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.     

Natural frequencies of orthotropic rectangular plates with stepped thickness

It is shown how one can derive an approximate formula for estimating a natural frequency of an orthotropic rectangular plate with stepped thickness by using several natural frequencies of the corresponding isotropic plate reduced from the orthotropic one. To justify the method, an orthotropic two-part rectangular plate with simply supported sides is discussed, and an approximate formula is proposed for estimating the fundamental natural frequency.     

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.     

The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic plates using Rayleigh's method

A simple approximate formula for the natural frequencies of flexural vibration of isotropic plates, originally developed by Warburton using characteristic beam functions in Rayleigh's method, is modified to apply to specially orthotropic plates and extended to include the effect of uniform, direct inplane forces. The initial buckling problem is treated simply by equating the frequency expression to zero. The approach permits the ready determination of reasonably accurate natural frequencies and/or buckling loads for a given plate involving any combination of free, simply supported or clamped edges, without requiring the aid of a sophisticated calculating device or a knowledge of plate, vibration or buckling theory. To illustrate the applicability and accuracy of the approach, numerical results for a number of specific plate problems are presented.     

Forced vibrations of a non-uniform thickness rectangular plate with two free sides

A forced vibration of a rectangular plate with thickness varying linearly in one direction is studied. The plate is simply supported along two opposite sides, and is free along the other sides. Approximate formulae are proposed for estimating the maximum deflection and surface stresses of a rectangular plate subject to a uniformly distributed harmonic lateral load. The accuracy of the formulae is discussed. The approximate formulae show the possibility that one can obtain solutions for the forced vibration of a plate with non-uniform thickness from those for the static bending of the same plate when the plate is subjected to a uniformly distributed load.     

Vibration of simply supported-clamped skew plates at large amplitudest

The large amplitude, free, flexural vibration of orthotropic skew plates simply supported along two opposite edges and clamped along the other two are investigated on the basis of an assumed mode shape. The relationship between the amplitude and period is studied for both isotropic and orthotropic skew plates for various aspect ratios and skew angles under two in-plane edge conditions. It is found that the modal equation reduces to the Dufling type equation from which the period of non-linear vibration is found to decrease with increasing amplitude, exhibiting hardening type of non-linearity. The validity of the Berger approximation is investigated for the problem under consideration and this approximation is shown to give reasonably good results.     

Vibrations of an orthotropic elliptic plate of non-uniform thickness and temperature

An analysis is presented of the free vibrations of an orthotropic elliptic plate whose temperature and thickness spatial variations both are parabolic along a line through the plate centre. An approximate but quite convenient frequency equation is derived by using Galerkin's method with a two-term deflection function. The upper bounds of frequencies corresponding to the first two modes of vibrations of a clamped orthotropic elliptic plate for various values of aspect ratio, taper constant and temperature constant are obtained.     

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.     

The flexural vibration of rectangular plates with point supports

Orthogonally generated polynomial functions are used in the Lagrangian multiplier method to study the free, flexural vibration problem of point supported, thin, flat, rectangular plates. The analysis applies to isotropic and specially orthotropic plates having any combination of clamped, simply supported or free edges with arbitrarily located point supports and to plates which are continuous over line supports parallel to the plate edges. Numerical results are presented for a number of specific problems, illustrating the accuracy and versatility of the approach, and which include natural frequencies and nodal patterns for a point supported plate which is continuous over two perpendicular line supports.     

Analysis of skew plate problems with various constraints

This paper presents the application of the modified Rayleigh-Ritz method with Lagrange multipliers to analyze skew plate problems with various constraints. By this procedure one can satisfy both geometric and natural boundary conditions of skew plates. To demonstrate the accuracy and versatility of the method, several examples of bending, vibration and buckling of skew plates are solved, and results are compared with those obtained by other approximate methods.     

Vibration and buckling of rectangular plates under in-plane hydrostatic loading

Numerical solutions are presented for the fundamental natural frequency and mode shape of a rectangular plate loaded by in-plane hydrostatic forces for a wide variety of aspect ratios, boundary conditions, and load magnitudes. All six possible combinations of simply supported and clamped edges are considered. The limiting conditions of unloaded vibration and buckling are discussed in detail, with emphasis on the preferred mode shape. Design curves and approximate formulae are presented which provide a simple means of determining the fundamental frequency parameter.     

Power flow model of flexural waves in finite orthotropic plates

In this paper, an approximate power flow model is developed for the analysis of the flexural waves in finite orthotropic plate transversely vibrating in the medium-to-high frequency ranges. The derived energy equation for the model is expressed with time- and locally space-averaged farfield wave energy density. It could be the more general form than that of the conventional power flow analysis model seen in the isotropic plate. With the derived model, dynamic characteristics varying with the direction can be expressed. To verify the validity and accuracy of the model, numerical analyses are performed for the case where a finite rectangular plate is excited by a harmonic point force, and the calculated results expressed with the energy levels are compared with classical modal solutions by changing the frequency and the damping loss factor of the plate. The dominant power transmission paths in the plate are also predicted from the distribution of the approximate intensity fields.     

