Harmonic response analysis of symmetric laminated composite beams with different boundary conditions

DOI 10.1515/secm-2013-0194 Sci Eng Compos Mater 2013; aop

Zeki K ı ral *

Harmonic response analysis of symmetric laminated composite beams with different boundary conditions Abstract: This study deals with the determination of the

harmonic response of symmetric laminated composite

beams by the finite element method. The structural stiff-

ness of the composite beam is determined by the classi-

cal laminated plate theory. Four different ply orientations,

namely, [0] 2s

, [0/90] s , [45/-45]

s , and [90]

2s are used to exam-

ine the effect of the stacking sequence on the harmonic

response of the beam. Proportional damping is used to

model the structural damping, and the damped harmonic

responses of the composite beams are obtained to show

the effect of the damping on the harmonic response.

The effect of the boundary conditions on the harmonic

response is also investigated. The displacement maps

calculated for varying excitation points are obtained for

different boundary conditions and damping ratios at dif-

ferent vibrational modes. The numerical results presented

in this study show that the magnitudes of the harmonic

response of the composite beam increase as the flexural

rigidity decreases, and the vibration magnitudes reduce

considerably with damping. The vibration patterns cre-

ated for varying excitation and observation locations

change as the damping ratio and excitation frequency

change.

Keywords: finite element method; harmonic response;

laminated composite beam; numerical integration; pro-

portional damping.

*Corresponding author: Zeki K ı ral, Faculty of Engineering,

Department of Mechanical Engineering, Dokuz Eyl ü l University,

35397 T ı naztepe-Buca, İ zmir, Turkey, e-mail: [email protected]

1 Introduction Engineering structures are generally subjected to time

varying excitations. Among them, harmonic excitation is

one of the most encountered loading types in which the

magnitude of the external load varies within a harmonic

envelope. The source of the harmonic excitation in engi-

neering structures is generally an unbalanced rotating

component. The frequency and location of the harmonic

excitation and the eigenfrequencies of the structure are

the main factors that affect the magnitude and the form

of the structural vibration. Determination of the har-

monic response of the lightweight structures in which the

composite members are involved is of great importance

especially at the design stage. Studies related to dynamic

response of the composite structures in the form of beams

and plates generally deal with the calculation of the

natural frequencies of the considered structure using ana-

lytical or numerical techniques. Contribution to the calcu-

lation of harmonic response of the composite structures

constitutes the main motivation of this study.

Because of the importance of the subject, research-

ers have paid attention to the calculation of the dynamic

response of the engineering structures. The studies on the

dynamic response analyses of engineering structures, which

are composed of isotropic or composite materials, have been

focused mainly on the calculation of free vibration frequen-

cies. Calculation of free vibration frequencies is important to

avoid the resonance phenomenon, but it is also important

to know the vibration levels in a broad range of harmonic

excitation frequency. Vaziri and Nayeb-Hashemi presented

the results of the dynamic response of repaired composite

beams subjected to harmonic peeling load [1] . They used the

finite element method to present the discrepancies between

the theoretical and the experimental results.

Rao and Ganesan used the finite element method in

order to study the harmonic response of tapered com-

posite beams [2] . Makhecha et al. studied the transient

dynamics and damping analysis of laminated sandwich

composite plates [3, 4] . Raja et al. [5] presented the free and

forced harmonic vibration control of composite beams by

active stiffening method using piezoelectric patches. They

reported that the active stiffening effect through displace-

ment control is more effective for smaller modes. Sun and

Huang proposed an analytical formulation to perform the

active vibration control of laminated composite beams

equipped with a piezoelectric sensor and actuator layers

under harmonic loading conditions [6] .

K ı ral investigated the harmonic response of a pinned-

pinned laminated composite beam using the finite element

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2 Z. K ı ral: Harmonic response analysis of symmetric laminated composite beams

method [7] and presented the effect of the damping on the

frequency response of the composite beam with the aid of

proportional damping assumption. Ribeiro [8] studied the

forced vibrations of composite laminated plates by using a

p version, hierarchical finite element including the effects

of rotary inertia, transverse shear, and geometrical non-

linearities. Patel et al. [9] studied the transient response of

anisotropic laminated composite plates by using the finite

element method. Parhi et al. [10] proposed a delamina-

tion model to analyze the dynamic behavior of laminated

composite plates possessing multiple delaminations by

using the finite element method. Bilasse and Oguamanam

[11] presented the forced harmonic response of large-scale

viscoelastic sandwich plates by combining the asymptotic

numerical method and reduction techniques based on the

modal analysis.

