## 2 System Model
### Dynamic Modeling of the Coupled System
The main reason is that the damping ratio of the flexible plate is far less than that of the joint mechanism. One can enlarge the physical displacement coordinates of the flexible beam using the assumed mode shapes,  
\[y(x,t)=\boldsymbol{\varphi}^{\mathrm{T}}(x)\boldsymbol{\mathrm{p}}(t); \boldsymbol{\varphi}^{\mathrm{T}}=[\varphi_{1},\varphi_{2},\ldots,\varphi_{n}] ;\boldsymbol{\mathrm{p}}^{\mathrm{T}}=[p_{1},p_{2},\ldots,p_{n}] \tag{12}\]  
where \(\boldsymbol{\varphi}(x)\) represents the eigenfunction of the clamped-free cantilever beam, \(\boldsymbol{\mathrm{p}}(t)\) is the generalized coordinate vector of the beam, and \(n\) refers to the number of assumed modes. In addition, we merely consider the first-order mode in the simulated calculation for the beam. For the sake of convenience, it is supposed that \(\varphi_{1}\rightarrow\varphi;\ p_{1}\to p\). For the state variables \(\{\theta,p\}\), the generalized force is expressed as \(\big{\{}Q_{\theta},Q_{p}\big{\}}\). Equation (12) can be rewritten as follows:  
\[y(x,t)=\sum_{i=1}^{N}\varphi_{i}(x)p_{i}(t)\approx\varphi_{1}(x)p_{1}(t)= \varphi(x)p(t) \tag{13}\]  
\[\varphi(x)=\varphi_{1}(x)=C_{1}\sin(\beta_{1}x)+C_{2}\cos(\beta_{1}x)+C_{3} \sinh(\beta_{1}x)+C_{4}\cosh(\beta_{1}x) \tag{14}\]  
\[p(x)=p_{1}(x)=A\cos(\omega_{1}t)+B\sin(\omega_{1}t) \tag{15}\]  
where the first resonant angular frequency of cantilever beam is \(\omega_{1}=\beta_{1}^{2}\sqrt{\frac{EI}{\rho A}}\), which can be obtained by the differential equation of free vibration of cantilever beam. The parameters \(A\) and \(B\) can be obtained from the initial conditions of vibration. The four parameters \(C_{1}\), \(C_{2}\), \(C_{3}\) and \(C_{4}\) can be obtained by the cantilever beam with one end constrained and one end free. For \(x=0\)  
\[y(0,\,t)= 0,\ \ \frac{\partial y(0,\,t)}{\partial x}=0 \tag{16}\]  
For \(x=l\)  
\[EI\frac{\partial^{2}y(l,t)}{\partial x^{2}}=0,\ \ EI\frac{\partial^{3}y(l,t)}{ \partial x^{3}}=0 \tag{17}\]  
According to reference [37], the first-order mode eigenfunction of cantilever beam can be expressed as follows:\[\varphi(x)=\varphi_{1}(x)=\sin(\beta_{1}x)-\sinh(\beta_{1}x)-\frac{\sinh(\beta_{1} l)+\sin(\beta_{1}l)}{\cosh(\beta_{1}l)+\cos(\beta_{1}l)}[\cos(\beta_{1}x)-\cosh( \beta_{1}x)] \tag{18}\]  
where \(\beta_{1}=\frac{1.875}{l}\). Therefore, based on the mode shapes together with the generalized coordinates in Equation (13), the kinetic energy \(T\) in Equation (9) and the potential energy \(V\) in Equation (10) can be rewritten as:  
\[ \begin{array}{l}T=J\dot{\theta}^{2}+\Big{\{}\Big{[}\frac{1}{2}\rho A\int_{0} ^{l}x^{2}dx\Big{]}+\Big{[}\frac{1}{2}\rho A\int_{0}^{l}\varphi^{2}(x)dx\Big{]} p^{2}\Big{\}}\dot{\theta}^{2}\\ \qquad\qquad+\Big{[}\rho A\int_{0}^{l}x\varphi(x)dx\Big{]}\dot{p}\dot{\theta} +\Big{[}\frac{1}{2}\rho A\int_{0}^{l}\varphi^{2}(x)dx\Big{]}\dot{p}^{2}\end{array}  \tag{19}\]  
\[V=\Bigg{\{}\frac{1}{2}EI\int_{0}^{l}\left[\frac{\partial^{2}\varphi(x)}{ \partial x^{2}}\right]^{2}dx\Bigg{\}}p^{2}+2\cdot\frac{1}{2}k\theta^{2}= \bigg{[}\frac{1}{2}\rho A\int_{0}^{l}\varphi^{2}(x)dx\bigg{]}\omega_{1}^{2}p^{ 2}+k\theta^{2} \tag{20}\]  
For generalized coordinate \(\theta\), based on the Lagrange equation Equations (3) and (4), we have:  
\[\frac{d}{dt}\left(\frac{\partial T}{\partial\dot{\theta}}\right)-\frac{ \partial T}{\partial\theta}+\frac{\partial V}{\partial\theta}+\frac{\partial D }{\partial\dot{\theta}}=Q_{\theta} \tag{21}\]  
where  
\[