Radiation impedance and equivalent circuit for piezoelectric ultrasonic composite transducers of vibrational mode-conversion

The piezoelectric ultrasonic composite transducer, which can be used in either gas or liquid media, is studied in this paper. The composite transducer is composed of a longitudinal sandwich piezoelectric transducer, a mechanical transformer, and a metal circular plate in flexural vibration. Acoustic radiation is produced by the flexural circular plate, which is excited by the longitudinal sandwich transducer and transformer. Based on the classic flexural theory of plates, the equivalent lumped parameters for a plate in axially symmetric flexural vibration with free boundary conditions are obtained. The radiation impedance of the plate is derived and the relationship between the radiation impedance and the frequency is analyzed. The equivalent circuits for the plate in flexural vibration and the composite transducer are given. The vibrational modes and the harmonic response of the composite piezoelectric transducer are simulated by the numerical method. Based on the theoretical and numerical analysis, two composite piezoelectric ultrasonic transducers are designed and manufactured, their admittance-frequency curves are measured, and the resonance frequency is obtained. The flexural vibrational displacement distribution of the transducer is measured with a laser scanning vibrometer. It is shown that the theoretical results are in good agreement with the measured resonance frequency and the displacement distribution.     

纵振激励频率对圆盘弯曲振动特性的影响

利用有限元方法,计算并分析了具有一定大小的激励面积、频率不同的纵向夹心式换能器激励圆盘中心时,产生一阶弯曲振动的规律.结果表明:随着激励源频率的逐渐提高,圆盘弯曲振动的基频在不断增大,节圆半径也在增大,圆盘圆心处的位移振幅先增大后减小.实验测试与有限元计算结果基本吻合.该研究为此类复合纵弯振动系统的设计和实际应用提供了一定的参考.     

Study on a new type of radial composite piezoelectric ultrasonic transducers in radial vibration

In this paper, a new type of radial composite ultrasonic transducers in radial vibration is presented and studied. The radial composite ultrasonic transducer consists of a solid piezoelectric ceramic thin disk and a hollow metal thin, circular ring in radial vibration; and they are connected together in the radial direction. The radial vibrations of a piezoelectric ceramic thin disk and a hollow, metal, thin, circular ring are analyzed, respectively. Their radial electromechanical equivalent circuits are obtained. Based on the electromechanical equivalent circuits and using the boundary conditions between the piezoelectric ceramic thin disk and the hollow, metal, thin ring, the electromechanical equivalent circuit of the radial composite ultrasonic transducer is derived and the resonance frequency equation is obtained. The theoretical results from the resonance frequency equation are in good agreement with the measured radial resonance frequencies, and they also are in a good agreement with the results from the numerical method.     

计及压电效应时矩形振子的三维振动研究

在计及压电效应的情况下，本文利用解析方法对有限尺寸压电陶瓷矩形振子的三维振动进行了研究，推出了振子耦合振动的频率方程，并对振子的振动模式进行了分析。理论研究表明，利用本文中的解析法研究振子的耦合振动，计算简单、物理意义明显。与传统的一维理论分析方法及数值方法相比，由于本研究考虑了振子的压电效应以及不同振动模式不同的相互耦合，因此，振子的理论计算频率与测量值更加符合。     

双迭片压电振子的振动研究

利用压电方程和小挠度弯振理论,推导了边界自由压电陶瓷双迭片薄板小挠度条件下弯曲振动位移的一般解及振型频率方程。由频率方程计算得到的各阶振型频率与用有限元方法计算的结果基本相符,并基本得到实验结果的验证。所得结论对边界自由双迭片压电振子弯曲振动的设计提供了理论参考。     

气介式功率超声复合换能器的声辐射特性研究

本文对气介式功率超声复合换能器的辐射声场进行了理论计算并与实验结果进行了对比。该复合换能器由夹心式纵向振动半波长换能器及弯曲振动的薄圆盘组成。从理论上推出了弯曲振动辐射圆盘在不同边界条件下的辐射声场分布及其共振频率方程。理论分析结果表明,固定边界弯曲振动圆盘辐射声场的指向性最佳,并且指向性的理论计算结果与实验结果符合很好。     

Sandwiched piezoelectric ultrasonic transducers oflongitudinal-torsional compound vibrational modes

Sandwiched piezoelectric ultrasonic transducers of longitudinal-torsional compound vibrational modes were studied. The transducers consist of coaxially segmented, longitudinally and tangentially polarized piezoelectric ceramic rings, a back metal cylinder, and a front exponential solid metal horn. Based on the plane-wave approximation, the equivalent circuits of the longitudinal and torsional vibrations in the sandwiched transducer were obtained and the resonance frequency equations of the transducer in longitudinal and torsional vibrations were derived. By means of choosing the radius decay coefficient of the front exponential horn, the longitudinal and torsional vibrations are made to resonate at the same frequency in the transducer. Sandwiched piezoelectric ultrasonic transducers of longitudinal-torsional compound modes were designed and fabricated according to the frequency equations. It is demonstrated that the measured resonance frequencies of the transducers are in good agreement with the theoretical results, and the measured resonance frequencies of the transducers in longitudinal and torsional vibration modes are also in good agreement with each other. Theoretical and experimental results show that this kind of transducer can be used in ultrasonic welding, ultrasonic machining, ultrasonic motors, and other ultrasonic applications which need large displacement amplitudes     