The aim of this study is to examine mainly the

harmonic response of a composite beam with differ-

ent boundary conditions, namely, clamped-clamped,

clamped-pinned, pinned-pinned, and clamped-free. The

finite element method is used in association with the

Newmark time integration method, which is a powerful

tool for structural dynamic analysis. In this study, four

different lay-up configurations, namely, [0] 2s

, [0/90] s , [45/-

45] s , and [90]

2s are considered in order to show the effect

of the layup orientation on the frequency response of the

beam. The effect of the location of the harmonic excitation

is also investigated and displacement maps are obtained

for undamped and damped cases. The structural damping

is considered using the proportional damping assump-

tion, and the effect of the damping on the frequency

response of the beam is presented.

2 Beam model In this study, a symmetric laminated composite beam with

a span length of L = 1 m and a rectangular cross section

is considered as shown in Figure 1 . The beam width b is

20 mm and thickness h is 5 mm. Four different boundary

conditions are considered while performing the harmonic

response analysis in order to show the effect of the end

conditions of the composite beam. The beam is subjected

to a harmonic load F ( t ) = F 0 sin(2 π ft ) with magnitude F

0 = 1 N

and the varying forcing frequency f in hertz. In the numeri-

cal analysis, location of the harmonic excitation is changed

and the numerical results of the dynamic analyses are pre-

sented as displacement contours to show the effect of the

source location on the dynamic displacement amplitudes.

Four different lay-up configurations, namely, [0] 2s

, [0/90] s ,

x z

h

L

F(t)=F0 sin(2πf t) F0XF

-F0

t

Figure 1 Clamped-free composite beam subjected to harmonic

loading.

Table 1 Material properties of AS4/3501-6.

E xx

147 GPa

E yy

, E zz

9 GPa

ν xy

, ν xz

0.3

ν yz

0.42

G xy

, G xz

5 GPa

G yz

0.3 GPa

ρ 1.58 g/cm 3

Ply thickness 1.25 mm

[45/-45] s , and [90]

2s are considered in order to investigate

the effect of the ply orientation on the harmonic response

of the laminated beam. The thickness of each layer is iden-

tical for all layers in the laminates. The material of each

lamina is assumed as an AS4/3501-6 material and the

mechanical properties of this material are given in Table 1 .

3 Method In this study, the classical laminated plate theory (CLPT)

is used in order to calculate the stiffness matrix of the

laminated composite beam. According to the CLPT that

was derived by Reddy [12] , the moment equation for unit

length is obtained in the case of plane strain and in-plane

forces of the symmetrical laminated beam as follows,

2

O

2

211 12 16

O

12 22 26 2

216 26 66O

-

2

xx

yy

xy

xM D D DM D D D

yD D DM

x y

ω

ω

ω

⎧ ⎫∂⎪ ⎪

∂⎧ ⎫ ⎪ ⎪⎡ ⎤⎪ ⎪ ∂⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬⎢ ⎥ ∂⎪ ⎪ ⎪ ⎪⎢ ⎥

⎣ ⎦⎩ ⎭ ⎪ ⎪∂⎪ ⎪∂ ∂⎩ ⎭

(1)

2

O

2* * *

11 12 162

* * *O

12 22 262

* * *2 16 26 66

O

-

2

xx

yy

xy

x D D D MD D D M

y MD D D

x y

ω

ω

ω

⎧ ⎫∂⎪ ⎪

∂ ⎧ ⎫⎡ ⎤⎪ ⎪⎪ ⎪∂ ⎢ ⎥⎪ ⎪

=⎨ ⎬ ⎨ ⎬⎢ ⎥∂⎪ ⎪ ⎪ ⎪⎢ ⎥⎣ ⎦⎩ ⎭⎪ ⎪∂

⎪ ⎪∂ ∂⎩ ⎭

(2)



Z. K ı ral: Harmonic response analysis of symmetric laminated composite beams 3

where *

ijD denotes the elements of the inverse of D ij . In the

derivation of the laminated beam theory, it is assumed

everywhere in the beam that

M yy

= M xy

= 0 (3)

To complete the theory, it is also assumed that the

laminated beam is long enough to disregard the effects

of the Poisson ratio and shear coupling on the deflection.

