CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 25,aNo. 5,a2012
·1 029·
DOI: 10.3901/CJME.2012.05.1029, available online at www.springlink.com; www.cjmenet.com; www.cjmenet.com.cn
Modeling and Free Vibration Behavior of Rotating Composite Thin-walled Closed-section Beams with SMA Fibers
REN Yongsheng1, *, YANG Shulian2, and DU Xianghong1
1 College of Mechanical and Electronic Engineering, Shandong University of Science and Technology, Qingdao 266510, China
2 Shandong Institute of Business and Technology, Yantai 264005, China
Received September 16, 2011; revised April 13, 2012; accepted April 19, 2012
Abstract: Smart structure with active materials embedded in a rotating composite thin-walled beam is a class of typical structure which is using in study of vibration control of helicopter blades and wind turbine blades. The dynamic behavior investigation of these structures has significance in theory and practice. However, so far dynamic study on the above-mentioned structures is limited only the rotating composite beams with piezoelectric actuation. The free vibration of the rotating composite thin-walled beams with shape memory alloy(SMA) fiber actuation is studied. SMA fiber actuators are embedded into the walls of the composite beam. The equations of motion are derived based on Hamilton’s principle and the asymptotically correct constitutive relation of single-cell cross-section accounting for SMA fiber actuation. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin’s method. The formulation for free vibration analysis includes anisotropy, pitch and precone angle, centrifugal force and SMA actuation effect. Numerical results of natural frequency are obtained for two configuration composite beams. It is shown that natural frequencies of the composite thin-walled beam decrease as SMA fiber volume and initial strain increase and the decrease in natural frequency becomes more significant as SMA fiber volume increases. The actuation performance of SMA fibers is found to be closely related to the rotational speeds and ply-angle. In addition, the effect of the pitch angle appears to be more significant for the lower-bending mode ones. Finally, in all cases, the precone angle appears to have marginal effect on free vibration frequencies. The developed model can be capable of describing natural vibration behaviors of rotating composite thin-walled beam with active SMA fiber actuation. The present work extends the previous analysis done for modeling passive rotating composite thin-walled beam. Key words: free vibration, thin-walled composite beams, shape memory alloy, rotating beams, pich angle, precone angle
1 Introduction
In recent year, high-performance composite materials have been widely used in advanced helicopter and wind turbine rotor blade, because composite structures have many desirable characteristics, such as high ratio of strength to weight and favorable elastic coupling through elastic tailoring, better fatigue life and damage tolerance. In dynamic analysis, the composite rotor blade is often idealized as rotating composite thin-walled beams with closed-section. In fact, the rotating composite thin-walled beams are much more complex compared with the non-rotating metallic one, in the sense that, in addition to the coupling between the rotating motion of rigid body and elastic deformations of beams, they also exhibit
* Corresponding author. E-mail: [email protected] This project is supported by National Natural Science Foundation of
cross-sectional warping and elastic coupling caused by material anisotropy. The accurate prediction of their free vibration is essential for the successful design of these structures. A review of the state-of-the-art of this problem was presented by HODGES, et al[1–2].
Recent research has shown that improvement in structural vibration problems can be achieved through the implementation of active control technology using smart materials and structures. Smart composite structures have received considerable attention due to the potential for designing adaptive structures that are both light in weight and possess adaptive control capabilities. Common materials used for such structures are piezoelectric, electrostrictive, magnetostrictive and shape memory alloys(SMA).
So far, several composite thin-walled beams theories have been reported for the analysis of thin-walled anisotropic beams. REHFIELD, et al[3–4], presented a linear thin-walled beam theory for the design analysis of composite rotor blades and discussed non-classical behavior of thin-walled composite beams with closed cross-sections. LIBRESCU, et al[5–7], developed models,
·1 030· REN Yongsheng, et al: Modeling and Free Vibration Behavior of Rotating Composite Thin-walled
Closed-section Beams with SMA Fibers
which were employed in the treatment of the statics and dynamics problems of composite thin-walled beam. In these models, the bending component of shear flexibility was taken into account but the warping torsion component was neglected. The variational asymptotic approach is asymptotically correct and includes both in-plane and out-of-plane warping effects of thin-walled composite beams with arbitrary cross sections[8]. In the variational asymptotic approach[9], the 3D properties of the beam are reduced to 1D beam properties (extension, twist and two bending terms) and the beam response is then approximated based on a 1D analysis.
ARMANIOS, et al[10], derived the equations of motion for free vibration analysis of anisotropic thin-walled closed-section beams by using the variational asymptotic approach and Hamilton’s principle. On the basis of governing equations presented by ARMANIOS, et al [10], DANCILA, et al[11], developed a closed form solution of natural frequencies and mode shapes and isolated the influence of coupling on free vibration of closed-section beams exhibiting extension-twist, bending-twist coupling.
