Effect of thermal gradient on frequencies of an orthotropic rectangular plate whose thickness varies in two directions

A simple model is presented for the use of research workers in space technology, mechanical sciences and nuclear energy where certain components of the structures have to operate under elevated temperatures. The effect of a constant thermal gradient on the free vibrations of an orthotropic rectangular plate whose thickness varies linearly in two directions is considered. An approximate but quite convenient frequency equation is derived by using Rayleigh-Ritz techniques with a two-term deflection function. The upper bound on the frequency corresponding to the first mode of vibrations is obtained for various boundary conditions, and for various values of the length to width ratio, two paper constants and the temperature gradient.     

Large amplitude free vibrations of annular plates of varying thickness

The non-linear (i.e., large deflection) free vibrations of thick, orthotropic annular plates with varying thickness are calculated. The formulation is based on the more general Reissner plate equations as well as the von Kármán plate equations for variable thickness annular plates. Numerical results for the ratio of the non-linear period to the linear period of natural vibration are compared with those existing in the literature. New results are also included for future comparisons.     

Transverse vibrations of thin,elastic plates with concentrated masses and internal elastic supports

The fundamental frequency of vibration of a plate carrying concentrated masses and with internal elastic supports is determined. The case of an orthotropic, rectangular plate elastically restrained against rotation along the four edges is tackled first by using simple polynomial approximations and the Galerkin method. Then, vibrations of clamped and simply supported isotropic plates of regular polygonal shape are studied by using the conformal mapping technique coupled with the variational method. Finally the case of a circular plate elastically restrained against translation and rotation is considered.     

EFFECT OF ELASTIC FOUNDATION ON ASYMMETRIC VIBRATION OF POLAR ORTHOTROPIC LINEARLY TAPERED CIRCULAR PLATES

Asymmetric vibrations of polar orthotropic circular plates of linearly varying thickness resting on an elastic foundation of Winkler type are discussed on the basis of the classical plate theory. Ritz method has been employed to obtain the natural frequencies of vibration using the functions based upon the static deflection of polar orthotropic plates, which has faster rate of convergence as compared to the polynomial co-ordinate functions. Frequency parameter of the plate with elastically restrained edge conditions are presented for various values of taper parameter, rigidity ratio and foundation parameter. A comparison of the results with those available in the literature obtained by finite element method, receptance method and polynomial co-ordinate functions shows an excellent agreement.     

On the non-linear vibrations of rectangular plates

A solution, based on a one-term mode shape, for the large amplitude vibrations of a rectangular orthotropic plate, simply supported on all edges or clamped on all edges for movable and immovable in-plane conditions, is found by using an averaging technique that helps to satisfy the in-plane boundary conditions. This averaging technique for satisfying the immovable in-plane conditions can be used to resolve many anisotropic and skew plate problems where otherwise, when a stress function is used, the integration of the u and v equations becomes difficult, if not impossible. The results obtained herein are compared with those available in the literature for the isotropic case and excellent agreement is found. Results available for the one-term mode shape solutions of these problems are compared and the non-linear effect is presented as functions of aspect ratio and of the orthotropic elastic constants of the plate. The results are further compared with those based on the Berger method and the detailed comparative studies show that the use of the Berger approximation for large deflection static and dynamic problems and its extension to anisotropic plates, skew plates, etc., can lead to quite inaccurate results.     

Thermal effect on frequencies of coupled vibrations of a rotating beam of linearly varying semi-circular cross-section

An analysis is presented of the effect of a constant thermal gradient on coupled vibrations of a beam of linearly varying semi-circular cross-section attached to a rotating disc. A method based on Rayleigh's quotient is used to obtain upper bounds of the frequencies corresponding to the first three modes of vibrations. The frequencies for various values of cross-section variation, hub-radius and temperature gradient are obtained.     

Axisymmetric vibrations of polar orthotropic mindlin annular plates of variable thickness

Analysis and numerical results are presented for the axisymmetric vibrations of polar orthotropic annular plates with linear variation in thickness, according to Mindlin's shear theory of plates. A chebyshev collocation technique has been employed to obtain the frequency equations for the transverse motion of such plates, for three different boundary conditions. Frequencies, mode shapes and moments for the first three modes of vibration have been computed for different plate parameters. A comparison of frequencies with the corresponding values obtained by classical plate theory leads to some interesting conclusions.     

