Effects of axial movements of the ends and aspect ratio of laminated composite beams on their non-dimensional natural frequencies

A. F. Ahmed Algarray1 , Hua Jun2

1, 2Department of Mechanical Design and Theory, College of Mechanical Engineering, Northeast Forestry University, Harbin, China

1Department of Mechanical Engineering, Faculty of Engineering, Red Sea University, Port Sudan, Sudan

2Corresponding author

Journal of Vibroengineering, Vol. 20, Issue 5, 2018, p. 2128-2136. https://doi.org/10.21595/jve.2018.19355
Received 4 November 2017; received in revised form 19 March 2018; accepted 20 April 2018; published 15 August 2018

Copyright © 2018 A. F. Ahmed Algarray, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

This study developed to solve the problem of prediction of the natural frequencies of free vibration for laminated beams. The study presented the natural frequencies of composite beams with four layered and different boundary conditions. In each boundary condition, two cases are assumed: movable ends and immovable ends. Numerical results are obtained for the same material to demonstrate the effects of the aspect ratio, fiber orientation, and the beam end-movements on the non-dimensional natural frequencies of beams. Two aspect ratios are given in the numerical results, one is for relatively short-thick beams, while the other is for slender beams. It was found that the results of the non-dimensional frequencies obtained from the short-thick beams are generally much less than those obtained from the other slender beams for same fiber orientation and generally, the frequencies of longitudinal vibration increase as the aspect ratio increased. It was also found the values of the non-dimensional frequencies of the transverse modes are not affected by the longitudinal movements of the ends since these modes are generated by lateral movements only. However, the values of the natural frequencies of longitudinal modes are found to be the same for all beams with movable ends since they are generated by longitudinal movements only.

Effects of axial movements of the ends and aspect ratio of laminated composite beams on their non-dimensional natural frequencies

Highlights
  • The study presented the natural frequencies of composite laminated beams.
  • First order shear deformation (FSDT) theory is used, and finite element method is employed.
  • The aspect or slenderness ratio has a considerable effect on all modes of vibration.
  • The natural frequencies of a laminated beam generally increase with the aspect ratio.
  • All beams with movable ends have equal longitudinal frequencies of vibration, while those of beams with immovable ends are different.

Keywords: composite materials, laminated beams, natural frequencies, aspect ratios and movements of the ends.

1. Introduction

Composite have been used in engineering structures over the last four decades or so. They could be seen in a variety of applications as in craft wings and fuselage, satellites helicopter blades, wind turbines boats and vessels, tubes and tanks etc. Their advantages over traditional materials are widely recognized and these are high strength to weight ratio, and their properties which can tailored according to need. Other advantages include high stiffness, high fatigue and corrosion resistance, good friction characteristics, and ease of fabrication. They are made of fiber such as glass, carbon, boron, etc. embedded in matrix or suitable resin that act as binding material. The increasing use of composites has been required a good understanding of composite mechanics and their behavior. Many mathematical models for laminates subjected to static and dynamic loading have been developed. This paper addresses free vibration. The knowledge of the few lower natural frequencies of a structure is utmost importance in order to save it in service from being subjected to unnecessary large amplitude of motion which can cause immediate collapse or ultimate failure by fatigue.

Free vibration analysis of laminated composite beams is presented by P. Subramanian, R. A. Jafari-Talookolaei et al. and A. Pagani [1-3] reference [1] used two higher order displacement based shear deformation theories, while references and [2, 3] used the first order shear deformation theory. M. Rueppel et al. [4] studied the damping of carbon fibre and flax fibre angle-ply composite laminate. Torabi K. et al. [5] Investigated on the effects of delamination size and its thickness-wise and lengthwise location on the vibration characteristics of cross-ply laminated composite beams. Analytical solutions for free vibration and buckling of composite beams using a higher order beam theory presented by He G. et al. [6]. Vibration prediction of thin-walled composite I-beams using scaled models analyzed by M. E. Asl et al. [7]. Within that study, which is an extension of Authors’ previous work on design of scaled composite models [8-10], similitude theory is applied to the governing equations of motion for vibration of a thin walled composite I-beam. Algarray et al. [11] studied the effects of end conditions of Cross-Ply laminated composite beams on their dimensionless natural frequencies

2. Modeling analysis

Fig. 1. Showed a composite laminated beam made up of n layers with different orientation, thickness, and properties. Where L is the length, b is breadth and h is depth.