Then,

2

*o

112- xxD M

xω∂

=∂

(4)

The following quantities can be used in order to write

Eq. (4) in the familiar form used in the classical Euler-Ber-

noulli beam theory

3 *

11

12, b

xx xxM b M Eh D

= =

(5)

The coordinate system, which is used in deriving the

layer stiffness, is shown in Figure 2 . In this study, vertical

translation ω and rotation θ within a beam element are

used as follows,

2 3 o

o 1 2 3 4( ) , ( )x a a x a x a x x

xω

ω θ∂

= + + + =∂

(6)

Figure 3 shows the elemental length of the beam,

where subscripts 1 and 2 denote the beam ends x = 0 and

x = d , respectively. Then the displacement vector of a beam

finite element is described as

e 1 1 2 2{ } { }Tq ω θ ω θ=

(7)

h/2

h/2 zk+1

z1z2

zk

z

x

Figure 2 Coordinate locations of plies in laminated composite beam.

ω1 ω2

θ2θ1

d

Figure 3 Elemental length of the beam.

The elastic strain energy and kinetic energy of an ele-

mental laminated beam, respectively, are given as follows,

2 22

o o

2

0 0

1 1 and

2 2

d dbxx yy

d dU E I dx T A dx

dtdxω ω

ρ⎛ ⎞ ⎛ ⎞

= =⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠∫ ∫

(8)

The stiffness [K e ] and mass matrices [M

e ] of an ele-

mental beam are obtained from elastic strain energy and

kinetic energy expressions.

e e e e e e

1 1{ } [ K ] { } , { } [ M ] { }

2 2

T TU q q T q q= = � �

(9)

As the basis of the finite element method, the overall

mass and stiffness matrices are obtained by assembling

the element matrices. The number of elements used in

the finite element vibration analysis is 20. The dynamic

response of the laminated composite beam is calculated

by using the procedure described in the Newmark integra-

tion method, which is widely used in structural dynamics.

By using the overall mass [M], damping [C], and stiffness

[K] matrices for the composite beam, the governing equa-

tions of the system are written in the matrix form as

[ M ] { } [ C ]{ } [ K ] { } { }t t t tq q q F+ + =��

(10)

where { } ,tq�� { } ,tq� and { q } t are the nodal acceleration,

velocity, and displacement vectors at time t , respectively.

{ F } t is the external excitation vector including the nodal

excitations at time t and its elements are zero except for

the vertical degree of freedom of the excitation node. For

the case of harmonic excitation applied at a node, the mag-

nitude of the corresponding element of the force vector

is changed with time in a sinusoidal manner. Eq. (10) is

solved for nodal displacement, velocity, and acceleration

by numerical time integration. The integration constants

are defined in the Newmark method as follows [13] ,

0 1 2 3 42

5 6 7

1 1 1, , , 1, 1,

2

2 , ( 1 ), 2

a a a a at tt

ta a t a t

γ γ

β Δ β Δ β ββ Δ

Δ γΔ γ γΔ

β

= = = = − = −

⎛ ⎞ ⎛ ⎞= − = − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(11)

where Δ t is the time increment used in the numeric analy-

sis. It is well known that the proper selection of the time

increment is very important in the structural dynamic

analysis in order to take into account the effect of the

higher modes in the dynamic response. The time incre-

ment is used as Δ t = T 10

/20, where T 10

is the 10th natural

period of the beam. The integration parameters β and γ are selected as 1/4 and 1/2, respectively, in order to obtain

a stable solution. In the Newmark method, the effective

stiffness matrix can be calculated as




[K ef

] = [K] + a 0 [M] + a

1 [C] (12)

The effective load vector F ef

is calculated for each time step

as

ef 0 2 3

1 4 5

{ } { } [ M ] ( { } { } { } )

