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Advances in Acoustics and Vibration
Volume 2011, Article ID 407470, 9 pages
http://dx.doi.org/10.1155/2011/407470
Research Article

Analytical Expressions for Frequency and Buckling of Large Amplitude Vibration of Multilayered Composite Beams

School of Mechanical Engineering, Sharif University of Technology, P.O. Box 14588-89694, Tehran, Iran

Received 9 May 2011; Accepted 24 May 2011

Academic Editor: K. M. Liew

Copyright © 2011 R. A. Jafari-Talookolaei 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.

Abstract

The aim of this paper is to present analytical and exact expressions for the frequency and buckling of large amplitude vibration of the symmetrical laminated composite beam (LCB) with simple and clamped end conditions. The equations of motion are derived by using Hamilton's principle. The influences of axial force, Poisson effect, shear deformation, and rotary inertia are taken into account in the formulation. First, the geometric nonlinearity based on the von Karman's assumptions is incorporated in the formulation while retaining the linear behavior for the material. Then, the displacement fields used for the analysis are coupled using the equilibrium equations of the composite beam. Substituting this coupled displacement fields in the potential and kinetic energies and using harmonic balance method, we obtain the ordinary differential equation in time domain. Finally, applying first order of homotopy analysis method (HAM), we get the closed form solutions for the natural frequency and deflection of the LCB. A detailed numerical study is carried out to highlight the influences of amplitude of vibration, shear deformation and rotary inertia, slenderness ratios, and layup in the case of laminates on the natural frequency and buckling load.

1. Introduction

The uses of composite materials in structural components are increasing due to their attractive properties such as high strength/stiffness, light weight, and facility to materialize fiber orientation, material and stacking sequence. Beam-like structures have numerous applications in the industries such as aerospace and robotics. The increased use of the LCB requires a better understanding of large amplitude vibration aspects of them.

Nonlinear dynamic analysis of the LCB on the basis of classical lamination theory (CLT) has received a good amount of attention in the literature [16] while relatively little investigations [7, 8] have been performed for dynamic investigation of such beams with low slenderness ratio (length of the beam to the radius of gyration of the cross-section i.e. = 𝐿/𝑟) which have to be taken into account the shear deformation and rotary inertia. It should be noted that for the later, the equation of motion is complex, and even obtaining an approximate solution is much more difficult.

In our previous work in [1], we presented analytical expressions for large amplitude-free vibration and postbuckling analysis of unsymmetrically LCB on elastic foundation by using variational iteration method. We used the CLT to obtain differential equations of motion. A differential quadrature approach for nonlinear free vibration analysis of symmetric angle-ply laminated thin beams on nonlinear elastic foundation with elastically restrained against rotation edges has been studied by Malekzadeh and Vosoughi [2]. Patel et al. [3] have extended a new three-nodded shear flexible beam element based on consistency approach to analyze the nonlinear flexural vibration and postbuckling behavior of simply supported laminated orthotropic composite beams on a two-parameter elastic foundation. The axial and transverse deformations of a geometrically nonlinear composite beam were considered to obtain a closed form solution for the postbuckling behavior of composite beams and to exactly solve for the free vibrations that take place around a buckled position by Emam and Nayfeh [4]. Kapania and Raciti [5] presented a two-nodded Timoshenko beam element with 10 degrees of freedom per node to examine the large amplitude free vibrations of LCB. The nonlinear free flexural vibrations of isotropic/laminated orthotropic straight/curved beams have been studied using a cubic B-spline shear flexible curved element [6]. The nonlinear governing equations are solved by employing Newmark’s numerical integration scheme coupled with modified Newton-Raphson iteration technique. Kiran et al. [7] had investigated the large amplitude free vibrations of generally layered laminated composite beams by developing a finite element model. A one-dimensional finite element formulation based on a higher-order displacement model has been developed by Obst and Kapania [8]. The model accounts for geometric nonlinearities, a parabolic shear strain distribution through the thickness, and satisfies the shear stress free boundary conditions at the upper and lower surfaces of the beam. The model has been applied to the nonlinear static and transient analysis, free vibration analysis, and impact analysis of laminated beams.

