## 3 Numerical Simulations
### Simulation Results
#### 3.2.2 Time Response of Certain Frequencies  
The transient time response of the system is also an important reference for investigating the control law effect. Time response curves of the 3-DOF system around the first resonance frequency of the aforementioned systems are depicted in Figure 5, and the excitation frequency is set to 13 Hz. Figure 4: Frequency response of active control force of system with active control and proposed hybrid control. Since friction dampers only exist between \(m_{1}\) and \(m_{2}\), the dynamic performance of these two masses is more sensible to friction forces. By comparing the time-domain responses of the system without control force and with active friction damping, it can be seen that active friction damping makes the amplitude of the two connected masses more even, and at the same time, it will slightly reduce the amplitude of \(m_{3}\). However, as concluded from Figure 3, the damping effect of active friction damping is not obvious at this frequency. The active control force can obviously reduce the amplitude of all these three masses. At the same time, the two curves without active control force both approach the steady state after two or three cycles, while the curve with active control force only needs one cycle to reach the steady vibration, so the active control force shows faster response speed. Under this excitation, the effects of active control and the proposed nonlinear velocity compensation control show similar response speed and characteristics. Figure 6 illustrates the time histories of active control force of pure active control and that of this hybrid control. When observing Figure 6, it can be found that the proposed nonlinear hybrid control method will slightly increase the active control force on both \(m_{1}\) and \(m_{2}\). The active control forces of the hybrid control applied on \(m_{1}\) and \(m_{2}\) are about 3% more than that of the pure active control, which accords with the analysis of Figure 4. Therefore, in the low-frequency band, the proposed nonlinear hybrid control method will increase the energy input of the external active control force. Figure 7 shows the displacement time histories of the three masses under the excitation frequency of the second resonance frequency; that is to say, the excitation force frequency is 25 Hz. Figure 5: Time histories of the system under 13 Hz excitation. (**a**) Time response of \(m_{1}\) (**b**) Time response of \(m_{2}\) (**c**) Time response of \(m_{3}\). Figure 6: Time histories of active control forces under 13 Hz excitation. (**a**) Time response of \(f_{c_{1}}\) (**b**) Time response of \(f_{c_{2}}\). By comparing the three figures in Figure 7, it can be seen that, around the second resonance, the amplitude of \(m_{1}\) in the noncontrolled system is much larger than that of \(m_{2}\) and \(m_{3}\). The amplitude of \(m_{1}\) and \(m_{2}\) are closer for the system with control. By comparing the time histories of the system with active friction damping and that with the proposed nonlinear velocity compensation control, it can be seen that the control effects of the system with active control for \(m_{2}\) and \(m_{3}\) are slightly better than that of the proposed hybrid control, while the advantage of the proposed nonlinear hybrid control for \(m_{1}\) can quite well compensate this disadvantage. The control effect of the active friction control is visibly worse than the proposed hybrid control. At the same time, the existence of friction force between \(m_{1}\) and \(m_{2}\) could make the amplitudes of the two masses more even. Similarly, Figure 8 has shown the advantage of the proposed nonlinear velocity compensation control in reducing the input energy of the active control force. From Figure 8a, it can be found that the proposed nonlinear hybrid control method can obviously reduce the active control force applied on \(m_{1}\). By inspecting Figure 8b, it can be seen that the control force of the proposed hybrid control is slightly larger than the pure active control.