## 5 Simulation and Experiment
### Simulation and Analysis  
Regarding the dynamic model described in Section 4, Equation (15) could be solved using MATLAB2015b software (MathWorks(tm), Natick, MA, USA), and different input parameters yielded different results. The simulation and experimental input parameters were set as follows. To mimic the influence of sea waves, which reportedly induce rocking motion periods of 3-12 s for ships [35], we set period T at 12 s and 3 s and amplitude \(A\) at \(15^{\circ}\) and \(3^{\circ}\). The linear spring damp chain parameters \(k\) and \(c\) are equal  
Figure 8: The influence of input disturbance amplitude on transmissibility. to 200 N/m and 10 Nm/s, respectively. Figure 10 shows the simulation output and the experimental output at a disturbance frequency \(f=0.0833\) and different amplitudes. Figure 10: Simulation and experimental outputs at \(f=0.0833\). (**a**) \(A=15^{\circ}\), simulation and experimental output; (**b**) \(A=3^{\circ}\), simulation and experimental output. Figure 9: Experimental vibration isolation system. Satisfactory vibration isolation is achieved with various inputted disturbance frequencies. In Figure 10a, the disturbance amplitude is theoretically expected to decrease by 66% at an input disturbance of \(A\sin(0.167\pi t)\), \(A=15^{\circ}\). However, the experimental results show that the vibration attenuation is about 53%, which has a deviation of 2\({}^{\circ}\) from theoretical value. Also, the phase angle is smaller compared with theoretic calculation. However, the experimental and simulation results exhibited the same trend. Figure 10b shows good theoretical vibration isolation at a maximum amplitude of 0.26\({}^{\circ}\) and an input disturbance of \(A\sin(0.167\pi t)\), \(A=3^{\circ}\). The experimental result varied greatly, with a maximum deviation of 0.62\({}^{\circ}\). In addition, the experimental results also show that the amplitude of the isolation is almost the constant over time. However, there exist slight center drifts, and the periodic characteristics are shown with the time span about 140 s (\(\approx\)12T), which corresponds to the results of the theoretical simulation. Overall, the isolation performance is good, despite the small deviation with the central position around 0\({}^{\circ}\). Figure 11 shows the results of a simulation and an experiment conducted at an input disturbance frequency of 0.3333 Hz and input amplitudes of 15\({}^{\circ}\) and 3\({}^{\circ}\). Figure 11: Simulation and experimental outputs at \(f=0.3333\). (a) \(A=15^{\circ}\), simulation and experimental output; (b) \(A=3^{\circ}\), simulation and experimental output. At an input disturbance of \(A\sin(0.6667\pi t)\), \(A=15^{\circ}\), 88% of the vibration is reduced. According to the simulation, the excitation amplitude is small when the output frequency fluctuated. The experimental results in Figure 11a demonstrate that the vibration attenuation is about 82%, and the actual deviation is quite small (\(\approx\)0.8\({}^{\circ}\)) compared to the theoretical value. The results show that the amplitude of the isolation is almost constant during the experiment time. However, there exist slight center drifts, and the periodic characteristics are shown with the time span of about 25 s (\(\approx\)8T), which corresponds to the results of the theoretical simulation. There are two reasons for the results. One is due to the joint friction caused by vibration isolation output from the center of the balance position; the other is because of the nonlinear characteristic caused by the inherent low-order harmonics of the system. At a low amplitude (3\({}^{\circ}\)), the maximum deviation was 0.27\({}^{\circ}\) (Figure 11b), indicating excellent theoretical vibration isolation. The experimental results demonstrated that the vibration amplitude was approximately 0.43\({}^{\circ}\) around the equilibrium position of 0\({}^{\circ}\).