## 3 Mathematical Model
### Non-Dimensional Equations of the System  
In order to transform the Equations (1) and (2) from the dimensional to the non-dimensional, we have introduced non-dimensional time as \(\tau=2\pi f_{\text{n}}t\). So, in the case of non-dimensional equations, the derivative has been done with respect to \(\tau\), where \(f_{\text{n}}\) is the natural linear frequency of oscillation of the considered system, \(f_{\text{n}}=\frac{\omega_{\text{n}}}{2\pi}\), and \(\omega_{\text{n}}=\sqrt{\frac{K_{1}}{m_{1}}}\).  
Now, we introduce the expression of the non-dimensional frequency as, \(\omega_{\text{nd}}=\frac{\omega_{\text{e}}}{\omega_{\text{n}}}\). The non-dimensional state-variables as \(y_{\text{i}}=\frac{x_{\text{i}}}{\delta}\); \(i=1,2\).  
\(\dot{x}_{\text{i}}=\frac{d\tau_{\text{i}}}{dt}=\frac{d\tau_{\text{i}}}{dt} \cdot\frac{d\tau}{dt}=\omega_{\text{n}}\delta y_{\text{i}}\); similarly, \(\dot{x}_{\text{i}}=\omega_{\text{n}}^{2}\delta\dot{y}_{\text{i}}\).  
So, from Equation (1),\[\begin{split}& m_{1}\omega_{\rm n}^{2}\delta\dot{y}_{1}+C_{1}\omega_{ \rm n}\delta\dot{y}_{1}+\frac{T_{1}\omega_{\rm n}\delta\dot{y}_{1}}{\omega_{\rm n }\delta\left\{\dot{y}_{1}^{2}+\left(\frac{\varepsilon}{\omega_{\rm n}\delta} \right)^{2}\right\}^{\frac{1}{2}}}+F_{\rm MO}\Bigg{[}\frac{1}{\{1+d_{1}( \delta-\delta y_{1})\}^{4}}-\frac{1}{\{1+d_{1}(\delta+\delta y_{1})\}^{4}} \Bigg{]}\\ &+K_{1}\delta\Bigg{\{}\left(\frac{1+\frac{K_{2}}{K_{1}}}{2} \right)+\left(\frac{1-\frac{K_{2}}{K_{1}}}{2}\right)\Bigg{\}}\cos(2\omega_{\rm nd }\tau)\cdot(y_{1}-y_{2})=0\end{split} \tag{3}\]  
Dividing both sides by \(m_{1}\omega_{\rm n}^{2}\delta\), we obtain  
\[\begin{split}&\ddot{y}_{1}+\frac{C_{1}}{m_{1}\omega_{\rm n}} \dot{y}_{1}+\frac{\frac{T_{1}}{m_{1}\omega_{\rm n}^{2}\delta}\dot{y}_{1}}{ \omega_{\rm n}\delta\left(\dot{y}_{1}^{2}+a_{1}^{2}\right)^{\frac{1}{2}}}+ \frac{F_{\rm MO}}{m_{1}\omega_{\rm n}^{2}\delta}\Bigg{[}\frac{1}{\{1+d_{1} \delta(1-y_{1})\}^{4}}-\frac{1}{\{1+d_{1}\delta(1+y_{1})\}^{4}}\Bigg{]}\\ &+\left(\frac{K_{1}\delta}{m_{1}\omega_{\rm n}^{2}\delta}\right) \cdot\left(\frac{1+\beta}{2}\right)\Bigg{\{}1+\left(\frac{1-\beta}{1+\beta} \right)\Bigg{\}}\cos(2\omega_{\rm nd}\tau)\cdot(y_{1}-y_{2})=0\end{split}\]  
From the above equation, we can write,  
\[\begin{split}&\dot{y}_{1}+2\zeta_{1}\dot{y}_{1}+\frac{t_{1}\dot{y}_ {1}}{\omega_{\rm n}\delta(\dot{y}_{1}^{2}+a_{1}^{2})^{\frac{1}{2}}}+b_{1} \Bigg{[}\frac{1}{\{1+D_{1}(\delta_{1}-y_{1})\}^{4}}-\frac{1}{\{1+D_{1}(\delta _{1}+y_{1})\}^{4}}\Bigg{]}\\ &+\delta\bigg{(}\frac{1+\beta}{2}\bigg{)}\Bigg{\{}1+\left(\frac{ 1-\beta}{1+\beta}\right)\Bigg{\}}\cos(2\omega_{\rm nd}\tau)\cdot(y_{1}-y_{2})= 0\end{split} \tag{4}\]  
where, \(\zeta_{1}=\frac{C_{1}}{2m_{1}\omega_{\rm n}}\), \(t_{1}=\frac{T_{1}}{m_{1}\omega_{\rm n}^{2}\delta}\), \(a_{1}=\frac{\varepsilon}{\omega_{\rm n}\delta}\), \(b_{1}=\frac{F_{\rm MO}}{K_{1}\delta}\), \(D_{1}=d_{1}\delta\), \(\beta=\frac{K_{2}}{K_{1}}\), and \(\delta_{1}=1\).  
The Equation (4) is the non-dimensional form of the Equation (1).  
Similarly, from the Equation (2), we get the non-dimensional form as  
\[\begin{split}&\ddot{y}_{2}+2\zeta_{2}\dot{y}_{2}+\frac{t_{2}\dot{y}_ {2}}{\omega_{\rm n}\delta(\dot{y}_{2}^{2}+a_{1}^{2})^{\frac{1}{2}}}+b_{2} \Big{[}\frac{1}{\{1+D_{1}(\delta_{1}-y_{2})\}^{4}}-\frac{1}{\{1+D_{1}(\delta _{1}+y_{2})\}^{4}}\Big{]}\\ &+\mu\bigg{(}\frac{1+\beta}{2}\bigg{)}\Bigg{\{}1+\left(\frac{1- \beta}{1+\beta}\right)\Bigg{\}}\cos(2\omega_{\rm nd}\tau)\cdot(y_{2}-y_{1})=0 \end{split} \tag{5}\]  
where, \(\zeta_{2}=\frac{C_{2}}{2m_{2}\omega_{\rm n}}\), \(t_{2}=\frac{T_{2}}{m_{2}\omega_{\rm n}^{2}\delta}\), \(\mu=\frac{m_{1}}{m_{2}}\), and \(b_{2}=b_{1}\mu\).