Buckling Induced Strongly Nonlinear Vibration of Supercritical Axially Moving Beam

Abstract

In this paper, the postbuckling problem and the free linear and nonlinear vibration surrounding postbuckling configuration are accurately analyzed. The placement of translating beams that undergo significant overall buckling deformations in this high-speed region, leading to strongly nonlinear vibration, is studied. Using the Galerkin method, we found that the partial differential equation is reduced to an ordinary differential equation with cubic and quadratic terms, which differs from the previous literature. The supercritical natural frequency is analytically obtained, the strongly nonlinear vibration problem is solved using the homotopy analysis method (HAM), and the solution is examined using numerical results. Results show that the obtained analytical solution is valid for both weakly and strongly nonlinear cases.

1. Introduction

Axially moving systems, such as threads, textiles, wires, magnetic tapes, belts and cables, are present in a wide class of engineering structures. The vibration of such devices has been studied extensively [1,2,3,4,5,6]. In general, one-dimensional string or beam theory can be used to model axially moving elements, depending on flexural stiffness [7,8,9,10,11,12,13,14].

It is known that the linear frequency of a moving beam decreases as its axial speed increases. At certain critical speeds, the linear frequency vanishes and buckling occurs [3,7]. In previous studies, the linear and nonlinear vibration of axially moving beams were studied in the subcritical regime, and referred to as subcritical axially moving beams. Herein, we refer to beams traveling in the supercritical regime as supercritical axially moving beams. Öz [3] studied the linear vibration of an axially moving beam with time-dependent velocity below critical value. The multi-scales method was used to investigate the resulting parametric resonance. Li and Lu [15,16] described three-dimensional parametric vibrations of a simply-supported pipe conveying pulsating fluid, and found that the jump phenomenon for the parametric resonance in the post-buckling state could be clearly observed. Chen and Yang [17,18] derived a new governing equation for the traveling beam using geometric nonlinearity, which was used to account for finite stretching of the beam; the nonlinear vibration problem was studied using the perturbation method. Ghayesh et al. [19,20,21] contributed to the field of forced dynamics of a subcritical axially moving viscoelastic beam and investigated the beam’s bifurcation and chaotic motions.

The literature regarding supercritical axially moving beams is rather limited. Hwang and Perkins [22,23] studied the influence of the initial curvature caused by the support wheel and pulley on the bifurcation and stability in the supercritical regime. Ghayesh [24,25] included the nonlinear stretching effects for a moving beam and analyzed both the subcritical and supercritical speed range. Ding et al. [26,27,28,29] utilized the Galerkin method and other numerical methods to calculate the supercritical natural frequency and equilibrium configurations for different boundary conditions. Zhang et al. [30] studied the stability of an axially moving beam subjected to harmonic excitation. They employed the multi-scales method to calculate the steady-state response amplitude for the first single mode; the strongly nonlinear state was shown in some situations.

In this paper, we focus on the dynamics of supercritical axially moving beams. In this high-speed region, the translating beam undergoes large overall buckling deformations leading to strongly nonlinear vibration. This situation is common; however, the solution is not available in the literature. Using Galerkin truncation, we found that the partial differential equation can be simplified and that the dynamic process can be described by an ordinary differential equation, which differs from the previous literature. Without using transformation, we directly used the homotopy analysis method to address the strongly nonlinear problem.

2. Governing Equations and Solutions

The equation of motion of an axially moving beam takes the form [3,18]

$$\rho A(\frac{{\partial }^{2}U}{\partial T^2}+2V\frac{{\partial }^{2}U}{\partial X\partial T}+V^2\frac{{\partial }^{2}U}{\partial X^2})-P_0\frac{{\partial }^{2}U}{\partial X^2}+EI\frac{{\partial }^{4}U}{\partial X^4}=\frac{EA}{L}{\displaystyle {{\displaystyle \mathrm{\int }}}_0^L(\frac{1}{2}{\left(\frac{\partial U}{\partial X}\right)}^2)}dX\frac{{\partial }^{2}U}{\partial X^2}$$

(1)

where ρ is linear density, U is transverse displacement, A is cross-sectional area, P₀ is initial tension, V is transport speed, and EI is flexural stiffness. The beam travels between two simple supports, which are separated by L.