[ C ] ( { } { } { } )t t t t t

t t t

F F a q a q a qa q a q a qΔ+= + + +

+ + +� ��

� ��

(13)

Then, the nodal displacement, acceleration, and velocity

responses at time t + Δ t can be obtained by using the fol-

lowing equations

{ q } t + Δ t = [K

ef ] -1 { F

ef } (14)

0 2 3{ } ({ } -{ } )- { } - { }t t t t t t tq a q q a q a q

Δ Δ+ +=��

(15)

6 7{ } { } { } { }t t t t t tq q a q a q

Δ Δ+ += + +� � ��

(16)

In this study, the structural damping is modeled as Ray-

leigh damping in which the damping matrix [C] of the

beam is formed by the linear combination of mass and

stiffness matrices as

[C] = c 0 [M] + c

1 [K] (17)

where c 0 is the mass proportional damping coefficient

and c 1 is the stiffness proportional damping coefficient.

Proportional damping assumption easily provides

information about the damped response of the consid-

ered structure, and therefore the Rayleigh damping is

commonly used in structural dynamics. If the damping

ratios ζ m

and ζ n are known for two specific natural frequen-

cies ω m

and ω n , Rayleigh damping coefficients c

0 and c

1 can

be calculated by the solution of the following equation [12] .

0

2 2

1

-2

-1 1-

n mm n m

nn mn m

cc

ω ωω ω ζ

ζω ωω ω

⎡ ⎤⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎢ ⎥=⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎩⎭ ⎭⎢ ⎥⎣ ⎦

(18)

In this study, first two fundamental free vibration modes

and the corresponding natural frequencies are considered

to calculate the damping coefficients c 0 and c

1 . The same

damping ratio ζ = 0.02 is used for both vibration modes in

order to calculate the damped response of the composite

beam following the numerical procedure described above.

This damping ratio is compatible with the damping ratios

presented in the literature [14, 15] .

4 Numerical results The dynamic response of the composite beam subjected

to the harmonic loading is calculated for different lay-ups,

Table 2 Comparison of first three natural frequencies obtained from Matlab code and LUSAS.

Natural frequencies (Hz)

f n1 f n2 f n3

Present LUSAS % Dif. a Present LUSAS % Dif. a Present LUSAS % Dif. a

Clamped-clamped

[0] 2s

47.931 48.010 0.164 130.290 132.400 1.593 250.010 260.100 3.879

[0/45] s 46.360 45.920 -0.958 126.020 126.700 0.536 241.820 248.800 2.805

[45/-45] s 16.380 16.260 -0.738 44.526 44.870 0.766 85.438 88.280 3.219

[90] 2s

12.141 11.880 -2.196 33.003 32.770 -0.711 63.327 64.380 1.635

Clamped-pinned

[0] 2s

33.048 33.080 0.097 105.750 107.300 1.444 216.230 224.100 3.511

[0/45] s 31.965 31.640 -1.027 102.290 102.600 0.302 209.150 214.300 2.403

[45/-45] s 11.294 11.140 -1.382 36.140 36.000 -0.389 73.895 75.220 1.761

[90] 2s

8.370 8.185 -2.260 26.787 26.540 -0.930 54.771 55.470 1.260

Pinned-pinned

[0] 2s

21.166 21.180 0.066 83.638 84.720 1.277 184.470 192.900 4.370

[0/45] s 20.472 20.270 -0.996 80.897 80.980 0.102 178.430 182.500 2.230

[45/-45] s 7.233 7.298 0.890 28.582 28.020 -2.005 63.042 63.560 0.815

[90] 2s

5.361 5.239 -2.328 21.185 20.960 -1.073 46.727 47.230 1.065

Clamped-free

[0] 2s

7.573 7.544 -0.384 47.201 47.281 0.169 130.49 132.500 1.517

[0/45] s 7.325 7.213 -1.552 45.654 45.210 -0.982 126.22 126.700 0.378

[45/-45] s 2.588 2.523 -2.576 16.130 15.820 -1.959 44.594 44.350 -0.550

[90] 2s

1.918 1.867 -2.731 11.956 11.700 -2.188 33.053 32.780 -0.832

a Natural frequency value calculated using LUSAS FE program is used as reference value.