Fig. 1. Composite laminated beam

 Composite laminated beam

Treat the beam as a plane stress problem and employ first-order shear deformation theory. The longitudinal displacement (U) and the lateral displacement (W) can be written as follows:

(1)
U x , z , t = u x , t + z ϕ x , t , W x , z , t = w x , t ,

where u and w are the mid-plane longitudinal and lateral displacements, ϕ is the rotation of the deformed section about the y-axis, z is the perpendicular distance from mid-plane to the layer plane, and t is time.

The Strain-Displacement Relations:

(2)
ε 1 = U x = u x + z ϕ x , ε 5 = W x + U z = w x + ϕ ,

where: ε1 is the longitudinal strain, and ε5 is the through-thickness shear strain.

Fig. 2. Composite laminated beam with 3-noded lineal element

 Composite laminated beam with 3-noded lineal element

By employing 3-noded lineal element as shown in Fig. 2.

The displacements can be expressed in terms of shape function Ni and nodal displacements:

(3)
u = N i u i ,       w = N i w i ,       = N i i .

The shape functions are: N1=-r21-r, N2=1-r2, N3=r21+r.

From Eqs. (2), (3), the strains can be written as:

(4)
ϵ = B a e ,

where:

B = d N i d r 0 z d N i d r 0 d N i d r N i ,         i = 1 ,   2 ,   3 .

And ae is the vector of nodal displacements ae=uiwiiT, i= 1, 2, 3.

The stress-strain relation:

(5)
σ = c ϵ ,

where σ=σ1σ5T, ϵ=ϵ1ϵ5T and the matrix containing the transformed elastic constants:

c = c 11 0 0 c 55 .

Substitute Eq. (4) in Eq. (5):

(6)
σ = c B a e .

The strain energy:

(7)
U S = 1 2 V   ϵ T σ d v ,
U S = 1 2 a e T b B T c B d x d z a e ,
(8)
U S = 1 2 a e T b K e a e ,

where:

K e = B T c B d x d z

The kinetic energy:

T = 1 2 ρ u ˙ + z ˙ 2 + w ˙ 2 d v ,

where ρ is density and the dot denotes differentiation with time:

(9)
T = - ω 2 a e T 2 ρ N T Z N a e d v ,       T = - 1 2 a e T b ω 2 M e a e ,

where:

M e = ρ N T Z N d x d z ,
N = N i 0 0 0 N i 0 0 0 N i ,       Z = 1 0 Z 0 1 0 Z 0 Z 2 .

In the above derivation it is assumed the motion is harmonic and ω is circular frequency.

In the absence of damping and external nodal load, the total energy is:

Ԉ = U S + T ,       Ԉ = 1 2 a e T b K e a e - 1 2 a e T b ω 2 M e a e .

The principle of minimum energy requires that:

Ԉ a e = 0 .

The condition yields the equation of motion:

K e a e - ω 2 M e a e = 0 ,
(10)
K - ω 2 M a = 0 ,

where:

K = e = 1 n K e ,         M = e = 1 n M e ,       a = e = 1 n a e ,

and n is number of elements. To facilitate the solution of Eq. (10), we introduce the following quantities:

A 11 , B 11 , D 11 = k = 1 n Z k - 1 Z k c 11 1 , Z , Z 2 d z ,         A 55 = K f k = 1 n Z k - 1 Z k c 55 d z ,

where Kf is the shear correction factor.

