# Nonlinear Dynamics of an Elastic Stop System and Its Application in a Rotor System
## 2 Model and Solving Method  
### Model  
In this paper, a two-Dof model with an elastic stop is established, as shown in Figure 1, and the influence of the stop on the dynamic characteristics of the system is investigated. Figure 1: Pedestal with an elastic stop and its dynamic model: (**a**) pedestal with an elastic stop; (**b**) dynamic model. A typical system with gaps is a pedestal with a stop in a rotating mechanical system, shown in Figure 1a. In general, its stiffness and damping characteristics caused by intermittent contact can be characterized by a piecewise linear function, and this paper establishes a dynamic model as shown in Figure 1b. The governing equations can be derived by Appendix A and are shown as follows:  
\[\left[ \begin{array}{cc}m_{1}&\\ m_{2}&\end{array} \right]\left\{ \begin{array}{cc}\ddot{x}_{1}\\ \ddot{x}_{2}\end{array} \right\}+\left[ \begin{array}{cc}c_{1}&\\ &c_{2}\end{array} \right]\left\{ \begin{array}{cc}\dot{x}_{1}\\ \dot{x}_{2}\end{array} \right\}+\left[ \begin{array}{cc}k_{1}&\\ &k_{2}\end{array} \right]\left\{ \begin{array}{cc}x_{1}\\ x_{2}\end{array} \right\}+\mathbf{f}_{nl}=\left\{ \begin{array}{cc}F_{0}\cos \omega t\\ \end{array} \right\} \tag{1}\]  
in which \(m_{1}\) and \(m_{2}\) are the mass of the two Dofs; \(c_{1}\) and \(c_{2}\) are the damping of the two Dofs; \(k_{1}\) and \(k_{2}\) are the stiffness of the two Dofs; \(F_{0}\) is the amplitude of the external excitational force; and the stop is present by a piecewise stiffness so that the nonlinear force \(\mathbf{f}_{nl}\) is as follows:  
\[\mathbf{f}_{nl} =k_{p}\left\{ \begin{array}{cc}x_{1}-x_{2}-\delta_{0}\\ -(x_{1}-x_{2}-\delta_{0})\\ \end{array} \right\}H(x_{1}-x_{2}-\delta_{0}) \tag{2}\] \[=k_{p}\mathbf{w}\Big{(}\mathbf{w}^{T}\left\{ \begin{array}{cc}x _{1}\\ x_{2}\end{array} \right\}-\delta_{0}\Big{)}H\Big{(}\mathbf{w}^{T}\left\{ \begin{array}{cc}x_{1}\\ x_{2}\end{array} \right\}-\delta_{0}\Big{)}\] \[=k_{p}\mathbf{w}f_{c}\Big{(}\mathbf{w}^{T}\left\{ \begin{array}{ cc}x_{1}\\ x_{2}\end{array} \right\}-\delta_{0}\Big{)}\]  
in which, the nonlinear force is denoted by the function \(f_{c}(\bullet)\) defined as follows,  
\[f_{c}(\bullet)=\bullet H(\bullet) \tag{3}\]  
where \(H(\bullet)\) is the Heaviside function, namely, \(H(\bullet)=\left\{ \begin{array}{cc}1,\bullet\geq 0\\ 0,\bullet<0\end{array} \right.\); \(k_{p}\) is the penalty stiffness of the elastic stop, \(\mathbf{w}=\left[ \begin{array}{cc}1&-1\end{array} \right]^{T}\). A nominal restoring force \(F^{*}\) is assumed to obtain dimensionless equations, and let  
\[\Omega^{2}=\frac{k_{1}}{m_{1}},\ \tau=\Omega t,\ \alpha=\frac{m_{2}}{m_{1}}, \beta=\frac{c_{2}}{c_{1}},\gamma=\frac{k_{2}}{k_{1}},\ \dot{\omega}=\frac{\omega}{\Omega},\zeta=\frac{c_{1}}{m_{1}\Omega},\ \hat{F}=\frac{F_{0}}{F^{*}},\ [y]=\frac{k_{1}}{F^{*}}[x],\ \delta= \frac{k_{1}\delta_matrices of the linear part, \(\mathbf{g}\big{(}\mathbf{x}(t),\dot{\mathbf{x}}(t)\big{)}\) denotes the nonlinear forces related to velocity and displacement, and \(\mathbf{F}_{ex}(t)\) is the external excitations.  
When the external excitation of the system is a single sinusoidal excitation, the external excitation can be specified as  
\[\mathbf{F}_{ex}(t)=\mathbf{F}_{0}\sin\
u\omega t \tag{7}\]  
where \(\mathbf{F}_{0}\) is the vector of excitation magnitude, \(\omega\) is the excitation frequency, and \(\
u\) is the coefficient of the excitation frequency. Based on the Fourier transform, the periodic solution of the response and the nonlinear forces are represented in the frequency domain by harmonic terms:  
\[\mathbf{x}(\theta)=\mathbf{A}_{0}+\sum_{k=1}^{l}(\mathbf{A}_{k}\cos k\theta+ \mathbf{B}_{k}\sin k\theta) \tag{8}\]  
\[\mathbf{g}(\theta)=\mathbf{P}_{0}+\sum_{k=1}^{l}(\mathbf{P}_{k}\cos k\theta+ \mathbf{Q}_{k}\sin k\theta) \tag{9}\]  
where \(l\) is the order truncated of harmonic terms, \(\theta=\omega t\), \(\mathbf{P}_{k}\), \(\mathbf{Q}_{k}\), \(\mathbf{A}_{k}\) and \(\mathbf{B}_{k}\) are the constant vectors to be determined.