The active control study of a damped two-level smart spring system

The vibration reduction performance of a damped two-level smart spring system is studied. The governing equations of the smart spring system are reduced to a first-order ordinary differential equation of matrix form. An active control algorithm employing the relative velocity relationships between the primary and auxiliary systems to prevent the movement of the controlled object is applied in the simulation. Different excitation types are considered to fully examine the validity of the control law. Responses of the controlled object under the control law are obtained and compared to the case of constant friction and the case of no control. The study indicates that the active control algorithm shows a good vibration reduction efficiency and a wide vibration reduction bandwidth.


Introduction
The vibration reduction technology based on smart spring controls [1] the action of the piezoelectric ceramic actuator by applying an electric field to change the dynamic characteristic parameters of the system, such as damping, stiffness characteristics, and suppress the transmission of vibration displacement or reaction force indirectly [2].Compared with the way of directly suppressing the excitation force by using a piezoelectric ceramic actuator, it does not require a complex displacement amplifying device, nor a high driving voltage.It is easier to implement and has good application prospects and research value.
The smart spring damping system contains a dry friction oscillator, which is a typical strong nonlinear system and contains complex stick-slip vibration phenomena.The response of the smart spring system with dry friction can be solved by the analytical method and numerical method.Smart springs can be used as passive or active vibration reduction devices.The advantages of passive vibration damping devices are that they are simple in structure and easy to install, but the vibration reduction frequency band is narrow.Therefore, it only has a good vibration reduction effect for a specific frequency or narrow frequency bandwidth.The active control or semi-active control smart spring has a good vibration reduction effect for a wide frequency band, and has a wide range of applications, especially for helicopters with multiple excitation frequencies, such as variable speed helicopters.
Experiments have been conducted on the dynamics of smart spring [3][4][5], and simulations are also performed to show the effect of smart spring in the IBC (individual blade control) control [6].Active control algorithms are also proposed to improve the vibration reduction effects.For example, Zhu et al. [5] conducted experiments on the smart spring system in a three-support shafting subcritical vibration control.Li et al. [7] analyzed the parameters that would influence the behavior of the smart spring system a step further.Wander et al. [8] proposed several semi-active control algorithms for smart spring systems, and simulations are carried out.Nitzsche et al. [9][10][11] performed the maximum energy extraction algorithm for a smart spring system.The controlled object is subjected to base chirp excitation; however, damping is not considered in the simulation.
The dynamic behavior of a damped two-level smart spring system is studied.The governing equations of the smart spring under certain excitation forces are given, and the order is reduced to first-order ordinary differential equations.According to the relationship between the movement speed and the relative speed of the main and auxiliary systems of the smart spring, an active control algorithm is employed.Various forms of excitation forces are considered, and passive control and nocontrol cases are also taken into consideration to test the validity of the control law.Responses of the system are obtained and the vibration reduction effects are analyzed.

Mathematical modelling of the smart spring system
The basic single-level smart-spring structure is given first, as the structure is relatively simple and the mechanism is easy to understand.The single-level smart spring system consists of a primary massspring system and an active mass-spring system, as shown in figure 1(a).The primary spring and the active spring are arranged in parallel, and the stiffnesses are K 1 and K 2 , respectively.The equivalent masses of the vibration structure and other components associated with the primary spring is m 1 , and the equivalent masses of the piezoelectric ceramic actuator structure and other components associated with the active spring is m 2 .For the convenience of description, the primary mass-spring system is called the main system and the active mass-spring system is the auxiliary system.The upper end of the main system is subjected to external excitation force F(t), and the lower end transmits the vibration energy to the target object, that is, the smart spring device is located in the transmission path from the vibration source to the target body.
The smart spring is an indirect active vibration control device.It adjusts the friction force F d through the actuator to control the degree of contact between the auxiliary system and the main system, and actively adjusts the dynamic impedance characteristics of the vibration control system to suppress the transmission of vibration energy.The main system in figure 1(a) is the controlled object, and the task of the auxiliary system is to suppress the vibration response of the main system and the transfer of vibration energy to other objects.There is an initial gap between the auxiliary system and the main system.When the actuator is not working, there is no contact between the friction surfaces.The auxiliary system has no contribution to the dynamic impedance characteristics of the vibration control system.The main system is separated from the auxiliary system, and the motions of the two systems are independent of each other.When the actuator works, the friction surfaces are in contact with each other.Now consider a more complex and more realistic situation.As shown in figure 1(b), the smart spring system is subjected to excitation force F(t), the mass of the object is m 1 , the displacement of m 1 is x 1 , and the stiffness of the spring connected to it is K 1 .The mass of the main system is m 2 , the displacement is x 2 , and the stiffness K 2 .The mass of the auxiliary system (the actuator) is m 3 , the displacement is x 3 , and the stiffness K 3 .The kinetic friction coefficient of the main and auxiliary systems is v, and the normal force acting on the contact surface is N.The equation of motion of the smart spring system can be written as, By introducing a new displacement vector , , , , , , the equations of motion of the three objects can be reduced as, where The above ordinary differential equation can be solved numerically.

