Vibration of plates with straight-line,clamped and free edges

This paper deals with vibration problems of thin plates having straight-line, mutually perpendicular, clamped and free edges and subjected to a load consisting of a set of transverse, arbitrarily located random forces. It is assumed that the number of edges of a plate forming recurring figures is optional but each of these edges is either clamped or free along its entire length. The procedure for solving the boundary problem based upon the R-functions method and for estimation of transverse displacements based upon the correlation analysis is presented. Numerical calculations are carried out for two example plates.     

An exact analytical approach to the free vibration analysis of rectangular plates with mixed boundary conditions

A comprehensive analytical technique is developed for the free vibration analysis of rectangular plates with discontinuities along the boundaries. For illustrative purposes a solution is obtained for plates with edges partially clamped and partially simply supported and plates with edges partially and partially simply supported. A vast array of first mode eigenvalues is provided for these families of plates. Solutions to the equations are obtained by exploiting a mathematical technique described by the author during an earlier publication. It is shown that eigenvalue matrices are easily generated for a wide range of plates with discontinuities in boundary conditions.     

Vibration transmission through plate/beam structures typical of lightweight buildings: Applicability and limitations of fundamental theories

This paper examines the modelling of vibration transmission through plate/beam structures typical of lightweight buildings. Key experiments have been carried out on simple structures to identify the applicability and limitations of fundamental theories. The systems tested included a single plate connected along its centre to a beam, two parallel plates attached along their centre to a beam (plates opposite or offset), and four plates connected along their edges to a beam. The analysis focused in particular on the applicability of modelling a beam as a one-dimensional element in point connected systems (widely spaced screws in terms of bending wavelength). Statistical energy analysis (SEA) was the framework of analysis used for all predictions, but the theories examined were independent from SEA. The results obtained indicate that simple point models are only applicable to the single plate and beam system, and to the parallel opposite plates connected along their centre to a beam; even then, the applicability of such models is limited to low and mid frequencies (below 2 kHz for the structures tested). Transmission between two parallel plates connected to a beam with screws closely spaced was also examined, and it was found that rigid and pinned line predictions can provide limits for transmission between panels on the same side of a wall (where junctions with shallow beams tend to behave rigidly, whilst junctions with deep beams are better modelled as pinned).     

Free vibration of rectangular plates with edges having different degrees of rotational restraint

A numerical method developed by the author has been used as a basis for determining natural frequencies of rectangular plates possessing different degrees of elastic restraints along the edges. The basic functions satisfying the boundary conditions along two opposite edges for such cases have been derived. Comparison of results with others that are available indicates excellent accuracy. Many new results have been presented.     

Refined vibration and damping analysis of multilayered rectangular plates

Refined vibration and damping analysis of a general multilayered rectangular plate consisting of an arbitrary number of layers of orthotropic materials has been developed by considering extension, bending, in-plane shear and transverse shear deformations in all the layers and taking into account the rotary and longitudinal translatory inertias along with the transverse inertia of the plate. The solution for a multilayered plate with simply supported edges has been taken in series summation form and resonating frequencies and associated loss factors for plates with alternate elastic and viscoelastic layers have been evaluated by application of the correspondence principle of linear viscoelasticity. Results for three-, five- and seven-layered plates obtained by the present refined analysis are compared with the results obtained by conventional analysis of multilayered plates.     

Natural frequencies of rectangular plates with free edges

This note presents vibration analysis of isotropic rectangular plates with free edges by the Rayleigh-Ritz method with B-spline functions. To show the accuracy of the present method, the results are compared with existing results based on other numerical methods and found to be in good agreement. Accurate frequencies of rectangular plates are analyzed for different aspect ratios and boundary conditions. The effects of Poisson's ratio on natural frequencies of square plates with free edges are also investigated.     

Vibrations of a polygonal plate having orthogonal straight edges by an extended rayleigh-ritz method

An extended Rayleigh-Ritz method is presented for solving vibration problems of a polygonal plate having orthogonal straight edges. The polygonal plate is considered as an assemblage of several rectangular plates. For each element rectangular plate, the transverse displacement is approximated by interpolation functions corresponding to unknown displacements and slopes at the discrete points which are chosen along the edges, and series of trial functions which satisfy homogeneous artificial boundary conditions. By minimizing the energy functional corresponding to the assumed displacement function, the dynamic stiffness matrix of the element rectangular plate, which is similar to that obtained in the finite element method, is derived. The dynamic stiffness matrix of the whole system is obtained by summing up those of the element rectangular plates. Numerical results are presented for the natural frequencies and mode shapes of cantilever L-shaped and T-shaped plates.     