To the best author's knowledge, there is no analytical expressions in the literature on the large amplitude free vibrations analysis of the symmetrically LCB on the basis of first shear deformation theory (FSDT). Thus, the contribution of this paper is to expand our previous work in [1] by studying the more complicated and important problem to obtain the closed-form expressions for natural frequency and deflection.

2. Equations of Motion and Boundary Condition

The considered LCB with simple end conditions is shown in Figure 1 with the coordinates ̂𝑥 along the axis of the beam and ̂𝑧 along the thickness direction. ̂̂𝑡𝐿,𝑃,𝑤(̂𝑥,𝑡), are length, compressive axial load, deflection of the beam, and time, respectively.

407470.fig.001
Figure 1: Geometry of a laminated composite beam.
2.1. Coupling Equations

Based on FSDT, the LCB is characterized by its bending rigidity EI, torsional rigidity GJ, bending-torsion coupling rigidity C, and shear rigidity kAG [9]. The derivation of these rigidities is briefly summarized in Appendix A [10].

The kinetic and potential energies of the LCB and the work done by the static lateral load per unit length of the beam 𝐹z can be written as1𝑇=2𝐿0𝐼1𝑤,̂𝑡2+𝐼2𝜃,̂𝑡2+𝐼𝑝𝜓,̂𝑡21𝑑̂𝑥,𝑈=2𝐿0EI𝜃,̂𝑥2𝑤𝑃,̂𝑥2+𝐼𝑝𝐼1𝜓,̂𝑥2+2𝐶𝜓,𝜃̂𝑥,𝑤̂𝑥+kAG,̂𝑥𝜃2+GJ𝜓,̂𝑥2𝑑̂𝑥,𝑊=𝐿0𝐹𝑧𝑤𝑑̂𝑥,(1) where comma denotes derivative with respect to 𝑥; 𝜃,𝜓, are the angle of rotation and torsional rotation, respectively; 𝐼1,𝐼2 and 𝐼p are the first and second moment of inertia about 𝑦-axis and the polar mass moment of inertia per unit length about the 𝑥-axis, respectively.

For the subsequent results to be general, we use the following nondimensional variables:𝑥=̂𝑥𝐿𝑤,𝑤=𝑟̂𝑡,𝑡=𝑏𝐷11/𝐼1𝐿4,𝑟=𝐼𝐴(2) which 𝑟,𝑏,𝐼, and 𝐴 are the radius of gyration of the cross-section, beam width, second moment of area, and the area of the cross-section, respectively, and 𝐷11 is defined in Appendix A.

Thus, the kinetic and potential energy and the work done can be rewritten as follows:𝐼𝑇=1𝐿52𝑏𝐷1110𝐼1𝑟2𝑤2,𝑡+𝐼2𝜃2,𝑡+𝐼𝑝𝜓2,𝑡1𝑑𝑥,𝑈=210EI𝐿𝜃2,𝑥𝑃𝐿𝑟2𝑤2,𝑥+𝐼𝑝𝐼1𝜓2,𝑥+2𝐶𝐿𝜓,𝑥𝜃,𝑥+kAG𝐿𝑟𝑤,𝑥𝐿𝜃2+GJ𝐿𝜓2,𝑥𝑑𝑥,𝑊=10𝐹𝑧𝑤𝑑𝑥.(3) Applying the principle of minimization of total potential energy (𝛿(𝑈𝑊)), we obtain the following equilibrium equations as follows:EI𝐿2𝜃𝑟+kAG𝐿𝑤+C𝜃𝐿2𝜓=0,(4)𝑟kAG𝐿21𝐿𝜃𝑃𝑟𝐿2𝑤+𝐹𝑧=0,(5)GJ𝐿2𝜓+C𝐿2𝜃𝑃𝐼𝑃𝐼1𝐿2𝜓=0(6) in which prime denotes derivative with respect to nondimensional coordinate 𝑥. These equations are coupled equations and can be solved to get the coupled displacement fields [11].