0 1 2 3-6

0

6

Time (s)

uz(m

m)

0 1 2 3-6

0

6

Time (s)

uz(m

m)

0 1 2 3 4-70

0

70

Time (s)

uz(m

m)

0 1 2 3 4-70

0

70

Time (s)

uz(m

m)

0 1 2 3-350

0

350

Time (s)

uz(m

m)

0 1 2 3-350

0

350

Time (s)

uz(m

m)

A

C

B

D

E F

ζ=0 .0 ζ=0 .02

Figure 4 Midpoint displacement responses of the clamped-clamped composite beam with [90] 2s

lay-up at (A, B) f = 5 Hz, (C, D) 11.5 Hz, and

(E, F) 12.14 Hz when the excitation is applied at the beam midpoint.

-50

-25

0

25

50

0 20 40 60 80 100Excitation frequency (Hz)

20 L

og 1

0(u z)

max

[0]2s [0/45]s[45/-45]s [90]2s

Figure 5 Frequency responses of the pinned-pinned composite

beams calculated for the point x = 0.25 L when the harmonic excita-

tion is applied at x F = 0.25 L , ζ = 0.02.

boundary conditions, damping ratios, and excitation loca-

tions by the numerical integration method. The results of

the numerical analyses are given in this section.

Before the harmonic response analysis, the natural

frequencies of the composite beam for considered lay-up

configurations and boundary conditions are calculated

using the developed Matlab code (The Mathworks Inc.,

Novi, MI, USA). For the comparison purpose, the first

three natural frequencies are also calculated by LUSAS

(LUSAS, Surrey, UK) commercial finite element software.

The results for three natural frequencies are given in

Table 2 . It is seen from the table that there is a good agree-

ment between the numerical results.

-40

-30

-20

-10

0

10

20

30

40

50

0 10 20 30 40 50Excitation frequency (Hz)

20 L

og10

(uz)

max

Clamped-clamped Pinned-pinnedClamped-pinned Clamped-free

Figure 6 Frequency responses of the composite beams with [45/-45]s lay-up calculated for the point x = 0.25 L when the harmonic excitation

is applied at x F = 0.25 L , ζ = 0.02.




xresp

(mm)

x F(m

m)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

0.5

1

1.5

2

2.2

zmax

(mm)A B

C D

E F

xresp

(mm)

x F(m

m)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0.3

0.6

0.9

1.2

1.5

1.8zmax

(mm)

0

xresp

(mm)

x F(m

m)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0.3

0.6

0.9

1.2

1.5

1.8

0

zmax

(mm)

xresp

(mm)

x F(m

m)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

zmax

(mm)

xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

0.1

0.2

0.3

0.4

0.5

zmax

(mm)

xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

0.025

0.05

0.075

0.1

0.125

0.15

zmax

(mm)

Figure 7 Displacement maps of the clamped-clamped composite beam with [45/-45] s lay-up (A) 5 Hz, ζ = 0.0, (B) 5 Hz, ζ = 0.02, (C) 50 Hz,

ζ = 0.0, (D) 50 Hz, ζ = 0.02, (E) 100 Hz, ζ = 0.0, and (F) 100 Hz, ζ = 0.02.

Figure 4 shows the harmonic response of the clamped-

clamped composite beam with [90] 2s

lay-up. Figure 4A and

B shows the undamped and damped harmonic response

of the composite beam at 5 Hz, respectively. As seen from

Figure 4B, damping removes the transient effects in short

time duration and the beam vibrates harmonically at the

excitation frequency. Figure 4C and D shows the harmonic

response at 11.5 Hz. This frequency value is very close to




xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

1

2

3

4

5A z

max (mm)

xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0.5

1

1.5

2

2.5

3

3.6

0

B zmax

(mm)

xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

0.2

0.4

0.6

0.8

1C z

max(mm)

xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35D z

max (mm)

xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

0.1

0.2

0.3

0.4

0.5E z

max(mm)

xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

0.02

0.04

0.06

0.08

0.1F z

max(mm)

Figure 8 Displacement maps of the clamped-pinned composite beam with [45/-45] s lay-up (A) 5 Hz, ζ = 0.0, (B) 5 Hz, ζ = 0.02, (C) 50 Hz,