The transformed elastic constants are:

c 11 = c 11 ' c 4 + 2 c 12 ' + 2 c 66 ' S 2 C 2 + c 22 ' S 4 ,           c 55 = c 44 ' S 2 + c 55 ' C 2 .

In which:

c 11 ' = E 1 1 - ν 12 ν 21 ,           c 12 ' = ν 12 E 21 1 - - ν 12 ν 21 = ν 21 E 11 1 - - ν 12 ν 21 ,         c 22 ' = E 21 1 - ν 12 ν 21 ,
c 66 ' = G 12 ,       c 55 ' = G 13 ,         c 44 ' = G 23 ,       S = s i n θ ,           C = c o s θ .  

And θ is the angle of orientation of the ply with respect to the beam axis:

I 1 , I 2 , I 3 = k = 1 n Z k - 1 Z k ρ 1 , Z , Z 2 d z .

Non-dimensional quantities used in the analysis are:

u - = L h u ,         w - = w h ,         - = L h ,         A - 11 = 1 E 1 h A 11 ,         B - 11 = 1 E 1 h 2 B 11 ,  
D - 11 = 1 E 1 h 3 D 11 ,         A - 55 = 1 E 1 h A 55 ,         I - 1 = 1 ρ h I 1 ,        
I - 2 = 1 ρ h 2 I 2 ,         I - 3 = 1 ρ h 3 I 3 ,           ω - = ω ρ L 4 E 1 h 2 .

The element stiffness matrix:

K e = B T c B d x d z ,
K e = A 11 d N i d x d N j d x 0 B 11 d N i d x d N j d x 0 A 55 d N i d x d N j d x A 55 d N i d x N j B 11 d N i d x d N j d x A 55 N i d N j d x D 11 d N i d x d N j d x + A 55 N i N j d x .

The mass matrix is 9×9 symmetrical matrix:

M e = ρ N T Z N d x d z ,             M e = ρ N i N j 0 Z N i N j 0 N i N j 0 Z N i N j 0 Z 2 N i N j d x d z ,
M e = I 1 N i N j 0 I 2 N i N j 0 I 1 N i N j 0 I 2 N i N j 0 I 3 N i N j d x .

3. Results and discussion

3.1. Effect of aspect ratios

Two aspect ratios are given in the numerical results, which are 10 and 50. The first one is for relatively short-thick beams, while the other is for slender beams. The results of the non-dimensional frequencies obtained from the aspect ratio 10 are generally much less than those obtained from the other aspect ratio 50 for same fiber orientation. For example, the fundamental mode of the non-dimensional natural frequencies for a symmetric [30/–30/–30/30] angle-ply hinged _hinged beam with immovable ends is 1.9918 for the aspect ratio 10, and 2.1947 for the aspect ratio 50 as can be seen in Table 1.

This observation can be seen in Fig. 3 to Fig. 5 for symmetric [45/–45/–45/45] angle-ply laminated beams. These figures show the variation of the non-dimensional frequencies with the aspect ratio range from 5 to 40 for the first three modes of vibration for all beams with immovable ends. It is obvious from the figure that the frequency increases rapidly for the range of aspect ratio from 5 to 20, and slows down beyond this range. When the aspect ratio is greater than 20, the beam is slender and consequently shear deformation and rotary inertia have small noticeable effects on the natural frequencies.

Table 2 shows the effect of aspect ratio in non-dimensional frequencies for symmetric [45/–45/–45/45] angle-ply beams. The percentage increase in the non-dimensional frequencies, for the first range of the aspect ratio, increases sharply as the mode order increased for all boundary conditions. For the second range, the percentage increase in frequencies is independent on the mode order. The longitudinal modes of free vibration are also affected by the change of aspect ratio. Generally, the frequencies of longitudinal vibration increase as the aspect ratio increased.

