# The Stability Analysis of a Vibrating Auto-Parametric Dynamical System Near Resonance
## 2 Formulation of the Dynamical System  
Let us consider the dynamical vibrations of an auto-parametric system consisting of an excited Duffing oscillator with normal length \(\ell_{0}\), static elongation \(Y_{s}\), stiffness \(k_{1}\), and mass \(M\) that is periodically forced to move vertically as a primary system. The secondary system is a damped spring pendulum with normal length \(\ell_{1}\), static elongation \(X_{s}\), stiffness \(k_{2}\), and mass \(m\). This spring is connected to the Duffing oscillator at a point \(O\). Let \(Y(t)\) denote the elongation of the Duffing oscillator and \(\phi\) represent the rotation angle of the spring that lies between the downward vertical \(Y_{1}\)-axis and the direction of the spring's elongation \(X(t)\) (see Figure 1). Moreover, we consider that the motion is influenced by external linear excitation forces, \(F_{1}(t)=F_{1}\cos(\Omega_{1}t)\) and \(F_{3}(t)=F_{3}\cos(\Omega_{3}t)\), and an excitation moment \(M_{\phi}(t)=F_{2}\cos(\Omega_{2}t)\) that are acting on the masses \(M\), \(m\) and at the point \(O\), respectively. Here, \(\Omega_{1},\ \Omega_{2}\), and \(\Omega_{3}\) are the frequencies of \(F_{1}(t)\), \(M_{\phi}(t)\), and \(F_{3}(t)\), respectively. The investigated auto-parametric system is damped by viscous forces and moments \(C_{1}\dot{Y},\ C_{2}\dot{\phi}\), and \(C_{3}\dot{X}\), in which \(C_{1}\), \(C_{2}\), and \(C_{3}\) are the respective coefficients of the viscous damping. Based on the above description, we can write the potential energy \(V\) and kinetic energy \(T\) as follows:  
\[ \begin{array}{l}V=\frac{1}{2}k_{1}Y^{2}+\frac{1}{2}k_{2}X^{2}-Mg(l_{0}+Y)-mg(l _{0}+Y)-mg(l_{1}+X)\cos\phi,\\ T=\frac{1}{2}(M+m)\dot{Y}^{2}+\frac{1}{2}m(l_{1}+X)2\dot{\phi}^{2}+\frac{1}{2 }m\dot{X}^{2}\\ -m(l_{1}+X)\dot{Y}\phi\sin\phi+m\dot{X}\dot{Y}\cos\phi,\end{array}  \tag{1}\]  
where \(g\) is the acceleration of Earth's gravitational force and the dots are time derivatives. The governing EOM can be derived using the following Lagrange's Equations:  
\[ \begin{array}{l}\frac{d}{dt}(\frac{\partial L}{\partial\dot{Y}})-(\frac{ \partial L}{\partial\dot{T}})=Q_{Y},\\ \frac{d}{dt}(\frac{\partial L}{\partial\dot{\phi}})-(\frac{\partial L}{\partial \dot{\phi}})=Q_{\phi},\\ \frac{d}{dt}(\frac{\partial L}{\partial\dot{X}})-(\frac{\partial L}{\partial \dot{X}})=Q_{X}.\end{array}  \tag{2}\]Here, \(L=T-V\) is the Lagrange's function, \((Y,\,\phi,X)\) and \((\dot{Y},\,\dot{\phi},\dot{X})\) are the system's generalized coordinates and their corresponding velocities, respectively, and \((Q_{Y},\,Q_{\phi},Q_{X})\) are the generalized forces which are formed as follows:  
\[ \begin{array}{l}Q_{Y}=F_{1}\cos(\Omega_{1}t)-C_{1}\dot{Y},\\ Q_{\phi}=F_{2}\cos(\Omega_{2}t)-C_{2}\dot{\phi},\\ Q_{X}=F_{3}\cos(\Omega_{3}t)-C_{3}\dot{X}.\end{array}  \tag{3}\]  
Consider the following definitions of the dimensionless parameters:  
\[ \begin{array}{l}\overline{\omega}_{1}{}^{2}=\frac{k_{1}}{m+M}\,,\quad \overline{\omega}_{2}{}^{2}=\frac{\xi}{l_{1}},\quad\overline{\omega}_{3}{}^{2 }=\frac{k_{2}}{m}\,,\quad\omega_{1}{}^{2}=\frac{\overline{\omega}_{1}{}^{2} }{\overline{\omega}_{2}{}^{2}},\\ \omega_{2}{}^{2}=\frac{\overline{\omega}_{2}{}^{2}}{\overline{\omega}_{2}{}^ {2}},\quad c_{1}=\frac{C_{1}}{(m+M)\overline{\omega}_{2}{}^{2}}\,,\quad c_{2 }=\frac{C_{2}}{ml_{1}\overline{\omega}_{2}{}^{2}},\quad c_{3}=\frac{C_{3}}{m \omega_{2}},\\ f_{1}=\frac{F_{1}}{(m+M)l_{1}\overline{\omega}_{2}{}^{2}},\quad f_{2}=\frac{F_ {2}}{ml_{1}\overline{\omega}_{2}{}^{2}}\,,\quad f_{3}=\frac{F_{3}}{ml_{1} \overline{\omega}_{2}{}^{2}},\\ p_{1}=\frac{\Omega_{1}}{\Omega_{2}},\quad p_{2}=\frac{\Omega_{2}}{\Omega_{2}}, \quad p_{3}=\frac{\Omega_{3}}{\Omega_{2}},\quad\beta=\frac{m}{m+M},\\ \tau=\omega_{2}f,\quad x=\frac{X-X_{s}}{l_{1}},\quad y=\frac{Y-X_{s}}{l_{1}}, \quad Y_{s}=\frac{(m+M)\xi}{k_{1}}\,,\;X_{s}=\frac{m\xi}{k_{2}},\,l_{0}=\ell_{0 }+Y_{s}\,,\\ l_{1}=\ell_{1}+X_{s}.\end{array}