ζ = 0.0, (D) 50 Hz, ζ = 0.02, (E) 100 Hz, ζ = 0.0, and (F) 100 Hz, ζ = 0.02.

the first natural frequency of the composite beam, and

the beating phenomenon in which the dynamic response

builds up and then ceases down continuously within a

harmonic envelope having the frequency f n - f

exc , occurs

as seen in Figure 4C. Moreover, it is seen from Figure 4D

that the beating phenomenon cannot be seen clearly after

introducing damping but the magnitude of the dynamic

response is still large. Figure 4E and 4F illustrates the




xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

5

10

15

20A z

max(mm)

xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

2

4

6

8

10

12B z

max(mm)

xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

0.2

0.4

0.6

0.8

1

1.2C z

max(mm)

xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

0.05

0.1

0.15

0.2

0.25

0.30.32

D zmax

(mm)

xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

0.1

0.2

0.3

0.4

0.5

0.6

0.70.75

E zmax

(mm)

xresp

(mm)

x F (

mm

)

50 200 350 500 650 800 95050

200

350

500

650

800

950

0

0.03

0.06

0.09

0.12

0.15F z

max(mm)

Figure 9 Displacement maps of the pinned-pinned composite beam with [45/-45] s lay-up (A) 5 Hz, ζ = 0.0, (B) 5 Hz, ζ = 0.02, (C) 50 Hz, ζ = 0.0,

(D) 50 Hz, ζ = 0.02, (E) 100 Hz, ζ = 0.0, and (F) 100 Hz, ζ = 0.02.

midpoint dynamic displacements at the resonance condi-

tion. The magnitude of the dynamic response gets larger

as time passes for the undamped case as expected, and

the dynamic response reaches a steady-state value for

damped case. The damping reduces the midpoint dynamic

displacements considerably at resonance.

Figure 5 shows the displacement frequency response

of the pinned-pinned composite beam with different stack-

ing sequences. Displacement responses are calculated for

the beam point located at x = 0.25 L in order to show two

or more resonance frequencies in the frequency response.

Three resonance frequencies for [45/-45] s and [90]

2s

lay-ups and two resonance frequencies for [0/45] s and

[0] 2s

lay-up configurations can be seen in this figure. The

anti-resonances for the identical excitation and observa-

tion point are also seen in this figure. As seen from the

figure, [90] 2s

lay-up has smaller resonance frequencies

and, consequently, larger resonance displacements due to




xresp

(mm)

x F (

mm

)

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0

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max(mm)

xresp

(mm)

x F (

mm

)

50 300 550 800 100050

300

550

800

1000

0

5

10

15

20

25

30

35

zmax

(mm)

xresp

(mm)

x F (

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)

50 300 550 800 100050

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10

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(mm)

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)

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x F (

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)

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)

50 300 550 800 100050

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0

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1.2

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max (mm)

xresp

(mm)

x F (

mm

)

50 300 550 800 100050

300

550

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1000

0

0.5

1

1.5

2

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xresp

(mm)

x F (

mm

)

50 300 550 800 100050

300

550

800

1000

0

0.05

0.1

0.15

0.2

0.25

0.3

0.36H z

max (mm)

Figure 10 Displacement maps of the clamped-free composite beam with [45/-45] s lay-up (A) 5 Hz, ζ = 0.0, (B) 5 Hz, ζ = 0.02, (C) 30 Hz, ζ = 0.0,

(D) 30 Hz, ζ = 0.02, (E) 50 Hz, ζ = 0.0, (F) 50 Hz, ζ = 0.02, (G) 100 Hz, ζ = 0.0, and (H) 100 Hz, ζ = 0.02.




its small bending rigidity. The [0] 2s

lay-up provides larger

flexural rigidity and, consequently, larger natural fre-

quencies. For this lay-up, the magnitudes of the harmonic

responses are smaller than those obtained for other lay-up

configurations.

Figure 6 shows the effect of the boundary condition

on the frequency response of the composite beam. The

frequency responses of the composite beam with [45/-45] s

lay-up are obtained for the point located at 0.25 L while

the harmonic excitation is applied at x F = 0.25 L . As seen

from the figure, the clamped-free beam has the smallest

first natural frequency due to its small bending rigidity.