The primary resonance of laminated piezoelectric rectangular plates

Based on Hamilton's principle and the Rayleigh- Ritz method, a model of a nonlinear dynamic laminated piezoelectric rectangular plate was established, and the governing equations were derived and solved for both the thin-plate and thick-plate models. In the present investigation, the nonlinear constitutive relations of piezoelectric materials were considered and the effects of the nonlinearity on the response of the plate were discovered. The primary resonance of rectangular plate is investigated with the use of the method of multiple scales. The results obtained in the present paper agree very well with the experiment results.     

压电陶瓷圆环与金属圆环复合振动系统的径向振动

对由压电陶瓷圆环与金属圆环组成的复合振动系统的径向振动特性进行了研究。首先分析了压电陶瓷圆环和金属圆环的径向振动,推出了其各自的机电等效电路。在此基础上,得出了压电陶瓷圆环与金属圆环复合振动系统的机电等效电路及其共振频率方程。探讨了系统的共振及反共振频率、有效机电耦合系数与其几何尺寸之间的关系。研究表明,当复合振动系统的壁厚比增大时,其共振及反共振频率升高。对于换能器的第一阶径向振动,其有效机电耦合系数随壁厚比的增大而单调减小;对于换能器的第二阶径向振动,其有效机电耦合系数随壁厚比的增大会出现一个极大值,而且,在一定的壁厚比范围内,换能器第二阶径向振动的有效机电耦合系数大于第一阶径向振动的有效机电耦合系数,这一规律与传统的有关压电换能器的分析理论及结果是有所不同的。     

Multi-pulse chaotic dynamics of non-autonomous nonlinear system for a laminated composite piezoelectric rectangular plate

The global bifurcations and multi-pulse chaotic dynamics of a simply supported laminated composite piezoelectric rectangular thin plate under combined parametric and transverse excitations are investigated in this paper for the first time. The formulas of the laminated composite piezoelectric rectangular plate are derived by using the von Karman-type equation, the Reddy’s third-order shear deformation plate theory and the Galerkin’s approach. The extended Melnikov method is improved to enable us to analyze directly the non-autonomous nonlinear dynamical system, which is applied to the non-autonomous governing equations of motion for the laminated composite piezoelectric rectangular thin plate. The results obtained here indicate that multi-pulse chaotic motions can occur in the laminated composite piezoelectric rectangular thin plate. Numerical simulation is also employed to find the multi-pulse chaotic motions of the laminated composite piezoelectric rectangular thin plate.     

大尺寸矩形截面幂函数复合变幅杆的研究

利用传输矩阵法,计算得到了幂函数复合变幅杆一维纵振动的频率方程,并在此基础上运用表观弹性法分析了大尺寸矩形截面幂函数复合变幅杆,得到了考虑横向耦合振动后的谐振频率方程,最后利用有限元软件ANSYS对三组变幅杆进行了模态分析。结果表明：变幅杆谐振频率的计算值与模拟值符合得很好,与一维理论相比,考虑横向耦合振动的二维理论具有更高的精确度。     

Flexural vibration behavior of piezoelectric heterogeneous bimorph beams

Bending oscillations of piezoelectric bimorph beams are effective sound sources in gases or fluids, and, therefore, of practical interest. On the basis of the piezoelectric constitutive relations and the elastodynamic equations, the differential equation of flexural vibrations of thin rectangular piezoelectric heterogeneous bimorph beams, consisting of a piezoelectric layer glued onto an elastic substrate, is derived. The piezoelectric layer is polarized in thickness direction and can be excited to thickness vibrations by an electric alternating current voltage applied to electrodes covering the upper and lower surface of the layer. This causes an oscillating transverse contraction in the piezoelectric layer but not in the substrate, and, thus, generates flexural vibrations of the beam. The differential equation is solved analytically for beams of finite length with both ends free, one clamped and one free end, as well as for both ends clamped. Their vibration behavior in viscous fluids is considered. For a piezo-ceramic composite layer joined to a steel plate vibrating in air and in water, the analytical results are evaluated numerically as function of frequency     

切向极化压电陶瓷细长圆棒的扭转振动研究

利用压电材料的压电及运动方程，本文研究了切向极化压电陶瓷细长棒的扭转振动，分析了扭转振动压电陶瓷细棒中扭转角及扭矩的分布规律，推出了扭转振动压电陶瓷振子的机电等效电路，并得出了振子的共振及反共振频率方程，为扭转振动超声换能器的设计及计算奠定了一定的理论基础。     