The control law for the smart spring system
The relationship between the speed and the relative of the main system m 2 and the auxiliary system m 3 , 2 x and 3 x   , is employed to determine whether the piezoelectric actuator applies frictional force to the main system.Specifically, the following formula is used as the criteria: x x x  could be positive, negative, or zero.
Occasion 1:   It is shown that the main system moves in the positive direction, and the relative speed of the auxiliary and main systems moves in the positive direction, indicating that the auxiliary system promotes the movement of main the system.Therefore, the piezoelectric actuator should be separated from the main system without providing friction forces.
Sub-occasion 2: It is shown that the main system moves in the negative direction, and the relative speed of the auxiliary and main systems moves also in the negative direction, indicating that the auxiliary system promotes the movement of the main system, and the vibration amplitude increases.Therefore, the piezoelectric actuator should be separated from the main system.
It is concluded from Occasion 1 that when   Sub-occasion 1: It is shown that the main system moves in the positive direction, and the relative speed of the auxiliary and main systems is in the negative direction, indicating that the auxiliary system hinders the movement of the main system and plays a role in reducing the amplitude of the controlled object.Thus, the piezoelectric actuator and the auxiliary system should continue to be combined to provide friction force.
Sub-occasion 2: It is shown that the main system moves in the negative direction, and the relative speed of the auxiliary and main systems moves in the positive direction, indicating that the main system hinders the movement of the auxiliary system and plays a role in reducing the amplitude of the controlled object.Therefore, the piezoelectric actuator and the auxiliary system should continue to be combined to provide friction forces.
It is concluded from Occasion 2 that when    , the piezoelectric actuator should be in contact with the main system, i.e., a high friction force F d =F high acts on the main system to prevent the movement of the main system.Therefore, in this control algorithm, the relationship between the speed and the relative speed of the main and the auxiliary system is the criterion for determining whether the piezoelectric actuator applies friction force.The control law could be written as, F last indicates the frictional force value of the previous round is used if the product two terms are zero.According to the above control algorithm and the first-order ODE matrix, simulations are performed employing the Matlab Simulink tool to verify the validity of the control law.The block diagram is shown in figure 2, and different types of excitations are considered.
The parameters used are as shown in Table 1 (with SI units): Table 1.parameters of the system (SI units).A chirp signal with a frequency range from 1 to 20 Hz is applied to the system at first, and the response with the active control law, a constant friction force control of F high , and a zero-friction force of F low are depicted in figure 3 (a)-(c).The constant friction force of F high in figure 3(b) represents the case that a constant voltage is given to the actuator, and the main and auxiliary system maintains a constant friction force.The zero-friction force of F low in figure 3(c) means the main and auxiliary systems are separated and no friction is acted on the main system, (i.e., no control).It is seen that the largest response is found in figure 3 (c) since there is no control, and the second largest response is found in figure 3 (b) with constant friction control, while the active control case in figure 3(a) shows the minimum response.Resonances are found in figure 3 (b)-(c) while the response in figure 3(a) remains at a low level throughout the chirp signal.The maximum response in figure 3(a) is merely 1/10 of that in figure 3(b).To have a better understanding of the dynamics of the system, a frequency sweep of the sine signal is acted on m 1 , with a frequency range from 1 to 20 Hz.The active control law, the constant friction force of F high , and the zero-friction force of F low are applied respectively for comparison.The amplitude of the response is depicted in figure 4. Resonance is observed at the frequency of 9.6 Hz for the case with friction force of F low , which is the natural frequency of the system made of m 1 and m 2 .Similarly, resonance frequency 10.7 Hz is observed for the case with friction force of F high , where m 2 and m 3 are combined together due to the friction force.It is concluded that a constant friction force would reduce the vibration in a certain bandwidth while creating another resonance point (for example 10.7 Hz in this case), and the active control law could maintain a relatively low amplitude in the whole frequency range.

