# Effective Frequency Range and Jump Behavior of Horizontal Quasi-Zero Stiffness Isolator
## 2 Isolator Subjected to Base Excitation
### Jump Behavior  
In order to verify the accuracy of this derivation, the semi-analytical solutions obtained via the complexification-averaging method and the numerical solutions obtained using the Runge-Kutta method were compared, the results are shown in Figure 3. The parameters that describe the system being modeled here are:  
\[M=10\;\mathrm{kg},\;\mu=10\;\mathrm{N}\cdot\mathrm{s}/\mathrm{m},\;K=5000\; \mathrm{N}/\mathrm{m},\;L=0.2\;\mathrm{m},\;\mathrm{and}\;l=0.201\;\mathrm{m}.\]  
Figure 3a,b shows the amplitude-frequency response curves of the system obtained for \(P=0.006\;\mathrm{m}\) and \(P=0.009\;\mathrm{m}\), respectively. The semi-analytical solutions are obtained by solving the steady-state response equations derived from Equation (4) using the least squares method, and the numerical solutions are obtained by solving Equation (1) directly via the Runge-Kutta method. The solution processes of the least squares method and Runge-Kutta method were both carried out in the Matlab software package. The results  
Figure 3: Amplitude–frequency responses of the isolator. Results are shown for **(a)**\(P=0.006\;\mathrm{m}\) and **(b)**\(P=0.009\;\mathrm{m}\); SSAS: stable semi-analytical solution, USAS: unstable semi-analytical solution, NS: numerical solution. indicate that the solutions obtained via the complexification-averaging method and the Runge-Kutta method are in good agreement under most excitation frequencies and the deviation between the solutions of the peak point is acceptable, which verifies the correctness of the theoretical derivation. In addition, the response of the vibration isolation system in Figure 3a is close to that of a linear system, and all solutions are stable. However, in Figure 3b, the frequency-response curves are clearly tilted to the right, and there exist three solutions within the frequency range considered; two of them are stable, and the third solution is unstable. The existence of multiple solutions is a necessary precondition for the jump phenomenon to occur. The response of the system will jump from a lower value to a higher value when the jump condition is met. In this work, the jump-up phenomenon is mainly studied (jump-down is not involved) because a nonlinear vibration isolator can lose its efficacy when the jump-up occurs. Typically, the different solutions that exist at the given frequency can be obtained by assigning different initial conditions to the system in the numerical simulations. In this paper, when the system is in a given harmonic excitation, an impact load is applied to induce the jump behavior in the system; that is, the system vibrates according to the solution in the lower branch before the jump occurs and vibrates according to the solution in the higher branch after the impact has occurred. This simulation method is more consistent with the failure process in the actual world. Figure 4 shows the displacement transmissibility of the vibration isolation system for two different input displacement amplitudes. The isolator is effective when the transmissibility is less than 1, and the isolator loses efficacy when the transmissibility is greater than 1. When the displacement transmissibility is large, the isolator amplifies the vibrations present in the system, which is not conducive to the protection of the isolated structure. Figure 5a shows the response of the isolator for \(P\) = 0.009 m, and \(\Omega\) = 4.5 rad/s. The isolator generates periodic vibrations under harmonic excitation. The amplitude of the isolator is significantly increased after the impact load is applied to the isolator. The expression of the impact load is as follows:  
\[F= \begin{cases}f\sin\bigl{(}\frac{2\pi t}{T}\bigr{)},&0\leq t\leq\frac{T}{2}\\ 0,&t>\frac{T}{2}\end{cases}