The effect of different combinations of boundary conditions on the average radiation efficiency of rectangular plates

The boundary conditions of a vibrating plate are known to have an influence on its sound radiation for frequencies below the critical frequency. To investigate this effect in a systematic way, the average radiation efficiency and radiated power are calculated for a rectangular plate set in an infinite baffle using a modal summation approach. Whereas analytical expressions exist for simply supported boundary conditions, a numerical approach is required for other cases. Nine combinations of boundary conditions are considered, consisting of simply supported, clamped and free edges on different plate edges. The structural vibration is approximated by using independent beam functions in orthogonal directions allowing simple approximate formulae for mode shapes and natural frequencies. This assumption is checked against a finite element model and shown to give reliable results. It is shown that a free plate has the lowest radiation efficiency and a clamped plate the highest for most frequencies between the fundamental panel natural frequency and the critical frequency. Other combinations of boundary condition give intermediate results according to the level of constraint introduced. The differences depend on frequency: excluding the extreme case of a fully free plate all the other boundary conditions give results within a range of 8 dB in the middle part of the short-circuiting region, decreasing towards the critical frequency. At low frequency the differences can be even greater, in some cases up to 20 dB. These conclusions are shown to hold for a range of plate thicknesses and dimensions.     

Vibration of moderately thick square orthotropic stepped thickness plates

Numerous studies that address the vibration of stepped thickness plates are reported in the literature. Predominately, classical plate theory has been used to formulate studies for both isotropic and anisotropic stepped plates. Mindlin plate theory has been employed to obtain results for thick isotropic stepped thickness plates. Exact solutions, Rayleigh-Ritz, differential quadrature and finite element methods have been employed to compute results for frequency of vibration. Results for frequency of vibration for thick orthotropic stepped thickness plates are presented here using orthorhombic material properties of aragonite. The finite element method has been used to compute frequencies and determine mode shapes for simply supported and clamped square Mindlin plates.     

Natural frequencies of completely free annular and circular plates having polar orthotropy

A simple approximate, yet quite accurate, Ritz method analysis is presented for dealing with vibration of completely free annular plates having polar orthotropic characteristics. It is shown that the method is readily applicable to the determination of approximate frequency values for solid circular plates. The natural frequencies of these plates are obtained for the various orthotropic parameters, and comparison is made with exact values for isotropic cases, showing excellent agreement.     

Vibration of skew plates resting on point supports

This paper deals with the vibrations of skew plates resting on point supports. The spline element method is used. To demonstrate the accuracy of the method, several examples are solved, and results are compared with those obtained by other approximate methods. The effects of skew angles and locations of point supports on natural frequencies of isotropic skew plates are also investigated.     

A variable kinematic Ritz formulation for vibration study of quadrilateral plates with arbitrary thickness

A new variable kinematic Ritz method applied to free vibration analysis of arbitrary quadrilateral thin and thick isotropic plates is presented. Carrera's unified formulation and the versatile pb-2 Ritz method are properly combined to build a powerful yet simple modeling and solution framework. The proposed technique allows to generate arbitrarily accurate Ritz solutions from a large variety of refined two-dimensional plate theories by expanding so-called Ritz fundamental nuclei of the plate mass and stiffness matrices. Theoretical development of the present methodology is described in detail. Convergence and accuracy of the method are examined through several examples on thin, moderately thick, and very thick plates of rectangular, skew, trapezoidal and general quadrilateral shapes, with an arbitrary combination of clamped, free and simply supported edges. Present results are compared with existing three-dimensional solutions from open literature. Maximum and average differences of various higher-order plate theories and three-dimensional results are also presented with the aim of providing useful guidelines on the choice of appropriate plate theory to get a desired accuracy on frequency parameters.     

On the vibration of skew plates of variable thickness

This paper is concerned with the determination of natural frequencies of a vibrating skew plate with variable thickness. Free and forced vibrations are treated for different ratios of the sides, skew angle and taper constant. The static deflection has also been obtained as a by product of the present solution.     

Dynamic response of orthotropic plate using BEM with approximate fundamental solution

In order to overcome the difficulty in finding the fundamental solution in the boundary element method for dynamic problems of orthotropic plates, approximate fundamental solutions are traced. Two types of approximate fundamental solutions are considered: i.e., the series solution and the superposition solution. The former is made up of Hermitian orthogonal polynomials and the latter consists of an analytical fundamental solution for the static bending problem of a generalized isotropic plate and a series solution for correction. Some numerical examples amply show the feasibility and effectiveness of the method offered.     

Free vibration of annular sector plates by an integral equation technique

A numerical method is presented for the free vibration analysis of polar orthotropic clamped annular sector plates. The results are compared with analytical and experimental values of other investigators. A parametric study has been done by varying the sector angle and radii ratio. The frequencies for isotropic and orthotropic cases are presented in the form of graphs.