The case V = 0 corresponds to a static beam with deformation, and the governing equation is reduced to

$$\rho A\frac{{\partial }^{2}U}{\partial T^2}-P_0\frac{{\partial }^{2}U}{\partial X^2}+EI\frac{{\partial }^{4}U}{\partial X^4}=\frac{EA}{L}{\displaystyle {\int }_0^L(\frac{1}{2}{\left(\frac{\partial U}{\partial X}\right)}^2)}dX\frac{{\partial }^{2}U}{\partial X^2}$$

(2)

Introduce the following non-dimensional quantities

$$x=\frac{X}{L},u=\frac{U}{L},t=\frac{T}{L}\sqrt{\frac{P_0}{\rho A}},\gamma =V\sqrt{\frac{\rho A}{P_0}},v_f^2=\frac{EI}{P_0{L}^2},k_f^2=\frac{EA}{P_0}$$

(3)

and Equation (4) is obtained by substituting Equation (3) into Equation (1).

$$\ddot{u}+2\gamma {\dot{u}}^{\prime }+({\gamma }^2-1)u^{″}+v_f^2{u}^{(4)}=\frac{1}{2}{k}_f^2{u}^{″}{\displaystyle \underset{0}{\overset{1}{\int }}{u^{\prime }}^{2}dx}$$

(4)

The pinned–pinned supported and the fixed–fixed boundary conditions are

$$u=0\mathrm{and}{u}^{″}=0\mathrm{at}x=0,1$$

(5)

$$u=0\mathrm{and}{u}^{\prime }=0\mathrm{at}x=0,1$$

(6)

Equation (7) is the supercritical equilibrium equation induced from Equation (4) and ψ is the supercritical configuration.

$$({\gamma }^2-1){\psi }^{″}+v_f^2{\psi }^{(4)}=\frac{1}{2}{k}_f^2{\psi }^{″}{\displaystyle \underset{0}{\overset{1}{\int }}{{\psi }^{\prime }}^{2}dx}$$

(7)

The integral of Equation (7) is a constant for a given ψ. Let

$$\mathrm{\Pi }=\frac{1}{2}{k}_f^2{\displaystyle \underset{0}{\overset{1}{\int }}{{\psi }^{\prime }}^{2}dx}$$

(8)

Next, Equation (7) can be rewritten as

$${\psi }^{(4)}+{\lambda }^2{\psi }^{″}=0$$

(9)

where

$${\lambda }^2=\frac{({\gamma }^2-1)-\mathrm{\Pi }}{v_f^2}$$

(10)

The characteristic equation is shown in Equation (11).

$$\sin \lambda =0\mathrm{or}\lambda =m\pi ,m=1,2,3\dots$$

(11)

The equilibrium configurations for the pinned–pinned supported case is

$$\psi (x)=c\sin (\lambda x)$$

(12)

where

$$c=\pm \frac{2}{\lambda k_f}\sqrt{({\gamma }^2-1)-v_f^2{\lambda }^2}$$

(13)

Note that the equilibrium configurations (12) were also obtained by Wickert [7] using a different method.

Figure 1 illustrates subcritical and supercritical equilibrium configurations. The blue dashed line indicates the equilibrium position of a beam in the subcritical regime. If the beam speeds up and exceeds critical speed, it will stand vertically as an arch, and be symmetrical about the x axis, as shown by the solid black curve.

It is noted that the configuration given by (12) is symmetric for the pinned–pinned case. Yang et al. [31] found two special configurations for the fixed–fixed boundary condition, which are shown below.

(1)

Symmetric configuration:

$$\psi (x)=c\left[1-\cos (2k\pi x)\right],\mathsf{\lambda }=2\mathsf{\pi },4\mathsf{\pi }$$

(14)

(2)

Anti-symmetric configuration:

$$\psi (x)=c\left[1-2x-\cos (\lambda x)+\frac{2}{\lambda }\sin (\lambda x)\right],\mathsf{\lambda }=8.9868,15.4505$$

(15)

In Equations (14) and (15), the coefficient c is defined by Equation (13).