Considerable work has been done on the modeling of beams with smart materials[9, 12–16]. However, these studies address modeling of the rotating thin-walled composite beams with piezoelectric actuators. To the best of the author’s knowledge, no literature on free vibration of rotating thin-walled composite beams with SMA fiber actuators is cited, though a few studies have been found in the static analysis of non-rotating smart beams[17–20]. GHOMSHEI et al[17], presented a mathematical model for analysis of SMA layerelastomer three-dimensional laminated composite box beam. Results demonstrate that significant changes occur in the actuator’s responses during phase transformation due to strain recovery. HE, et al[18], presented an analytical model of thin tube with SMA wires winded and pasted on the outside surfaces of the thin wall tube. The mixed deformations of compression and torsion of the tube were analyzed and the predicted results were validated with experiments. REN, et al[19–20], developed a formulation for the active deformation analysis of closed-section composite beams with SMA fibers embedded. An analytical model describing the force- deformation constitutive relationship of induced-strain of SMA fiber hybrid closed-section composite beams due to SMA phase transformation effect is presented. The effects of the volume of the SMA fibers, the martensitic residual strain and ply angle are investigated.
The asymptotic formulation for the cross-sectional analysis of beams with thin-walled, single-cell was presented by ARMANIOS, et al[10] and has become the underlying basis for comprehensive active and passive thin-walled composite beam modeling developed and used by several researchers [9, 16, 19–23].
In this paper, we extend the analysis in Ref. [10] to include rotating composite the thin-walled closed-section beams with SMA fiber actuation. The structural dynamic
modeling is split into two parts: a two-dimensional analysis over the cross section, and a linear analysis of a beam along the beam span. The cross-sectional analysis employs the asymptotically correct constitutive relation of single-cell cross-section accounting for SMA fiber actuation[19–20]. The equations of motion and associated boundary conditions of the beams are derived using Hamilton’s principle. The formulation for free vibration analysis includes anisotropy, pitch and precone angle, centrifugal force and SMA actuation effect. The Galerkin’s method is employed in order to solve the coupled differential equations. Numerical results are obtained for two cantilevered box beam: circumferentially uniform stiffness (CUS) and circumferentially antisymmetric stiffness(CAS), the effects of SMA fiber actuation, rotating speeds, fiber orientations, pitch and precone angle are investigated.
2 Basic Theory
2.1 Displacement and strain field
Consider the straight thin-walled composite beam with an arbitrary cross-section shown in Fig. 1. The elastic axis of the undeformed beam x is inclined to the rotational plane at the precone angle βp. The length of the beam is R with the constant rotating speed Ω. Three sets of coordinate systems will be used in the theoretical developments presented in this paper. The origin of the rotating systems of coordinates ( , ,x y z ) is located at the beam root O. The second coordinate system is the local coordinates ( , ,x η ς ), where η and ς are the principal axes of the beam cross-section. In addition, a local coordinate systems ( , , )x s ξ is defined. The geometric configuration of the beam cross-section, along with the associated systems of coordinate is presented in Fig. 2.
Fig. 1. Coordinate systems and geometry
of thin-walled beam
Fig. 2. Cross-section and local coordinate systems
Coordinate transformation between the rotating systems of coordinates and the principal systems of coordinates of the beam is given as follows:
0 0
0 0
,
( ) cos sin ,
( ) sin cos ,
x x
y s
z s
η θ ς θ
η θ ς θ
(1)
where 0θ represents the pitch angle of the beam.
The displacement components with respect to local coordinate axis ( , , )x s ξ are determined as follows, which include the warping field function ( , ),g s x associated with extension, torsion and bending out-of-plane warping of the cross-section[9]:
1 3( ) ( ) ( ) ( ) ( ) ( , ),
( ) ( ) ( ) ,
( ) ( ) ( ) ,
s n
t
u u x y s v x z s w x g s xy zv v x w x x rs sz yv v x w x x rs sξ
φ
φ
d d
d dd d
d d
(2)
where the displacements in the axial, tangential and normal directions are denoted by 1, , ,su v vξ respectively, and the average displacements along the coordinates are denoted by
( ), ( ), ( )u x v x w x and ( )xφ is the twist angle. The primes in the Eq. (2) represent differentiation with respect to .x .
From continuity of the displacement field and the vanishing of hoop stress resultant and constant shear flow, the out-of-plane warping function ( , )g s x can be obtained as follows:
1( , ) ( ) ( ) ( ) ( )g s x G s x g s u xφ
2 3( ) ( ) ( ) ( ),g s v x g s w x (3)
where the function ( )G s and ( )ig s are the warping functions associated with torsion, extension and two bending measures. The components of the displacement along the coordinate ( , ,x y z ) are expressed as follows[10]:
u x s u x y s v x z s w x g s xu x s v x z s xu x s w x y s x
φφ
(4)
The in-plane strain field associated with Eq. (2) is as
follows:
11
112
32
22
( ) ( ) ( ) ( ) ( ),
2 ( ) ( )
( ) ( ),
0.
n
u x y s v x z s w xgG r x u x
s sgg v x w x
s s
γ
γ φ
γ
dd
d d
dd
d d
(5)
The relationship between the in-plane strain tensor 11 12,γ γ and engineering strain ,xx xsε ε is represented as follows:
11 121, .2xx xsγ ε γ ε (6)
2.2 Equations of motion and boundary conditions
The equations and associated boundary conditions of the undamped free vibration are derived by using Hamilton principle:
1
0( ) 0,
t
tU T W tδ δ δ d (7)
where U is the strain energy, T is the kinetic energy , and W is the virtual work of the external forces.
