## 4 Dynamics and Isolation Performance of the Vibration Isolator
### Vibration Isolation Performance under Different Excitations  
#### 4.4.1 Displacement Transmissibility to Sinusoidal Excitation  
When subjected to a base displacement excitation, the vibration isolation capability of an isolator can be evaluated by displacement transmissibility \(T\). For a linear isolation system, its absolute displacement transmissibility can be given by the ratio of the response amplitude of the isolated object to the excitation amplitude of the baseplate, and is described as \(T_{1}=\Omega^{2}/\sqrt{\left(1-\Omega^{2}\right)^{2}+4\xi^{2}\Omega^{2}}\). The transmissibility of a nonlinear vibration isolation system is more complicated because it is related to the level of excitation. According to the relationship of motion, the non-dimensional absolute displacement of the isolated mass is \(\overline{y}=\overline{u}+\overline{z}\). Then, the absolute displacement transmissibility \(T_{\mathrm{n}}\) of the nonlinear vibration isolator is obtained as:  
\[T_{\mathrm{n}}=\frac{\sqrt{\overline{A}^{2}+\overline{Z}^{2}+2\overline{AZ} \cos\theta}}{\overline{Z}} \tag{27}\]  
From the previous analysis, the response of the vibration isolator with QCRM-NSC under QZS condition is related to the nonlinear stiffness coefficient \(4\overline{a}^{2}r\), damping ratio \(\xi\), and excitation amplitude \(\overline{Z}\). Since \(\overline{a}^{2}\) and \(r\) play the same role in vibration response, here only \(\overline{a}\) is selected to represent the nonlinear stiffness coefficient. Considering the bounded condition of the response and the practical working condition, the calculations for dynamic characteristics of the vibration isolator have been conducted. The parameter values listed in Table 1 can clearly reflect the system response changes with the parameters, and are used in the following discussion. Figure 10: Comparisons between FRCs about original and approximate stiffness computed by Runge–Kutta method (RKM) and the harmonic balance method (HBM). The quadratic coefficient \(\overline{a}\) is a key parameter of the negative stiffness structure. When \(\xi=0.10\) and \(\overline{Z}=0.25\), the effects of \(\overline{a}\) on the absolute displacement transmissibility of the QZS isolator are shown by the solid line in Figure 11. With a small \(\overline{a}\), the transmissibility curve is gentle and there is no resonance. The effective vibration isolation starts at \(\Omega=0.23\). When \(\overline{a}\) reaches to 0.425, a resonance peak with a tendency to the right occurs on the curve, but the transmissibility does not present obvious nonlinear characteristics. As \(\overline{a}\) increases, the peaks of the transmissibility become larger and move to higher frequency, which results in the shrink of the vibration isolation range. At the same time, the curves begin to show strong hardening nonlinearity with the nonlinear phenomena, such as bend and jump. This is consistent with the cases in Figures 5 and 7, in which increasing \(\overline{a}\) strengthens the nonlinearity of stiffness. In addition, according to Figure 11a, it can be found that when \(\overline{a}\) is large, the resonance response is sensitive to the change of \(\overline{a}\). A small change in \(\overline{a}\) will cause a large difference in response. Therefore, \(\overline{a}\) should be adjusted to make the QZS isolator have a suitable nonlinearity to guarantee the vibration isolation performance. Compared with the transmissibility curves of the equivalent linear vibration isolator with the same supporting stiffness drawn by dashed line, the resonance response of the linear system is eliminated. Even though the value of \(\overline{a}\) is large, the resonance for the QZS isolator still has a lower frequency and smaller amplitude. Figure 12 shows the effects of damping ratio \(\xi\) on absolute displacement transmissibility, when \(\overline{a}=0.45\) and \(\overline{Z}=0.25\).