# Study of the Lifetime Testing of Spherical Rubber Isolators
## 4 Result Analysis and Prediction  
This section mainly analyzes the results of the vibration experiment and the accelerated lifetime experiment and establishes the life prediction model of the rubber isolator. Figure 6 illustrates the experiment results of the frequency response function of the rubber isolator subjected to harmonic and random vibrations. Figure 6a plots the result of the harmonic vibration FRF, in which the resonance peak value is reduced from around 220 Hz under a low excitation level to 110 Hz under a high excitation level. Figure 6b indicates that as the power spectral density of the excitation increases, the resonant frequency at resonance decreases (from 80 Hz under low excitation to around 50 Hz), whereas the amplification factor of the amplitude shows a rising trend. The experimental results demonstrated that the rubber isolator demonstrated nonlinear stiffness and damping characteristics when it was excited. According to the stiffness and damping algorithm scheme shown in Figure 5, as well as the results of the harmonic vibration experiment, the relationship among nonlinear stiffness, nonlinear damping, and relative displacement amplitude can be established. Figure 7 illustrates the nonlinear dynamic characteristics of the spherical rubber isolators. Notably, both stiffness and damping are negatively correlated with the displacement amplitude, and the correlation is nonlinear. Figure 8 shows the sine frequency sweep results of the accelerated lifetime test when the load mass is [50; 60; 70] g. As indicated by the figure, for the accelerated lifetime test with constant load, the resonance peak value and the amplitude amplification factor show a downward trend as the random vibration repeats. The root-mean-squared value of acceleration can be calculated by Equation (9), which also shows a downward trend. In addition, as the load mass increases, the initial resonance frequency decreases. isolator stiffness were established. Figure 9a illustrates the results of repeated accelerated lifetime test when the load mass is 50 g, indicating that stiffness decreases as the time of the experiments increases. It is illustrated that under the same load mass, the stiffness decay of the rubber isolator remains basically the same every time a random vibration experiment is repeated. Therefore, the arithmetic mean value of the stiffness decay after each random vibration experiment of repeated experiments is calculated to represent the stiffness value corresponding to the acceleration mean value under the load. In the early stage of the accelerated lifetime test, the stiffness decreases significantly. As the random vibration experiment repeats, the stiffness declines stably. Finally, when the stiffness approaches the experimental cut-off threshold, the stiffness drops sharply. Figure 9b shows that as the load mass increases, the isolator stiffness decreases even more. Notably, the stiffness value at the end of the experiment decreases, indicating that the load mass of the random vibration experiment produces a profound effect on the fatigue lifetime of the rubber isolator. After the stiffness results of the experiment were converted into logarithm form, a correlation function relationship was established between the stiffness and the mean value of the random vibration acceleration [16]. This correlation can be used to predict the fatigue damage of the rubber isolator. The polynomial function was used to fit the correlation between the stiffness and acceleration mean value. The equation is as follows:  
\[Log(K)=\alpha_{1}\alpha_{rms}^{3}+\alpha_{2}\alpha_{rms}^{2}+\alpha_{3}\alpha _{rms}+\alpha_{4} \tag{10}\]  
Table 1 shows the coefficient fitting values under different loads and the Root-Mean-Squared Error (RMSE) calculated by Equation (10). The RMSE represents the deviation between the predicted value of stiffness and the experimental test value. A smaller RMSE value demonstrates a more accurate stiffness prediction value. In this paper, the calculated RMSE value under different loads is very