2.2. Coupled Displacement Fields

Equations (4) and (6) are used to couple the rotation 𝜃, torsion 𝜓, and deflection of the beam 𝑤 [11]. These equations can be rewritten as 𝜓=𝐴1𝜃,𝜃+𝐴2𝐴𝜃=2𝜂𝑤,(7) where𝐴1=𝐶GJ𝑃𝐼𝑃/𝐼1,𝐴2=kAG𝐿2C𝐴1EI,(8) and 𝜂 is the slenderness ratio, that is, 𝜂=𝐿/𝑟.

2.2.1. Simple End Conditions

The boundary conditions for the simple end conditions beam are as follows:(𝑤=0,𝑀=0,𝑇=0)at𝑥=0,1𝑤=0,𝜃=0,𝜓=0at𝑥=0,1.(9) The transverse displacement 𝑤 is assumed as𝑤(𝑥,𝑡)=𝑢(𝑡)sin(𝜋𝑥),(10) where 𝑢(t) is the amplitude parameter and only is a function of 𝑡. Substituting the function 𝑤 in (7), we obtain the coupled rotation 𝜃 and torsion 𝜓 as𝜃(𝑥,𝑡)=𝐷1𝑢(𝑡)cos(𝜋𝑥),𝜓=𝐴1𝐷1𝑢(𝑡)cos(𝜋𝑥),(11) where𝐷1=𝜋𝐴2𝜂𝐴2𝜋2.(12)

2.2.2. Clamped End Conditions

In the case of the clamped-clamped beam, the boundary conditions are as follows:(𝑤=0,𝜃=0,𝜓=0)at𝑥=0,1.(13) The admissible function for the 𝑤 is taken as follows:𝑤=𝑢(𝑡)(1cos(2𝜋𝑥)).(14) From (7) and (14), the coupled displacement fields for 𝜃 and 𝜓 are obtained as follows:𝜃=𝐷2𝑢(𝑡)sin(2𝜋𝑥),𝜓=𝐴1𝐷2𝑢(𝑡)sin(2𝜋𝑥),(15) where:𝐷2=2𝜋𝐴2𝜂𝐴24𝜋2.(16)

2.3. Large Amplitude Vibrations

Consider that there is no external load on the beam, that is, 𝐹𝑧=0. The total energy for free oscillation of a system without damping is constant (energy conservation principle):𝑇+𝑈+𝑊1=Constant,(17) where 𝑊1 is the work done by the tension developed in the LCB due to the large amplitudes and is given by [11, 12]𝑊1=EI𝑟28𝐿310𝑤2,𝑥𝑑𝑥2.(18) Substituting the expressions for 𝑤,𝜃, and 𝜓, the expressions for 𝑈,𝑇, and 𝑊1 can be obtained as𝑇=𝐶1̇𝑢2𝐶,𝑈=2𝐶3𝑃𝑢2,𝑊1=𝐶4𝑢4(19) in which coefficients 𝐶𝑖(𝑖=14) for simple and clamped end conditions are expressed as

simple end conditions: 𝐶1=𝐼1𝐿5𝐼1𝑟2+𝐼𝑃𝐴12𝐷12+𝐼2𝐷124𝑏𝐷11,𝐶2=𝜋2EI𝐷122𝐶𝐷12𝐴1+4𝐿kAG𝐿2𝐷122kAG𝑟𝜋𝐿𝐷1+kAG𝑟2𝜋2+GJ𝐴12𝐷12𝜋2,𝐶4𝐿3=𝜋2𝐼𝑃𝐴12𝐷12+𝑟2𝐼14𝐿𝐼1,𝐶4=EI𝑟2𝜋432𝐿3,(20) clamped end conditions:𝐶1=𝐼1𝐿53𝐼1𝑟2+𝐼𝑃𝐴12𝐷22+𝐼2𝐷224𝑏𝐷11,𝐶2=𝜋2EI𝐷222C𝐷22𝐴1𝐿+kAG𝐿2𝐷224kAG𝑟𝜋𝐿𝐷2+4kAG𝑟2𝜋2+4GJ𝐴12𝐷22𝜋2,𝐶4𝐿3=𝜋2𝐼𝑃𝐴12𝐷22+𝑟2𝐼1𝐿𝐼1,𝐶4=EI𝑟2𝜋42𝐿3.(21) Substituting the expressions for 𝑇,𝑈, and 𝑊1 in (17), we get the following equation:𝐶1̇𝑢2+𝐶2𝐶3𝑃𝑢2+𝐶4𝑢4=Constant.(22) Differentiating (22), we obtain the nonlinear ordinary differential equations as follows:𝐶̈𝑢+2𝐶3𝑃𝐶1𝑢+2𝐶4𝐶1𝑢3=0̈𝑢+𝛼𝑢+𝛽𝑢3=0.(23) In order to obtain the postbuckling load-deflection relation, one can set all time-derivative terms in (23) equal to zero which yields𝑃NL=2𝐶4𝑢2+𝐶2𝐶3.(24) Neglecting the contribution of 𝑢 in (24), the critical buckling load can be determined as𝑃cr=𝐶2𝐶3.(25)