The usual expression for strain energy and kinetic energy are defined as follows, respectively:
0
0
1 ( )d d d ,2
1 ( )d d d .2
R
xx xx xs xsA
R
A
U x
T x
σ ε σ ε η ς
ρ η ς
V V (8)
where xxσ and xsσ represent the engineering stress associated with the engineering strain xxε and ,xsε V represents the velocity vector of the arbitrary point for the deformed beam, and can be determined as VR, with
1 2 3( ) ( ) ( ) ,x u y u z u R i j k which represents the position vector of a point, the superposed dots represents time t derivatives, i, j, k are unit vector associated with the systems of coordinates ( , ,x y z ). The material density is represented by ρ.
By substituting Eq. (8) into Eq. (7), and the integrations in the circumferential and thickness direction, one get
where
The variation of the kinetic energy can be written as follows:
·1 032· REN Yongsheng, et al: Modeling and Free Vibration Behavior of Rotating Composite Thin-walled
Closed-section Beams with SMA Fibers
where
2
21 0 0 2 0 0
21 0 2 0
1 0 2 0 1 0 2 02
1 0 0 2 0 0
1
2 ,cos sin sin cos2 2 2 cos sin
sin cos sin coscos sin sin cos
2
u
v
p
v
T x u v uT e e
v w u v e ew e e v e e
T x e ev e
Ω ΩΩ θ φ θ θ φ θΩ Ω β Ω Ω θ θ
θ θ φ θ θΩ θ φ θ θ φ θ
Ω
0 2 02
1 0 2 02
1 0 0 2 0 0
1 0 2 02 2 2 2m m2 m1 0 0
20 0 1 0 2
cos sin2 cos sin
sin cos cos sin2 sin cos
cos sinsin cos cos sin
w p p
w
eT x v w e eT x e e
v e eT k k k
x w e eφ
θ θΩ β Ω β φ θ θ
Ω θ φ θ θ φ θΩ θ θ
φ Ω θ φ θ
θ φ θ Ω θ
02
1 0 2 0 1 0 2 02
1 0 2 0 1 0 2 0
1 0 2 0
sin cos sin coscos sin sin cos
cos sinp
v e e v e ex e e v e e
w e e
θθ θ Ω θ θ
Ω β θ θ θ θθ θ
(12)
m is the beam mass per unit length, 1e and 2e are the centers of mass offset from the elastic axis, 2
mmk is polar mass moment of inertia; 2
m1mk and 2m2mk are mass
moment of inertia . The variation of the virtual work of the external forces
can be written as
(13)
where , ,x y zF F F and xM are the distributed loads that act in the x, y, z directions and a twisting moment about the elastic axis, respectively.
By substituting Eqs. (9), (11), and (13) into Eq. (7), one obtain the total variational equation in terms of
( ), ( ), ( )u x v x w x and ( ).xφ For arbitrary, admissible variations uδ , vδ , wδ , ,φδ the coefficients of the variations must vanish in the integrant for all x from 0 to R and also must vanish in the remaining terms evaluated at 0 and R.
The resulting equations of motion are as follows:
1
1 3
1 2
1
ˆ ,ˆˆ( ) ( ) ,
ˆˆ( ) ( ) ,ˆ .
u x
v yv
w zw
x
F mT F
m T T v F M F
m T T w F M F
mT M Mφ
(14)
The boundary conditions become
( ) ( ) 0,b U b T (15)
where the generalized beam forces 1F , 1M , 2M and 3M are axial force, torsional moment, bending moments in the y and z directions, respectively, defined as follows[9]:
( )
1 11 11 12 13 14 1
( )1 1 12 22 23 24 2
( )2 11 13 23 33 34 3
( )3 11 14 24 34 44 4
ˆ d ,
ˆ d ,
ˆ d ,
ˆ d ,
a
as n
a
a
F N s c u c c w c v L
M N r s c u c c w c v L
M N z s c u c c w c v L
M N y s c u c c w c v L
φ
φ
φ
φ
(16)
where ( )aiL represent the forces or moments associated with
the temperature change and the SMA actuation due to the phase transformation[19–20].
The cross sectional stiffness denoted ijc can be expressed in terms of their cross sectional principal axes ς and η counterparts, ijk , and pitch angle θ0:
11 11
12 12
13 14 0 13 0
14 14 0 13 0
22 22
23 24 0 23 0
24 24 0 23 02 2
33 44 0 33 0 34 0 02 2
34 0 0 34 0 0 44 33
44 4
,,sin cos ,cos sin ,,sin cos ,cos sin ,
sin cos 2 sin cos ,
( cos sin ) sin cos ( )
c kc kc k kc k kc kc k kc k k
c k k k
c k k k
c k
θ θθ θ
θ θθ θ
θ θ θ θ
θ θ θ θ
2 24 0 33 0 34 0 0cos sin 2 sin cos ,k kθ θ θ θ
(17)
where ijk can be determined as described in Ref. [10].
