## 2 An Isolator against Harmonic Base Excitation  
The governing equation of the SDOF isolator with a quasi-zero stiffness, a damper and an inerter in parallel connection, subjected to base excitation, as shown in Figure 1, is expressed as  
\[m\ddot{x}+b\big{(}\ddot{x}-\ddot{x}_{b}\big{)}+c\big{(}\dot{x}-\dot{x}_{b} \big{)}+\big{(}k_{p}-k_{n}\big{)}(x-x_{b})=0, \tag{1a}\]or  
\[(m+b)\Big{(}\ddot{x}-\ddot{x}_{b}\Big{)}+c\Big{(}\dot{x}-\dot{x}_{b}\Big{)}+ \Big{(}k_{p}-k_{n}\Big{)}(x-x_{b})=-m\ddot{x}_{b}, \tag{1b}\]  
where \(x\) and \(x_{b}\) are the absolute displacements of the sprung mass and base excitation, respectively; \(m,c\) and \(k_{p}\) are the mass, damping and positive stiffness coefficients of the isolator, respectively; \(b\) and \(k_{n}\) are the inertance and absolute value of the negative stiffness coefficient of the isolator, respectively. For brevity, this isolator is denoted as \((m,c,k_{p},k_{n},b)\). The introduction of non-dimensional parameters is expressed as  
\[\omega_{0}=\sqrt{\frac{k_{p}}{m}},\ \tau=\omega_{0}t,\ \zeta=\frac{c}{2m \omega_{0}},\ X=\frac{x}{x_{bmax}},\ X_{b}=\frac{x_{b}}{x_{bmax}},\ \alpha=\frac{b}{m},\ \beta=\frac{k_{n}}{k_{p}},\]  
where \(x_{bmax}\) is the amplitude of the base motion \(x_{b}\). Equation (1) can be written as  
\[(1+\alpha)X^{\prime\prime}+2\zeta X^{\prime}+(1-\beta)X=\alpha X_{b}^{\prime \prime}+2\zeta X_{b}^{\prime}+(1-\beta)X_{b},\] (2a) where the primes denote derivatives with respect to non-dimensional time \[\tau\]. Introducing the relative displacement \[Y=X-X_{b}\], Equation ( 2a ), can also be written as \[(1+\alpha)Y^{\prime\prime}+2\zeta Y^{\prime}+(1-\beta)Y=-X_{b}^{\prime\prime}. \tag{2b}\]  
The transfer function from the base excitation to the absolute displacement of the sprung mass can be obtained from Equation (2a) as  
\[H(s)=\frac{X(s)}{X_{b}(s)}=\frac{\alpha s^{2}+2\zeta s+(1-\beta)}{(1+\alpha)s ^{2}+2\zeta s+(1-\beta)}, \tag{3}\]  
where \(s\) is the complex frequency parameter in the Laplace transform. Let \(x_{b}=x_{b0}\cos\omega t\) be the harmonic displacement of the base, \(X_{b0}=x_{b0}/x_{bmax}\) and \(\Omega=\omega/\omega_{0}\), then \(X_{b}=X_{b0}\cos\Omega\tau\), and the steady-state response of the sprung mass can be represented by \(X=X_{0}\cos(\Omega\tau+\varphi)\). Consequently, the frequency response can be calculated by replacing \(s\) in the transfer function by j\(\Omega\), i.e.,  
\[H(\mathrm{j}\Omega)=\big{|}H(\mathrm{j}\Omega)\big{|}\mathrm{e}^{\mathrm{j} \varphi}, \tag{4}\]  
where j is the imaginary unit, and  
\[\big{|}H(\mathrm{j}\Omega)\big{|}=\frac{X_{0}}{X_{b0}}=\frac{\sqrt{\left(1- \beta-\alpha\Omega^{2}\right)^{2}+\left(2\zeta\Omega\right)^{2}}}{\sqrt{ \left(1-\beta-(1+\alpha)\Omega^{2}\right)^{2}+\left(2\zeta\Omega\right)^{2}}}. \tag{5}\]It should be noted here that the displacement transmissibility \(T_{d}\) of the isolator is identical to \(\left|H(\mathrm{j}\Omega)\right|\), i.e.,\(T_{d}=\left|H(\mathrm{j}\Omega)\right|\). Additionally, Equation (2b) can be rewritten as  
\[(m+b)\Big{(}\ddot{x}-\ddot{x}_{b}\Big{)}+c\Big{(}\dot{x}-\dot{x}_{b}\Big{)}+ \Big{(}k_{p}-k_{n}\Big{)}(x-x_{b})=-m\ddot{x}_{b}, \tag{1b}\]0  
where \(\Omega_{n}\)(or \(\omega_{n}\)) and \(\eta\) are the undamped natural frequency and damping ratio of the isolator, respectively; they have the following expressions as  
\[(m+b)\Big{(}\ddot{x}-\ddot{x}_{b}\Big{)}+c\Big{(}\dot{x}-\dot{x}_{b}\Big{)}+ \Big{(}k_{p}-k_{n}\Big{)}(x-x_{b})=-m\ddot{x}_{b}, \tag{1b}\]1 \[(m+b)\Big{(}\ddot{x}-\ddot{x}_{b}\Big{)}+c\Big{(}\dot{x}-\dot{x}_{b}\Big{)}+ \Big{(}k_{p}-k_{n}\Big{)}(x-x_{b})=-m\ddot{x}_{b}, \tag{1b}\]2  
For the isolator with the quasi-zero stiffness discussed here, one has \(1-\beta\approx 0\), and \(1-\beta>0\) in order to maintain the stability of the isolator in reality.