## 2 An Isolator against Harmonic Base Excitation
It can be seen that, for the under-damped case, (1) the peak value of \(T_{d}\) is a decreasing function of \(\alpha\) and \(\beta\) (see the curves in \[\omega_{0}=\sqrt{\frac{k_{p}}{m}},\ \tau=\omega_{0}t,\ \zeta=\frac{c}{2m \omega_{0}},\ X=\frac{x}{x_{bmax}},\ X_{b}=\frac{x_{b}}{x_{bmax}},\ \alpha=\frac{b}{m},\ \beta=\frac{k_{n}}{k_{p}},\]0 black solid line --, black dashed line -, blue dashed line - with \(\alpha=0.5,0.8,1.0\), respectively; and blue solid line --, blue dashed line - with \(\beta=0.8,0.9\), respectively; and refer to Equation(13)); (2) In certain low-frequency regions (e.g., from \(\Omega\approx 0.3\) to \(0.7\) for \(\alpha=0.5,\beta=0.9\)), located on the right of the resonance frequency, the isolator \((m,c,k_{p},k_{n},b)\) normally has a better isolation performance in terms of values of \(T_{d}\), compared with the corresponding isolators \((m,c,k_{p})\), \((m,c,k_{p},k_{n})\) and \((m,c,k_{p},b)\), which are denoted by red solid, green dashed and green solid lines, respectively. When \(\Omega\) becomes larger, the \(T_{d}\) of isolators \((m,c,k_{p},k_{n},b)\) and \((m,c,k_{p},b)\) will not tend to zero, as also shown in Figure 2, and referred to in Equation(15) below, compared with isolators \((m,c,k_{p})\) and \((m,c,k_{p},k_{n})\). These results indicate that the addition of the inerter has an adverse effect on vibration suppression at high frequencies. In the case of smaller damping ratios, there is a particular frequency \(\Omega_{b}\), also called an anti-resonant frequency, where the minimum magnitude of \(T_{d}\) is obtained. This is a peculiar feature presenting among the SDOF isolators. For the undamped case, the anti-resonant frequency can be obtained as \(\Omega_{b}=\sqrt{(1-\beta)/\alpha}\). The magnitudes of \(T_{d}\) at the natural frequency \(\Omega_{n}\), the anti-resonant frequency \(\Omega_{b}\) and infinity can be derived as  
\[\omega_{0}=\sqrt{\frac{k_{p}}{m}},\ \tau=\omega_{0}t,\ \zeta=\frac{c}{2m \omega_{0}},\ X=\frac{x}{x_{bmax}},\ X_{b}=\frac{x_{b}}{x_{bmax}},\ \alpha=\frac{b}{m},\ \beta=\frac{k_{n}}{k_{p}},\]1  
\[\omega_{0}=\sqrt{\frac{k_{p}}{m}},\ \tau=\omega_{0}t,\ \zeta=\frac{c}{2m \omega_{0}},\ X=\frac{x}{x_{bmax}},\ X_{b}=\frac{x_{b}}{x_{bmax}},\ \alpha=\frac{b}{m},\ \beta=\frac{k_{n}}{k_{p}},\]2  
\[\omega_{0}=\sqrt{\frac{k_{p}}{m}},\ \tau=\omega_{0}t,\ \zeta=\frac{c}{2m \omega_{0}},\ X=\frac{x}{x_{bmax}},\ X_{b}=\frac{x_{b}}{x_{bmax}},\ \alpha=\frac{b}{m},\ \beta=\frac{k_{n}}{k_{p}},\]3Note that for the isolator \((m,c,k_{p},b)\) with addition of the parallel connected inerter only, the values of \(T_{d}\) at \(\Omega_{n}\) and \(\Omega_{b}\) can be found by replacing \(1-\beta\) with \(1\) in Equations (13) and (14), and the \(T_{d}\), when \(\Omega\to\infty\), is the same as Equation (15) [2]. \[\omega_{0}=\sqrt{\frac{k_{p}}{m}},\ \tau=\omega_{0}t,\ \zeta=\frac{c}{2m \omega_{0}},\ X=\frac{x}{x_{bmax}},\ X_{b}=\frac{x_{b}}{x_{bmax}},\ \alpha=\frac{b}{m},\ \beta=\frac{k_{n}}{k_{p}},\]4 Transmissibility \(T_{d}\) of the corresponding isolators with some specific values of \(\alpha\), \(\beta\) and the damping ratio \(\zeta=0.1\). In what follows, the threshold value of the damping ratio \(\zeta\) or \(\eta\), below which the minimum of \(T_{d}\) occurs, is sought by using the limit value of \(T_{d}\) when \(\Omega\rightarrow\infty\) (Equation (15)).