# Effective Frequency Range and Jump Behavior of Horizontal Quasi-Zero Stiffness Isolator
## 2 Isolator Subjected to Base Excitation  
### Dynamic Model  
The model of the horizontal QZS isolator subject to a base displacement excitation is shown in Figure 1. The system is composed of a mass, a nonlinear spring, and a damping element. The QZS property of the system is achieved by two opposing linear springs, as shown in Figure 2. The mass element comprises the isolator platform and the isolated structure, which can translate along the guide rail. The dynamic equation describing the vibration isolation system can be expressed as,  
\[M\ddot{x}+\mu\big{(}\dot{x}-P\Omega\cos\Omega t\big{)}+k_{1}(x-P\sin\Omega t)+k_ {2}(x-P\sin\Omega t)^{3}=0, \tag{1}\]  
where \(k_{1}=2K-\frac{2 KL}{l}\) and \(k_{2}=\frac{KL}{l^{3}}\)[32, 33]. \(k_{\text{qZS}}\) represents the stiffness of the isolator, \(M\) is the total mass of the isolator platform and the isolated structure, \(x\) is the displacement of the mass element, \(K\) is the stiffness of the linear springs, \(L\) is the original length of the springs, \(l\) is the distance between the fixed hinge support and the mass element, \(\mu\) is the damping coefficient, \(P\) is the input displacement amplitude, and \(\Omega\) is the excitation frequency.  
In this study, a semi-analytical method and a numerical method were applied. The semi-analytical method was mainly used to study the effective frequency range and the numerical method was used to study the jump behaviors. The results obtained from the two methods were compared for mutual verification.  
For obtaining the first-order equation that describes the system, the complexification-averaging method [34, 35] was adopted. The transformation \(\dot{x}+i\Omega x=\alpha e^{i\Omega t}\) is considered, which yields:  
\[x=\frac{\alpha e^{i\Omega t}-\overline{\alpha}e^{-i\Omega t}}{2 i\Omega},\ \dot{x}=\frac{\alpha e^{i\Omega t}+\overline{\alpha}e^{-i\Omega t}}{2},\] \[\ddot{x}=\frac{1}{2}\Bigg{(}\dot{\alpha}e^{i\Omega t}+i\Omega \alpha e^{i\Omega t}+\dot{\overline{\alpha}}e^{-i\Omega t}-i\Omega\overline{ \alpha}e^{-i\Omega t}\Bigg{)},\]  
where \(\overline{\alpha}\) is the conjugate of \(\alpha\), and \(i\) represents the symbol of imaginary number (\(i=\sqrt{-1}\)). Substituting the above expressions into Equation (1) and retaining the slowly varying part, then obtaining:  
Figure 1: Mechanical model of a horizontal QZS isolator subject to a base excitation.  
Figure 2: The realization of the QZS property using two linear springs.  
\[\dot{\alpha}+i\Omega\alpha+\frac{\mu}{M}(\alpha-P\Omega)+\frac{k_{1}}{M}\bigg{(} \frac{\alpha}{i\Omega}-\frac{P}{i}\bigg{)}+\frac{3k_{2}(\alpha-P\Omega)^{2} \bigg{(}\widetilde{\alpha}-P\Omega\bigg{)}}{4M\Omega^{3}i}=0. \tag{2}\]  
Introducing \(\alpha=a+ib\) into Equation (2) yields:  
\[\dot{a}+i\dot{b}+i\Omega(a+ib)+\frac{\mu}{M}(a+ib-P\Omega)+\frac{k_{1}(a+ib-P \Omega)}{iM\Omega} \tag{3}\] \[+\frac{3k_{2}\Big{[}(a-P\Omega)^{2}+b^{2}\Big{]}(a-P\Omega+ib)}{4 iM\Omega^{3}}=0.\]  
The real and imaginary parts of Equation (3) can then be separated giving:  
\[\dot{a}=b\Omega-\frac{\mu}{M}(a-P\Omega)-\frac{k_{1}b}{M\Omega}-\frac{3k_{2} \big{[}(a-P\Omega)^{2}+b^{2}\big{]}b}{4M\Omega^{3}},\]  
\[\dot{b}=-a\Omega-\frac{\mu}{M}b+\frac{k_{1}(a-P\Omega)}{M\Omega}+\frac{3k_{2} \big{[}(a-P\Omega)^{2}+b^{2}\big{]}[a-P\Omega]}{4M\Omega^{3}}. \tag{4}\]  
In order to obtain the steady-state response of the system, \(\dot{a}=0\) and \(\dot{b}=0\) are set. The amplitude of the isolator can then be obtained by solving Equation(4):  
\[A=\frac{\sqrt{a^{2}+b^{2}}}{\Omega}.\]