## 4 Dynamic Modeling of the Vibration Isolator
### Dynamic Model Based on a Lagrangian Equation  
Although the vibration isolator rotates, rods \(l_{2}\) and \(l_{3}\) in the parallelogram mechanism exhibit coupled translation and rotation motions. Therefore, a dynamic model can be derived using the motions of all rods, or simultaneous translation and rotation under the input disturbance. We established a dynamic model based on the Lagrangian equation, assuming that the input was harmonic excitation (\(\theta=A\sin(\omega t)\)), where \(A\) and \(\omega\) represent the amplitude and frequency, respectively. The displacement of the mass center of each rod is deduced in Section 2.2 (\(p_{i}=(x_{i},y_{i},\phi_{i})\)). Accordingly, the corresponding velocity of each can be described as:  
\[\dot{p}_{i}=\left(\frac{dx_{i}}{dt},\frac{dy_{i}}{dt},\frac{d\varphi_{i}}{dt} \right)i=1,2,...,7 \tag{8}\]  
Two independent variables \(\theta\) and \(\phi\) are chosen as generalized variables of the system, then the generalized velocity can be expressed as:  
\[\dot{q}_{i}=\left(\dot{\theta},\dot{\phi}\right) \tag{9}\]  
According to the Lagrangian equation:  
\[\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_{i}}\right)-\frac{ \partial L}{\partial q_{i}}+\frac{\partial F}{\partial\dot{q}_{i}}=Q_{i},i=1, 2,\ldots,n \tag{10}\]  
Figure 6: Comparison curves of equivalent torsional stiffness.  
where \(L=T-V-V_{k}\), \(F\) is energy consumption, \(Q\) is external excitation.  
The whole-system kinetic energy comprises both the translational and rotational kinetic energy of the rods. The total kinetic energy can be written as:  
\[T=\frac{1}{2}\sum\limits_{i=0}^{7}m_{i}v_{i}^{2}+\frac{1}{2}\sum\limits_{i=0}^ {7}l_{i}\dot{\varphi}^{2},\ i=\ 1,2,\ldots,7 \tag{11}\]  
where \({{v_{i}}^{2}}=\left(\frac{dx_{i}}{dt}\right)^{2}+\left(\frac{dy_{i}}{dt} \right)^{2}\) is the translational velocity of the center of mass of each rod.  
When \(\theta\) and \(\phi\) are equal to zero, the system is at the central position, and this static equilibrium position can be set as the overall system point with zero potential energy. \((x_{i0},y_{i0},\phi_{i0})\) represents the generalized coordinates of each rigid body. Therefore, the gravitational potential energy of the whole system can be expressed as:  
\[V=\sum\limits_{i=0}^{7}m_{i}g(y_{i}-y_{i0}) \tag{12}\]  
Note that the initial spring length is \(l_{5}\). When the spring length is changed to \(l_{5}\)', the elastic energy \(V_{k}\) of the system is:  
\[V_{k}=\frac{1}{2}k\left(l_{5}-l_{5}^{\prime}\right)^{2} \tag{13}\]  
Energy is mainly lost because of damping of the spring \(F_{1}\) and friction in the joint \(F_{2}\), which is proportional to the normal force and frictional coefficient. The symmetry of the system allows the grouping of pairs and consequent omission of the internal force. \(F_{1}\) and \(F_{2}\) can be described as  
\[\left\{\begin{array}{l}F_{1}=\frac{1}{2}cv^{2}\\ F_{2}=\sum\limits_{i=1}^{7}F_{N_{1}}\mu\beta_{i}\text{sgn}(\dot{\beta}_{i}) \end{array}\right. \tag{14}\]  
where \(F_{Ni}\) is the normal force of each joint, \(\beta_{i}\) is the rotational angular of each joint, and \(c\) is damping coefficient, and \(\mu\) is friction factor, set to \(\mu=8\times 10^{-4}\).  
Furthermore, \(v\) is the generalized velocity, and \(v^{2}=\left(\frac{dx_{M}-dx_{N}}{dt}\right)^{2}+\left(\frac{dy_{M}-dy_{N}}{dt }\right)^{2}\).  
Based on Equations (11)-(14) and the Lagrange equation, we can obtain  
\[\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_{i}}\right)-\frac{ \partial L}{\partial\dot{q}_{i}}+\frac{\partial F}{\partial\dot{q}_{i}}=M \tag{15}\]  
where, \(M\) is input disturbance torque. When the system is affected by sinusoidal disturbances, \(M=\overline{M}\sin(\theta)\).  
Normally, non-linear vibration isolation research involves analytical solutions to non-linear equations. However, we selected a numerical method to account for the large numbers of rods and relative complexity of our dynamic model.