As described in Ref. [19], ( )aiL has the following
expressions:
(18)
CHINESE JOURNAL OF MECHANICAL ENGINEERING
·1 033·
where ,i ijΓ Λ ( ,i j 1, 2, 3, 4) represent the coefficients of
the forces or moments associated with the SMA fiber actuation, and ΔT
iL ( i 1, 2, 3, 4) represent the forces or moments associated with the temperature change. Note that for beams under rotational condition[24], the torsional stiffness term 22c become 2
22 1+ ,Ac F K where 2e AA K
2 2( + )d d .A
ς η η ς Since the axial stiffness is much
larger as compared to the ones associated with horizontal bending and vertical bending, the effect of the axial displacement u , inertia u and Coriolis term 2 uΩ are discarded [13]. Moreover, for CUS or CAS configuration beams, the first moment of mass 1 2 0e e . As a result, Eq. (12) reduces to as follows:
2
2p
2p p
2 2 2 2m m2 m1 0 0
2 20 0
,
2 ,
0,
2 ,
0,
( )[ cos sin
(cos sin )],
u
v
v
w
w
T x
T v w v
T
T x v w
T
T k k kφ
Ω
Ω Ω β
Ω β Ω β
φ Ω θ θ
φ θ θ
(19)
where
2 2m1
2 2 2 2 2m2 m m1 m2
d d , d d ,
d d , ,
A A
A
m mk
mk k k k
ρ η ς ρς η ς
ρη η ς
Substituting Eq. (19) into Eq. (14), droppings external
force xF and integrating the first equation of the resulting equations from 0 to R, yields
2
2 21 ( )
2mF R x Ω (20)
It should be noticed that in the cases of discarding the
SMA actuation, rotating effect and the pitch angle from the Eqs. (12), (14) and (16), one can easily obtain the motion equations derived by Armanios and Badir[10] as follows:
11 12 13 142
12 22 23 24 m 1 2
13 23 33 34 1
14 24 34 44 2
0,
0,
0,
0.
k u k k w k v mu
k u k k w k v mk me w me v
k u k k w k v me mw
k u k k w k v me mv
φ
φ φ
φ φ
φ φ
(21) Substituting Eqs. (16), (18)–(20) into the last three
equations of Eqs. (14) and dropping external forces, one obtain
2p 1 24 42
(4) (4)34 43 44 44
2p p 1 23 32
(4) (4)33 33 34 34
2 2 2 2 2m m2 m1 0 0 0
20
ˆ( 2 ) ( ) ( )
( ) ( ) 0,ˆ( 2 ) ( ) [( )
( ) ( ) ] 0,( )[ cos sin (cos
sin )] [(
m v w v v F cc w c v
m x v w w F cc w c v
m k k k
Ω Ω β Λ φ
Λ ΛΩ β Ω β Λ φ
Λ Λφ Ω θ θ φ θ
θ
22 22 23 23
224 24 1
) ( )ˆ( ) ( ) ] 0.A
c c w
c v K F
Λ φ Λ
Λ φ
(22) The solutions of Eq. (22) can be expressed as follows:
d
s d
s d
,,
,
v vw w wφ φ φ
(23)
where subscript s and d represent the static and dynamic components of the displacement, respectively. And the free vibration can only be determined by the dynamic systems.
Substitute the form of solution Eq. (23) into Eq. (22), the resulting free vibration equations are as follows:
2
d d p d d 1 24 42 d(4) (4)
34 43 d 44 44 d
d p d d 1 23 32 d(4) (4)
33 33 d d 34 34 d2 2 2 2 2 2m d m2 m1 0 0 d
ˆ( 2 ) ( ) ( )
( ) ( ) 0,ˆ(2 ) ( ) [( )
( ) ( ) ] 0,[ ( )( cos sin ) ]
[(
m v w v v F cc w c v
m v w w F cc w c v
m k k kc
Ω Ω β Λ φ
Λ ΛΩ β Λ φ
Λ Λφ Ω θ θ φ
22 22 d 23 23 d 24 24 d2
1 d
) ( ) ( )ˆ( ) ] 0.A
c w c vK F
Λ φ Λ Λφ
(24) In order to convert the free vibration equations into a
dimensionless form, the following dimensionless parameters are introduced:
3 Solution Procedure
3.1 CUS configuration beams Governing Eq. (24) is represented in terms of
displacement quantities, it involves the horizontal bending, vertical bending and twisting vibration coupling.
The present model is applied two lamination scheme configuration, including CUS and CAS to illustrate the
·1 034· REN Yongsheng, et al: Modeling and Free Vibration Behavior of Rotating Composite Thin-walled
Closed-section Beams with SMA Fibers
influence SMA fiber actuation, rotating speeds, fiber orientations, pitch and precone angle on the natural frequencies. For a CUS configuration, the ply lay-up is
( ) ( ),y yθ θ ( ) ( ),z zθ θ where θ represents the orientation of the fibers measured from the s-positive axis the x-positive direction. As a result, the stiffness coefficients and the actuation forcemoment coefficients for a rectangular cross-section have the following properties:
13 14 23 24 0C C C C ,
13 14 23 24 34 31
41 32 42 43 0.Λ Λ Λ Λ Λ Λ
Λ Λ Λ Λ
Thus, from Eq. (24), the following equations can be obtained:
2d4 4
d d2 74 4
2d d
d p 2
2d4 4
d d1 74 4
2d d
p 2
2d2 2
2 2d3 32
(1 )12
2 0,
(1 )12
2 0,
(1 )12
v xv w x
xx xw v
v
w xw v x
xx xv w
xx
xx
Λ Λ
βψ ψ
Λ Λ
βψ ψ
φφ
Λ µ µ
d2
2 2 2 22 1 0 0 d( )( cos sin ) 0.
