## 4 An Alternative Configuration of the Isolator \((m,c,k_{p},k_{n},b)\)  
Connecting the inerter directly between the mass and the base cannot attenuate the system response at high frequencies, as shown in Figures 2 and 3. To overcome this weakness, other simple inerter-based configurations are adopted in this paper, which uses a damper (or a spring) in series connection with the inerter, as shown in Figure 11. This isolator is denoted as \((m,c,k_{p},k_{n},b,c_{1})\) and has an additional degree of freedom \(x_{1}\), which has an equation of motion as follows. Figure 10: The effect of \(\beta\) on SDR and RDR of the isolator \((m,c,k_{p},k_{n},b)\) with the damping ratio \(\zeta=0.1\). Figure 9: The effect of \(\alpha\) on SDR and RDR of the isolator \((m,c,k_{p},k_{n},b)\) with the damping ratio \(\zeta=0.1\). \[(m+b)\ddot{x}+c\dot{x}+\left(k_{p}-k_{n}\right)x=b\ddot{x}_{1}+c\dot{x}_{b}+ \left(k_{p}-k_{n}\right)x_{b}, \tag{27}\]  
\[b\left(\ddot{x}-\ddot{x}_{1}\right)=c_{1}\left(\dot{x}_{1}-\dot{x}_{b}\right). \tag{28}\]  
Eliminating \(x_{1}\) from Equations (27) and (28) yields  
\[ \begin{split}&\ddot{x}+2a_{0}c\left(1+\varepsilon+\frac{ \varepsilon}{\alpha}\right)\ddot{x}+\left(\omega_{0}^{2}(1-\beta)+\frac{ \varepsilon}{\alpha}(2a_{0}\zeta)^{2}\right)\dot{x}+2\omega_{0}^{3}\zeta(1- \beta)\frac{\varepsilon}{\alpha}x\\ &=2\omega_{0}\zeta(1+\varepsilon)\ddot{x}_{b}+\left(a_{0}^{2}(1- \beta)+\frac{\varepsilon}{\alpha}(2a_{0}\zeta)^{2}\right)\ddot{x}_{b}+2\omega _{0}^{3}\zeta(1-\beta)\frac{\varepsilon}{\alpha}x_{b},\end{split}  \tag{29}\]  
or  
\[ \begin{split}& X^{\prime\prime\prime}+2\zeta\left(1+\varepsilon+ \frac{\varepsilon}{\alpha}\right)X^{\prime\prime}+\left((1-\beta)+\frac{ \varepsilon}{\alpha}(2\zeta)^{2}\right)X^{\prime}+2\zeta(1-\beta)\frac{ \varepsilon}{\alpha}X\\ &\quad=2\zeta(1+\varepsilon)X^{\prime\prime}_{b}+\left((1-\beta) +\frac{\varepsilon}{\alpha}(2\zeta)^{2}\right)X^{\prime}_{b}+2\zeta(1-\beta) \frac{\varepsilon}{\alpha}X_{b}.\end{split}  \tag{30}\]  
where the non-dimensional parameter \(\varepsilon=c_{1}/c\), and other non-dimensional parameters are the same as those of Equation (2a). Obviously, this isolator makes a third-order SDOF system. Instead of seeking the analytical solutions of undamped natural frequencies for this new isolator, the characteristic roots or poles of this isolator are numerically calculated as a function of the non-dimensional parameter \(\varepsilon\) by utilizing the root-locus method. Figure 12 shows the root-locus plot with respect to different \(\beta\) for the given \(\alpha=1.0\) and \(\zeta=0.1\) as \(\varepsilon=0\rightarrow\infty\). There are three branches of the root locus for each set of different \(\beta\), \(\alpha\), one of which is located on the negative real axis while the other two represent a pair of conjugate complex poles. They all lie in the left-half \(s\) plane. The starting poles are represented by \(\times\) and the ending poles by \(\omega\), as a convention when \(\varepsilon\) is increased from \(0\) to \(\infty\). The value of \(\varepsilon\) and the corresponding undamped natural frequency of this isolator at an arbitrary point on the root locus can thus be obtained. It is also shown from the rootlocus plot that a large dynamic stiffness, i.e., a small \(\beta\) value, can lead to a lower damping ratio this isolator when the same \(\varepsilon\) value is taken, which is beneficial for high frequency vibration isolation; meanwhile, this makes the undamped natural frequency increase, and a small \(\alpha\) value has the same effect.