ
2
119888
2
+ 2ℎ
1198611
1198601
+ 2
1198631
1198601
)
1198883= minus
1198973
3ℎ3
119862
1198601
1198884=
1198972
3ℎ2
119862
1198601
1198885=
21198972
3ℎ2
1198886=
21198972
3ℎ2
1198887=
119897
3ℎ1198601
1198888=
1198972
3ℎ21198601
1198889=
1198972
3ℎ21198601
(211986801+ 11986802)
(A4)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)
References
[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
12 Shock and Vibration
It is one of the effective methods to suppress the structurevibration by modifying the structural parameters especiallyin the complex working circumstanceThismethod can avoidintroducing new influence factors to the original systemThus it is very useful for the truss core sandwich structureswhich are usually used in aerospace vehicles The free vibra-tion and hardening behaviors under different resonances ofthe sandwich beamare studied in this paperTheparametricalstudy can provide the theoretical and data support for thedesign and vibration suppression of the truss core sandwichstructures from the point of view of dynamics
Appendix
The relations between the in-plane displacements and trans-verse displacement are
119906119899= 1198781198991199082
119899
119878119899= minus
1198991198861120587
8
(A1)
The relations between the rotation angles and transversedisplacement are
120601119899= 119870119899119908119899
119870119899=
1198991198873120587 minus 1198993
11988721205873
119899211988711205872minus 1
(A2)
The coefficients in (20) are given as follows
120574119899=
1198792119899
1198791119899
1205962
119899= minus
1198793119899
1198791119899
120572119899= minus
1198794119899
1198791119899
120573119899=
1198795119899
1198791119899
1198791119899=
1198889
2
1198792119899=
1198888
2
1198793119899=
1198994
1205874
1198882
2
minus
1198992
1205872
1198884
2
minus
1198991205871198701198991198883
2
+
1198993
1205873
1198701198991198881
2
1198794119899= minus
1198994
1205874
8
+
1198993
1205873
1198781198991198886
2
minus 1198993
1205873
1198781198991198885
1198795119899=
1198887
2
(A3)
where
1198861=
ℎ2
1198972
1198871= minus
ℎ2
ℎ
2
119888
21198972
1198601
119862
1198872=
ℎ3
ℎ
2
119888
21198973
1198601
119862
+
ℎ3
ℎ119888
1198973
1198611
119862
1198873= minus
ℎ
119897
1198881=
119897
3ℎ
(
ℎ119888
2
+ ℎ
1198611
1198601
)
1198882= minus
1
3
(
ℎ
2
119888
2
+ 2ℎ
1198611
1198601
+ 2
1198631
1198601
)
1198883= minus
1198973
3ℎ3
119862
1198601
1198884=
1198972
3ℎ2
119862
1198601
1198885=
21198972
3ℎ2
1198886=
21198972
3ℎ2
1198887=
119897
3ℎ1198601
1198888=
1198972
3ℎ21198601
1198889=
1198972
3ℎ21198601
(211986801+ 11986802)
(A4)
Competing Interests
The authors declare that they have no competing interests
Acknowledgments
This research was supported by the National Natural ScienceFoundation of China (nos 11402170 and 11402165)
References
[1] A G Evans J W Hutchinson N A Fleck M F Ashby andH N G Wadley ldquoThe topological design of multifunctionalcellular metalsrdquo Progress in Materials Science vol 46 no 3-4pp 309ndash327 2001
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 13
[2] V S Deshpande and N A Fleck ldquoCollapse of truss coresandwich beams in 3-point bendingrdquo International Journal ofSolids and Structures vol 38 no 36-37 pp 6275ndash6305 2001
[3] J C Wallach and L J Gibson ldquoMechanical behavior of a three-dimensional truss materialrdquo International Journal of Solids andStructures vol 38 no 40-41 pp 7181ndash7196 2001
[4] S Hyun A M Karlsson S Torquato and A Evans ldquoSimu-lated properties of Kagome and tehragonal truss core panelsrdquoInternational Journal of Solids and Structures vol 40 no 25 pp6989ndash6998 2003
[5] T KimH PHodson andT J Lu ldquoFluid-flow and endwall heat-transfer characteristics of an ultralight lattice-frame materialrdquoInternational Journal of Heat and Mass Transfer vol 47 no 6-7pp 1129ndash1140 2004
[6] J-H Lim and K-J Kang ldquoMechanical behavior of sandwichpanels with tetrahedral and Kagome truss cores fabricated fromwiresrdquo International Journal of Solids and Structures vol 43 no17 pp 5228ndash5246 2006
[7] R Biagi and H Bart-Smith ldquoImperfection sensitivity of pyra-midal core sandwich structuresrdquo International Journal of Solidsand Structures vol 44 no 14-15 pp 4690ndash4706 2007