Semi-Analytical Solution for Functionally Graded Solid Circular and Annular Plates Resting on Elastic Foundations Subjected to Axisymmetric Transverse Loading

In this paper, the static analysis of functionally graded (FG) circular plates resting on linear elastic foundation with various edge conditions is carried out by using a semi-analytical approach. The governing differential equations are derived based on the three dimensional theory of elasticity and assuming that the mechanical properties of the material vary exponentially along the thickness direction and Poisson's ratio remains constant. The solution is obtained by employing the state space method (SSM) to express exactly the plate behavior along the graded direction and the one dimensional differential quadrature method (DQM) to approximate the radial variations of the parameters. The effects of different parameters (e.g., material property gradient index, elastic foundation coefficients, the surfaces conditions (hard or soft surface of the plate on foundation), plate geometric parameters and edges condition) on the deformation and stress distributions of the FG circular plates are investigated.     

Natural vibration of rectangular plates with an internal line hinge using the first order shear deformation plate theory

This paper presents exact solutions for vibration of rectangular plates with an internal line hinge. The rectangular plate is simply supported on two parallel edges and the remaining two edges may take any combination of support conditions. The line hinge is perpendicular to the two simply supported parallel edges. The Lévy type solution method and the state-space technique are employed in connection with the first order shear deformation plate theory (FSDT) to study natural vibration of rectangular plates with an internal line hinge. In particular, exact vibration frequencies are obtained for rectangular plates of different aspect ratios and edge support conditions. The influence of the internal line hinge on the vibration behavior of rectangular plates is studied.     

A comprehensive approach to the free vibration analysis of rectangular plates by use of the method of superposition

The authors have found the above techniques to constitute a powerful means for solving rectangular plate problems. At the time of writing, solutions for plates with two adjacent simply supported edges and two adjacent free edges have been obtained. The first 20 eigen-values for plates with all edges clamped have also been determined for a full range of aspect ratio and they are shown to be accurate to within less than one half of one percent. It will be appreciated that solutions for any combination of clamped-simply supported edge conditions can easily be obtained from the all-clamped solution by simply deleting appropriate solutions from the all-clamped combination. In Figure 2 contour lines for first mode vibration of a plate with two adjacent clamped and two adjacent simply supported edges is presented. The higher density of the contour lines along the simply supported edges will be noted.The method of superposition is currently being used by the authors to good advantage to obtain solutions of any desired degree of accuracy to all of the problems discussed. It is found to be easily utilized and unlike more complicated methods is readily comprehensible to the practicing engineer. Eigenvalues for all modes, aspect ratios, and boundary conditions are readily obtained. Modal shapes are expressed in terms of familiar analytic functions. Results of these studies will be made available in future publications.     

Free vibration of two identical circular plates coupled with bounded fluid

This paper examines the hydroelastic vibration of two identical circular plates coupled with a bounded fluid. An analytical method based on the finite Fourier-Bessel series expansion and the Rayleigh-Ritz method is suggested. In the theory, it is assumed that a rigid cylindrical container is filled with the ideal fluid and the two plates are clamped along the container edges. The proposed method is verified by finite element analysis using commercial software with excellent accuracy. Two transverse vibration modes, in-phase and out-of-phase, are observed alternately in the fluid-coupled system when the number of nodal circles increases for the fixed nodal diameter. It is found that the normalized natural frequency of the system monotonically increases with an increase in the number of nodal diameters and circles by virtue of a decrease in relative hydrodynamic mass.     

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.     

Resonance frequencies and mode shapes of elastically restrained,prestressed annular plates attached together by flexible cores

The free vibrations of annular plates attached together by flexible cores are studied analytically. Both axisymmetric and non-axisymmetric vibrations are considered. The plates are elastically constrained against rotation at the inner and outer edges. At the same time, the plates are subjected to initial radial tensions. Detailed analysis is worked out for systems consisting of five through two identical plates with identical boundary conditions and a uniform radial tension. General frequency equations and mode shapes are developed. The first nine eigenvalues are calculated for a plate system having identically constrained inside and outside edges and are tabulated as functions of the initial tension parameter, the elastic edge constraint parameter and the ratio of inner to outer radius. The orthogonality property of the mode function is also discussed.     

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.     