Parameters
Defining the displacement transfer rate as the ratio of amplitude under control law (active control law or constant friction control) to the amplitude of no control (i.e., the case with friction force of F low ), the displacement transfer rates under active control law and constant friction control are shown in figure 5 (a)-(b).The displacement transfer rate of the active control law is less than 50% when the excitation frequency is between 4 to 10 Hz, while the constant friction control only shows a good vibration reduction effect near 10 Hz.
Figure 6 shows the time history comparison of main system m 2 under three cases: in the first 5 seconds no control is added (zero-friction force F low ); 5-10s, a constant friction force F high is added; in 10-15s, the constant friction force F high is removed and active control law is applied.The response decreases gradually under the three stages.Figure 7 (a)-(b) show the time history and velocity of the auxiliary system m 3 in another simulation: no control (zero-friction force F low ) in the first 10 seconds and then active control law is added after 10s.The amplitude and velocity of the auxiliary system increase immediately as the control law is applied, which means the energy has been transferred from the main system to the auxiliary system.Then other types of excitations are applied to the system, as shown in Figure 8. Figure 8 (a)-(c) shows the excitations of a random wave, square wave, and pulse wave, with maximum amplitude F(t)=500, and the response under each excitation is displayed in figure 8 (d)-(f).The active control law is applied in the last half-time of figure 8 (d)-(f).Results show a good vibration reduction effect in all three kinds of excitations, which indicates that the control law could be useful for various vibration occasions.

Conclusions
The dynamic performance of a damped two-level smart spring system under active control law is studied.The governing equation of the system is given and reduced to a first-order ordinary differential equation matrix form.According to the relationship between the speed and the relative speed of the main system and the auxiliary system, an active control algorithm is employed.And simulation is performed by applying the Matlab Simulink tool to obtain the vibration reduction performance.Different kinds of excitation forms are considered, and constant friction control and no control are also included to examine the validity of the control law.The following conclusions could be drawn from the study: (1) The active control law shows good vibration reduction performance for a wide range of frequencies, and the response of the main system (the controlled object) maintains a relatively low level in the whole frequency band.The constant friction control is effective in a certain frequency while causing another resonance point.
(2) When the active control law is applied, the energy is transferred from the controlled object to the auxiliary system (the piezoelectric actuator), thus the velocity and displacement of the auxiliary system increase rapidly as the control law is applied.
(3) The control law shows a good vibration reduction effect for different kinds of excitation forms including chirp wave, sine wave, random wave, square wave, and pulse wave.Therefore, it is of great potential for industrial usage for vibration reduction.

Figure 1 .
Figure 1.Schematic diagram of the smart spring system.
 , the piezoelectric actuator should be separated from the main system, i.e. a low friction force F d =F low acts on the main system.

Figure 2 .
Figure 2. Block diagram of the smart spring system with the control law.

Figure 3 .
Figure 3. Response of the system with chirp excitation force under three control cases.

Figure 4 .
Figure 4. Frequency sweep response of the system under three control cases.

Figure 6 .
Figure 6.Time history under three different cases.

Figure 7 .
Figure 7. Displacement and velocity of the active mass m 3 .

Figure 8 .
Figure 8. (a)-(c): Random wave excitation, square wave excitation, and pulse wave excitation; (d)-(f):Responses of the main system m 2 under the three different types of excitations, respectively.