Plate equations and equivalent circuit for micromachined piezoelectrically actuated flextensional transducers

This paper presents novel micromachined two-dimensional array piezoelectrically actuated flextensional transducers that can be used to generate sound in air or water. Micromachining techniques to fabricate these devices are also presented. Individual unimorph array elements consist of a thin piezoelectric annular disk and a thin, fully clamped, circular plate. We manufacture the transducer in two-dimensional arrays using planar silicon micromachining and demonstrate ultrasound transmission in air at 2.85 MHz with 0.15 /spl mu/m/V peak displacement. The devices have a range of operating resonance frequencies starting from 450 kHz up to 4.5 MHz. Such an array could be combined with on-board driving and addressing circuitry for different applications. Classical thin plate theory and Mindlin plate theory are applied to derive two-dimensional plate equations for the transducer, and to calculate the coupled electromechanical field variables such as mechanical displacement and electrical input impedance. In these methods, the variations across the thickness direction vanish by using the bending moments per unit length or stress resultants. Thus, two-dimensional plate equations for a step-wise laminated circular plate are obtained as well as two different solutions to the corresponding systems. An equivalent circuit of the transducer is also obtained from these solutions.

9 - Relationships between vibration frequencies of higher-order plate theories and classical thin plate theory

Analytical, three-dimensional elasticity solutions to some plate problems, and some observations on Mindlin's plate theory

The exact value of Sanders’ path-independent, energy-release rate integral I for an infinite, bent elastic slab containing an elliptic hole is shown to be approximated by its value from classical plate theory to within a relative error of O(h/c)F(e), where h is the thickness, c is the semimajor axis of the ellipse, and F is a function of the eccentricity e. This result is based on Golden’veiser’s analysis of three-dimensional edge effects in plates, as developed by van der Heijden. As the elliptic hole approaches a crack, F(e)~In (1−e). However, this limit is physically meaningless, because Golden’veiser’s analysis assumes that h is small compared to the minimum radius of curvature of the ellipse. Using Knowles and Wang’s analysis of the stresses in a cracked plate predicted by Reissner’s theory, we show that the relative error in computing I from classical plate theory is only O(h/c)In(h/c), where c is the semicrack length. Our results suggest that classical plate and shell theories are entirely adequate for predicting crack growth, within the limitations of applying any elastic theory to an inherently inelastic phenomenon.

An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression

Higher order Shear Defamation Plate Theories (HSDPT) are improved theories over Mindlin plate theory because their assumptions are closer to reality. However, they are seldom used in solving ordinary engineering works. This is due to the fact that mathematical formulations and computations are so lengthy that time and efforts required are close to solving a exact 3-D model. For problems involving sharp stress variation, higher order theories are anticipated to give better results. The combination of HSDPT and Finite Element Modelings are especially attractive because a finite element modeling is much simpler. The current research develops a finite element model on the higher order shear deformation theory. A plate vibration problem was solved. The plates contain square interior cutout. Stress distributions are much complicated than whole plates. Results of HSDPT are compared with FSDPT (First order Shear Deformation Plate Theories) and CPT (Classical Plate Theory). Better accuracies are obtained by using the HSDPT.

Based on Mindlin’s theory of elastic thick plates and Hamiltonian formulism to structural vibration of cantilever plates, the solution of the problem is given. Using Lagrangian density function and Hamiltonian function, the flexural wave equation of plates is derived, and the relation of the transverse eigenvalue and longitudinal eigenvalue is given. Different from traditional engineering analysis, dispersion equations of the propagation mode of the Mindlin plate are deduced from the eigenfunction expansion method. By satisfying the boundary conditions of plates, the wavenumber of vibration modes in the plate structure can be obtained. The independent modal space control is applied to study the active vibration control of cantilever plates. Feedback gain is adjusted to change the eigenstructure of the original vibration system in order to improve the modal damping and stiffness of the system. The applicability of plates based on two theories (Mindlin’s theory and classical thin plate theory) is studied. Finally, the results are analyzed and discussed by numerical simulations.

An improved model for unsymmetric plates

Vibration of moderately thick square orthotropic stepped thickness plates

Acoustic radiation from a fluid-loaded laminated composite plate with two sets of periodic stiffeners is investigated, in which the expressions for the stress tensors due to the transverse shear deformation are included on the basis of Reddy’s layerwise plate theories. By employing the Fourier transform method and the stationary phase approximation, far field pressure expressions are presented to address the acoustic characteristics of the stiffened laminated plate. Sound pressure obtained from the layerwise theory are compared with existing results from the classic thin plate theory, and increasingly remarkable differences among them are found from medium to high frequency range. Numerical calculations are subsequently carried out to explore the influences of the spacing of the stiffeners, the thickness of the plate, and the lamination schemes on the sound pressure. Practical concerns are addressed which would be beneficial to the optimization design of stiffened plates.

Exact relationships between the bending solution of the classical (Kirchhoff) thin plate theory and the sinusoidal shear deformation thick plate theory for homogeneous and functionally graded material plates are derived by using the mathematical similarity of governing equations of the two theories, and the basis of load equivalence. It is assumed in the analysis that the mechanical properties of the functionally graded plates vary continuously through the thickness of the plate and obey a simple power law distribution of the volume fraction of the constituents. Effects of material gradient property and shear deformation on the bending of functionally graded plates are discussed in the framework of the sinusoidal shear deformation plate theory. The boundary conditions at two edges of the functionally graded plate are simply supported while the remaining two edges having the same boundary conditions are, namely, simply supported, clamped, or free. Several examples are presented to illustrate the use and accuracy of these relationships. The effects of many parameters on the deflections and stresses are investigated.

Within the limitations of the classical small deflexion theory of thin plates and using complex variable methods, exact expressions are obtained in series form for the deflexion at any point of a thin isotropic circular plate simply supported along a concentric circle and subject to loading symmetrically distributed over an eccentric circular patch which lies inside the circle of support. In special and limiting cases the solutions reduce to those obtained before.

Natural modes of vibration of linear viscoelastic circular plates with free edges

Deflection of concentrically supported and eccentrically loaded thin circular plate

In this paper, an analytical three-dimensional (3D) bending characteristic of an isotropic rectangular thick plate with all edges simply supported (SSSS) and carrying uniformly distributed transverse load using the energy technique is presented. The three-dimensional constitutive relations which involves six stress components were used in the established, refined shear deformation theory to obtain a total potential energy functional. This theory obviates application of the shear correction factors for the solution to the problem. The governing equation of a thick plate was obtained by minimizing the total potential energy functional with respect to the out of plane displacement. The deflection functions which are in form of trigonometric were obtained as the solution of the governing equation. These deflection functions which are the product of the coefficient of deflection and shape function of the plate were substituted back into the energy functional, thereafter a realistic formula for calculating the deflection and stresses were obtained through minimizations with respect to the rotations and deflection coefficients. The values of the deflections and stresses obtained herein were tabulated and compared with those of previous 3D plate theory, refined plate theories and, classical plate theory (CPT) accordingly. It was observed that the result obtained herein varied more with those of CPT and RPT by 25.39% and 21.09% for all span-to-thickness ratios respectively. Meanwhile, the recorded percentage differences are as close as 7.17% for all span-to-thickness ratios, when compared with three dimensional plate analysis. This showed that exact 3D plate theory is more reliable than the shear deformation theory which are quite coarse for thick plate analysis. Doi: 10.28991/esj-2021-01320 Full Text: PDF