The nonlinear analysis of elastic wave of piezoelectric crystal plate with perturbation method

When investigating or designing acoustic wave sensors, the behavior of piezoelectric devices is supposed to be linear. However, if the sensors are subjected to a strong elastic field, the amplitude of the elastic strain induced in the piezoelectric material is so large that the nonlinearity, which affects the stability and performance of the piezoelectric sensors, can no longer be ignored. In this paper, we perform a theoretical analysis on nonlinear anti-symmetric thickness vibration of thin-film acoustic wave resonators made from quartz. Using Green’s identity, under the usual approximation of neglecting higher time harmonics, a perturbation analysis is performed from which the resonator frequency–amplitude (A–F) relation is obtained. Numerical calculations are made. Furthermore, the validity of the method is examined.     

On the optimum shape of apertures for a perforated plate subject to bending

The problem of finding the optimum shape of the holes in a perforated plate weakened by a triangular or square lattice of holes and subject to bending is considered by methods based on the theory of functions of a complex variable. The criterion determining the optimum shape of the hole is based on the condition that no stress concentration should occur on the hole contour or, alternatively, that a plastic region should be created around the whole contour of the hole at exactly the same instant.     

A perturbation solution to a plate postbuckling problem

The modal coupling behavior of a compressed, imperfect plate is examined as a six degree of freedom structural system. A direct variational solution procedure is developed by the use of the functional for the total potential energy of the system. The non-linear algebraic equations generated are solved by use of a continued perturbation technique. The segmented deflection history is then developed for a variety of imperfection compositions and aspect ratios.     

3D stress local nondestructive analysis in the middle plane of a plate

The Isodyne technique based on the scattered light scanning from a thin laser beam can be used to obtain the information inside the loaded object in the room temperature nondestructively, so it is a very powerful technique in 3D stress analysis. The problems are how to interpret the information and how to get sufficient information from the few interference fringe. Birefringence phase shift technique can distinguish the fringe orders automatically and enrich the information in 256 gray levels between maxim and minim light intensity. In the paper the Isodyne birefringence phase shift method with an oblique incidence and equilibrium equation is presented, by which the 3D stresses in the middle plane of a plate with U shape notch are separated successfully. The project supported by the National Natural Science Foundation of China under the Grant No. 1380345     

Conditions for dynamical symmetry of a nonideal hemispherical shell with perturbation of the shape of the rim

Kiev Polytechnic Institute. Translated from Prikladnaya Mekhanika, Vol. 25, No. 9, pp. 102–108, September, 1989.     

局部荷载作用下有梁矩形板的弹性分析

写出了任意局部荷载作用下各种不同边界条件矩形板的解的表达式．通过梁与板的边界协调分析，求出不同荷载作用下的有梁矩形板解析解，并通过改变其中参数EI与GIt的数值，可以得出局部荷载作用下各种不同边界条件下矩形板的解．     

On the local analysis of continuum deformation

Summary The role of the cartesian representation of the deformation gradient in the non-linear theory of continuum deformation is studied. Indeed, explicit formulae for the local analysis of deformation are obtained in terms of such a representation. Further, some connections with the multiplicative strain and rotation measures are examined.

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Sommario Si studia il ruolo della rappresentazione cartesiana del gradiente di deformazione nella teoria non-lineare della deformazione di un corpo continua tridimensionale. In termini di tale rappresentazione, si ottengono formule esplicite per le variazioni locali di lunghezza, angolo, area e volume. Si esaminano inoltre alcune connessioni con le misure multiplicative di deformazione pura e di rotazione.

     

On a vibrating plate with a concentrated mass

We consider the vibrations of an elastic plate that contains a small region whose size depends on a small parameter ε. The density is of order O(ε^−m) in the small region, the concentrated mass, and it is of order O(1) outside; m is a positive parameter. Depending on the value of m (m<2, m=2 and m>2) we describe the asymptotic behaviour, as ϵ→0, of the eigenvalues and eigenfunctions of the corresponding spectral problem. For m>2 the vibrations associated with the low frequencies affect asymptotically only a neighbourhood of the concentrated mass; we also consider the asymptotic behaviour of the eigenfunctions associated with the high frequencies.     

