# Enhancing Vibration Isolation Performance by Exploiting Novel Spring-Bar Mechanism
## 2 Mathematical Modelling
### Governing Equation  
With reference to Equation (4), the governing equation of motion of the vibration isolation system can be written as  
\[m\ddot{x}+c\dot{x}+k_{1}x+2k_{2}\Big{(}\delta_{0}+l-\sqrt{l^{2}-x^{2}}\Big{)} \frac{x}{\sqrt{l^{2}-x^{2}}}=f_{\rm e}\cos\omega t,\ |x|\ <l. \tag{6}\]  
To assist analysis, non-dimensional parameters and variables are introduced as  
\[\omega_{0}=\sqrt{\frac{k_{1}}{m}},\ \xi=\frac{c}{2m\omega_{0}},\ X=\frac{x}{l}, \ F_{0}=\frac{f_{e}}{k_{1}l},\ \Omega=\frac{\omega}{\omega_{0}},\ \tau=\omega_{0}f,\ \alpha=\frac{\delta_{0}}{l},\ \beta= \frac{2k_{2}}{k_{1}}, \tag{7}\]  
where \(\omega_{0}\) is the natural frequency of the linear system without adding the SBM; \(\xi\) is non-dimensional damping ratio; \(X,F,\Omega\) and \(\tau\) represent the non-dimensional displacement, excitation amplitude, excitation frequency, and time, respectively; \(\alpha\) denotes non  
Figure 2: A schematic representation of the SBM system with (**a**) the geometrical deflection and (**b**) a free-body diagram of the bars. -dimensional deflection of the horizontal springs when the bars are in the horizontal direction. Non-dimensional parameter \(\beta\) characterizes the relative stiffness of the horizontal springs against the vertical one and thus can only be positive. Using these parameters and variables, Equation (6) is written into a dimensionless form  
\[X^{\prime\prime}+2\xi X^{\prime}+(1-\beta)X+\beta(1+\alpha)\frac{X}{\sqrt{1-X^{ 2}}}=F_{0}\cos\Omega\tau,\ |X|\ <1 \tag{8}\]  
where the primes denote differentiations with respect to the non-dimensional time \(\tau\). Equation (8) shows that the SBM introduces a linear term \(-\beta X\) in the system and also a possible nonlinear one \(\beta(1+\alpha)X/\sqrt{1-X^{2}}\) when \(\alpha\
eq-1\). This nonlinearity is affected by different combinations of design parameters \(\alpha\) and \(\beta\). Consequently, for a better application of the SBM, the dynamics of the integrated system should be investigated thoroughly. In comparison, there is only one design variable to control the stiffness in the previous study, and nonlinearity and some undesirable nonlinear phenomena cannot be avoided [14]. Therefore, the current SBM design has a larger degree of control over the dynamic behaviour of the system. The non-dimensional restoring force of the system is  
\[G(X)\ =\ (1-\beta)X+\beta(1+\alpha)\frac{X}{\sqrt{1-X^{2}}}. \tag{9}\]  
The overall stiffness of the system is obtained by differentiating \(G(X)\) with respect to \(X\):  
\[H(X)\ =\ (1-\beta)+\frac{\beta(1+\alpha)}{(1-X^{2})\sqrt{1-X^{2}}}. \tag{10}\]  
As shown in Figure 3, when \(\alpha=-1\), i.e., \(\delta_{0}=-l\), the restoring force will be proportional to the displacement \(X\), and the system will be linear, with a constant stiffness of \(1-\beta\). When \(\alpha\
eq-1\), the restoring force is a nonlinear function of the displacement, and the system becomes nonlinear. When \((\alpha,\beta)\ =\ (-0.5,\ 1.5)\) or \((-0.4,\ 5)\), the stiffness \(H(X)\) increases with the deflection \(|X|\). For the former case, the stiffness remains positive so that the restoring force increases with \(X\). In comparison, the stiffness of the latter system with \((\alpha,\beta)\ =\ (-0.4,\ 5)\) may be negative in the neighbourhood of the static equilibrium position \(X=0\). For a system with \((\alpha,\beta)\ =\ (-1.5,\ 0.5)\), it can be seen that the stiffness \(H(X)\) decreases with an increasing deflection and that the system exhibits a softening stiffness characteristic. Clearly, the parameters \(\alpha\) and \(\beta\) of the SBM should be chosen properly, as they determine the stiffness characteristics of the system. Figure 3: **(a)** Restoring force and **(b)** stiffness characteristics of the system. Solid, dashed, dash–dot, and dotted lines represent systems with \((\alpha,\ \beta)\ =\ (-1,0.75)\), \((-1.5,0.5)\), \((-0.5,1.5)\), and \((-0.4,5)\), respectively.