受机械力作用金属／陶瓷功能梯度板主共振奇异性

研究机械力作用下金属，陶瓷功能梯度薄板主共振奇异性问题。按照功能梯度薄板的非线性动力学方程，得到金属，陶瓷功能梯度薄板受横向机械力作用的非线性振动方程。应用非线性振动的多尺度法得到系统主共振幅频响应分岔方程并进行奇异性分析，求得幅频响应分岔方程在开折参数平面的转迁集和分岔图。     

Vibration characteristics of a corrugated cylindrical shell piezoelectric transducer

We study coupled extensional and flexural cylindrical vibrations of a corrugated cylindrical shell piezoelectric transducer consisting of multiple pieces of circular cylindrical surfaces smoothly connected along their generatrices. Using the classical shell theory, a theoretical solution is obtained. Based on the solution, basic vibration characteristics of resonant frequencies, mode shapes, and internal forces are calculated and examined for the corrugated transducers, consisting of a few circular pieces each about 50 μm thick, with radius of 5 mm and span angle of 120°. For these transducers the first resonance is of the order of 10-100 Hz.     

The determination of the optimal length of crystal blanks in quartz crystal resonators

It is well understood that the strong coupling of thickness-shear and flexural vibrations in piezoelectric crystal plates only occurs at specific length at which the vibration mode conversion, like the flexural mode gradually converting to thickness-shear mode while the thickness-shear mode converting to higher-order flexural mode, happens. It is important to avoid the strong coupling of modes in a crystal resonator that uses thickness-shear vibrations to enhance the energy trapping. To achieve such a design goal, the length of a crystal blank should be carefully chosen such that the coupling is at its weakest, which usually is in the middle of two strong coupling points. Through a closer examination of the frequency spectra, or the frequency-length relationship in this study, we can see that the strong coupling points appear periodically. This implies that we can find exact locations with the plate theory that predicts the resonance frequency. Based on this observation, we first use the first-order Mindlin plate theory with the precise thickness-shear frequency, which is normalized to one, to find corresponding wavenumbers. Then the length as a variable is solved from the coupled frequency equation for exact coupling points in a crystal plate of AT-cut quartz. The optimal length of a crystal blank in the simplest resonator model is calculated for the coupled thickness-shear, flexural, and extensional vibrations. The solutions and the method will be important in the determination of optimal length of a crystal blank in the resonator design process.     

Fundamental thermo-elasticity equations for thermally induced flexural vibration problems for inhomogeneous plates and thermo-elastic dynamical responses to a sinusoidally varying surface temperature

Equations of motion governing thermally induced vibration of plates with inhomogeneous material properties through the thickness direction are presented. Equations of motion for thermally induced flexural vibration for inhomogeneous rectangular plates in which the material properties are given in the form of a power of the thickness coordinate are derived from the above-mentioned fundamental equations. An exact solution of the one-dimensional temperature change is presented for an inhomogeneous plate in which one surface is exposed to a sinusoidally varying temperature, and the other is kept at zero temperature change. The associated quasi-static and dynamic solutions pertinent to deflection and thermal stresses in the inhomogeneous rectangular plate are derived under the condition of simply supported edges. Numerical calculations are performed, and the effects of material inhomogeneity such as Young’s modulus, coefficient of linear thermal expansion, and mass density, angular frequency in cyclic heating, and aspect ratio on the thermo-elastic response of the rectangular plate are shown in graphical form. Comparing the dynamic solutions with quasi-static ones, the effect of inertia on the thermo-elastic response of the inhomogeneous rectangular plate is evaluated.     

基于布局优化的压电分流阻尼抑振实验研究

建立了带压电分流阻尼系统的四边固支正方形弹性薄板振动控制的实验模型。针对其前五阶模态振动响应的抑制问题,以压电元件存储的电能最大化为优化目标,对采用多个压电元件的压电分流阻尼系统进行了布局优化分析。将压电元件布局优化后的压电分流阻尼系统应用于弹性薄板的多阶模态振动响应的抑制实验,分别研究了不同压电元件数量和布局对抑振效果的影响。在压电分流阻尼抑振实验中,与分流电路开路时相比,电路闭路后弹性薄板的频响函数在对应的模态频率处的幅值分别降低了11.90dB、16.94dB、16.94dB、19.91dB和16.77dB,说明经过布局优化后的压电分流阻尼系统使弹性薄板的前五阶稳态响应都得到了很好的抑制。同时也进行了非优化布局的压电分流阻尼抑振实验,实验结果对比表明,对压电元件进行布局优化可以明显提高压电分流阻尼系统的抑振效果。     

受机械力作用金属／陶瓷功能梯度板的超谐共振研究

研究机械力作用下金属／陶瓷功能梯度薄板3次超谐共振问题．按照功能梯度薄板的非线性动力学方程，得到金属／陶瓷功能梯度薄板受横向机械力作用的非线性振动方程。应用非线性振动的多尺度法得到系统3次超谐共振近似解并进行数值计算。分析阻尼、激励、几何尺寸等参数对系统3次超谐共振幅频响应曲线的影响．