3. Method of Solution: Implementation of the HAM in Beam Vibrations

For convenience of the reader, we present a brief description of the HAM in Appendix B.

Under the transformation 𝜏=𝜔𝑡 and 𝑊(𝜏)=𝑢(𝑡), (23) becomes as follows:𝜔2̈𝑊+𝛼𝑊+𝛽𝑊3=0.(26) The zero-order deformation equation can be written as below 𝜑(1𝑞)𝐿(𝜏;𝑞)𝑊0[𝜑](𝜏)=𝑞(𝜏)𝑁(𝜏;𝑞)(27) in which𝑁[]𝜑(𝜏;𝑞)=𝜔2𝜕2𝜑(𝜏;𝑞)𝜕𝜏2+𝛼𝜑(𝜏;𝑞)+𝛽𝜑(𝜏;𝑞)3=0.(28) We chose the following auxiliary linear operator as:𝐿[]𝜑(𝜏;𝑞)=𝜔02𝜕2𝜑(𝜏;𝑞)𝜕𝜏2+𝜑(𝜏;𝑞).(29) We employ Taylor expansion series for 𝜑(𝑡;𝑞) and𝜔(𝑞) as𝜑(𝜏;𝑞)=𝜑(𝜏;0)+𝑚=11𝜕𝑚!𝑚𝜑(𝜏;𝑞)𝜕𝑞𝑚||||𝑞=0×𝑞𝑚=𝑊0(𝜏)+𝑚=1𝑊𝑚(𝜏)𝑞𝑚,(30)𝜔(𝑞)=𝜔0+𝑚=11𝜕𝑚!𝑚𝜔(𝑞)𝜕𝑞𝑚||||𝑞=0𝑞𝑚=𝜔0+𝑚=1𝜔𝑚𝑞𝑚.(31) In order to satisfy the initial conditions, the initial guess of 𝑊(𝜏) is chosen as follows:𝑊0(𝜏)=𝑊maxcos(𝜏).(32) In our case, to obtain the first-order approximation, the function of 𝑊1(t) can be expressed as (see Appendix B)𝐿𝑊1[𝜙]||(𝑡)=(𝑡)𝑁(𝑡;𝑝)𝑝=0,(33)𝑊1(0)=0,𝑑𝑊1(0)𝑑𝑡=0.(34) Assuming =1,(t)=1 and after substituting (32) in (33), one would get𝜔02̈𝑊1+𝑊1=𝑊max𝜔cos(𝜏)023𝛼4𝛽𝑊2max𝛽𝑊3max4cos(3𝜏),(35)𝑊1̇𝑊(0)=0,1(0)=0.(36) Eliminating the secular term, we have𝜔0=3𝛼+4𝛽𝑊2max.(37) Solving (35) and (36), the 𝑊1(𝜏) is obtained as follows:𝑊1𝐿(𝜏)=18𝜔02(cos(𝜏)cos(3𝜏)),(38) where𝐿11=4𝛽𝑊3max.(39) Thus the first-order approximation of the 𝑊(𝜏) yields to,𝑊(𝜏)=𝑊0(𝜏)+𝑊1(𝜏)(40) in which𝜏=𝜔𝑡,𝜔=𝜔0.(41)

4. Numerical Results and Discussion

To validate the presented relations, the results of the natural frequency for the single isotropic lamina and the LCB with simple and clamped end conditions are compared with the available results in the literatures.

