# Buckling Induced Strongly Nonlinear Vibration of Supercritical Axially Moving Beam
## 2 Governing Equations and Solutions  
The equation of motion of an axially moving beam takes the form [3,18]  
\[\rho A(\frac{\partial^{2}U}{\partial T^{2}}+2V\frac{\partial^{2}U}{\partial X \partial T}+V^{2}\frac{\partial^{2}U}{\partial X^{2}})-P_{0}\frac{\partial^{2 }U}{\partial X^{2}}+EI\frac{\partial^{4}U}{\partial X^{4}}=\frac{EA}{L}\int_{0 }^{L}(\frac{1}{2}\bigg{(}\frac{\partial U}{\partial X}\bigg{)}^{2})dX\frac{ \partial^{2}U}{\partial X^{2}} \tag{1}\]  
where \(\rho\) is linear density, \(U\) is transverse displacement, \(A\) is cross-sectional area, \(P_{0}\) 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  
\[\rho A\frac{\partial^{2}U}{\partial T^{2}}-P_{0}\frac{\partial^{2}U}{\partial X ^{2}}+EI\frac{\partial^{4}U}{\partial X^{4}}=\frac{EA}{L}\int_{0}^{L}(\frac{1} {2}\bigg{(}\frac{\partial U}{\partial X}\bigg{)}^{2})dX\frac{\partial^{2}U}{ \partial X^{2}} \tag{2}\]  
Introduce the following non-dimensional quantities  
\[x=\frac{X}{L},u=\frac{U}{L},t=\frac{T}{L}\sqrt{\frac{P_{0}}{\rho A}},\gamma=V \sqrt{\frac{\rho A}{P_{0}}},\upsilon_{f}^{2}=\frac{EI}{P_{0}L^{2}},k_{f}^{2}= \frac{EA}{P_{0}} \tag{3}\]  
and Equation (4) is obtained by substituting Equation (3) into Equation (1). \[\ddot{u}+2\gamma\dot{u}^{\prime}+(\gamma^{2}-1)u^{\prime\prime}+\upsilon_{f}^ {2}u^{(4)}=\frac{1}{2}k_{f}^{2}u^{\prime\prime}\int\limits_{0}^{1}u^{\prime 2}dx \tag{4}\]  
The pinned-pinned supported and the fixed-fixed boundary conditions are  
\[u=0\text{ and }u^{\prime\prime}=0\text{ at }x=0,\ 1 \tag{5}\]  
\[u=0\text{ and }u^{\prime}=0\text{ at }x=0,\ 1 \tag{6}\]  
Equation (7) is the supercritical equilibrium equation induced from Equation (4) and \(\psi\) is the supercritical configuration. \[(\gamma^{2}-1)\psi^{\prime\prime}+\upsilon_{f}^{2}\psi^{(4)}=\frac{1}{2}k_{f} ^{2}\psi^{\prime\prime}\int\limits_{0}^{1}{\psi^{\prime}}^{2}dx \tag{7}\]  
The integral of Equation (7) is a constant for a given \(\psi\). Let  
\[\Pi=\frac{1}{2}k_{f}^{2}\int\limits_{0}^{1}{\psi^{\prime}}^{2}dx \tag{8}\]Next, Equation (7) can be rewritten as  
\[\psi^{(4)}+\lambda^{2}\psi^{\prime\prime}=0 \tag{9}\]  
where  
\[\lambda^{2}=\frac{(\gamma^{2}-1)-\Pi}{v_{f}^{2}} \tag{10}\]  
The characteristic equation is shown in Equation (11). \[\rho A\frac{\partial^{2}U}{\partial T^{2}}-P_{0}\frac{\partial^{2}U}{\partial X ^{2}}+EI\frac{\partial^{4}U}{\partial X^{4}}=\frac{EA}{L}\int_{0}^{L}(\frac{1} {2}\bigg{(}\frac{\partial U}{\partial X}\bigg{)}^{2})dX\frac{\partial^{2}U}{ \partial X^{2}} \tag{2}\]0  
The equilibrium configurations for the pinned-pinned supported case is  
\[\rho A\frac{\partial^{2}U}{\partial T^{2}}-P_{0}\frac{\partial^{2}U}{\partial X ^{2}}+EI\frac{\partial^{4}U}{\partial X^{4}}=\frac{EA}{L}\int_{0}^{L}(\frac{1} {2}\bigg{(}\frac{\partial U}{\partial X}\bigg{)}^{2})dX\frac{\partial^{2}U}{ \partial X^{2}} \tag{2}\]1  
where  
\[\rho A\frac{\partial^{2}U}{\partial T^{2}}-P_{0}\frac{\partial^{2}U}{\partial X ^{2}}+EI\frac{\partial^{4}U}{\partial X^{4}}=\frac{EA}{L}\int_{0}^{L}(\frac{1} {2}\bigg{(}\frac{\partial U}{\partial X}\bigg{)}^{2})dX\frac{\partial^{2}U}{ \partial X^{2}} \tag{2}\]2  
Note that the equilibrium configurations (12) were also obtained by [PERSON] [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. [PERSON] et al. [31] found two special configurations for the fixed-fixed boundary condition, which are shown below.