Table 1. Non-dimensional natural frequencies ω¯=ωρL4/E1h2 [30/–30/–30/30] composite beams with different aspect ratio

Mode No.
Beam type aspect ratio (L/h= 10)
CF
HH
CC
HC
HF
FF
1
0.7465
1.9918
3.4380
2.7113
3.0503
4.3728
2
3.7279
6.4128
7.4386
6.9645
7.9206
9.5329
3
8.4193
11.4744
12.0720
11.7816
12.1545*
14.9573
4
12.1545*
16.5865
16.9085
16.7518
13.1707
20.2046
5
13.4194
21.6353
21.8199
21.7278
18.3742
24.3090*
6
18.5147
24.3090*
24.3090*
24.3090*
23.4730
25.3532
7
23.5463
26.6166
26.7237
26.6709
28.4805
30.3492
8
28.5288
31.5448
31.6114
31.5778
33.4158
35.3042
9
33.4400
36.4344
36.4743
36.4548
36.4635*
40.0877
10
36.4635*
41.2968
41.3227
41.3093
38.2900
44.9060
11
38.3118
46.1413
46.1540
46.1482
43.0983
48.6181*
12
43.1022
48.6181*
48.6181*
48.6181*
47.7785
48.9418
Mode No.
Beam type aspect ratio (L/h= 50)
CF
HH
CC
HC
HF
FF
1
0.7837
2.1947
4.8908
3.4027
3.4251
4.9666
2
4.8503
8.6633
13.1481
10.8187
10.9431
13.4813
3
13.3186
19.0823
24.9877
21.9844
22.3328
25.8305
4
25.4001
32.9800
39.8324
36.3924
37.1019
41.4535
5
40.6240
49.8113
57.1574
53.4997
54.6965
59.7934
6
58.4417
69.0255
76.4775
72.7861
60.7725*
80.3045
7
60.7725*
90.1153
97.3759
93.7899
74.5680
102.4978
8
78.3401
112.6436
119.5087
116.1227
96.2177
121.5450*
9
99.8689
121.5450*
121.5450*
121.5450*
119.2209
125.9612
10
122.6521
136.2519
142.5999
139.4697
143.2325
150.3635
11
146.3874
160.6577
166.4325
163.5836
167.9831
175.4481
12
170.8386
185.6456
190.8379
188.2740
182.3175*
201.0230
(*) Modes with predominance of longitudinal vibration

Table 2. The effect of aspect ratio in non-dimensional frequencies for symmetric [45/–45/–45/45] angle-ply beams

Beam
type
Approximate % increase in non-dimensional frequencies
Aspect ratios from 5 to 20
Aspect ratios from 20 to 40
1st. mode
2nd. mode
3rd. mode
1st. mode
2st. mode
3rd. mode
CF
25
50
100
10
6
9
HH
25
62
100
10
6
9
CC
68
120
150
10
10
15
HC
45
85
125
10
9
14
HF
22
70
220
8
6
10
FF
85
115
145
12
8
12

3.2. Effect of axial movements of the ends

From the results of Table 1, that the values of the non-dimensional frequencies of the transverse modes are not affected by the longitudinal movements of the ends since these modes are generated by lateral movements only (at the yellow shaded). However, the values of the natural frequencies of longitudinal modes are found to be the same for all beams with movable ends since they are generated by longitudinal movements only. Table 3 shows this observation for symmetric [60/–60/–60/60] laminated beams with aspect ratio of 10.