Non-linear axisymmetric vibration of orthotropic thin circular plates on elastic foundations

This study deals with the large amplitude axisymmetric free vibrations of cylindrically orthotropic thin circular plates resting on elastic foundations. Geometric non-linearity due to moderately large deflections has been included. Movable and immovable simply supported plates and immovable clamped plates resting on Winkler, Pasternak and non-linear Winkler foundations have been considered. The von Kármán type governing equations have been employed. Harmonic vibrations are assumed and the time t is eliminated by the Kantorovich averaging method. An orthogonal point collocation method is used for spatial discretization. Numerical results are presented for the linear natural frequency of the first axisymmetric mode and for the ratio of the non-linear period to the linear period of natural vibration. The effects of foundation parameters, the orthotropic parameter and the edge conditions on the non-linear vibration behaviour have been investigated.     

Large amplitude vibrations of rectangular plates with non-uniform elastic edge supports

Large amplitude vibrations of a rectangular plate with parabolically varying rotational constraints on two opposite edges are studied. The problem has been solved by both Berger's approach and the more rigorous von Kármán approach. Galerkin's method is used to obtain an ordinary differential equation in the modal function, the solution to which is given in terms of elliptic functions.     

Analysis of a monolithic crystal plate acoustic wave filter

We study thickness–shear and thickness–twist vibrations of a finite, monolithic, AT-cut quartz plate crystal filter with two pairs of electrodes. The equations of anisotropic elasticity are used with the omission of the small elastic constant c₅₆. An analytical solution is obtained using Fourier series from which the resonant frequencies, mode shapes, and the vibration confinement due to the electrode inertia are calculated and examined.     

T-matrix method formulation applied to the study of flexural waves scattering from a through obstacle in a plate

Elastic wave scattering in a flat thin plate hosting a through obstacle of arbitrary closed form is examined using a numerical technique based on the T-matrix approach, which is applied to describe of flexural waves in plates. The limiting cases of a hole and a rigid obstacle are considered. The vibrations of the plate are described by the Kirchhoff model. The far field backscattered amplitude as a function of wave frequency for inclusions of elliptic, triangular and square form with rounded corners is analysed numerically. Comparison of present results for circular obstacles with the analytical solutions obtained by other authors show excellent agreement.     

Multi-frequency excitation of stiffened triangular plates for large amplitude oscillations

Free and forced vibrations of triangular plate are investigated. Diverse types of stiffeners were attached onto the plate to suppress the undesirable large-amplitude oscillations. The governing equation of motion for a triangular plate, based on the von Kármán theory, is developed and the nonlinear ordinary differential equation of the system using Galerkin approach is obtained. Closed-form expressions for the free undamped and large-amplitude vibration of an orthotropic triangular elastic plate are presented using the two well-known analytical methods, namely, the energy balance method and the variational approach. The frequency responses in the closed-form are presented and their sensitivities with respect to the initial amplitudes are studied. An error analysis is performed and the vibration behavior, as well as the accuracy of the solution methods, is evaluated. Different types of the stiffened triangular plates are considered in order to cover a wide range of practical applications. Numerical simulations are carried out and the validity of the solution procedure is explored. It is demonstrated that the two methods of energy balance and variational approach have been quite straightforward and reliable techniques to solve those nonlinear differential equations. Subsequently, due to the importance of multiple resonant responses in engineering design, multi-frequency excitations are considered. It is assumed that three periodic forces are applied to the plate in three specific positions. The multiple time scaling method is utilized to obtain approximate solutions for the frequency resonance cases. Influences of different parameters, namely, the position of applied forces, geometry and the number of stiffeners on the frequency response of the triangular plates are examined.     

A reduction method for problems of vibration of orthotropic plates

It is shown that the problem of vibration of an orthotropic plate can be reduced to that of another orthotropic plate by a simple co-ordinate transformation, and reduction formulae are obtained. To justify the reduction formulae, fundamental natural frequencies of orthotropic rectangular plates with various boundary conditions and of a clamped orthotropic elliptical plate are discussed. As an example, an exact natural frequency of a simply supported generally orthotropic skew plate with special flexural rigidities is obtained from that of a simply supported isotropic rectangular plate.     