To obtain the natural frequency of a supercritical axially moving beam subjected to buckling, we introduce a small disturbance as follows

$$u=\psi (x)+v(x,t)$$

(16)

where v(x,t) is a small dynamic disturbance around the buckling configuration $\psi (x)$. Substituting Equation (16) into Equation (4) leads to

$$\begin{array}{l}\ddot{v}+2\gamma {\dot{v}}^{\prime }+({\gamma }^2-1)v^{″}+v_f^2{v}^{(4)}=\\ \begin{array}{cc}& \end{array}{k}_f^2\left({\psi }^{″}{\displaystyle \underset{0}{\overset{1}{\int }}{\psi }^{\prime }{v}^{\prime }dx}+\frac{1}{2}{\psi }^{″}{\displaystyle \underset{0}{\overset{1}{\int }}{v^{\prime }}^{2}dx}+v^{″}{\displaystyle \underset{0}{\overset{1}{\int }}{\psi }^{\prime }{v}^{\prime }dx}+\frac{1}{2}{v}^{″}{\displaystyle \underset{0}{\overset{1}{\int }}{v^{\prime }}^{2}dx}\right)\end{array}$$

(17)

Note that the first term, $k_f^2{\psi }^{″}{\displaystyle \underset{0}{\overset{1}{\int }}{\psi }^{\prime }{v}^{\prime }dx}$, on the right side of Equation (17) is linear. Using the configuration relation (7) and dropping the nonlinear terms of Equation (17), we obtain the linear vibration equation

$$\ddot{v}+2\gamma {\dot{v}}^{\prime }+v_f^2{\lambda }^2{v}^{″}+v_f^2{v}^{(4)}-\frac{1}{2}{k}_f^2{\psi }^{″}{\displaystyle \underset{0}{\overset{1}{\int }}{\psi }^{\prime }{v}^{\prime }dx}=0$$

(18)

Equation (19) corresponds to the case V = 0.

$$\ddot{v}+v_f^2{\lambda }^2{v}^{″}+v_f^2{v}^{(4)}-\frac{1}{2}{k}_f^2{\psi }^{″}{\displaystyle \underset{0}{\overset{1}{\int }}{\psi }^{\prime }{v}^{\prime }dx}=0$$

(19)

in which $\psi (x)$ is the linear equilibrium configuration defined by Equations (12), (14) and (15), depending on different boundary conditions. Supercritical linear vibration properties can be studied using Equation (18), which significantly differs from previous linear subcritical models [3,5].

We let

$$v(x,t)={\displaystyle \sum _{j=1}^N{\varphi }_j(x)e^{i\omega t}}$$

(20)

where ${\varphi }_j(x)$ is the j-th supercritical linear vibration mode around the buckled shape and ω is the corresponding natural frequency. Substituting Equation (20) into Equation (18) leads to

$$v_f^2{\varphi }_j{}^{(4)}+2\gamma i\omega {{\varphi }^{\prime }}_j+v_f^2{\lambda }^2{{\varphi }^{″}}_j-\frac{1}{2}{k}_f^2{\psi }^{″}{\displaystyle \underset{0}{\overset{1}{\int }}{\psi }^{\prime }{{\varphi }^{\prime }}_{j}dx}={\omega }^2{\varphi }_j$$

(21)

Multiplying Equation (21) by ${\varphi }_i(x)$ and integrating the result over [0, 1] yields

$${\displaystyle \underset{0}{\overset{1}{\int }}L({\varphi }_j)}{\varphi }_{i}dx={\omega }^2$$

(22)

where

$$L({\varphi }_j)=v_f^2{\varphi }_j{}^{(4)}+v_f^2{\lambda }^2{{\varphi }^{″}}_j+2\gamma i\omega {{\varphi }^{\prime }}_j-\frac{1}{2}{k}_f^2{\psi }^{″}{\displaystyle {\int }_0^1{\psi }^{\prime }{{\varphi }^{\prime }}_{j}dx}$$

(23)

which denotes the linear operator. Evaluating Equation (23) yields

$$v_f^2{\lambda }_i^4{\delta }_{ij}-v_f^2{\lambda }_i^4{C}_{ij}+i{\omega }_i^2{B}_{ij}-\frac{1}{2}{k}_f^2{\lambda }_i^4{c}^2={\omega }_i^2$$

(24)

where

$$B_{ij}={\displaystyle {\int }_0^1{\varphi }_i}{{\varphi }^{\prime }}_{j}dx,C_{ij}={\displaystyle {\int }_0^1{\varphi }_i}{{\varphi }^{″}}_{j}dx$$

Evaluations of B and C are shown in Appendix A.

