## 3 Numerical Simulations
### Simulation Results
#### 3.2.1 Frequency Response  
For a system with passive friction dampers, the frequency responses of the 3-DOF system with different friction force amplitude are shown in Figure 2.  
It can be seen from Figure 2 that dry friction can alter the amplitude and frequency around the resonances. As the amplitude of dry friction increases, the frequency of the first resonance peak increases, and the frequency of the second resonance peak decreases. With the increase in the dry friction amplitude, the peak value of these two resonances peak will at first decrease and then increase, as shown in Figure 2. According to reference [30], the peak value of a frictional system would at first decrease and then increase, which is in accordance with the results shown in Figure 2. In order to achieve a better passive friction damping effect, the normal force applied to the damper needs to be adapted to the system requirements  
The frequency response of the initial noncontrolled system, the system involving only active friction damper, the system involving only active control force, and the system involving the proposed hybrid control strategy are, respectively, shown in Figure 3.  
Figure 3: Frequency response of system without control, with active friction, with active force, and proposed hybrid control.  
Figure 2: Frequency response with friction of different fixed amplitude. (**a**) Frequency response with friction from 0 N to 30 N. (**b**) Frequency response with friction from 0 N to 100 N.  
It can be seen from Figure 3 that for the system using only an active friction damper, the kinetic energy amplitude of the first resonance is slightly reduced, and the frequency is slightly increased. The kinetic energy amplitude of the second resonance is visibly reduced. By comparing the blue line with Figure 2, it can be concluded that active friction damping and passive friction damping have similar characteristics for the 3-DOF oscillator; that is to say, the damping effect for low-frequency vibration is poor, while the damping effect for high-frequency vibration is better. For the system containing only an active control force, the system can achieve a better control effect in the low-frequency band. The peak value of the first resonant is greatly reduced. However, the control effect becomes worse for high-frequency excitation. Another observation is that the system with pure active control force provides no significant shift in the resonant frequencies, which is in accordance with theoretical formulations. For the system with the proposed nonlinear hybrid control, the kinetic energy amplitude around both resonances is greatly reduced. Although the performance around the first resonance frequency is slightly worse than the control effect that only includes the active control force, the performance around the second resonance still justifies its obvious advantages. The proposed control law can greatly attenuate the system's kinetic energy amplitude for both resonance zones, resulting in a global better dynamic performance in the interested frequency band.  
In order to verify the effect of the control law in reducing the active control force, the amplitudes of control forces introduced in the control law for the system with pure active control and the proposed nonlinear velocity compensation control were calculated, as shown in Figure 4.  
It can be seen from Figure 4 that around the first resonance frequency, the nonlinear velocity compensation control law proposed in this manuscript increases the amplitude of the active control force by a maximum of 4%, while around the second resonance frequency, the amplitude of active control force decreases by about 6%. Therefore, the proposed nonlinear hybrid control law can achieve the initial purpose of reducing the external energy input for frequency above 20 Hz for this 3-DOF system.