Bending of a stretched plate containing an eccentrically plate reinforced hole of arbitrary shape

In this paper we consider the problem of a stretched plate containing a hole of arbitrary shape which is reinforced by thickening the plate, on one side only, in a region surrounding the hole. Due to the eccentricity of the reinforcement a bending boundary layer occurs in the neighbourhood of the junction between the plate and the reinforcement. The equations for the moments at the junction are found to be identical to those for the circular hole in Ref. [1]. The boundary layer occurring at a clamped edge of arbitrary shape is also discussed.     

On the impulsive motion of a flat plate with suction

Summary The impulsive motion of a flat plate through a viscous incompressible fluid with time-dependent suction applied normally at the plate is considered. An approximate solution is obtained which describes the motion of the fluid during the initial stage of the motion of the plate, and also one which describes the motion at later stages when the suction velocity is large. The particular case when the suction velocity depends linearly on the time is considered in detail.     

Stability of an annular plate of a shape memory alloy

Analytical solutions using various hypotheses are obtained for the problem of buckling of an annular circular plate of a shape memory alloy in the forward or reverse thermoelastic phase transformations under uniform radial compression. It is established that, despite the axial symmetry of the body geometry, loads, and boundary conditions, the minimum critical loads for annular plates correspond to nonaxisymmetric buckling modes, in contrast to continuous plates.     

On the singular perturbation of a nonlinear ordinary differential equation with two parameters

In this paper, the method of differential inequalities has been applied to study the boundary value problems of nonlinear ordinary differential equation with two parameters. The asymptotic solutions have been found and the remainders have been estimated.Project Supported by the Science Fund of the Chinese Academy of Sciences     

On the optimal shape of a column with partial elastic foundation

By using Pontryagin's principle we study the optimal shape of an elastic column with clamped ends and positioned on elastic foundation of Winkler type. Two problems were treated. In the first one, which is a generalization of our previous work, we consider the case of a partially supported column. In the second problem we determine the optimal shape of a column on elastic foundation subjected to restrictions on minimal cross-sectional area. It is shown that in this case the optimization can be both bimodal and unimodal. We determine the transition value between unimodal and bimodal optimization for specified values of parameters.     

Photoviscoelastic stress analysis of a plate with a central hole

An epoxy resin containing excessive plasticizer was developed and characterized. The material, which deforms viscously at room temperature, has optical properties that depend on stress and strain. A tensile specimen was prepared from the epoxy resin so that the mechanical and optical properties of the epoxy resin could be characterized. The elastic and plastic behavior was determined at 37°C using tensile stresses between 4 and 26 MPa. The birefringence was also recorded as a function of time and stress. From these results, a photoviscoelastic constitutive equation was constructed to describe the dependence of the birefringence on stress and strain. The constitutive equation was then applied to study the deformation of a tensile specimen containing a central circular hole. By using the isochromatic fringes in combination with the isoclinic, the time-dependent variation of the stress field in the specimen was solved.     

Observation of the shape of a water drop on an oscillating Teflon plate

This is an experimental investigation of the dynamics of water drops under large-amplitude forcing within a frequency range of 10–700 Hz. Water drops on an oscillating horizontal plate displayed various behaviors, such as bouncing, dividing, ejecting, axisymmetric polygonal vibration, and polygonal but not axisymmetric vibration. These last two were alike in that the capillary waves of a water drop gave rise to various polygons. However, the planar directions of the polygons of the non-axisymmetric vibrations seemed stable, and their frequencies were the same as the induced frequencies of the oscillating plate. These two features make non-axisymmetric polygonal vibration different from the axisymmetric polygonal vibration reported to date, and the phenomenon has not yet been successfully analyzed. Trials of simulating some shapes of non-axisymmetric polygonal vibrations according to Rayleigh’s theory are described in this paper.