If m = 1, the first supercritical natural frequency for a pinned–pinned case is

$${\omega }_1^2\approx \frac{1}{2}{k}_f^2{c}^2{\lambda }^4=\frac{1}{2}{k}_f^2\frac{4{\pi }^4}{k_f^2}\left(\frac{({\gamma }^2-1)}{{\pi }^2}-v_f^2\right)\approx 2{\pi }^4\left(\frac{({\gamma }^2-1)}{{\pi }^2}-v_f^2\right)$$

(25)

Equation (26) shows that if V = 0, ω will become a constant.

$${\omega }_1^2\approx -2{\pi }^2-v_f^2$$

(26)

The first supercritical natural frequency for a fixed–fixed case is

$${\omega }_1^2=v_f^2{\lambda }_1^4-v_f^2{\lambda }_1^4{C}_{11}-\frac{1}{2}{k}_f^2{\lambda }_1^4{c}^2$$

(27)

Ding and Chen [32] adopted the finite difference method to calculate the natural frequency of an axially moving beam in the supercritical regime. They numerically showed that supercritical natural frequency for a pinned–pinned case is independent of the nonlinear coefficient, k_f. The supercritical natural frequency increases regardless of the magnitude of the nonlinear coefficient, k_f. In this paper, Equations (25) and (27) are the analytical approximation of a supercritical axially moving beam, in which k_f disappears. Moreover, we analytically obtained the supercritical natural frequency for a fixed–fixed case and found that, in contrast, it is highly dependent on k_f.

First, to facilitate comparison with classical results, we plot the natural frequencies in the subcritical region in Figure 2 and Figure 3, where ω is the axial speed and γ is the natural frequency. Note that the subcritical natural frequency decreases as the axial speed increases for both the pinned–pinned and fixed–fixed cases. If V = 0, these series of curves in Figure 2 and Figure 3 will reduce to a series of points on the y-axis, which means that the subcritical natural frequency becomes a fixed value.

In Figure 4 and Figure 5, variations in supercritical natural frequency with axial speed increases are depicted by Equations (25) and (27). Note that the supercritical natural frequencies increase as axial speeds increase, which is quite different from a subcritical case. Moreover, a beam with greater flexural stiffness has a larger vibration natural frequency and starts to vibrate at a higher axial speed. This effect of flexural stiffness on the supercritical natural frequency is the same as that of a subcritical case. The basis for this finding is quite reasonable; similar to the one-degree-of-freedom system, stiffness is proportional to natural frequency.

After analyzing nature frequencies, the nonlinear vibration Equation (17) is studied. We used standard Galerkin-type projections to approximate Equation (17). Assume the displacement expansion is

$$v={\displaystyle \sum _{n=1}^N{\varphi }_n(x){\eta }_n(q)}$$

(28)

where ${\varphi }_n(x)$ is the eigenfunction of the buckled beam. For a pinned–pinned case, ${\varphi }_n(x)=\sin (n\pi x)$. For a fixed–fixed case,

$${\varphi }_n(x)=\cos ({\beta }_{n}x)-\cosh ({\beta }_{n}x)-\frac{\cos ({\beta }_{n}x)-\cosh ({\beta }_{n}x)}{\sin ({\beta }_{n}x)-\sinh ({\beta }_{n}x)}\left[\sin ({\beta }_{n}x)-\sinh ({\beta }_{n}x)\right]$$

where β_n is the n-th root of the equation

$$\cos ({\beta }_{n}x)-\cosh ({\beta }_{n}x)=1$$

In this section, we only take the pinned–pinned boundary condition as an example. η_n(t) is the generalized coordinates of the discretized system, and N is the number of retained modes. Substituting Equation (28) into Equation (17) yields

$${\ddot{\eta }}_m+{\omega }_m^2\eta =k_f^2\left[c{\displaystyle \sum _{i,j=1}^N{A}_{mij}}+{\displaystyle \sum _{i,j=1}^N{B}_{mij}}\right]$$

(29)

where

$$A_{mij}=\frac{1}{2}{n}^2{\pi }^2\left({\displaystyle {\int }_0^1{\varphi }_i{\varphi }_{m}dx}\right)\left({\displaystyle {\int }_0^1{{\varphi }^{\prime }}_i{{\varphi }^{\prime }}_{m}dx}\right)+n\pi \left({\displaystyle {\int }_0^1{{\varphi }^{\prime }}_i{{\varphi }^{\prime }}_{m}dx}\right)\left({\displaystyle {\int }_0^1{{\varphi }^{″}}_i{\varphi }_{m}dx}\right)$$

