# Buckling Induced Strongly Nonlinear Vibration of Supercritical Axially Moving Beam
## 2 Governing Equations and Solutions
(1) Symmetric configuration:  
\[\rho A\frac{\partial^{2}U}{\partial T^{2}}-P_{0}\frac{\partial^{2}U}{\partial X ^{2}}+EI\frac{\partial^{4}U}{\partial X^{4}}=\frac{EA}{L}\int_{0}^{L}(\frac{1} {2}\bigg{(}\frac{\partial U}{\partial X}\bigg{)}^{2})dX\frac{\partial^{2}U}{ \partial X^{2}} \tag{2}\]3  
(2) Anti-symmetric configuration:  
\[\rho A\frac{\partial^{2}U}{\partial T^{2}}-P_{0}\frac{\partial^{2}U}{\partial X ^{2}}+EI\frac{\partial^{4}U}{\partial X^{4}}=\frac{EA}{L}\int_{0}^{L}(\frac{1} {2}\bigg{(}\frac{\partial U}{\partial X}\bigg{)}^{2})dX\frac{\partial^{2}U}{ \partial X^{2}} \tag{2}\]4  
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  
\[\rho A\frac{\partial^{2}U}{\partial T^{2}}-P_{0}\frac{\partial^{2}U}{\partial X ^{2}}+EI\frac{\partial^{4}U}{\partial X^{4}}=\frac{EA}{L}\int_{0}^{L}(\frac{1} {2}\bigg{(}\frac{\partial U}{\partial X}\bigg{)}^{2})dX\frac{\partial^{2}U}{ \partial X^{2}} \tag{2}\]5  
\[\rho A\frac{\partial^{2}U}{\partial T^{2}}-P_{0}\frac{\partial^{2}U}{\partial X ^{2}}+EI\frac{\partial^{4}U}{\partial X^{4}}=\frac{EA}{L}\int_{0}^{L}(\frac{1} {2}\bigg{(}\frac{\partial U}{\partial X}\bigg{)}^{2})dX\frac{\partial^{2}U}{ \partial X^{2}} \tag{2}\]6 The supercritical equilibrium configuration of an axially moving beam. where \(v(x,t)\) is a small dynamic disturbance around the buckling configuration \(\psi(x)\). Substituting Equation (16) into Equation (4) leads to  
\[ \begin{split}\ddot{v}+2\gamma\dot{v}^{\prime}+(\gamma^{2}-1)v^{ \prime\prime}+v_{f}^{2}v^{(4)}=\\ k_{f}^{2}\Bigg{(}\psi^{\prime\prime}\int\limits_{0}^{1}\psi^{ \prime}v^{\prime}dx+\frac{1}{2}\psi^{\prime\prime}\int\limits_{0}^{1}v^{\prime 2 }dx+v^{\prime\prime}\int\limits_{0}^{1}\psi^{\prime}v^{\prime}dx+\frac{1}{2}v^{ \prime\prime}\int\limits_{0}^{1}v^{\prime 2}dx\Bigg{)}\end{split}  \tag{17}\]  
Note that the first term, \(k_{f}^{2}\psi^{\prime\prime}\int\limits_{0}^{1}\psi^{\prime}v^{\prime}dx\), 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  
\[\ddot{v}+2\gamma\dot{v}^{\prime}+v_{f}^{2}\lambda^{2}v^{\prime\prime}+v_{f}^{ 2}v^{(4)}-\frac{1}{2}k_{f}^{2}\psi^{\prime\prime}\int\limits_{0}^{1}\psi^{ \prime}v^{\prime}dx=0 \tag{18}\]  
Equation (19) corresponds to the case \(V=0\). \[\ddot{v}+v_{f}^{2}\lambda^{2}v^{\prime\prime}+v_{f}^{2}v^{(4)}-\frac{1}{2}k_{ f}^{2}\psi^{\prime\prime}\int\limits_{0}^{1}\psi^{\prime}v^{\prime}dx=0 \tag{19}\]  
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  
\[v(x,t)=\sum\limits_{j=1}^{N}\phi_{j}(x)e^{i\omega t} \tag{20}\]  
where \(\phi_{j}(x)\) is the \(j\)-th supercritical linear vibration mode around the buckled shape and \(\omega\) is the corresponding natural frequency. Substituting Equation (20) into Equation (18) leads to  
\[v_{f}^{2}\phi_{j}(^{4})+2\gamma i\omega\phi^{\prime}_{j}+v_{f}^{2}\lambda^{2} \phi^{\prime\prime}_{j}-\frac{1}{2}k_{f}^{2}\psi^{\prime\prime}\int\limits_{ 0}^{1}\psi^{\prime}\phi^{\prime}_{j}dx=\omega^{2}\phi_{j} \tag{21}\]  
Multiplying Equation (21) by \(\phi_{i}(x)\) and integrating the result over [0, 1] yields  
\[\int\limits_{0}^{1}L(\phi_{j})\phi_{i}dx=\omega^{2} \tag{22}\]  
where  
\[L(\phi_{j})=v_{f}^{2}\phi_{j}(^{4})+v_{f}^{2}\lambda^{2}\phi^{\prime\prime}_{ j}+2\gamma i\omega\phi^{\prime}_{j}-\frac{1}{2}k_{f}^{2}\psi^{\prime\prime} \int_{0}^{1}\psi^{\prime}\phi^{\prime}_{j}dx \tag{23}\]  
which denotes the linear operator.