[8] C-H Lim I Jeon and K-J Kang ldquoA new type of sandwichpanel with periodic cellular metal cores and its mechanicalperformancesrdquo Materials and Design vol 30 no 8 pp 3082ndash3093 2009
[9] J S Park J H Joo B C Lee and K J Kang ldquoMechanicalbehaviour of tube-woven Kagome truss cores under compres-sionrdquo International Journal of Mechanical Sciences vol 53 no 1pp 65ndash73 2011
[10] J Lou L Ma and L-Z Wu ldquoFree vibration analysis of simplysupported sandwich beams with lattice truss corerdquo MaterialsScience and Engineering B Solid-State Materials for AdvancedTechnology vol 177 no 19 pp 1712ndash1716 2012
[11] M Xu and Z Qiu ldquoFree vibration analysis and optimizationof composite lattice truss core sandwich beams with intervalparametersrdquo Composite Structures vol 106 pp 85ndash95 2013
[12] J Lou L Z Wu L Ma J Xiong and B Wang ldquoEffects of localdamage on vibration characteristics of composite pyramidaltruss core sandwich structurerdquo Composites Part B Engineeringvol 62 pp 73ndash87 2014
[13] F-M Li and X-X Lyu ldquoActive vibration control of latticesandwich beams using the piezoelectric actuatorsensor pairsrdquoComposites Part B Engineering vol 67 pp 571ndash578 2014
[14] Z-G Song and F-M Li ldquoAeroelastic analysis and activeflutter control of nonlinear lattice sandwich beamsrdquo NonlinearDynamics vol 76 no 1 pp 57ndash68 2014
[15] Y Frostig and O T Thomsen ldquoHigh-order free vibration ofsandwich panels with a flexible corerdquo International Journal ofSolids and Structures vol 41 no 5-6 pp 1697ndash1724 2004
[16] H Biglari and A A Jafari ldquoHigh-order free vibrations ofdoubly-curved sandwich panels with flexible core based on arefined three-layered theoryrdquo Composite Structures vol 92 no11 pp 2685ndash2694 2010
[17] J E Jam B Eftari and S H Taghavian ldquoA new improved high-order theory for analysis of free vibration of sandwich panelsrdquoPolymer Composites vol 31 no 12 pp 2042ndash2048 2010
[18] E Nilsson and A C Nilsson ldquoPrediction and measurement ofsome dynamic properties of sandwich structures with honey-comb and foam coresrdquo Journal of Sound and Vibration vol 251no 3 pp 409ndash430 2002
[19] Y Q Li and D W Zhu ldquoFree flexural vibration analysis ofsymmetric rectangular honeycomb panels using the improvedReddyrsquos third-order plate theoryrdquo Composite Structures vol 88no 1 pp 33ndash39 2009
[20] J H Zhang and W Zhang ldquoMulti-pulse chaotic dynamics ofnon-autonomous nonlinear system for a honeycomb sandwichplaterdquo Acta Mechanica vol 223 no 5 pp 1047ndash1066 2012
[21] X-D Yang W Zhang and L-Q Chen ldquoTransverse vibrationsand stability of axially traveling sandwich beam with soft corerdquoJournal of Vibration and Acoustics vol 135 no 5 Article ID051013 2013
[22] R Sahoo and B N Singh ldquoA new trigonometric zigzag theoryfor buckling and free vibration analysis of laminated compositeand sandwich platesrdquo Composite Structures vol 117 no 1 pp316ndash332 2014
[23] R K Khare T Kant and A K Garg ldquoFree vibration ofcomposite and sandwich laminates with a higher-order facetshell elementrdquo Composite Structures vol 65 no 3-4 pp 405ndash418 2004
[24] T Sakiyama H Matsuda and C Morita ldquoFree vibrationanalysis of sandwich beam with elastic or viscoelastic core byapplying the discrete green functionrdquo Journal of Sound andVibration vol 191 no 2 pp 189ndash206 1996
[25] J R Banerjee ldquoFree vibration of sandwich beams using thedynamic stiffness methodrdquo Computers and Structures vol 81no 18-19 pp 1915ndash1922 2003
[26] M Cetkovic and D Vuksanovic ldquoBending free vibrations andbuckling of laminated composite and sandwich plates using alayerwise displacement modelrdquo Composite Structures vol 88no 2 pp 219ndash227 2009
[27] A J M Ferreira ldquoAnalysis of composite plates using a lay-erwise theory and multiquadrics discretizationrdquo Mechanics ofAdvanced Materials and Structures vol 12 no 2 pp 99ndash1122005
[28] H G Allen Analysis and Design of Structural Sandwich PanelsPergamon Oxford UK 1969
[29] A H Nayfeh and D T Mook Nonlinear Oscillations Wiley-Interscience New York NY USA 1979
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of