The variables used in the tables and plots are the nonlinear to linear frequency ratio, 𝜔NL/𝜔L, the nondimensional amplitude of the vibration, 𝑊max, and the postbuckling load to critical load, 𝑃NL/𝑃cr, (henceforth referred to as the frequency ratio, the amplitude ratio, and the buckling load ratio, resp.).

The materials steel and AS4/3501 Graphite-Epoxy, having the following mechanical properties, are used for the study.

Steel:𝐸=206.8Gpa,𝐺=79Gpa,𝜐=0.3,𝜌=7800kg/m3.(42)

AS4/3501 graphite/epoxy:𝐸11=144.8Gpa,𝐸22𝐺=9.65Gpa,12=4.14Gpa,𝐺13𝐺=4.14Gpa,(43)23=4.14Gpa,𝜐12=0.3,𝜌=1389.23kg/m3.(44)

The shear correction factor 𝑘 is taken as 5/6, as commonly used in the literature. All the layers are of equal thickness. Fiber orientation is measured from 𝑥-axis.

The validity of the results is established by Tables 1 and 3 for simple end conditions and Table 2 for clamped end conditions. Table 1 shows the frequency ratio for the single isotropic beam along with the results from literatures. It can be seen that the frequency obtained from two different beam models (i.e., Bernoulli-Euler [1, 13, 14] and Timoshenko [15] beams) yields the same value provided that the beam’s slenderness ratio be at least 100. This finding is also reported by [16] in which only the linear analysis was studied. The same results from [17] are included in Table 2 for the sake of comparison, and the agreement of the values of frequency ratio is good for various values of the amplitude and slenderness ratios.

tab1
Table 1: Comparison of the frequency ratio of the isotropic single layer beam with simple end conditions for different amplitude ratio.
tab2
Table 2: Comparison of the frequency ratio of the isotropic single-layer beam with clamped end conditions for different amplitude ratio.
tab3
Table 3: Comparison of the frequency ratio of the LCB with simple end conditions for different amplitude ratio (𝜂=34.6410).

Also, comparison of the frequency ratio for four-layer symmetric cross-ply, angle-ply and general lay-up composite beams with the finite element solutions from [7] for different amplitude ratio is reported in Table 3. As can be seen from these tables, the agreement between the results is quite good.

Figure 2 illustrates the effects of shear deformation and rotary inertia on the frequency ratio for three different amplitude ratios. It is clear that the CLT predicts the frequency ratio lower than the one obtained by FSDT. For thin beams and based on the two considered beam models, namely, CLT and FSDT, almost no difference is seen for different slenderness ratio.

407470.fig.002
Figure 2: Effects of shear deformation and rotary inertia on the frequency ratio versus slenderness ratio for the [0/90/90/0] configuration.

The same layers, that is, layers oriented at 0, 90, 90, and 0, in two different configurations are considered in Figure 3 to examine the stacking sequence on the frequency ratio. As can be observed, the frequency ratio for [0/90/90/0] configuration is higher than the [90/0/0/90] configuration. Furthermore, as the amplitude ratio increases, the frequency ratio increases for two considered lamination schemes.

407470.fig.003
Figure 3: Effect of ply orientation on the frequency ratio of the LCB for 𝜂=10.

Figure 4 displays the frequency ratio versus the angle of orientation for the symmetric angle-ply (𝜃/𝜃/𝜃/𝜃) thin and moderately thick LCB and for three different amplitude ratios and two considered boundary conditions. It is evident that all figures for thin and thick beams reach their maximum values at the orientation around (15°–20°). This result is also reported by [18] in which only the linear study was conducted. In the intermediate region (60𝜃90), the frequency ratio remains almost unchanged.

fig4
Figure 4: Variation of the frequency ratio versus the angle of orientation for the angle-ply LCB[𝜃,𝜃,𝜃,𝜃] at two slenderness ratios. (𝑤max=1,𝑤max=2,𝑤max=5).

Variation of the buckling load ratio versus amplitude ratio is shown in Figure 5. As can be observed for thick beams, there are significant differences between buckling load ratio predicted by CLT and FSDT. This difference is more considerable at larger amplitudes.