(30)

$$B_{mij}=\frac{1}{2}\left({\displaystyle {\int }_0^1{{\varphi }^{\prime }}_i{{\varphi }^{\prime }}_{m}dx}\right)\left({\displaystyle {\int }_0^1{{\varphi }^{″}}_i{\varphi }_{m}dx}\right)$$

(31)

The single-mode approximation can be obtained by letting N = 1

$$\ddot{\eta }+{\omega }_1^2\eta +k_f^2(c{a}_2{\eta }^2+a_3{\eta }^3)=0$$

(32)

Equation (32) is the key equation in this paper, which contains both a quadratic term and a cubic term at the same time. Evaluating a₂ and a₃ for N = 1, the results are

$$a_2=-\frac{1}{2}{\pi }^2\left({\displaystyle {\int }_0^1\sin \pi x{\varphi }_{1}dx}\right)\left({\displaystyle {\int }_0^1{({{\varphi }^{\prime }}_1)}^2}\right)-\pi \left({\displaystyle {\int }_0^1\cos \pi x{{\varphi }^{\prime }}_{1}d}x\right)\left({\displaystyle {\int }_0^1{{\varphi }^{″}}_1{\varphi }_1}dx\right)=\frac{1}{8}{\pi }^4$$

(33)

$$a_3=-\frac{1}{2}\left({\displaystyle {\int }_0^1{{\varphi }^{″}}_1{\varphi }_{1}dx}\right)\left({\displaystyle {\int }_0^1{({{\varphi }^{\prime }}_1)}^{2}dx}\right)=\frac{1}{2}{\left({\displaystyle {\int }_0^1{({{\varphi }^{\prime }}_1)}^{2}dx}\right)}^2=\frac{1}{8}{\pi }^4$$

(34)

The nonlinear free vibration problem of Equation (32) will be investigated in the next section. First, the parameters are discussed in detail. Recalling the non-dimensional flexural stiffness and nonlinear coefficient in Equation (3)

$$v_f^2=\frac{EI}{P_0{L}^2},k_f^2=\frac{EA}{P_0}$$

(35)

it follows from Equation (35) that

$$0\le k_f^2=12v_f^2{(\frac{L}{h})}^2$$

(36)

In general, note that the non-dimensional flexural stiffness is in the range [0, 1], and for a slender beam is in the range $L/h\gg 1$. The nonlinear coefficient, $k_f^2$, is not a small parameter; therefore, Equation (32) exhibits as strongly nonlinear, and the traditional perturbation method is not valid. In the next section, we employ the homotopy analysis method to solve Equation (32).

3. Principle of Homotopy Analysis

Liao [33] introduced the basic principle of homotopy analysis. Assume that Equation (34) is a nonlinear differential equation

$$Fu(x,t)=g(x,t)$$

(37)

where F is a nonlinear ordinary or partial differential operator, u(x,t) is an unknown function, and g is the source term.

$$F=L+N$$

(38)

where L and N indicate the linear and nonlinear terms, respectively. Equation (39) is the zero-order deformation equation:

$$(1-q)L[\mathrm{\Phi }(t;q)-u_0]=q\hslash [F\mathrm{\Phi }(t;q)-g]$$

(39)

The convergence range can be well controlled by $\hslash$, and the constraint conditions for constructing auxiliary linear operator, L, are relatively loose.

When q = 0 and q = 1, it holds that

$$\mathrm{\Phi }(t;0)=u_0,\begin{array}{cc}& \end{array}\mathrm{\Phi }(t;1)=u(q,t)$$

(40)

As q changes from 0 to 1, the solution of Equation (39) also changes from u₀ to u(x,t). Equations (41) and (42) are the Taylor series expansion of Φ(t;q), the exact solution of Equation (37).

$$\mathrm{\Phi }(t;q)=u_0(t;q)+{\displaystyle \sum _{m=1}^{+\infty }{u}_m}(t;q)q^m$$

(41)

$$u_m(t;q)=\frac{1}{m!}\frac{{\partial }^m\mathrm{\Phi }(t;q)}{\partial q^m}$$

(42)

