# Analyzing the Stability for the Motion of an Unstretched Double Pendulum near Resonance
## 2 Dynamical Modelling  
Nonlinear vibration of the 2-DOF dynamical paradigm consisting of a double pendulum is investigated in the present study. It is supposed that the supported point \(Q\) of the modelling system travels along an elliptic track with lengths \(2b\) and \(2a\) of its major and minor axes, respectively. This point is connected with the double pendulum with constant lengths \(\updownarrow_{1}\) and \(\updownarrow_{2}\), see Figure 1. Consider \(N\) to be the compatibility point of \(Q\) on the spare circle \(b\), in which it moves with a constant angular speed \(\Omega\). The kinematics Cartesian coordinates of \(Q\) are written in the following form:  
\[x_{Q}=a\cos(\Omega t),\quad y_{Q}=b\sin(\Omega t). \tag{1}\]  
The kinetic and potential energies of the system can be expressed as follows:  
\[ \begin{array}{l}T=\frac{1}{2}(m_{1}+m_{2})\Omega^{2}\{a^{2}\sin^{2}(\Omega t )+b^{2}\cos^{2}(\Omega t)\}+(m_{1}+m_{2})\Omega\updownarrow_{1}\dot{\theta}_ {1}\\ \quad\times\{a\sin(\Omega t)\sin\theta_{1}+b\cos(\Omega t)\cos\theta_{1}\}+m_{ 2}\Omega\updownarrow_{2}\dot{\theta}_{2}\{a\sin(\Omega t)\sin\theta_{2}\\ \quad+b\cos(\Omega t)\cos\theta_{2}\}+m_{2}\updownarrow_{1}\updownarrow_{2} \dot{\theta}_{1}\dot{\theta}_{2}\cos(\theta_{1}-\theta_{2})+\frac{1}{2}m_{2} \updownarrow_{2}^{2}\dot{\theta}_{2}{}^{2}\\ \quad+\frac{1}{2}(m_{1}+m_{2})\updownarrow_{1}^{2}\dot{\theta}_{1}{}^{2},\\ V=-(m_{1}+m_{2})g\{a\cos(\Omega t)+\updownarrow_{1}\cos\theta_{1}\}-m_{2}g \updownarrow_{2}\cos\theta_{2}\,.\end{array}  \tag{2}\]  
Here, \(g\) is the gravity acceleration, \(\theta_{1}\) and \(\theta_{2}\) are the angles formed by the vertical and the line directed through \(QA\) and \(AB\), respectively, and the dots refer to the differentiation regarding the time \(t\). Consider a moment \(M_{\theta_{1}}(t)=M_{1}\cos(\Omega_{1}t)\) at the point \(Q\), besides an external moment \(M_{\theta_{2}}(t)=M_{2}\cos(\Omega_{2}t)\) that acts on the mass \(m_{2}\), in which \((\Omega_{1},\,\Omega_{2})\) and \((M_{1},\,M_{2})\) are the forcing frequencies and the amplitudes of \(M_{\theta_{1}}(t)\) and \(M_{\theta_{2}}(t)\), respectively. Using [PERSON]'s functions \(L=T-V\), one can derive the governing EOM by using the following [PERSON]'s equations of the second type:  
\[ \begin{array}{l}\frac{d}{dt}(\frac{\partial L}{\partial\dot{\theta}_{1}})-( \frac{\partial L}{\partial\dot{\theta}_{1}})=Q_{\theta_{1}}\,,\\ \frac{d}{dt}(\frac{\partial L}{\partial\dot{\theta}_{2}})-(\frac{\partial L}{ \partial\dot{\theta}_{2}})=Q_{\theta_{2}}\,.\end{array}  \tag{3}\]  
Figure 1: Description of the model. Here, \(\theta_{1}\) and \(\theta_{2}\)  are the system's generalized coordinates, and \(Q_{\theta_{1}}\)  and \(Q_{\theta_{2}}\)  are the generalized forces that have the following forms:  
\[ \begin{array}{l}Q_{\theta_{1}}=M_{1}\cos(\Omega_{1}t),\\ Q_{\theta_{2}}=M_{2}\cos(\Omega_{2}t).\end{array}  \tag{4}\]  
Let us consider the following frequencies and parameters:  
\[ \begin{array}{l}{\omega_{1}}^{2}=\frac{\delta}{\omega_{1}},\ \ {\omega_{2}}^{2}=\frac{ \delta}{\frac{1}{2}},\ \ p=\frac{\Omega}{\omega_{1}}\,,\ \ p_{1}=\frac{\Omega_{1}}{\omega_{1}},\ \ p_{2}=\frac{ \Omega_{2}}{\omega_{1}}\,,\\ {\omega^{2}}=\frac{\omega_{1}^{2}}{\omega_{1}^{2}}\,,\ \ \mu=\frac{\eta_{2}\gamma_{2}}{( \eta_{1}+m_{2})\zeta_{1}}\,,\ \ \alpha=\frac{\delta}{\zeta_{1}},\ \ \beta=\frac{b}{\zeta_{1}},\ \ \tau= \omega_{1}\,t,\\ f_{1}=\frac{M_{1}}{(m_{1}+m_{2})\omega_{1}^{2}\zeta_{1}^{2}}\,,\ \ f_{2}=\frac{M_{2}}{m \omega_{1}^{2}\zeta_{1}^{2}\zeta_{2}^{2}}.\end{array} \]  
Using (2), (3) and (4), we can obtain the EOM in the following dimensionless forms:  
\[ \begin{array}{l}\ddot{\theta}_{1}+p^{2}[\alpha\cos(pt)\sin\theta_{1}-\beta \sin(pt)\cos\theta_{1}]+\sin\theta_{1}+\mu\ddot{\theta}_{2}\cos\theta_{1}\cos \theta_{2}\\ +\dot{\mu}\ddot{\theta}_{2}\sin\theta_{1}\sin\theta_{2}+\mu\dot{\theta}_{2}^{2 }\sin\theta_{1}\cos\theta_{2}-\mu\dot{\theta}_{2}^{2}\sin\theta_{2}\cos\theta _{1}=f_{1}\cos(p_{1}t),\end{array}  \tag{5}\]  
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