φψ
µ µ θ θ φ
(25)
As one can see from Eq. (25) for CUS configuration
beams, the free vibration equations reduce to coupling horizontal-vertical bending equations and one decoupled twisting equation. It also becomes evident that the decoupled modes are independent of the pitch and precone angle.
3.2 CAS configuration beams
For a CAS configuration, the ply lay-up is For a CUS configuration, the ply lay-up is ( ) ( ),y yθ θ
( ) ( ).z zθ θ As a result, for a rectangular cross-section, the stiffness coefficients and the actuation forcemoment coefficients are characterized by, respectively:
12 13 14 0,C C C
2d4 3 4
d d d2 6 74 3 4
2d d
d p 2
(1 )12
2 0,
v xv w x
xx x xw vv
φΛ Λ Λ
βψ ψ
2d4 3 4
d d d1 5 74 3 4
2d d
p 2
2d2 3 3
2d d d3 4 6 32 3 3
22 2 2 2 2d
2 1 0 0 d2
(1 )12
2 0,
(1 )12
( )(cos sin ) 0.
w xw v x
xx x xv w
xw v x
xx x x
φΛ Λ Λ
βψ ψ
φφ
Λ Λ Λ µ
φµ µ µ θ θ φ
ψ
(26)
In this case, the following equations can be obtained:
2d4 3 4
d d d2 6 74 3 4
2d d
d p 2
2d4 3 4
d d d1 5 74 3 4
2d d
p 2
2 3 3d d d
3 4 62 3 3
(1 )12
2 0,
(1 )12
2 0,
v xv w x
xx x xw v
v
w xw v x
xx x xv w
w vx x x
φΛ Λ Λ
βψ ψ
φΛ Λ Λ
βψ ψ
φΛ Λ Λ
2d
23
22 2 2 2 2d
2 1 0 0 d2
(1 )12
( )( cos sin ) 0.
xx
x
φ
µ
φµ µ µ θ θ φ
ψ
(27)
It is obviously that in the case of CAS configuration beams, the free vibration equations result in a completely coupled form. To obtain ordinary differential equations of free vibration, deformation variables are approximated by using the Galerkin’s method as follows:
d d1 1
( ) ( ), ( ) ( ) ,N N
j j j jj j
v V x w W xψ φ ψ φ
d1
( ) ( ) ,N
j jj
xφ φ ψ θ
(28)
where ( ),jV ψ ( )jW ψ and ( )jφ ψ represent the generalized coordinates. The spatial mode functions ( )j xϕ and ( )j xθ for the bending and torsion deformations are the standard nonrotating, uncoupled mode shape for a uniform cantilever beam[25]:
CHINESE JOURNAL OF MECHANICAL ENGINEERING
·1 035·
where constant jβ is the roots of the transcendental characteristic equation of cantilever beam. Substituting these expressions into Eq. (27) and by using Galerkin’s method, one can obtain to 3N ordinary equations in terms of ( ), ( )j jV Wψ ψ
and ( )jφ ψ :
,MX CX KX 0 (30)
where
It should be noticed that if let 4 5 6 0,Λ Λ Λ Eq. (30) reduces to the free vibration equations of CUS configuration rotating beams which are same as Eq. (30) in form, with
Eq. (30) can be easily simplified as the generalized eigenvalue problem and take the form as
,λAy y (31)
where
T
1 1
( ) ,
.
y x x0 I
ΑM K M C
(32)
The decoupled torsional free vibration equation in terms
of the generalized coordinates for CUS configuration beams is
(33)
The marix form of Eq. (33) is also similar to Eq. (30),
and the associated terms in Eq. (30) become
4 Applications and Numerical Results
4.1 Code verifications
In order to check the accuracy for numerical simulation in this study, the numerical results obtained by using the proposed modeling method are compared to those presented in Refs. [10–11].
As a first example, the vertical bending (VB) and horizontal bending (HB) fundamental natural frequencies for CUS configuration beams are compared with the previous results. The stacking sequence of the composite box beam is 6[ ]θ along the entire circumference of the cross-section. Furthermore, the beam is made of graphite-epoxy. The geometry properties of the cantilevered beam are provided in Ref. [10]. Fig. 3 show the variation of the first VB and HB natural frequencies with ply angle for
·1 036· REN Yongsheng, et al: Modeling and Free Vibration Behavior of Rotating Composite Thin-walled
Closed-section Beams with SMA Fibers
CUS configuration beams. It clearly shows that the results obtained by the present modeling method are in almost equal to the closed form solutions of Ref. [10]. It is also observed that in this case, using only 2 modes (N2) will obtain quite accurate natural frequencies.
