## 4 Dynamics and Isolation Performance of the Vibration Isolator
### Frequency Response Characteristics and Stability  
From Equation (15), the positive solutions for non-dimensional excitation frequency \(\Omega\) can be obtained as:  
\[\Omega_{1,2}=\sqrt{\frac{(3\overline{a}^{2}r\overline{A}^{4}-2\xi^{2} \overline{A}^{2})\pm\overline{A}^{2}\sqrt{4\xi^{4}-12\xi^{2}\overline{a}^{2}r \overline{A}^{2}+9\overline{Z}^{2}\overline{a}^{4}r^{2}\overline{A}^{2}}}{ \overline{A}^{2}-\overline{Z}^{2}}} \tag{16}\]  
Equation (16) implies the relationships between \(\Omega\) and \(\overline{A}\) during the steady-state response which are often plotted by frequency response curve (FRC). Based on this relationship, the following response features of the vibration isolation system can be predicted in the form of formulas. The peak amplitude of vibration response occurs when \(\Omega\) has a unique solution. Hence, the peak amplitude \(\overline{A}_{\mathrm{peak}}\) and the corresponding excitation frequency \(\Omega_{\mathrm{peak}}\) under QZS condition are:  
\[\overline{A}_{\mathrm{peak}}=\frac{2\xi^{2}}{\sqrt{12\xi^{2}\overline{a}^{2}r -9\overline{Z}^{2}\overline{a}^{4}r^{2}}} \tag{17}\]  
and  
\[\Omega_{\mathrm{peak}}=\sqrt{\frac{18\xi^{6}\overline{a}^{2}r\overline{Z}^{2} -12\xi^{8}}{12\xi^{6}-45\xi^{4}\overline{a}^{2}r\overline{Z}^{2}+54\xi^{2} \overline{a}^{4}r^{2}\overline{Z}^{4}-\frac{81}{4}\overline{a}^{6}r3 \overline{Z}^{6}}} \tag{18}\]  
In resonance response, the backbone curve, drawn by the peak amplitude points (\(\Omega_{\mathrm{peak}}\), \(\overline{A}_{\mathrm{peak}}\)), can be formulated as:  
\[\Omega_{\mathrm{b}}=\sqrt{\frac{3\overline{a}^{2}r\overline{A}^{4}-2\xi^{2} \overline{A}^{2}}{\overline{A}^{2}-\overline{Z}^{2}}} \tag{19}\]When the square root of the denominator in Equation (17) is not positive, \(\overline{A}_{\text{peak}}\) does not exist. To ensure the maximum response of the system is bounded, the parameters should be satisfied that:  
\[\sqrt{\frac{3\overline{a}^{2}r\overline{Z}^{2}}{4}}<\xi \tag{20}\]  
In a nonlinear system, the jump phenomenon is always unavoidable. To investigate the stability of the steady-state response in Equation (13), a small disturbance \(\overline{e}(\tau)\) is added in the harmonic solution. Then, \(\overline{A}\cos(\Omega+\theta)+\overline{e}(\tau)\) is substituted into Equation (12) without considering the higher terms of \(\overline{e}\), resulting in:  
\[\overline{e}^{\prime\prime}(\tau)+2\xi\overline{e}^{\prime}(\tau)+6\overline{ a}^{2}r\overline{A}^{2}(1+\cos 2(\Omega\tau+\theta))\overline{e}(\tau)=0 \tag{21}\]  
Therefore, the unstable solution of Equation (12) at steady state is given as:  
\[\Omega^{4}+(4\xi^{2}-12\overline{a}^{2}r\overline{A}^{2})\Omega^{2}+27 \overline{a}^{4}r^{2}\overline{A}^{4}=0 \tag{22}\]  
Two FRCs of the vibration isolation system are plotted in Figure 8 to illustrated the transition of response from bounded to unbounded. \(\overline{Z}\) is chosen as the variable parameter, and the maximum response is bounded when \(Z~{}<\)3.079 with \(\overline{a}\)\(=0.45\), \(r=1\) and \(\xi=0.12\). At \(\overline{Z}=0.3\), the FRC bends to the right as \(\Omega\) grows, which denotes the hardening stiffness characteristic of the system. The unstable response region between the jump-down and the jump-up frequencies appears on the inclined side of the resonant response. As \(\overline{Z}\) increases to 0.31, the FRC still has right bend and unstable region, but the maximum response of the system changes to unbounded. In order to prove the vibration isolation capability of the QZS isolator, the equivalent linear vibration isolator is also given in this paper for comparison.