## 2 An Isolator against Harmonic Base Excitation
Obviously, from Equation (7a) the isolator \((m,c,k_{p},k_{n},b)\) has a much lower natural frequency than that of the corresponding isolator with the introduction of the negative stiffness only, denoted as \((m,c,k_{p},k_{n})\) or the introduction of the inerter only, denoted as \((m,c,k_{p},b)\), which has the natural frequency of \(\omega_{0}\sqrt{1-\beta}\) or \(\omega_{0}\sqrt{1/(1+\alpha)}\). Therefore, this isolator is more suitable for the isolation of low-frequency or ultra-low frequency vibration. It is clearly shown from Equations (7a) and (7b) that the undamped natural frequency \(\Omega_{n}\) is a decreasing function with respect to \(\alpha\) and \(\beta\),and \(\eta\) is a decreasing function of \(\alpha\) and an increasing function of \(\beta\). Figure 2a-d depict the transmissibility \(T_{d}\) with respect to different \(\alpha\) and \(\zeta\) for a given \(\beta=0.95\). It is shown that, in each graph, there exists a particular frequency, \(\Omega_{i}\), also called an invariant point, where the value of \(T_{d}\) is independent of \(\zeta\) for any given \(\alpha\) and \(\beta\). This invariant point is \(\Omega_{i}=\sqrt{2}\) for the corresponding linear viscously damped isolator, denoted as \((m,c,k_{p})\), without the addition of the parallel connected negative stiffness and inerter, and \(\Omega_{i}=\sqrt{2/(1+2\alpha)}\) for the isolator \((m,c,k_{p},b)\)[2]. To find the invariant point \(\Omega_{i}\) in this paper, the transmissibility \(T_{d}\) in Equation (5) can be rewritten as  
\[(m+b)\Big{(}\ddot{x}-\ddot{x}_{b}\Big{)}+c\Big{(}\dot{x}-\dot{x}_{b}\Big{)}+ \Big{(}k_{p}-k_{n}\Big{)}(x-x_{b})=-m\ddot{x}_{b}, \tag{1b}\]3  
\[(m+b)\Big{(}\ddot{x}-\ddot{x}_{b}\Big{)}+c\Big{(}\dot{x}-\dot{x}_{b}\Big{)}+ \Big{(}k_{p}-k_{n}\Big{)}(x-x_{b})=-m\ddot{x}_{b}, \tag{1b}\]4 An isolator \((m,c,k_{p},k_{n},b)\) (the configuration 1). where \(A=4\Omega^{2}\), \(B=\left(1-\beta-\alpha\Omega^{2}\right)^{2}\), \(C=4\Omega^{2}\), and \(D=\left(1-\beta-(1+\alpha)\Omega^{2}\right)^{2}\). As \(T_{d}\) is independent of \(\zeta\) at \(\Omega_{i}\), one has  
\[(m+b)\Big{(}\ddot{x}-\ddot{x}_{b}\Big{)}+c\Big{(}\dot{x}-\dot{x}_{b}\Big{)}+ \Big{(}k_{p}-k_{n}\Big{)}(x-x_{b})=-m\ddot{x}_{b}, \tag{1b}\]5  
that is,  
\[(m+b)\Big{(}\ddot{x}-\ddot{x}_{b}\Big{)}+c\Big{(}\dot{x}-\dot{x}_{b}\Big{)}+ \Big{(}k_{p}-k_{n}\Big{)}(x-x_{b})=-m\ddot{x}_{b}, \tag{1b}\]6  
Then, the nonzero invariant point \(\Omega_{i}\) is  
\[(m+b)\Big{(}\ddot{x}-\ddot{x}_{b}\Big{)}+c\Big{(}\dot{x}-\dot{x}_{b}\Big{)}+ \Big{(}k_{p}-k_{n}\Big{)}(x-x_{b})=-m\ddot{x}_{b}, \tag{1b}\]7  
Note that \(T_{d}=1\) at \(\Omega=\Omega_{i}\) for all values of \(\zeta\). Additionally, for the isolator \((m,c,k_{p},k_{n})\), the invariant point is \(\Omega_{i}=\sqrt{2(1-\beta)}\) (obtained by letting \(\alpha=0\) in Equation (11)). Therefore, to enable the effective isolation of the base excitation, i.e., \(T_{d}<1\), one has  
\[(m+b)\Big{(}\ddot{x}-\ddot{x}_{b}\Big{)}+c\Big{(}\dot{x}-\dot{x}_{b}\Big{)}+ \Big{(}k_{p}-k_{n}\Big{)}(x-x_{b})=-m\ddot{x}_{b}, \tag{1b}\]8  
In addition, \(T_{d}\) varies with \(\zeta\) or \(\eta\) in the same way as that of the conventional isolator \((m,c,k_{p})\). For \(\Omega<\Omega_{i}\), smaller damping ratios lead to larger values of \(T_{d}\); on the other hand, for \(\Omega>\Omega_{i}\), smaller values of damping ratio lead to smaller values of \(T_{d}\), as shown in \[(m+b)\Big{(}\ddot{x}-\ddot{x}_{b}\Big{)}+c\Big{(}\dot{x}-\dot{x}_{b}\Big{)}+ \Big{(}k_{p}-k_{n}\Big{)}(x-x_{b})=-m\ddot{x}_{b}, \tag{1b}\]9  
Figure 3 gives the \(T_{d}\) curves for several specific values of \(\alpha\) and \(\beta\) with a fixed damping ratio \(\zeta=0.1\).