Fig. 3. Fundamental natural frequencies vs. ply angle for the vertical (VB) and horizontal (HB) bending mode of CUS beam In the second example, the natural frequencies associated
with bending-twist coupling for CAS configuration beams are examined. The CAS stacking sequence used by this example consists of 6[ ]θ in the top wall, 6[ ]θ in the bottom wall and 3[ ]θ θ in the vertical walls. The CAS configuration beam properties are given in Ref. [11]. According to Ref. [11], the first and second bending-dominated natural frequency are represented by symbols B1 and B2, respectively, whereas T1 represents twisting-dominated natural frequency. In Fig. 4, it can obviously be seen that the present numerical results are in good agreement with the exact results presented in Ref. [11]. Moreover, based on mode convergence examination, it is found that in this case, N5 gives suitably converged eigenvalues.
Fig. 4. CAS natural frequencies vs. ply angle
4.2 CUS thin-walled beam Consider the CUS box beam shown in Fig. 5. The beam
is made of graphiteepoxy composite matrix with NiTi fibers embedded. The geometry and material properties are given in Table 1.
Fig. 5. Lay-up of CUS box-beam cross section
Table 1. Material properties of NiTigraphiteepoxy
Graphite epoxy [11] 55–Nitinol [17] Tensile modulu E11 GPa
142 Tensile modulu EsGPa
26.3
Tensile modulu E22GPa
9.8 Austenitic start temperature As
34.5
Shear modulu G12GPa
6.0 Austenitic finish Temperature Af
49
Shear modulu G23GPa
4.83 Stress induced transf. const. CA(MPa·(–1))
13.8
Possion’s ratio ν12 0.42 Thermal coefficient θ 0.55 Possion’s ratio ν23 0.50 Possion’s ratio νs 0.33 Thermal expansion coefficient α1m–1
–0.110–6 Thermal expansion coefficient αs–1
10.310–6
Thermal Expansion coefficient α2m–1
3010–6 Phase transformation tensor ΩMPa
-1 762
Density ρm(kg • m–3)
1 601.1 Density ρs(kg • m–3) 6 500
Fig. 6 shows that the variations of the first two coupled
V-H natural frequencies of the CUS rotating beam as a function of the ply-angle for different SMA fiber volumes ( Ω 200 rads, 0 5θ , 0ε 0.067, pβ 0.1).
Fig. 6. First two coupled V-H natural frequencies vs. ply
angle of CUS beam for different SMA fiber volume
In this study, assume that NiTi fibers are orientated along
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the same direction to graphite fibers, and the chosen temperature is T50 under which NiTi fibers are actuated and the phase transformation from martensite to austenite occurs[19–20]. Actived NiTi fibers can generate the actuated moments about the y and z axes, whereas to decrease the bending stiffness due to the existence of these actuated moments. The results show that lower natural frequency can be achieved for the active thin-walled composite beams as compared with the passive one (vs 0) and the decrease in natural frequency becomes more significant as SMA fiber volume increases. Fig. 6 also shows that as the ply-angle increases, the coupled V-H natural frequencies decrease. It can be also noted that the effect of SMA fiber volume is the highest when the beam has the fiber-orientation of θ 0°.
Fig. 7 shows that the variations of the first two coupled V-H natural frequencies with ply-angle for different initial strains of SMA fiber (Ω 200 rads, θ05°, vs0.1,
pβ 0.1). As it appears from these figures, the initial strains of SMA fiber do not influence the coupled V-H natural frequencies significantly.
Fig. 7. First two coupled V-H natural frequencies vs. ply
angle of CUS beam for different initial strains of SMA fiber
Fig. 8 shows that the variations of the first two coupled V-H natural frequencies with rotating speed for different SMA fiber volumes ( 0θ 5°,θ 45°, 0ε 0.067, pβ 0.1). Natural frequencies increase as the rotating speed is increased, because of the existence of a larger centrifugal force at a higher revmin, which is called as the dynamic stiffening. The significant effect of the SMA fiber volume can also be seen clearly, as shown in Fig. 8. However, from Fig. 8 it is evident that the effect of SMA fiber volume on the coupled V-H natural frequencies, especially the
higher-mode ones is significant at lower rotational speeds.
Fig. 8. First two coupled V-H natural frequencies vs. rotational
speeds of CUS beam for different SMA fiber volume
Fig. 9 presents that the variations of the first two decoupled twist natural frequencies with ply-angle of CUS configuration beam for different SMA fiber volumes ( Ω 200 rads, 0 5θ , 0ε 0.067, pβ 0.1).
