Research on the radial vibration of piezoelectric hollow cylinder with radial polarization

Combined with piezoelectric plane strain and three-dimensional piezoelasticity theory, the vibration characteristics of a piezoelectric ceramic thick hollow cylinder with radial polarization has been studied, and closed-type solutions are obtained for the mechanical radial displacement and electric potential. The electric displacement and electric field strength are derived form the charge equation of electrostatics, which solves the problems of nonlinear relationship between voltage and electric field strength. Based on maple software, the equivalent admittance of the thick hollow cylinder is studied for the first time, and corresponding exact resonance and anti-resonance frequency equations are also obtained. By means of numberical method, the resonance and anti-resonance frequencies of different size tubular oscillators are calculated. Accuracy and precision of this theory are verified by finite element analysis. All these provide the basis for theoretical research and design of the piezoelectric ceramic thick oscillators.

Piezoelectric ceramic round tube is a commonly used for acoustic transducer. 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 for 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 vibrator electrodes are different from the polarized electrodes, and the polarization and voltage has the ratio of the axially polarized vibrator .The polarization is much higher, and there is almost no application in engineering, 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 tubular vibrator, previous studies have mostly adopted the theory of thin film or thin shell The thin film theory ignores the shear stress and radial stress in the equation of motion, and the thin shell theory retains the shear, stress, and the above theory is only applicable to special-sized vibrators, such as thin walls, and the ideal situation where the longitudinal and radial dimensions are many orders of magnitude larger than the thickness, thus causing inconvenience to the application. Previous studies have also studied the radial vibration mode of thick-walled vibrators.

However, different approximations are used. For example, piezoelectric ceramics are regarded as isotropic materials, and the series of truncated approximations are taken during the operation. The piezoelectric ceramics and motion equations of the radially-polarized acoustic piezoelectric tubes thick-walled slender vibrators are derived from the radial polarization. Starting with the electrostatic charge equation of the vibrator, the radial vibration is studied, and the electric admittance expression is obtained. The resonance and anti-resonance frequency equations of the vibrator are derived. The modal analysis is performed by ANSYS finite element. The results show that the theoretical calculation results are limited. The meta-simulation results are in the good agreement.

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 it is the inner radius of the vibrator. b is the outer radius of the vibrator, and the elongated tube is 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, and the piezoelectric ceramic is subjected to the radial polarization treatment,which 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

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 piezo tube stack belongs to the plane strain problem, and the displacement and electric field components exist only in the orθ plane.

Mechanical vibration characteristics

Cylindrical piezoelectric tube are mostly harmonic excitations in use. The electric field and steady-state displacement distributions are subject to the harmonics. The theoretical calculations and finite element numerical simulation values of the radial vibration resonance or 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 above theoretical derivation method for the radial vibration of the slender tube. Table shows the variation of the resonance frequency of the vibrator with the thickness. It can be seen from the data in the table that the resonance or anti-resonance frequency of the vibrator with the same length and the same inner diameter becomes smaller with the increasing of the thickness, and the vibrators 2 and 3 can be clearly seen. it 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 of the resonance anti-resonance frequency of 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 or anti-resonance frequencies.

in conclusion