## 3 Numerical Simulation  
To confirm the dimensional sizes of the framed piezoelectric actuator and to analyze its dynamic behavior, we conducted the numerical simulation in this section by using the commercial software COMSOL based on the finite element method. The four straight beams are made of titanium alloy (TC4), the PZT-8 type piezoelectric ceramic is selected as the piezoelectric plates, and their material parameters are listed in Table 1. Due to the fact that the selected piezoelectric plate bonded on the long straight beams has a dimension size of 14 mm \(\times\) 8 mm \(\times\) 2 mm, the width of the long straight beam is determined to be 8 mm. Additionally, piezoelectric plates with a dimension size of 10 mm \(\times\) 5 mm \(\times\) 1 mm bonded on the short straight beams confirm the height of the short straight beam in 5 mm. The other geometric parameters of the framed piezoelectric actuator were optimized in order to minimize the difference between the two operating resonant frequencies and to isolate other interference modes. The finite element model of the framed piezoelectric actuator was created, as shown in Figure 3. The type of all elements is free tetrahedral, and this finite element model was divided into 49,777 elements. The MUMPS (Multifrontal Massively Parallel Sparse Direct Solver) eigensolver was adopted for this finite element simulation. During calculation, all mechanical boundary conditions were free, and the zero voltage was applied to all piezoelectric plates as the electrical boundary condition. After a series of modal analysis simulations, the optimized geometrical sizes of the framed piezoelectric actuator were determined. The length of the short straight beam is 34 mm, and its cross-section is 5 mm \(\times\) 6.5 mm. The long straight beam is 104 mm in length and has a cross-section of 8 mm \(\times\) 8 mm. The two operating vibration modes were obtained, as shown in Figure 4. In the mode A, the two short straight beams vibrate at the first order bending vibration along the \(x\)-direction, and its corresponding resonant frequency is 20.498 kHz, as illustrated in Figure 4a. The resonant frequency of the mode B is 20.508 kHz when the third order bending vibration and first order bending vibration both along the \(y\)-direction are stimulated in the long and short straight beams, respectively, as shown in Figure 4b. The frequency  
Figure 2: Operating principle of the proposed framed piezoelectric actuator. (**a**) Mode A: the first order bending vibration along the \(x\)-direction excited in the two short piezoelectric composite beams. (**b**) Mode B: the first order bending vibration along the \(y\)-direction stimulated in the short piezoelectric composite beams while the third order bending vibration along the \(y\)-direction excited in the long piezoelectric composite beams. difference of the two simulated resonant frequencies for the two operating vibration modes is 10 Hz, indicating that the simulation results meet the design requirement of the framed piezoelectric actuator. Based on the optimized geometrical sizes of the framed piezoelectric actuator, the harmonic response analysis was carried out to study its dynamic behaviors. At first, we simulated the vibration shape variation during one excitation period. Two electric signals with a temporal phase difference of  
\begin{table}
\begin{tabular}{c|c c  
\(\pi\)/2 were applied to the piezoelectric plates located in the long and short straight beams, respectively, and an exciting frequency of 20.503 kHz was selected during the simulation. During one excitation period, the vibration shape of the framed piezoelectric actuator can be divided into four stages (from time \(t_{0}\) to \(t_{4}\), in which the period \(T=t_{4}-t_{0}\)), as shown in Figure 5. It indicates that the vibration shapes of the framed piezoelectric actuator varied from (1) to (4) during one excitation period, in which the two driving tips moved with a completed elliptical trajectory. Then, the motion characteristics of the two driving tips were calculated.