407470.fig.005
Figure 5: Effects of shear deformation and rotary inertia on the buckling load ratio versus amplitude ratio for the [0/90/90/0] configuration.

5. Conclusions

Analytical expressions for the frequency and deflection equations of the LCB at large amplitude on the basis of FSDT with simple end conditions have been derived. The applicability of the theory is demonstrated by numerical results, which show good agreement with published results.

Limited numerical studies are conducted to examine the effect of slenderness ratio, fiber orientation, and amplitude of vibration on the vibration and buckling characteristics of the LCB. The present study can serve as a quick and accurate reference to predict the vibration and postbuckling response of the composite beam at any amplitude.

Appendices

A. Derivation of Bending (EI), Torsional (GJ), Bending-Torsion Coupling (C), and Shear (kAG) Rigidities

Based on CLT, the constitutive equations of the laminate relating the stress resultant and the curvatures can be written as𝑀𝑥𝑀𝑦𝑀𝑥𝑦=𝐷11𝐷12𝐷16𝐷12𝐷22𝐷26𝐷16𝐷26𝐷66𝑘𝑥𝑘𝑦𝑘𝑥𝑦,(A.1) where 𝐷𝑖𝑗 are anisotropic stiffness coefficients which are defined by𝐷𝑖𝑗=𝑡/2𝑡/2𝑄𝑖𝑗𝑧2𝑑𝑧,(𝑖,𝑗=1,2,6)(A.2) where 𝑄𝑖𝑗 and 𝑡 are the transformed material constants [19] and thickness of the beam. The moment 𝑀𝑦 can be considered to be zero. Then 𝑘𝑦 can be eliminated from (A.1) to give𝑀𝑥𝑀𝑥𝑦=𝐷11𝐷212𝐷22𝐷16𝐷12𝐷26𝐷22𝐷16𝐷12𝐷26𝐷22𝐷66𝐷226𝐷22𝑘𝑥𝑘𝑥𝑦.(A.3) The bending 𝑀𝑥 and twisting 𝑀𝑥𝑦 moments can be related to the resultant bending (𝑀) and torsional (𝑇) moments as follows [20]:𝑀=𝑏𝑀𝑥,𝑇=2𝑏𝑀𝑥𝑦.(A.4) Also, the curvatures 𝑘𝑥 and 𝑘𝑥𝑦 can be related to the bending slope 𝜃 and twist rate 𝜓 so that [20]𝑘𝑥=𝜃,𝑘𝑥𝑦=2𝜓.(A.5) Thus, the expressions for 𝑀 and 𝑇 are given by𝑀𝑇𝐷=𝑏11𝐷212𝐷222𝐷16𝐷12𝐷26𝐷222𝐷16𝐷12𝐷26𝐷224𝐷66𝐷226𝐷22𝜃𝜓.(A.6) The common relationship between the bending 𝑀 and torsional 𝑇 moments with the bending slope 𝜃 and twist rate 𝜓 are as follows:𝑀𝑇=𝜃EI𝐶𝐶GJ𝜓.(A.7) An element-by-element comparison of (A.6) and (A.7), we obtain𝐷EI=𝑏11𝐷212𝐷22𝐷,𝐶=2𝑏16𝐷12𝐷26𝐷22,𝐷GJ=4𝑏66𝐷226𝐷22.(A.8) The transverse shear force-strain relation for the LCB can be expressed as𝑄𝑥𝑧=kb𝑡/2𝑡/2𝑄55𝛾𝑑̂𝑧𝑥𝑧=kAG𝛾𝑥𝑧,(A.9) where:kAG=kb𝑡/2𝑡/2𝑄55𝑑̂𝑧.(A.10)

B. An Overview of Homotopy Analysis Method (HAM)

For convenience of the reader, we will first present a brief description of the HAM [21]. Consider the following nonlinear homogeneous differential equations:𝑁[]𝑢(𝑡)=0,(B.1) where 𝑁 is nonlinear operators, 𝑡 denotes the independent variable, and 𝑢(𝑡)=0 are unknown functions.

