## 4 An Alternative Configuration of the Isolator \((m,c,k_{p},k_{n},b)\)
Additionally, using a similar procedure as that derived from Equation (5), the displacement transmissibility of this isolator is given by [25]  
\[T_{d}=\frac{\sqrt{\left(2\zeta(1-\beta)\frac{\varepsilon}{\alpha}-2\zeta(1+ \varepsilon)\Omega^{2}\right)^{2}+\left(\left((1-\beta)+\frac{\varepsilon}{ \alpha}(2\zeta)^{2}\right)\Omega\right)^{2}}}{\sqrt{\left(2\zeta(1-\beta)\frac {\varepsilon}{\alpha}-2\zeta\left(1+\varepsilon+\frac{\varepsilon}{\alpha} \right)\Omega^{2}\right)^{2}+\left(\left((1-\beta)+\frac{\varepsilon}{\alpha} (2\zeta)^{2}\right)\Omega-\Omega^{3}\right)^{2}}}. \tag{31}\]It can be seen that when \(\Omega\to\infty\), the \(T_{d}\) tends zero, which is greatly different from the result of the isolator \((m,c,k_{p},k_{n},b)\) (Equation (15)). Figure 13a-d depict the transmissibility \(T_{d}\) of the isolator \((m,c,k_{p},k_{n},b,c_{1})\) with respect to different \(\varepsilon\) and \(\zeta\) for two sets of \(\alpha\), \(\beta\) values, i.e., \(\alpha=1.0,\ \beta=0.95\) and \(\alpha=0.5,\ \beta=0.90\). It can be seen that, compared with results of the isolator \((m,c,k_{p},k_{n},b)\), as shown in Figure 2, the configuration of a damper in series connection with the inerter can effectively overcome the weakness of the isolator \((m,c,k_{p},k_{n},b)\) in the suppression of high frequency-excited vibration. Moreover, it is shown from Figure 13a-d that variations in \(\alpha\) and \(\beta\) values have the same effects on the damping ratio and the undamped natural frequency as those shown in Figure 12. Figure 14 depicts the effect of \(\varepsilon\) on \(T_{d}\) of the isolator \((m,c,k_{p},k_{n},b,c_{1})\) with \(\alpha=1.0\), \(\beta=0.95\) and \(\zeta=0.1\). Figure 15 shows the transmissibility \(T_{d}\) of the corresponding linear isolators with damping ratio \(\zeta=0.1\) for some specific \(\alpha,\beta\) and \(\varepsilon\) values, which indicates that the isolator \((m,c,k_{p},k_{n},b,c_{1})\) can gain a better isolation performance in a very wide excitation frequency range. Additionally, the effect of \(\alpha\) on \(T_{d}\) of this isolator seems unremarkable. Figure 16 shows the SAR of the corresponding linear isolators with the damping ratio \(\zeta=0.1\) and some specific \(\alpha,\beta\) and \(\varepsilon\) values under the base displacement excitation of the half-cycle sine pulse. It can be seen that the isolator \((m,c,k_{p},k_{n},b,c_{1})\) can achieve better isolation performance in terms of the SAR than those of other corresponding isolators, when appropriately choosing \(\alpha,\beta\) and \(\varepsilon\) values (see the black solid line in Figure 16). Figure 17 presents the effect of \(\alpha\) on the SAR of two isolators, \((m,c,k_{p},k_{n},b,c_{1})\) and \((m,c,k_{p},b,c_{1})\). Figures 18 and 19 show the SDR and RDR of the corresponding linear isolators with the damping ratio \(\zeta=0.1\) and some specific \(\alpha,\beta\) and \(\varepsilon\) values, respectively. It can be seen from the numerical results that keeping a high positive stiffness relative to the negative stiffness, i.e., taking a small \(\beta\) value, is beneficial for reductions in the SDR and RDR for the isolator \((m,c,k_{p},k_{n},b,c_{1})\), and makes its displacement isolation better than that of the corresponding isolator \((m,c,k_{p},k_{n},b)\); however, the isolator \((m,c,k_{p},b,c_{1})\) has the best displacement isolation performance in terms of the SDR and RDR for the majority of severity factor \(\mu\) values, compared with other corresponding linear isolators. Figure 20 shows the time responses of non-dimensional absolute displacement \(X\) under the base displacement excitation of a half-cycle sine pulse (severity factor \(=4\)) for some isolators, as shown in Figure 18. It should be specifically noted for two isolators, \((m,c,k_{p},k_{n},b,c_{1})\) and \((m,c,k_{p},b,c_{1})\), shown in Figure 20, that taking a small \(\alpha\) valueis also beneficial for reductions in the SDR, and the time response of isolator \((m,c,k_{p},b,c_{1})\) has the shortest time duration under pulse excitation.