Fig. 9. First two decoupled twist natural frequencies vs. ply angle of CUS beam for different SMA fiber volume
The results in Fig. 9 show that the effect of the SMA
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Closed-section Beams with SMA Fibers
fiber actuation on the decoupled twist natural frequency is significant. There is a obvious decrease in the decoupled twist natural frequency as the SMA fiber volume is increased. The reason is that the actuated NiTi fibers can generate the twist moments about the x axe, which reduce the twist stiffness of the beam. In addition, Fig. 9 shows that the trend of variation of decoupled twist frequencies with ply-angle is different from that of coupled V-H mode. Moreover, as the results reveal, the effect of SMA fiber volume on the decoupled twist natural frequencies is substantial in the neighborhood of 30 .θ
Fig. 10 presents that the variations of the first two decoupled twist natural frequencies with ply-angle of CUS configuration beam for different initial strains of SMA fiber ( Ω 200 rads, 0 5 ,θ vs0.1, pβ 0.1). It can be observed that the initial strain of SMA fiber has much more influence on the decoupled twist model than the coupled V-H mode. As it clearly appears from Fig.10, the initial strain of SMA fiber has the maximum effect on the decoupled twist natural frequencies at ply-angle 30 .θ
Fig. 10. First two decoupled twist natural frequencies vs. ply
angle of CUS beam for different initial strains of SMA fiber
Fig. 11 shows that the variations of the first two decoupled twist natural frequencies with rotational speed of CUS configuration beam, for various SMA fiber volumes ( 0 5 ,θ 45 ,θ 0ε 0.067, pβ 0.1). For the range of the rotational speed considered, the decoupled twist frequencies are mostly unchanged. But, the significant effect of the SMA fiber volume can be seen evidently.
Fig. 12 shows that the variations of the first two decoupled twist natural frequencies with rotational speed of CUS configuration beam, for various initial strains of SMA fiber ( 0 5 ,θ 45 ,θ 0ε 0.067, pβ 0.1). As the results reveal, the decoupled twist model is much stronger affected by the initial strain of SMA fiber, than the coupled V-H mode.
Fig. 11. First two decoupled twist natural frequencies vs.
rotational speeds of CUS beam for different SMA fiber volume
Fig. 12. First two decoupled twist natural frequencies vs. rotational speeds of CUS beam for different initial strains
of SMA fiber
The variations of the first two coupled V-H natural frequencies of the CUS beam with the ply-angle and rotational speed for different precone angles are presented in Fig. 13 ( 0 5 ,θ 45 ,θ vs0.1, 0ε 0.067) and Fig. 14 ( Ω 200 rads, 0 5 ,θ vs0.1, 0ε 0.067), respectively. Generally, rotating beams like rotor blade, the precone angle is small, so in this study, only the case of βp0°, 3°, 6° are considered. As the precone angle
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increases, the first and third coupled V-H mode decrease and the second coupled V-H mode increases. On the other hand, it can be noted that the effect of the precone angle appears to be not significant for the higher-mode ones.
Fig. 13. First two coupled V-H natural frequencies vs.
rotational speed of CUS beam for different precone angle
Fig. 14. First two coupled V-H natural frequencies vs. ply angle of CUS beam for different precone angle
Fig. 15 shows the effect of pitch angle of the CUS beam
( Ω 200 rads, 0ε 0.067, vs0, pβ 0°). It can be seen from these numerical simulation results that the increase of
the pitch angle causes either a decrease or increase of the coupled V-H frequency, depending on the odd or even mode number. Moreover, the effect of the pitch angle appears to be more significant for the lower-mode ones.
Fig. 15. First two coupled V-H natural frequencies vs.
ply angle of CUS beam for different pitch angle 4.3 CAS thin-walled beam
Consider NiTi graphite-epoxy composite box beam with CAS configuration as shown in Fig. 16. The geometry and material properties of the beam are given in Table 1.
Fig. 16. Lay-up of CAS box-beam cross section
Fig. 17 ( Ω 200 rads, 0 5 ,θ 0ε 0.067, pβ 0.1)
and Fig. 18 ( Ω 200 rads, 0 5 ,θ vs0.1, pβ 0.1) show the variations of the first two bending-dominated natural frequencies with ply-angle of CAS beam for different SMA fiber volumes and initial strains, respectively. It is evident that the reduction of frequencies can be caused under SMA fiber actuation. When the SMA fiber volume or the initial strain of SMA fiber increase, the bending-dominated frequencies decrease, and this trend is similar to the case of coupled V-H for CUS configuration beam. In addition, these figures reveal a number of trends related to the influence of the ply-angle upon each bending-dominated frequency. These trends are consistent with that presented in the case of
·1 040· REN Yongsheng, et al: Modeling and Free Vibration Behavior of Rotating Composite Thin-walled
Closed-section Beams with SMA Fibers
CUS configuration beams. However, it can be observed that the initial strain of SMA fiber seems to reveal more pronounced effect, as compared to the coupled V-H mode in the case of CUS configuration beam.
Fig. 17 First two bending-dominated natural frequencies vs.
ply angle of CAS beam for different SMA fiber volume
Fig. 18. First two bending-dominated natural frequencies vs. ply
angle of CAS beam for different initial strains of SMA fiber The influence of the SMA fiber volume and initial strain
upon the first two bending-dominated natural frequencies of
CAS beam are presented in Fig. 19 ( 45 ,θ 0 5 ,θ 0ε 0.067, pβ 0.1) and Fig. 20 ( 45 ,θ 0 5 ,θ vs0.1, pβ 0.1), respectively. As shown in these
figures, the SMA fiber volume has a clear influence on the modes, while the initial strain of SMA fiber appears to have a marginal effect on the modes. At the same time, it can also be noted that the variation trends of the bending-dominated natural frequencies with rotational speed are similar to that in the case of CUS configuration beams.
