# Buckling Induced Strongly Nonlinear Vibration of Supercritical Axially Moving Beam
## 2 Governing Equations and Solutions
Evaluating Equation (23) yields  
\[v_{f}^{2}\lambda_{i}^{4}\delta_{ij}-v_{f}^{2}\lambda_{i}^{4}C_{ij}+i\omega_{i} ^{2}B_{ij}-\frac{1}{2}k_{f}^{2}\lambda_{i}^{4}c^{2}=\omega_{i}^{2} \tag{24}\]  
where  
\[B_{ij}=\int_{0}^{1}\phi_{i}\phi^{\prime}_{j}dx,\ C_{ij}=\int_{0}^{1}\phi_{i} \phi^{\prime\prime}_{j}dx\]  
Evaluations of \(B\) and \(C\) are shown in Appendix A. If \(m=1\), the first supercritical natural frequency for a pinned-pinned case is  
\[\omega_{1}^{2}\approx\frac{1}{2}k_{f}^{2}c^{2}\lambda^{4}=\frac{1}{2}k_{f}^{2} \frac{4\pi^{4}}{k_{f}^{2}}\bigg{(}\frac{(\gamma^{2}-1)}{\pi^{2}}-v_{f}^{2} \bigg{)}\approx 2\pi^{4}\bigg{(}\frac{(\gamma^{2}-1)}{\pi^{2}}-v_{f}^{2} \bigg{)} \tag{25}\]  
Equation (26) shows that if \(V=0\), \(\omega\) will become a constant. \[\omega_{1}^{2}\approx-2\pi^{2}-v_{f}^{2} \tag{26}\]  
The first supercritical natural frequency for a fixed-fixed case is  
\[\ddot{v}+v_{f}^{2}\lambda^{2}v^{\prime\prime}+v_{f}^{2}v^{(4)}-\frac{1}{2}k_{ f}^{2}\psi^{\prime\prime}\int\limits_{0}^{1}\psi^{\prime}v^{\prime}dx=0 \tag{19}\]0  
[PERSON] and [PERSON] [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 Figures 2 and 3, where \(\omega\) is the axial speed and \(\gamma\) 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 Figures 2 and 3 will reduce to a series of points on the y-axis, which means that the subcritical natural frequency becomes a fixed value. In Figures 4 and 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. \[\ddot{v}+v_{f}^{2}\lambda^{2}v^{\prime\prime}+v_{f}^{2}v^{(4)}-\frac{1}{2}k_{ f}^{2}\psi^{\prime\prime}\int\limits_{0}^{1}\psi^{\prime}v^{\prime}dx=0 \tag{19}\]1 The subcritical natural frequencies for simple supports. 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  
\[\ddot{v}+v_{f}^{2}\lambda^{2}v^{\prime\prime}+v_{f}^{2}v^{(4)}-\frac{1}{2}k_{ f}^{2}\psi^{\prime\prime}\int\limits_{0}^{1}\psi^{\prime}v^{\prime}dx=0 \tag{19}\]2  
where \(\phi_{n}(x)\) is the eigenfunction of the buckled beam. For a pinned-pinned case, \(\phi_{n}(x)=\sin(n\pi x)\).