Effects of axial movements of the ends and aspect ratio of laminated composite beams on their nondimensional natural frequencies
A. F. Ahmed Algarray^{1} , Hua Jun^{2}
^{1, 2}Department of Mechanical Design and Theory, College of Mechanical Engineering, Northeast Forestry University, Harbin, China
^{1}Department of Mechanical Engineering, Faculty of Engineering, Red Sea University, Port Sudan, Sudan
^{2}Corresponding author
Journal of Vibroengineering, Vol. 20, Issue 5, 2018, p. 21282136.
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
JVE Conferences
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 endmovements on the nondimensional natural frequencies of beams. Two aspect ratios are given in the numerical results, one is for relatively shortthick beams, while the other is for slender beams. It was found that the results of the nondimensional frequencies obtained from the shortthick 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 nondimensional 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.
 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. JafariTalookolaei et al. and A. Pagani [13] 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 angleply composite laminate. Torabi K. et al. [5] Investigated on the effects of delamination size and its thicknesswise and lengthwise location on the vibration characteristics of crossply 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 thinwalled composite Ibeams 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 [810], similitude theory is applied to the governing equations of motion for vibration of a thin walled composite Ibeam. Algarray et al. [11] studied the effects of end conditions of CrossPly 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
Treat the beam as a plane stress problem and employ firstorder shear deformation theory. The longitudinal displacement ($U$) and the lateral displacement ($W$) can be written as follows:
where $u$ and $w$ are the midplane longitudinal and lateral displacements, $\varphi $ is the rotation of the deformed section about the $y$axis, $z$ is the perpendicular distance from midplane to the layer plane, and $t$ is time.
The StrainDisplacement Relations:
where: ${\epsilon}_{1}$ is the longitudinal strain, and ${\epsilon}_{5}$ is the throughthickness shear strain.
Fig. 2. Composite laminated beam with 3noded lineal element
By employing 3noded lineal element as shown in Fig. 2.
The displacements can be expressed in terms of shape function ${N}_{i}$ and nodal displacements:
The shape functions are: ${N}_{1}=\frac{r}{2}\left(1r\right)$, ${N}_{2}=1{r}^{2}$, ${N}_{3}=\frac{r}{2}\left(1+r\right)$.
From Eqs. (2), (3), the strains can be written as:
where:
And ${a}^{e}$ is the vector of nodal displacements ${a}^{e}={\left[\begin{array}{ccc}{u}_{i}& {w}_{i}& {\varnothing}_{i}\end{array}\right]}^{T}$, $i=$ 1, 2, 3.
The stressstrain relation:
where $\sigma ={\left[\begin{array}{cc}{\sigma}_{1}& {\sigma}_{5}\end{array}\right]}^{T}$, $\u03f5={\left[\begin{array}{cc}{\u03f5}_{1}& {\u03f5}_{5}\end{array}\right]}^{T}$ and the matrix containing the transformed elastic constants:
Substitute Eq. (4) in Eq. (5):
The strain energy:
where:
The kinetic energy:
where $\rho $is density and the dot denotes differentiation with time:
where:
In the above derivation it is assumed the motion is harmonic and $\omega $ is circular frequency.
In the absence of damping and external nodal load, the total energy is:
The principle of minimum energy requires that:
The condition yields the equation of motion:
where:
and $n$ is number of elements. To facilitate the solution of Eq. (10), we introduce the following quantities:
where ${K}_{f}$ is the shear correction factor.
The transformed elastic constants are:
In which:
${c}_{66}^{\text{'}}={G}_{12},{c}_{55}^{\text{'}}={G}_{13},{c}_{44}^{\text{'}}={G}_{23},S=\mathrm{s}\mathrm{i}\mathrm{n}\theta ,C=\mathrm{c}\mathrm{o}\mathrm{s}\theta .$
And $\theta $ is the angle of orientation of the ply with respect to the beam axis:
Nondimensional quantities used in the analysis are:
${\stackrel{}{D}}_{11}=\left(\frac{1}{{E}_{1}{h}^{3}}\right){D}_{11},{\stackrel{}{A}}_{55}=\left(\frac{1}{{E}_{1}h}\right){A}_{55},{\stackrel{}{I}}_{1}=\left(\frac{1}{\rho h}\right){I}_{1},$
${\stackrel{}{I}}_{2}=\left(\frac{1}{\rho {h}^{2}}\right){I}_{2},{\stackrel{}{I}}_{3}=\left(\frac{1}{\rho {h}^{3}}\right){I}_{3},\stackrel{}{\omega}=\omega \sqrt{\frac{\rho {L}^{4}}{{E}_{1}{h}^{2}}}.$
The element stiffness matrix:
The mass matrix is 9×9 symmetrical matrix:
${M}^{e}=\int \left[\begin{array}{ccc}{I}_{1}{N}_{i}{N}_{j}& 0& {I}_{2}{N}_{i}{N}_{j}\\ 0& {I}_{1}{N}_{i}{N}_{j}& 0\\ {I}_{2}{N}_{i}{N}_{j}& 0& {I}_{3}{N}_{i}{N}_{j}\end{array}\right]dx.$
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 shortthick beams, while the other is for slender beams. The results of the nondimensional 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 nondimensional natural frequencies for a symmetric [30/–30/–30/30] angleply 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] angleply laminated beams. These figures show the variation of the nondimensional 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 nondimensional frequencies for symmetric [45/–45/–45/45] angleply beams. The percentage increase in the nondimensional 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. Nondimensional natural frequencies $\left[\overline{\omega}=\omega \sqrt{\rho {L}^{4}/{E}_{1}{h}^{2}}\right]$ [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 nondimensional frequencies for symmetric [45/–45/–45/45] angleply beams
Beam
type

Approximate % increase in nondimensional 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 nondimensional 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] crossplay clampedfree beam
Fig. 4. Effect of aspect ratio on natural frequencies of a symmetric [45/–45/–45/45] crossplay clampedclamped beam
Fig. 5. Effect of aspect ratio on natural frequencies of a symmetric [45/–45/–45/45] crossplay freefree 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 nondimensional natural frequencies of the longitudinal vibration for the clampedfree and hingedfree beams are equal, and those of the other beams are also the same. This phenomenon occurs since both clampedfree and hingedfree 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 nondimensional 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 clampclamp beams.
A thirdorder 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 nondimensional 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] crossply 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. Nondimensional 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. Nondimensional 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 firstorder shear deformation theory was applied in the analysis. A finite element model has been formulated to predict the nondimensional 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 clampedfree, hingedhinged, clampedclamped, hingedclamped, hingedfree, and freefree 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, clampedfree and hingedfree beams with immovable ends have equal longitudinal frequencies, and the other beams have also equal longitudinal frequencies.
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