Liao constructed the so-called zero-order deformation equation as [21]𝜙(1𝑝)𝐿(𝑡;𝑝)𝑢0[𝜙](𝑡)=𝑝(𝑡)𝑁(𝑡;𝑝),(B.2) where 𝑝[0,1] is an embedding parameter, are nonzero auxiliary functions, (t) are nonzero auxiliary function, 𝐿 is an auxiliary linear operator, 𝑢0(𝑡) are initial guesses of 𝑢(t), and 𝜙(t;p) are unknown functions. As 𝑝 increases from 0 to 1, the 𝜙(𝑡;𝑝) varies from the initial approximation to the exact solution. In other words,𝜙(𝑡;0)=𝑢0(𝑡) and𝜙(𝑡;1)=𝑢(𝑡).

The deformation equation (B.2) can provide us with a family of approximation series whose convergence region depends upon the auxiliary parameter and the auxiliary function (t). More importantly, this provides us with a simple way to adjust and control the convergence regions and rates of approximation series.

Differentiating once more from (B.2) with respect to the embedding parameter 𝑝 and then setting 𝑝=0, we obtain the first-order deformation equation as𝐿𝑢1[𝜙]=(𝑡)𝑁(𝑡;𝑝)𝑝=0(B.3) which gives the first-order approximation for the 𝑢(𝑡) [21]. The higher order approximation of the solution can be obtained by calculating the 𝑚-order (𝑚>1) deformation equation which can be calculated by differentiating (B.3) 𝑚 times with respect to the 𝑝 and then setting 𝑝=0 and finally dividing them by 𝑚! [21]. Therefore, the 𝑚th-order deformation equation can be obtained as follows:𝐿𝑢𝑚(𝑡)𝑢𝑚1(𝑡)=(𝑡)𝑅𝑚𝑢𝑚1,,𝑢𝑚1,(B.4) where (𝑢𝑚1,,𝑢𝑚1) and 𝑅𝑚(𝑢𝑚1,,𝑢𝑚1) are defined as𝑅𝑚𝑢𝑚1,,𝑢𝑚1=1𝑑(𝑚1)!𝑚1𝑁[]𝜙(𝑡;𝑝)𝑑𝑝𝑚1||||𝑞=0𝑢𝑚1=𝑢0,𝑢1,𝑢2,,𝑢𝑚1,(B.5) subjected to the following initial conditions:𝑢𝑚(0)=0,̇𝑢𝑚(0)=0.(B.6) To solve (B.2), we employ Taylor expansion series for 𝜙(𝑡;𝑝) as𝜙(𝑡;𝑝)=𝜙(𝑡;0)+𝑚=11𝜕𝑚!𝑚𝜙(𝑡;𝑝)𝜕𝑝𝑚||||𝑝=0𝑝𝑚𝑢(𝑡)=𝑢0(𝑡)+𝑚=1𝑢𝑚(𝑡)𝑝𝑚,(B.7) where 𝑢𝑚(𝑡) is called the 𝑚th-order derivative of unknown function 𝜙(𝑡;𝑝).

Acknowledgment

The authors are grateful for the encouragement and useful comments of the reviewers.