正交各向异性弹性平面应力的近似计算方法

本文在列赫尼茨基方法的基础之上提供一个正交各向异性弹性平面应力的近似计算方法。利用这个方法,我们处理了下面三种问题:(1)具有圆孔梁的纯弯曲;(2)具有圆形空洞的板的拉伸;(3)具有椭圆空洞的无限平面。     

Effect of an elastic foundation on axisymmetric vibrations of polar orthotropic annular plates of variable thickness

Free axisymmetric vibrations of a polar orthotropic annular plate of linearly varying thickness resting on an elastic foundation of Winkler type are studied on the basis of classical theory of plates. The fourth order linear differential equation with variable coefficients governing the motion is solved by using the quintic spline interpolation technique for three different combinations of boundary conditions. The effect of the elastic foundation together with the orthotropy on the natural frequencies of vibration is illustrated for different values of the radii ratio and the thickness variation parameter for the first three modes of vibration. Transverse displacements and moments are presented for a specified plate. The validity of the spline technique is demonstrated by presenting a comparison of present results with those available in the literature.     

FREE VIBRATION OF ORTHOTROPIC SQUARE PLATES WITH A SQUARE HOLE

An approximate method is extended for analyzing the free vibration problem of orthotropic square plate with a square hole. In this paper, a square plate with a square hole is transformed into an equivalent square plate with non-uniform thickness by considering the hole as an extremely thin part of the equivalent plate. Therefore, the dynamic characteristics of a plate with a hole can be obtained by analyzing the equivalent plate. The Green function, which is the discrete solution for the deflection of the equivalent plate, is used to obtain the characteristic equation of the free vibration. The effects of the side to thickness ratio, hole side to plate side ratio and the variation of the thickness on the frequencies are considered. Some numerical analyses are carried out for the simply supported orthotropic square plate with a square hole. The efficiency and accuracy of the numerical solutions by the present method are investigated.     

Large amplitude flexural vibration of eccentrically stiffened plates

The large amplitude free flexural vibrations of thin, orthotropic, eccentrically and lightly stiffened elastic rectangular plates are investigated. Clamped boundary conditions with movable in-plane edge conditions are assumed. A simple modal form of one-term transverse displacement is used and in-plane displacements are made to satisfy the in-plane equilibrium equations. By using Lagrange's equation, the modal equations for the nonlinear free vibration of stiffened plates are obtained for the cases when the stiffeners are assumed to be smeared out over the entire surface of the plate, and when the stiffeners are located at finite intervals. Numerical results are obtained for various possibilities of stiffening and for different aspect ratios of the plate. By particularizing the problem to different known cases, the results obtained here are compared with available analytical and experimental results, and the agreement is good.     

不同材料参数薄板振动中的热力耦合效应

 利用有限Hankel变换法,导出了周界等温-弹性支撑圆薄板在激光束辐照下的轴对称耦合热弹性弯曲振动近似解;针对具有不同弹性模量和热膨胀系数的薄板进行了热力耦合和非耦合弯曲振动的解析和有限元计算与分析。结果表明：热力耦合效应使薄板振动的振幅和周期都有所减小,其程度与材料的性能参数（如弹性模量和热膨胀系数等）密切相关,材料弹性模量和热膨胀系数越大,板振动中的热力耦合效应就越明显。     

Non-linear flexural vibrations of orthotropic rectangular plates

This study is an analytical investigation of free flexural large amplitude vibrations of orthotropic rectangular plates with all-clamped and all-simply supported stress-free edges. The dynamic von Karman-type equations of the plate are used in the analysis. A solution satisfying the prescribed boundary conditions is expressed in the form of double series with coefficients being functions of time. The model equations are solved by expanding the time-dependent deflection coefficients into Fourier cosine series. As obtained by taking the first sixteen terms in the double series and the first two terms in the time series, numerical results are presented for non-linear frequencies of various modes of glass-epoxy, boron-epoxy and graphite-epoxy plates. The analysis shows that, for large values of the amplitude, the effect of coupling of vibrating modes on the non-linear frequency of the fundamental mode is significant for orthotropic plates, especially for high-modulus composite plates.     

Damping in micro-scale generalized thermoelastic circular plate resonators

The out-of-plane vibrations of a generalized thermoelastic circular plate are studied under different environmental temperature, plate dimensions and boundary conditions. The analytical expressions for thermoelastic damping of vibration and phase velocity of circumferential surface wave modes are obtained. It is noticed that the damping of vibrations and phase velocities of circumferential surface wave modes significantly depend on thermal relaxation time in addition to thermoelastic coupling in circular plates under resonance conditions. The surface conditions also impose significant effects on the vibrations of such resonators. The expressions for displacement and temperature fields in the plate resonator are also derived and obtained. Some numerical results have also been presented for illustration purpose in case of silicon material plate.