However, clamped-clamped beam has the largest first

natural frequency reflecting its high bending rigidity.

The changes in the vibration displacement magnitudes

due to the varying end conditions are clearly seen in this

figure.

The displacement map is a 2D graph showing the dis-

placement levels for different excitation ( x F ) and obser-

vation points ( x resp

) constructed at a specific excitation

frequency. In this study, displacement maps of the com-

posite beam with [45/-45] s lay-up are plotted for four dif-

ferent boundary conditions for considering undamped

and damped cases. The displacement maps are plotted

using 50-mm increment for both excitation and response

points. These displacement maps can be used to find a

suitable location insensitive to structural vibration in

the component placement procedure. These maps have

comprehensive information about the vibration levels

recorded at different observation points for varying exci-

tation points.

The displacement maps given in Figures 7 – 10 have

been plotted to cover the forced vibrations up to 100 Hz.

Figure 7A – F shows the displacement maps of the

clamped-clamped composite beam for 5, 50, and 100 Hz.

Figure 7A, C, and E shows the displacement values for

the undamped case. As shown in these figures, distribu-

tion of the vibration displacement directly reflects the

related mode shape of the composite beam. In Figure 7A,

displacement values get larger when the excitation point

gets closer to the middle of the beam at 5 Hz. This result

is compatible with the first mode shape of the beam.

The same results are valid for the excitation frequencies

50 Hz (Figure 7C) and 100 Hz (Figure 7E). The symmetry

in the displacement maps is due to the boundary condi-

tion. The displacement magnitudes reduce as the excita-

tion frequency increases. Figure 7B, D, and F shows the

displacement maps for the damped case. Two percent

damping ratio is used to show the effect of the damping

on the displacement distribution. The relation between

the excitation-observation points remains the same but

the displacement magnitudes reduce considerably when

damping is introduced.


clamped-pinned composite beam with [45/-45] s lay-up.

This boundary condition is asymmetric, and it is expected

that the displacement distribution has an asymmetric

form. For the clamped-pinned boundary condition, the

displacement values increase when the excitation point is

slightly close to the pinned end of the beam (Figure 8A, C,

and F). The displacement values reduce considerably with

the damping.


pinned-pinned composite beam with [45/-45] s lay-up.

Similar to the clamped-clamped beam, displacement

values increase as the excitation location moves to the

middle of the beam. Symmetry in the displacement distri-

bution also remains the same for three different excitation

frequencies (Figure 9A, C, and F), and damping reduces

the vibration displacements considerably especially at

higher frequencies (Figure 9B, D, and E).

Figure 10A – H show the displacement maps of the

clamped-free composite beam with [45/-45] s lay-up. This

boundary condition is also asymmetric and the displace-

ment distribution also has an asymmetric form. This

boundary condition has the smallest bending rigidity and

four natural frequencies exist in the 0 – 100 Hz frequency

range. Thus, four different displacement maps at 5, 30, 50,

and 100 Hz are drawn to give the change in the displace-

ment pattern. Similar to the clamped-pinned boundary

condition, the displacement values increase as the excita-

tion point moves to the free end of the beam (Figure 10A,

C, F, and H). As observed for the other boundary condi-

tions, the displacement values reduce considerably with

the damping.

5 Conclusions Determination of the vibration response of the engineer-

ing structures for harmonic excitations is an important

step in structural design. Numerical methods are widely

used for this purpose. The displacement maps of a sym-

metric laminated composite beam having different ply

orientations and boundary conditions are investigated

in this study. The effect of the damping on the frequency

response of the symmetric laminated composite beam

is presented. Stacking sequence and boundary condi-

tion has a decisive effect on the free and forced vibration

characteristics of the symmetric laminated composite

beams. Damping ratio reduces the vibration displacement




magnitudes in the harmonic excitation case. The rate of

the reduction is greater for higher vibration modes. In

the case of harmonic excitation, displacement maps can

be obtained using the numerical procedure described in

this study and they can be used in the structural design

of engineering structures, which work under harmonic

excitations.

Received August 16 , 2013 ; accepted September 29 , 2013

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Harmonic response analysis of symmetric laminated composite beams with different boundary conditions

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