**3. Active-passive composite vibration suppression system based on piezoelectric actuator**

The passive vibration suppression system has a simple structure and good highfrequency vibration suppression performance, but it is powerless to low-frequency vibration, and the resonance peak suppression conflicts with high-frequency vibration suppression. The vibration suppression performance of the active vibration suppression system is good, but due to the limitation of the working bandwidth and power of the actuator, it is difficult to realize the active vibration suppression only. Therefore, active actuators are usually used in combination with passive elements to form an active-passive composite vibration suppression system to achieve the best vibration suppression effect [22–24].

**Figure 4.** *The relationship between voltage and displacement of piezoelectric actuator.*

#### **Figure 5.**

*Hysteresis adaptive inverse compensation control block diagram of piezoelectric actuator.*

#### **Figure 6.**

*Single-degree-of-freedom vibration suppression system: (a) three-dimensional model diagram; (b) simplified model diagram.*

The active control element based on the piezoelectric actuator has high stiffness. In order to reduce the overall stiffness of the vibration suppression system, thereby reducing the natural frequency of the system and increasing the vibration suppression bandwidth, the active control element and the leaf spring with relatively low stiffness are connected in series to form a vibration suppression system with an active-passive composite structure in this paper, as shown in **Figure 6a**. The vibration suppression system is a single-degree-of-freedom system. The modular design reduces the mass and size of the system to the greatest extent, which makes it more suitable for applications in those fields with strict space and mass constraints, such as aerospace, ships, etc. In addition, according to the needs of the application, it can be conveniently used as the branch chain of the multi-degree-of-freedom vibration suppression

platform through the flexible hinges [13, 25–27]. Its simplified model is shown in **Figure 6b**.

According to Newton's second law, the dynamic equation of the system can be obtained as:

$$\begin{cases} M\_p \ddot{\mathbf{x}}\_o + c\_{ini}(\dot{\mathbf{x}}\_o - \dot{\mathbf{x}}\_m) + k\_{ini}(\mathbf{x}\_o - \mathbf{x}\_m) = F\_x \\ c\_{ini}(\dot{\mathbf{x}}\_o - \dot{\mathbf{x}}\_m) + k\_{ini}(\mathbf{x}\_o - \mathbf{x}\_m) = k\_{add}(\mathbf{x}\_m - \mathbf{x}\_i) \end{cases} \tag{12}$$

where *Mp* is the load mass, *cini* is the initial damping of the system, *kini* is the initial stiffness of the system, *kadd* is the additional stiffness of the leaf spring, *xo* is the load displacement, *xm* is the displacement of the connection point between the displacement amplifying mechanism and the leaf spring, *xi* is the base displacement, *Fz* is the output force of the active control element.

By combining the two equations in Eq. (12) and eliminating the relevant variables at the intermediate connection point, the dynamic model of the single-degree-offreedom vibration suppression system is obtained as follows:

$$M\_p \ddot{\mathbf{x}}\_o + c(\dot{\mathbf{x}}\_o - \dot{\mathbf{x}}\_i) + k\_d(\mathbf{x}\_o - \mathbf{x}\_i) = F\_\mathbf{z} \tag{13}$$

where *c* ¼ *kaddcini=*ð Þ *kadd* þ *kini* is the equivalent damping of the system and *kd* ¼ *kaddkini=*ð Þ *kadd* þ *kini* is the equivalent stiffness of the system.

It can be seen from Eq. (13) that the active control element in series with the leaf spring can effectively reduce the overall stiffness of the vibration suppression system, and the establishment of the structure and dynamic model of the active-passive composite vibration suppression system provides the object and theoretical basis for the subsequent active vibration suppression control.