References

  1. M. Baghani, R. A. Jafari-Talookolaei, and H. Salarieh, “Large amplitudes free vibrations and post-buckling analysis of unsymmetrically laminated composite beams on nonlinear elastic foundation,” Applied Mathematical Modelling, vol. 35, no. 1, pp. 130–138, 2011. View at Publisher · View at Google Scholar · View at Scopus
  2. P. Malekzadeh and A. R. Vosoughi, “DQM large amplitude vibration of composite beams on nonlinear elastic foundations with restrained edges,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 906–915, 2009. View at Publisher · View at Google Scholar · View at Scopus
  3. B. P. Patel, M. Ganapathi, and M. Touratier, “Nonlinear free flexural vibrations/post-buckling analysis of laminated orthotropic beams/columns on a two parameter elastic foundation,” Composite Structures, vol. 46, no. 2, pp. 189–196, 1999. View at Google Scholar · View at Scopus
  4. S. A. Emam and A. H. Nayfeh, “Postbuckling and free vibrations of composite beams,” Composite Structures, vol. 88, no. 4, pp. 636–642, 2009. View at Publisher · View at Google Scholar · View at Scopus
  5. R. K. Kapania and S. Raciti, “Nonlinear vibrations of unsymmetrically laminated beams,” AIAA Journal, vol. 27, no. 2, pp. 201–210, 1989. View at Google Scholar · View at Scopus
  6. M. Ganapathi, B. P. Patel, J. Saravanan, and M. Touratier, “Application of spline element for large-amplitude free vibrations,” Composites Part B, vol. 29B, no. 1, pp. 1–8, 1998. View at Google Scholar · View at Scopus
  7. M. Kiran, I. Bangera, and K. Chandrashekhara, “Nonlinear vibration of moderately thick laminated beams using finite element method,” Finite Elements in Analysis and Design, vol. 9, no. 4, pp. 321–333, 1991. View at Google Scholar · View at Scopus
  8. A. W. Obst and R. K. Kapania, “Nonlinear static and transient finite element analysis of laminated beams,” Composites Engineering, vol. 2, no. 5–7, pp. 375–389, 1992. View at Google Scholar · View at Scopus
  9. J. R. Banerjee and F. W. Williams, “Exact dynamic stiffness matrix for composite timoshenko beams with applications,” Journal of Sound and Vibration, vol. 194, no. 4, pp. 573–585, 1996. View at Publisher · View at Google Scholar · View at Scopus
  10. J. R. Banerjee, “Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method,” Computers and Structures, vol. 69, no. 2, pp. 197–208, 1998. View at Google Scholar · View at Scopus
  11. G. V. Rao, K. M. Saheb, and G. R. Janardhan, “Fundamental frequency for large amplitude vibrations of uniform Timoshenko beams with central point concentrated mass using coupled displacement field method,” Journal of Sound and Vibration, vol. 298, no. 1-2, pp. 221–232, 2006. View at Publisher · View at Google Scholar · View at Scopus
  12. S. Woinowsky-Krieger, “The effect of an axial force on the vibration of hinged bars,” Journal of Applied Mechanics, vol. 17, pp. 35–36, 1950. View at Google Scholar
  13. L. Azrar, R. Benamar, and R. G. White, “A semi-analytical approach to the non-linear dynamic response problem of S-S and C-C beams at large vibration amplitudes part I: general theory and application to the single mode approach to free and forced vibration analysis,” Journal of Sound and Vibration, vol. 224, no. 2, pp. 183–207, 1999. View at Google Scholar · View at Scopus
  14. M. I. Qaisi, “Application of the harmonic balance principle to the nonlinear free vibration of beams,” Applied Acoustics, vol. 40, no. 2, pp. 141–151, 1993. View at Google Scholar · View at Scopus
  15. G. V. Rao, I. S. Raju, and K. K. Raju, “Nonlinear vibrations of beams considering shear deformation and rotary inertia,” AIAA Journal, vol. 14, no. 5, pp. 685–687, 1976. View at Google Scholar · View at Scopus
  16. S. M. Han, H. Benaroya, and T. Wei, “Dynamics of transversely vibrating beams using four engineering theories,” Journal of Sound and Vibration, vol. 225, no. 5, pp. 935–988, 1999. View at Google Scholar · View at Scopus
  17. G. V. Rao and K. K. Raju, “A numerical integration method to study the large amplitude vibration of slender beams with immovable ends,” Journal of the Institution of Engineers, vol. 83, pp. 42–44, 2002. View at Google Scholar · View at Scopus
  18. E. Chandrasekaran, K. Jayaraman, and S. M. Nazeer, “Effects of symmetric and antisymmetric fiber orientations on the natural frequencies of FRP aircraft panel boards,” Journal of Reinforced Plastics and Composites, vol. 23, no. 8, pp. 831–841, 2004. View at Publisher · View at Google Scholar · View at Scopus
  19. R. M. Jones, Mechanics of Composite Materials, McGraw-Hill, NewYork, NY, USA, 1975.
  20. T. A. Weisshaar and B. L. Foist, “Vibration tailoring of advanced composite lifting surfaces,” Journal of Aircraft, vol. 22, no. 2, pp. 141–147, 1985. View at Google Scholar · View at Scopus
  21. S. Liao, Beyond Perturbation—Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, 2004.