# Enhancing Vibration Isolation Performance by Exploiting Novel Spring-Bar Mechanism
## 2 Mathematical Modelling
### Stability Analysis  
To assist in further analysis, Equation (8) is written in the form of state-space representation as  
\[\left(\begin{array}{c}X^{\prime}\\ Y^{\prime}\end{array}\right)=\left(\begin{array}{c}Y\\ F_{0}cos\ \Omega\tau-2\xi Y-(1-\beta)X-\beta(1+\alpha)\frac{X}{\sqrt{1-X^{2}}} \end{array}\right). \tag{12}\]  
The equilibrium points of the unforced system \((F_{0}=0)\) can be obtained by letting \(X^{\prime}=Y^{\prime}=0\):  
\[(X_{0},Y_{0})\ =\ (0,\ 0)\ \text{and}\ (X_{1,2},Y_{1,2})\ =\ \left(\pm\sqrt{1- \beta^{2}\Big{(}\frac{1+\alpha}{1-\beta}\Big{)}^{2}},0\right),\ (\beta\
eq 1). \tag{13}\]  
While \((X_{0},Y_{0})\) always exists for varying \(\alpha\) and \(\beta\), the latter two points \((X_{1,2},Y_{1,2})\) exist only when \(0<\frac{\beta(1+\alpha)}{1-\beta}<1\). The stability of these equilibria can be assessed by studying the corresponding Jacobian matrix of Equation (12):  
\[A=\left[\begin{array}{cc}0&1\\ -\beta(1+\alpha)\big{(}1-X^{2}\big{)}^{-\frac{3}{2}}-(1-\beta)&-2\xi\end{array} \right]. \tag{14}\]  
Figure 4: Natural frequency of the linearized system; (**a**) 3–dimensional and (**b**) 2–dimensional plots.  
The characteristic equation of matrix \(A\) is  
\[\lambda^{2}+2\xi\lambda+\beta(1+\alpha)\Big{(}1-X^{2}\Big{)}^{-\frac{3}{2}}+(1- \beta)\ =0. \tag{15}\]  
By investigating the signs of the real part of the solutions to this equation, the stability of the equilibrium points can be determined. Using this and the total number of equilibria, the bifurcation sets of the unforced system are obtained and shown in Figure 5, where the entire parameter region is divided into four sub-regions by lines ACE and BCD. The former line is defined by \(\alpha\beta=-1\) with \(\Omega_{n}=0\) and the latter by \(\alpha=-1\), representing linear systems. Bifurcation occurs when crossing either of these two lines. Table 1 shows different types of systems categorized by parameters \(\alpha\) and \(\beta\). As shown previously in Equation (11), as \(\beta\) remains positive, a linearized natural frequency lower than 1 requires \(\alpha<0\). However, systems satisfying this condition may either be due to softening stiffness in Region I to hardening stiffness in Region II, to a double-well potential in Region III, or to a single hump in the potential energy in Region IV. In contrast, the previous NSM study only exhibited the softening stiffness characteristic. Therefore, with a prescribed design natural frequency or a fixed product \(\alpha\beta\), the relative values of \(\alpha\) and \(\beta\) should still be selected carefully with careful consideration of the stiffness characteristics. For example, for the softening stiffness systems, the mass should not deflect beyond the unstable equilibrium points \((X_{1,2},Y_{1,2})\) so as to avoid reaching \(X=\pm 1\)[14]. Similarly, the systems in Region IV are unstable and thus not useful for vibration isolation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Locations of \((\alpha,\beta)\)** & **Parameter Values** & **System Characteristics** \\ \hline Line ACE & \(\alpha\beta=-1\) & \(\Omega_{n}=0\) \\ Line BCD & \(\alpha=-1\) & Linear system \\ Region I & \(\alpha\beta>-1\), \(\alpha<-1\) & Softening stiffness system \\ Region II & \(\alpha\beta>-1\), \(\alpha>-1\) & Hardening stiffness system \\ Region III & \(\alpha\beta<-1\), \(\alpha>-1\) & Double-well potential system \\ Region IV & \(\alpha\beta<-1\), \(\alpha<-1\) & Single-hump potential system \\ \hline \hline \end{tabular}
\end{table}
Table 1: Different categories of the system.  
Figure 5: The bifurcation set of the unforced system.