# Study of the Lifetime Testing of Spherical Rubber Isolators
## 3 Modeling  
This study mainly investigated the vibration mode of the rubber isolator in the Z direction, and the rubber isolator was modeled as a single degree-of-freedom system [13]. The schematic diagram of the model is shown in Figure 4. M represents the load mass of the isolator, and K and C represent the stiffness and damping of the system, respectively.  
For a given input acceleration excitation of the shaking table, the dynamic equation of a single degree-of-freedom system is as follows:  
\[M\ddot{x}(\mathrm{t})+C\dot{x}(\mathrm{t})+Kx(\mathrm{t})=F(\mathrm{t}) \tag{1}\]  
\(y(\mathrm{t})\) is the input displacement of the system. The equation can be rewritten as:  
\[M\ddot{x}+C(\dot{x}-\dot{y})+K(x-y)=0 \tag{2}\]  
where \(\omega_{0}=\sqrt{K/M}\), \(\xi=C/2M\omega_{0}\) are respectively defined as the undamped circular frequency and damping ratio.  
Figure 4: Single degree-of-freedom system model.  
The simplified kinetic equation is as follows:  
\[\ddot{x}+2\omega_{0}\xi\dot{x}+\omega_{0}{}^{2}x=\omega_{0}{}^{2}y+2\omega_{0} \xi\dot{y} \tag{3}\]  
The transfer function \(T(\omega)\) is obtained by the application of the Laplace transform on the left and right of the equation.  
\[T(\omega)=\frac{1+2i\frac{\omega}{\omega_{0}}\xi}{1-\left(\frac{\omega}{\omega _{0}}\right)^{2}+2i\frac{\omega}{\omega_{0}}\xi} \tag{4}\]  
Accordingly, when \(\omega_{\max}=\omega_{0}\sqrt{1-2\xi^{2}}\), the transfer function \(T(\omega)\) amplitude reaches the maximum value \(\left|\left|T(\omega)\right|\right|_{\max}\approx 1/2\xi\).  
Figure 5 illustrates the algorithm flow for solving the stiffness and damping of the rubber isolator at the resonance peak. The FRF with respect to a given acceleration excitation could be obtained through the harmonic experiment [14]. Then, the maximum amplitude \(\hat{X}_{\max}\) could be determined, and the damping ratio \(\xi\) could be calculated by \(\hat{X}_{\max}=\left|\left|T(\omega)\right|\right|_{\max}\approx 1/2\xi\). The undamped circular frequency \(\omega_{0}\) could be calculated by \(\omega_{0}=\omega_{\max}/\sqrt{1-2\xi^{2}}\). Through the formula derivation \(\omega_{0}=\sqrt{K/M}\xi=C/2M\omega_{0}\), the damping and stiffness under a given acceleration excitation could be obtained. Through the calculation of the accelerated lifetime test, the stiffness decay of the rubber isolator, i.e., the failure data, could be obtained.  
According to [15], the mean-squared response of acceleration can be defined as:  
\[D_{A}=S_{0}\frac{\pi^{2}\omega_{0}}{\psi_{0}} \tag{5}\]  
Figure 5: Stiffness and damping algorithm scheme.  
where \(S_{0}\) (g\({}^{2}s/rad\)) defines the acceleration power spectral density and \(\psi_{0}\) defines the energy dissipation coefficient at the resonance peak. The following acceleration root-mean-squared response can be obtained:  
\[\text{a}_{rms}=\pi\sqrt{\frac{S_{0}\omega_{0}}{\psi_{0}}} \tag{6}\]  
At the resonance peak, the energy dissipation coefficient corresponds to the acceleration transfer rate in the form of:  
\[\psi_{0}=2\pi/\eta_{0} \tag{7}\]  
Substituting Formula (7) into (6), the following formula can be obtained:  
\[\text{a}_{rms}=\sqrt{\frac{\pi S_{0}\omega_{0}\eta_{0}}{2}} \tag{8}\]  
After converting the angular frequency to frequency, it becomes:  
\[\text{a}_{rms}=\sqrt{\frac{\pi Sf_{0}\eta_{0}}{2}} \tag{9}\]  
where \(S\) (g\({}^{2}\)/Hz) is the acceleration power spectral density and \(\eta_{0}\) is the transfer efficiency at the resonance peak, i.e., the amplitude amplification factor. According to the resonance frequency and amplitude amplification factor obtained by the sweep frequency experiment, Equation (9) can be used to estimate the value of the root-mean-squared acceleration when the power spectral density under vibration equals \(S\).