If we perform differentiation on Equation (41) m times, the corresponding deformation equation can be obtained as Equation (43) and represented as a set of linear ordinary equations.

$$L[u_m(t;q)-{\chi }_m{u}_{m-1}(t;q)]=\hslash R_m({\overrightarrow{u}}_{m-1})$$

(43)

where ${\overrightarrow{u}}_{m-1}$ is a vector defined as

$${\overrightarrow{u}}_m=\{u_0(x,t),u_1(x,t),u_2(x,t),\cdots u_m(x,t)\}$$

(44)

and

$$R_m({\overrightarrow{u}}_{m-1})={\frac{1}{(m-1)!}\frac{{\partial }^{m-1}F[\mathrm{\Phi }(t;q)]}{\partial q^{m-1}}|}_{q=0}$$

(45)

$${\chi }_m=\{\begin{array}{cc}0& m=1\\ 1& m>1\end{array}$$

4. Application of the Homotopy Analysis Method

Under the transformation $\tau =\mathrm{\Omega }t$, Equation (32) can be written as

$${\mathrm{\Omega }}^2{\eta }^{″}+{\omega }_1^2\eta +k_f^2(c{a}_2{\eta }^2+a_3{\eta }^3)=0$$

(46)

with the following initial conditions:

$$\eta (0)=A,{\eta }^{\prime }(0)=0$$

(47)

Thus, with these initial conditions, the initial guess of η(τ) for a zero-order deformation equation is chosen as follows.

$${\eta }_0=A\cos \tau$$

(48)

According to (46), we should choose the auxiliary linear operator

$$L[\eta (\tau ;q)]={\mathrm{\Omega }}^2\left(\frac{{\partial }^2\eta (\tau ;q)}{\partial {\tau }^2}+\eta (\tau ;q)\right)$$

(49)

which has the property

$$L(C_1\sin \tau +C_2\tau )=0$$

(50)

for any integration constants C₁ and C₂. Define the following nonlinear operator

$$N[\eta (\tau ;q),\mathrm{\Omega }]={\mathrm{\Omega }}^2\frac{{\partial }^2\eta (\tau ;q)}{\partial {\tau }^2}+{\omega }_1^2\eta (\tau ;q)+k_f^2\left[c{a}_2{\eta }^2(\tau ;q)+a_3{\eta }^3(\tau ;q)\right]$$

(51)

The zero-order deformation is constructed as

$$(1-q)L[\mathrm{\Phi }(\tau ;q)-{\eta }_0(\tau ;q)]=q\hslash N[\mathrm{\Phi }(\tau ;q)]$$

(52)

with the following initial conditions:

$$\mathrm{\Phi }(\tau ;0)={\eta }_0(t),\mathrm{\Phi }(\tau ;1)=\eta (\tau ;q)$$

(53)

The functions $\mathrm{\Phi }(\tau ;q)$ and $\mathrm{\Omega }(q)$ can be expanded as power series of the q using Taylor’s theorem as

$$\mathrm{\Phi }(\tau ;q)={\eta }_0(\tau ,q)+{\displaystyle \sum _{m=1}^{+\infty }{\eta }_m(\tau ,q)}{q}^m$$

(54)

$$\mathrm{\Omega }(q)={\mathrm{\Omega }}_0+{\displaystyle \sum _{m=1}^{+\infty }{\mathrm{\Omega }}_m(\tau ,q)}{q}^m$$

(55)

where η_m and Ω_m are the m-order deformation derivatives.

$$N_m({\eta }_{m-1},{\omega }_{m-1})={\frac{1}{(m-1)!}\frac{d^{m-1}N[\mathrm{\Phi },\omega ]}{d{q}^{m-1}}|}_{q=0}$$

(56)

For free vibration without damping, periodic motion can be expressed by the following base functions

$$\{\cos (m\tau ),m=1,2,3,...\}$$

(57)

For the first-order approximation of HAM, N₁ is as follows:

$$N_1=({\omega }_1^{2}A-{\mathrm{\Omega }}^{2}A+\frac{3}{4}{k}_f^2{a}_3{A}^3)\cos \tau +\frac{1}{2}{k}_f^{2}c{a}_2{A}^2\cos 2\tau +\frac{1}{4}{k}_f^2{a}_3{A}^3\cos 3\tau +\frac{1}{2}{k}_f^{2}c{a}_2{A}^2$$

(58)

Thus, Ω₀ can be written as

$${\mathrm{\Omega }}_0=\sqrt{{\omega }_1^2+\frac{3}{4}{k}_f^2{a}_3{A}^2}$$

(59)

Equation (59), does not contain the coefficient of quadratic term $k_f^{2}c{a}_2$, which means that the nonlinear frequency is independent of a quadratic term. The cubic term of Equation (32) plays the dominant role in the nonlinear vibration process.

