# Enhancing Vibration Isolation Performance by Exploiting Novel Spring-Bar Mechanism
## 2 Mathematical Modelling
### Frequency Response Function  
For periodic oscillations around the equilibrium position \(X=0\), the method of averaging [38] can be used for a first-order approximation of the response in the steady state. For its implementation, it is first assumed that the steady-state displacement is \(X=r\cos\,(\Omega\tau+\phi)\) and that the corresponding velocity is \(Y=-r\Omega\,\sin(\Omega\tau+\phi)\), where \(r\) and \(\phi\) are the response amplitude and the phase angle, respectively. Equation (12) is then transformed into  
\[\left( \begin{array}{cc}\cos(\Omega\tau+\phi)&-r\sin(\Omega\tau+\phi)\\ -\Omega\sin(\Omega\tau+\phi)&-r\Omega\cos(\Omega\tau+\phi)\end{array} \right) \left( \begin{array}{c}r^{\prime}\\ \phi^{\prime}\end{array} \right)=\left( \begin{array}{c}0\\ f_{1}\end{array} \right), \tag{16}\]  
where  
\[f_{1}=F_{0}\!\cos\Omega\tau+2r\xi\Omega\sin(\Omega\tau+\phi)\ +\ r\Omega^{2}\cos( \Omega\tau+\phi)-r\cos(\Omega\tau+\phi)\Delta, \tag{17}\]  
\[\Delta=1-\beta+\frac{\beta(1+\alpha)}{\sqrt{1-r^{2}\cos^{2}(\Omega\tau+\phi)}}. \tag{18}\]  
A manipulation of Equations (17) and (18) leads to the following expressions of the derivatives of the response amplitude and the phase angle:  
\[r^{\prime}=-\frac{1}{\Omega}f_{1}\sin(\Omega\tau+\phi), \tag{19}\]  
\[\phi^{\prime}=-\frac{1}{r\Omega}f_{1}\cos(\Omega\tau+\phi). \tag{20}\]  
According to the averaging method, the values of \(r^{\prime}\) and \(\phi^{\prime}\) can be approximated by their average values over a period of the excitation load so that  
\[r^{\prime}=-\frac{\Omega}{2\pi}\int_{0}^{\frac{\pi\Omega}{2\Omega}}\frac{1}{ \Omega}f_{1}\sin(\Omega\tau+\phi)\mathrm{d}\tau, \tag{21}\]  
\[\phi^{\prime}=-\frac{\Omega}{2\pi}\int_{0}^{\frac{\pi\Omega}{2\Omega}}\frac{1} {r\Omega}f_{1}\cos(\Omega\tau+\phi)\mathrm{d}\tau. \tag{22}\]  
By further simplifying Equations (21) and (22), we obtain  
\[r^{\prime}=-\frac{1}{2\Omega}F_{0}\sin\phi-r_{0}^{\pi}, \tag{23}\]  
\[r\phi^{\prime}=-\frac{1}{2\Omega}F_{0}\cos\phi-\frac{r\Omega}{2}+\frac{(1- \beta)r}{2\Omega}+\frac{2\beta(1+\alpha)}{\pi\Omega r}[K(r)-E(r)], \tag{24}\]  
where \(K(r)\) and \(E(r)\) represent the first and second complete elliptic integrals, respectively [39]. By setting \(r^{\prime}=\phi^{\prime}=0\) in Equations (23) and (24) and by cancelling out the phase angle \(\phi\), the relationship between the system parameters and response amplitude is obtained as  
\[F_{0}^{2}=(2\xi\Omega r)^{2}+\left\{\left(\Omega^{2}-1+\beta\right)r-\frac{4 \beta(1+\alpha)}{\pi r}[K(r)-E(r)]\right\}^{2} \tag{25}\]  
When the system parameters are known, Equation (25) can be solved by using a bisection method [40]. Figure 6 compares the obtained results based on the averaging approximations with those obtained using the numerical time-marching Runge-Kutta method. A good agreement between the analytical and numerical results is observed; thus, the averaging formulations are verified. In Figure 6a, four sets of parameters with \((\alpha,\,\beta)=(-1.5,\,0.5),\,(-1.25,\,0.6),\,(-1,\,0.75)\), and \((-0.5,\,1.5)\) are considered. The product \(\alpha\beta\) is kept as \(-0.75\). The other parameters are set as \(\xi=0.015,\ F_{0}=0.01\). Essentially, the variations in the combinations of \(\alpha\) and \(\beta\) change the system from being that of a softening stiffness system for the first two parameter sets to a linear system and then finally to a hardening stiffness system. Correspondingly, the figure shows that the resonant peak bends to the low-frequency range for the softening stiffness systems and to high frequencies for the hardening stiffness system. It shows that the response peak of the softening systemsis larger than that of the linear and the hardening systems. Away from the resonance region, the amplitude \(r\) is not sensitive to the variations in \(\alpha\) and \(\beta\). Figure 6b examines the system responses when \((\alpha,\ \beta)\ =\ (-1.2,\ 0.7),\ (-1.6,\ 0.4),\ (-1,\ 0.36),\) and \((0.2,\ 2.2)\). The product \(\alpha\beta\) varies from \(-0.84\) to \(-0.64\) to \(-0.36\) and then to \(0.44\).