Research on the radial vibration of piezoelectric hollow cylinder with radial polarization(2)

Piezoelectric ceramic round tube is a commonly used for the acoustic transducer unit. It has simple structure, stable performance, convenient layout, uniform directivity along the radial direction and high sensitivity. Therefore, it is mostly used in the fields of underwater acoustics, geology and petroleum exploration. The vibration characteristics of the vibrator directly affect the dynamic performance of the transducer. The study of its vibration mode is the basis of designing such a transducer. Therefore, this work has important theoretical and practical significance. The circular tubular vibrator is divided into three types: axial, tangential and radial polarization. The axial and tangentially polarized oscillator excitation electrodes are different from the polarized electrodes, and the polarization and excitation voltage ratio of the axially polarized oscillator .

The polarization is much higher, and there is almost no application in the engineering. They are radially polarized, radially excited oscillator, the polarized electrode and the excitation electrode can be combined into one, and the polarization and excitation voltage are also low, which is more in the manufacturing process. There are advantages and practical applications. Regarding the radial vibration mode of the radially polarized piezoelectric tube transducer vibrator, the previous studies have mostly adopted the theory of thin film or thin shell. The thin film theory ignores the shear stress and the radial normal stress in the equation of motion, and the thin shell theory retains the shear stress. The theory applies only to special-sized vibrators, such as thin walls, and ideal situations where the longitudinal and radial dimensions are orders of magnitude greater than the thickness.This brings inconvenience to the application. The predecessors have also studied the radial vibration modes of thick-walled vibrators, but they have adopted different approximations. For example, piezoelectric ceramics are regarded as isotropic materials, and the series is cut off in the operation process. Starting from the piezoelectric and motion equations of the thick-walled slender vibrator of piezoelectric ceramics, combining with the electrostatic charge equation of the vibrator, the radial vibration is studied to obtain the electrical admittance expression, and the resonant and anti-resonant frequency equations of the vibrator are derived. The finite element performs modal analysis. The results show that the theoretical calculation results are in good agreement with the finite element simulation results.

The figure shows a piezoelectric ceramic thick-walled slender tube. For the convenience of research, this paper adopts the cylindrical coordinate system and takes the order of θ -1, z-2, r-3, 2L is the length of the vibrator, and a is the inner radius of the vibrator. , b is the outer radius of the vibrator, and the elongated acoustic piezoelectric cylinder tubes are infinitely long in the z direction, so the piezoelectric vibrator makes an axisymmetric vibration.

In the figure, the polarization direction and the excitation direction of the vibrator are both in the radial direction, that is, the r direction of the piezo tube stack piezoelectric ceramic is subjected to the radial polarization treatment is an isotropic material (isotropic in the θ z direction) perpendicular to the polarization direction, E-type piezoelectric process of axisymmetric vibration of slender tube under cylindrical coordinates

(2)Since the vibration of the slender tube is symmetric about the z-axis, the displacement and electric field components are satisfied: the slender tube is very long, so the study of the slender tube belongs to the plane strain problem, and the displacement and electric field components exist only in the orθ plane.

Mechanical vibration characteristics

Piezoelectric ceramic vibrators are mostly harmonic excitations to use. The electric field and steady-state displacement distributions are subject to the harmonics. The theoretical calculations and finite simulation values of the radial vibration resonance and anti-resonance frequency of the slender tube vibrator are the theoretical calculated values of the effective electromechanical coupling coefficient are in the good agreement with the finite element numerical simulation values, which explains the rationality of the theoretical derivation method for the radial vibration of the slender tube. Table 1 shows the variation of the resonance resonance frequency of the piezoelectric ceramics vibrator with the thickness. It can be seen from the data in the table that the resonance anti-resonance frequency of the vibrator with the same length and the same inner diameter becomes smaller with the increase of the thickness, and the vibrators 2 and 3 can be clearly seen. 4 is a thick-walled vibrator. From the comparison of the calculation results in the table, the theory is applicable to thick-walled vibrators with small errors. Table shows the variation law of the resonance anti-resonance frequency of piezoelectric tubes element vibrators with different lengths. It can be seen from the comparison of the data in the table that the model is satisfied. Under the premise, the resonators with the same inner and outer diameters have different resonance anti-resonance frequencies.

in conclusion

In this paper, combined with the plane strain and three-dimensional piezoelectric elastic mechanics theory, the radial vibration of the radially polarized thick-walled piezoelectric ceramic slender tube is carried out.The study obtains the exact solution of the displacement function and the potential function. For the first time, the equivalent conductance nano-hole of the thick-walled vibrator is studied, and the exact resonant anti-resonance frequency equation is obtained and verified by finite element analysis. The above analysis shows that the thick-wall theory is used in this paper. In summary, the following conclusions are drawn by aspects. 

(1) The radial vibration of the radially polarized thick-walled slender tube belongs to axisymmetric vibration; 

(2) The slender tube problem is simplified to the plane strain problem when the model is established. The relative error term can be seen that the wall thickness has no effect on the calculation results when the model requirements of piezoceramic tube transducer are met, and the accuracy is higher than the film and thin shell theory;

(3) In the actual use, the slender tube only needs to satisfy . If it is not satisfied, there may be some error;

(4) The resonance frequency of the vibrator is analyzed by numerical method. The results show that the resonance frequency and numerical method derived from the theory are used to obtain the resonance frequency and numerical method.