Fig. 3. Effect of aspect ratio on natural frequencies of a symmetric [45/–45/–45/45] cross-play clamped-free beam

 Effect of aspect ratio on natural frequencies of a symmetric [45/–45/–45/45] cross-play  clamped-free beam

Fig. 4. Effect of aspect ratio on natural frequencies of a symmetric [45/–45/–45/45] cross-play clamped-clamped beam

 Effect of aspect ratio on natural frequencies  of a symmetric [45/–45/–45/45] cross-play  clamped-clamped beam

Fig. 5. Effect of aspect ratio on natural frequencies of a symmetric [45/–45/–45/45] cross-play free-free beam

 Effect of aspect ratio on natural frequencies of  a symmetric [45/–45/–45/45] cross-play free-free beam

Table 1 also shows the fundamental modes of longitudinal vibration for various beams with immovable ends for the symmetric case [30/–30/–30/30] and for two aspect ratios, 10, and 50. It could be noticed that the values of non-dimensional natural frequencies of the longitudinal vibration for the clamped-free and hinged-free beams are equal, and those of the other beams are also the same. This phenomenon occurs since both clamped-free and hinged-free beams with immovable ends are the same when restricted from executing longitudinal motion at the ends. Similarly, the rest of beams with immovable ends have the same longitudinal end conditions.

Table 3. The first two non-dimensional modes of longitudinal free vibration of [60/–60/–60/60] laminated beams with aspect ratio 10

Beam ends
Mode No.
Beam type
CF
HH
CC
HC
HF
FF
Immovable
1
5.6426
11.2852
11.2852
11.2852
5.6426
11.2852
2
16.9279
22.5705
22.5705
22.5705
16.9279
22.5705
Movable
1
11.2852
11.2852
11.2852
11.2852
11.2852
11.2852
2
22.5705
22.5705
22.5705
22.5705
22.5705
22.5705

3.3. Verification

The natural frequencies results which obtained by this study are closer with Abramovich [12] results, as show in Table 4, and difference between two results less than 0.6 % for cantilever and clamp-clamp beams.

A third-order shear deformation theory was used by Kant et al. [13] in the analysis of the free vibration of composite and sandwich simply supported beams. Two comparisons of non-dimensional natural frequencies between the present method (using FSDT) and the results of this reference are presented in Table 5, which presented a comparison for symmetric [0/90/90/0] cross-ply laminated beams respectively, with aspect ratio of (L/h= 5), where the shear effect is significant. The comparison shows a difference of less than 3.3 % associated with the fundamental frequency and less than 4.5 % for higher modes. These differences are due to the employment of different shear theories as stated bellow.

Table 4. Non-dimensional frequencies of [0/90/90/0] composite beams with immovable ends and aspect ratio 10

Mode No.
Cantilever
Clamp- clamp
Present
Ref. [11]
Present
Ref. [11]
1
0.8866
0.8819
3.6855
3.7576
2
4.1062
4.0259
7.7244
7.8718
3
8.9536
9.1085
12.381
12.573
4
11.504
12.193
17.192
17.373
5
13.924
14.080
22.119
22.200
6
18.980
18.980
23.007*
23.007

Table 5. Non-dimensional frequencies of [0/90/90/0] composite beams with simple support ends and aspect ratio 5

Mode No.
Present
Ref. [13]
1
1.7619
1.820
2
4.2749
4.528
3
6.7214
7.201
4
9.1414
9.814
5
11.5783*
(*) Mode with predominance of longitudinal vibration

It is clear, from the above comparisons, that the differences are very small even for higher modes. This confirms the accuracy of the method of analysis and the computer program.

4. Conclusions

In this paper, free vibration of four layered composite beams has been studied. Both secondary effects of transverse shear deformation and rotary inertia were included in the analysis. A first-order shear deformation theory was applied in the analysis. A finite element model has been formulated to predict the non-dimensional natural frequencies and to study the influence of aspect ratio and movable ends of fibers on the natural frequencies. Different end conditions were studied which are clamped-free, hinged-hinged, clamped-clamped, hinged-clamped, hinged-free, and free-free beams with immovable and movable ends. The main conclusion is the natural frequencies of a laminated beam generally increase with the aspect ratio and all beams with movable ends have equal longitudinal frequencies of vibration, while those of beams with immovable ends are different. Namely, clamped-free and hinged-free beams with immovable ends have equal longitudinal frequencies, and the other beams have also equal longitudinal frequencies.

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