Fig. 19. First two bending-dominated natural frequencies vs.
rotational speeds of CAS beam for different SMA fiber volume
Fig. 20. First two bending-dominated natural frequencies vs.
rotational speeds of CAS beam for different initial strains of SMA fiber
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Fig. 21 ( Ω 200 rads, 0 5 ,θ vs0.1, 0ε 0.067) and Fig. 22 ( 45 ,θ 0 5 ,θ vs0.1, 0ε 0.067) show the variation of the first two bending-dominated natural frequencies with ply-angle and rotational speed, respectively, for selected precone angles. As seen in these figures, the effect of the precone angle on the natural frequencies is similar to that previously presented for the pitch angle.
Fig. 21. First two bending-dominated natural frequencies vs.
ply angle of CAS beam for different precone angle
Fig. 22. First two bending-dominated natural frequencies vs.
rotational speeds of CAS for different precone angle
Fig. 23 shows the variation of the first two bending-dominated natural frequencies with ply-angle of CAS beam, for different pitch angles ( Ω 200 rads, pβ 0, 0ε 0.067, vs0). The general effect of the pitch angle is similar to those of the CUS configuration beam indicate d in the paper.
Fig. 23. First two bending-dominated natural frequencies vs.
ply angle of CAS beam for different pitch angle
The effect of the SMA fiber volume and initial strain on
the first two twisting-dominated natural frequencies of CAS beam are presented in Fig. 24 ( Ω 200 rads,
0 5 ,θ 0ε 0.067, pβ 0.1) and Fig. 25 ( Ω 200 rads, 0 5 ,θ vs0.1, pβ 0.1), respectively. From these figures,
it can be seen that the SMA fiber volume and initial strain influence the twisting-dominated natural frequencies significantly. Also, the effect of SMA fiber volume and initial strain is maximum when the beams have the fiber ply-angleθ 30°.
Fig. 26 ( 45 ,θ 0 5 ,θ 0ε 0.067, pβ 0.1) and Fig. 27 ( 45 ,θ 0 5 ,θ vs0.1, pβ 0.1) show the variation of the first three bending-dominated natural frequencies with rotational speed, for selected SMA fiber volumes and initial strains, respectively. The significant effects of the SMA fiber volume and initial strain on the twisting-dominated frequencies are observed again in these figures, same as in the case of decoupled twist natural frequency of CUS configuration beam. Moreover, we also note that in the rotating condition, the twisting-dominated natural frequency vs. ply-angle curves corresponding to different pitch angles are almost coincidence and the effect of precone angle is not significant. So the results are not presented in this paper, for the sake of simplicity.
·1 042· REN Yongsheng, et al: Modeling and Free Vibration Behavior of Rotating Composite Thin-walled
Closed-section Beams with SMA Fibers
Fig. 24. First two twisting-dominated natural frequencies vs.
ply angle of CAS beam for different SMA fiber volume
Fig. 26. First two twisting-dominated natural frequencies vs.
rotational speeds of CAS beam for different SMA fiber volume
Fig. 25. First two twisting-dominated natural frequencies vs.
ply angle of CAS beam for different initial strains of SMA fiber
Fig. 27. First two twisting-dominated natural frequencies vs.
rotational speeds of CAS beam for different initial strains of SMA fiber
5 Conclusions
(1) The developed model provides means of predicting of the natural frequency of rotating composite thin-walled with SMA fiber actuation.
(2) Active actuation effect can be obtained through SMA
phase transformation. (3) SMA fiber volume has significant impact on the
bending or twist vibration frequencies of the rotating beam associated with general elastic coupling.
(4) The twist vibration frequencies are much stronger affected by the initial strain of SMA fiber than the bending vibration frequencies.
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(5) Rotational speed and ply-angle affect the actuation performance of SMA fibers significantly.
(6) The effect of the pitch angle appears to be more significant for the lower-bending mode ones.
(7) Precone angle appears to have marginal effect on free vibration frequencies.
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Biographical notes REN Yongsheng, born in 1956, received his BS and MS degrees from Taiyuan University of Technology, China, in 1982 and Southeast University, China, in 1989, respectively. He received his PhD degree from Nanjing University of Aeronautics and Astronautics, China, in 1992. He is currently a professor at College of Mechanical and Electronic Engineering, Shandong University of Science and Technologys, China. His research interests include nonlinear dynamics, smart materials, vibration and shock control and aeroelasticity, etc. He has published several scientific papers. E-mail: [email protected] YANG Shulian born in 1966, she received his BS and MS degrees from Taiyuan University of Technology, China, in 1988 and 1991, respectively. She received his PhD degree from Naval Aeronautical and Astronautical University, China, in 2011. She is currently an associate professor at Shandong Institute of Business and Technology, China. Her research interests include vibration and shock control, etc. E-mail: [email protected] DU Xianghong, born in 1984, is currently a master candidate at College of Mechanical and Electronic Engineering, Shandong University of Science and Technologys, China. E-mail: [email protected]