Free vibration of two elastically coupled rectangular plates with uniform elastic boundary restraints

An analytical method is derived for determining the vibrations of two plates which are generally supported along the boundary edges, and elastically coupled together at an arbitrary angle. The interactions of all four wave groups (bending waves, out-of-plane shearing waves, in-plane longitudinal waves, and in-plane shearing waves) have been taken into account at the junction via four types of coupling springs of arbitrary stiffnesses. Each of the transverse and in-plane displacement functions is expressed as the superposition of a two-dimensional (2-D) Fourier cosine series and several supplementary functions which are introduced to ensure and improve the convergence of the series representation by removing the discontinuities that the original displacement and its derivatives will potentially exhibit at the edges when they are periodically expanded onto the entire x-y plane as mathematically implied by a 2-D Fourier series. The unknown expansions coefficients are calculated using the Rayleigh-Ritz procedure which is actually equivalent to solving the governing equation and the boundary and coupling conditions directly when the assumed solutions are sufficiently smooth over the solution domains. Numerical examples are presented for several different coupling configurations. A good comparison is observed between the current results and the FEA models. Although this study is specifically focused on the coupling of two plates, the proposed method can be directly extended to structures consisting of any number of plates.     

Dynamic responses of Reissner–Mindlin plates with free edges resting on tensionless elastic foundations

Dynamic response analysis is presented for a Reissner–Mindlin plate with four free edges resting on a tensionless elastic foundation of the Winkler-type and Pasternak-type. The mechanical loads consist of transverse partially distributed impulsive loads and in-plane static edge loads while the temperature field is assumed to exhibit a linear variation through the thickness of the plate. The material properties are assumed to be independent of temperature. The two cases of initially compressed plates and of initially heated plates are considered. The formulations are based on Reissner–Mindlin first-order shear deformation plate theory and include the plate–foundation interaction and thermal effects. A set of admissible functions is developed for the dynamic response analysis of moderately thick plates with four free edges. The Galerkin method, the Gauss–Legendre quadrature procedure and the Runge–Kutta technique are employed in conjunction with this set of admissible functions to determine the deflection-time and bending moment–time curves, as well as shape mode curves. An iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region. The numerical illustrations concern moderately thick plates with four free edges resting on tensionless elastic foundations of the Winkler-type and Pasternak-type, from which results for conventional elastic foundations are obtained as comparators. The results confirm that the plate will have stronger dynamic behavior than its counterpart when it is supported by a tensionless elastic foundation.     

Vibration of edge reinforced annular plates

The free transverse vibration of annular plates reinforced by circular rings along the simply supported outer and free inner boundaries is analyzed. A closed form solution is obtained by applying the appropriate forces, moments and motions of the circular edge-beams as boundary conditions on the differential equation for the free transverse vibration of plates. A study of the resulting natural frequencies compared with previous results for unreinforced edges indicates that edge-beams with even a relatively small stiffness have a significant effect upon the natural frequencies of the system.     

Vibration and buckling of tapered rectangular plates with two opposite edges simply supported and the other two edges elastically restrained against rotation

The paper describes an application of a method of power series expansions to the free vibration and buckling problems of isotropic rectangular plates with linear thickness variation. The plates are simply supported on the two opposite edges parallel to the direction of thickness variation and the other two edges are elastically restrained against rotation. By the present method, one can solve exactly the governing equation with variable coefficients. The choice of the origin for the power series expansion plays an important role in obtaining rapid convergence and accurate results. The effects of thickness variation and rotational stiffness of the elastic spring on the eigenvalues and mode shapes are shown numerically and graphically on the basis of new results obtained by the present exact analysis.     

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.     

Influence of boundary stress singularities on the vibration of clamped and simply-supported sectorial plates with arbitrary radial edge conditions

This paper extends previous studies made for sectorial plates having re-entrant (i.e., interior) corners causing stress singularities, to provide accurate frequencies when the circular edge is either clamped or simply-supported. An extensive review of the literature is also given herein spanning nearly the past two decades explaining the free vibration characteristics of sectorial plates. In this work, the classical Ritz method is employed with two sets of admissible functions assumed for the transverse vibratory displacements. These sets include: (1) mathematically complete algebraic-trigonometric polynomials which guarantee convergence to exact frequencies as sufficient terms are retained and (2) corner functions which account for the bending moment singularities at the re-entrant vertex corner of the radial edges having arbitrary edge conditions. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of non-dimensional frequencies for sectorial plates having either a clamped or hinged circumferential edge and various combinations of clamped, hinged, and free conditions on the radial edges. Accurate (to at least four significant figure) frequencies and normalized contours of the transverse vibratory displacement are presented for the spectra of sector angles [90°, 180° (semi-circular), 270°, 300°, 330°, 350°, 355°, 360° (complete circular)] causing a re-entrant vertex corner of the radial edges. For sector angles of 360°, a clamped-clamped, clamped-hinged, clamped-free, hinged-free or free-free radial crack ensues. One general observation is the substantial reduction in the first six frequencies as the sector angle increases for all plates, except in the first two modes of plates having free-free radial edges.