Solving Equation (52) for the first order solution, η₁, is obtained as

$$\begin{array}{l}{\eta }_1=\frac{1}{96{\mathrm{\Omega }}_0^2}[(48{\mathrm{\Omega }}_0^{2}A-48{\omega }_1^{2}A-32k_f^{2}c{a}_2{A}^2-33k_f^2{a}_3{A}^3)\cos \tau \\ \begin{array}{cccc}& & & \end{array}-16k_f^{2}c{a}_2{A}^2\cos 2\tau -3k_f^2{a}_3{A}^3\cos 3\tau +48k_f^{2}c{a}_2{A}^2]\end{array}$$

(60)

Finally, the approximate solution of Equation (32) is

$$\begin{array}{l}\eta ={\eta }_0+{\eta }_1=A\cos \mathrm{\Omega }t+\frac{\hslash }{96{\mathrm{\Omega }}_0^2}[(48{\mathrm{\Omega }}_0^{2}A-48{\omega }_1^{2}A-32k_f^{2}c{a}_2{A}^2-33k_f^2{a}_3{A}^3)\cos \tau \\ \begin{array}{cccc}& & & \end{array}-16k_f^{2}c{a}_2{A}^2\cos 2\tau -3k_f^2{a}_3{A}^3\cos 3\tau +48k_f^{2}c{a}_2{A}^2]\end{array}$$

(61)

5. Numerical Comparison

In order to verify the effectiveness of the proposed method, numerical comparison and discussion are presented in this section. In Figure 6, the curves at different axial speeds are plotted for v_f = 0.8 k_f = 1, where Ω is nonlinear frequency and A is amplitude. It is shown that the increasing trend of frequency is consistent with the axial speed. A lower axial speed yields a smaller nonlinear frequency.

Figure 7 shows the dependence of the nonlinear frequency on the amplitude at a fixed axial speed, γ = 3. The dimensionless stiffness is chosen as v_f = 0.8. Moreover, we plot the nonlinear frequency for k_f =1 and k_f = 100, respectively. It is seen that the nonlinear coefficient, k_f, is independent of nonlinear frequency. This finding is the same as that for linear frequency, which is independent of nonlinear coefficient k_f.

Figure 8 depicts the relationship between nonlinear frequency and flexural stiffness, v_f, at a fixed axial speed for γ = 3 and k_f = 1. Increasing flexural stiffness causes an increase in nonlinear frequency.

When we compare the analytical solution (61) in Figure 9, let ħ = −1, $k_f^2=1$, γ = 3, A = 0.1, and v_f = 0.8, which indicates a weakly nonlinear case. In this figure, the exact and approximate solutions are represented by a solid line and a dashed line, respectively, and they are calculated using the fourth-order Runge Kutta method and the average method. They are in good agreement, indicating that solution (61) is accurate enough for the weakly nonlinear case.

Figure 10 can prove the accuracy of the solutions (61) and the effectiveness of the homotopy analysis method. The parameters chosen in this section are γ = 3, A = 0.1, v_f = 0.8, and the nonlinear coefficient is chosen as a larger value, $k_f^2=10$. The approximation, (61), is consistent with the numerical results. Note that the method in this paper is valid for weak and relatively large cases; it does not work for the extreme case where v_f approaches infinity.

6. Conclusions

In this paper, we present analytical solutions of natural frequency and response for beams traveling in the supercritical regime. Postbuckling shapes were derived for pinned–pinned and fixed–fixed boundary conditions. The linear vibration around the equilibrium configuration (buckling shape) was analyzed and the corresponding natural frequency and modes were analytically derived. We found that supercritical natural frequency increases as axial speed increases. The equation of motion for a buckling axially moving beam processes quadratic and cubic nonlinearity. Homotopy analysis was employed to solve the nonlinear governing equation of a buckled axially moving beam. Numerical results show that the obtained solution is valid for both weakly and strongly nonlinear vibration cases.