Mode Shapes Of Plate Research Articles

Exact solutions for the free in-plane vibration of rectangular plates with two opposite edges simply supported

Two distinct types of simple support boundary conditions are formulated for rectangular plates undergoing in-plane free vibration. Each type of simple support is shown to be analogous to the well known simple support edge conditions encountered in the free transverse vibration analysis of rectangular plates. It is shown that exact solutions may be obtained for plate free vibration eigenvalues and mode shapes when two opposite plate edges are given either type of simple support, the other two edges being given any combination of classical edge conditions. A complete analysis is provided for plate in-plane vibration with pairs of clamped or free edge conditions imposed on the non-simply supported boundaries and exact eigenvalues are tabulated for a large range of plate aspect ratios. This appears to be the first thorough study of these in-plane vibration problems with exact solutions.

Local and Global Damped Vibrations of Plates with a Viscoelastic Soft Flexible Core: An Improved High-order Approach

An improved high-order theory is presented to investigate the dynamic behavior of thin and thick fiber-reinforced plastic (FRP) plates with a soft viscoelastic flexible core. Shear deformation theory is used for the face sheets while three-dimensional elasticity theory is used for the soft core. The analysis permits nonlinear distortions of the cross-sectional plane of the core as well as changes in its height. The analysis determines the damped natural frequencies, loss factors, and local and global mode shapes of plates. Some of the results are hitherto not reported in the literature based on a higher-order plate theory (HSAPT). Transverse shear and rotary inertia effects of face sheets are taken into consideration. For simply supported boundary condition, closed-form solutions are obtained by Navier’s technique. Numerical results are presented and compared with the experimental and theoretical results found in literature.

A state-space coupling method for fluid-structure interaction analysis of plates

A state-space coupling method is presented for the direct solution of fluid-loaded natural frequencies and mode shapes of plates. This method expands the frequency-dependent term in the Rayleigh integral in a power series on circular frequency. After factoring out the frequency term from the integrands, integration involved in setting up the acoustic impedance coefficient matrices is confined to a frequency-independent part. The acoustic impedance coefficient matrices are therefore frequency-independent and can be coupled directly with the structural matrix into a canonical state-space form to yield the fluid-loaded modes via the direct eigenvalue analysis. A fluid-loaded stiffened plate is involved to demonstrate the method. Numerical results of the fluid-loaded natural frequencies, mode shapes, and modal damping ratios are given to show the efficacy of the state-space coupling method.

Measurement of transverse vibrations/visualization of mode shapes in square plate by using digital speckle pattern interferometry and wavelet transform

New filtering scheme is investigated and implemented on digital speckle pattern interferometric fringes to enhance the signal-to-noise ratio (SNR) in the speckle interferograms. To establish the potential of new filtering scheme the experiment was conducted on the vibrating cantilever beam. Experimental results revealed that the new filtering scheme is more powerful than other known filtering schemes (Kumar et al. Opt Laser Eng 2004;41:81–93. Kumar et al. Opt Laser Technol 2001;33:567–571. Shakher et al. Opt Eng 2002;41:176–180. Shakher et al. Proceedings ICICS, Singapore, 9–12 September 1997, p. 953–956. Bowler et al. Proceedings of IEE second international conference on image processing, London, 1986, p. 24–26. Lim. Opt Eng 1981;20:670–678. Devila et al. J Mod Opt 1995;42:1795–1804; Kaufmann et al. Opt Eng 1996;35:9–14). The new scheme reduced the speckle noise and improved SNR in speckle interferograms. Further, the new investigated filtering scheme is implemented to study the mode shapes of square plates under two different boundary conditions. In the first condition all the edges of the square plate were fixed while in the second condition two adjacent edges were fixed and the remaining two edges were free. From the experimental results it is clear that the mode shapes are more distinctly visible with the implementation of the new filtering scheme as compared to other known schemes. Under both the boundary conditions the recorded resonance frequencies were compared with the calculated values of resonance frequencies based on classical theory. The results obtained form DSPI show good agreement with classical theory.

Analysis of Bending Vibration of Rectangular Plates Using Two-Dimensional Plate Modes

A higher order assumed modes (Ritz) method is developed for rectangular plate analysis using two-dimensional plate mode shape functions. These two-dimensional plate mode shape functions were determined using the extended Kantorovich‐Krylov method. The rectangular plate is assumed isotropic and uniform, and has each edge of its span (root and tip) clamped and remaining edges free (clamped‐free‐clamped‐free boundary conditions). Natural frequencies, mode shape functions, and frequency responses of the plate were calculated and compared to the results of traditional analysis using one-dimensional beam modes to approximate plate modes in both x and y directions. The updated two-dimensional plate mode shape functions substantially reduce the computational cost compared to the case of using one-dimensional beam mode shape functions. Fewer modes are needed in the proposed method to yield the same accuracy. Experiments were conducted to validate our predictions and the data agree with our predictions.

Free in-plane vibration analysis of rectangular plates with elastic support normal to the boundaries

The superposition method is employed to analyze the effects of elastic edge support on the in-plane free vibration frequencies and mode shapes of rectangular plates. The elastic edge support is considered to be uniformly distributed along the plate edges and symmetrically distributed with respect to the plate central axes. In addition, exact solutions are also obtained for a family of simply supported plates whose eigenvalues constitute upper limits for those of the elastically supported plates. An array of highly accurate eigenvalues for the completely free plate is also computed in order to provide lower limit eigenvalues. Eigenvalues are computed and tabulated for elastically supported plates of various aspect ratios and dimensionless elastic support coefficients.

An unusual exact, closed-form solution for axisymmetric vibration of inhomogeneous simply supported circular plates

Apparently for the first time in the literature, an exact closed-form solution is derived for an axisymmetrically vibrating inhomogeneous circular plate that is simply supported at its boundary. The solution is characterized here as an unusual one since for its counterpart—the homogeneous plate, transcendental functions are called for whereas here a solution is found in elementary functions, namely, polynomials. The analysis is based on an inverse vibration problem: Given a candidate mode shape and density distribution, we calculate the plate stiffness so that the governing equation for the plate mode shape is identically satisfied. We also are able to obtain the expression for the corresponding natural frequency.

Flexural vibration of rectangular plates subjected to sinusoidally distributed compressive loading on two opposite sides

Natural frequencies and mode shapes of rectangular plates subjected to sinusoidally distributed in-plane compressive loading on two opposite edges were considered in this work. Although vibration analysis of rectangular plates was investigated by numerous authors, flexural vibration of plates with nonlinearly distributed in-plane loading received much less attention. However, certain problems involving thermal stresses may have such nonlinear loading conditions in the plane of the plate. The analysis procedure for the title problem involves first finding a plane elasticity solution for the in-plane problem satisfying all boundary conditions. Then using this in- plane solution, flexural vibration analysis has to be carried out. In a related work investigated recently by the present authors, an in-plane elasticity solution was developed and was compared with existing analytical solutions in the literature and with the finite element method. One aspect of this study was the stress diffusion (i.e., reduction of stress) phenomenon along the length of the plate as well as the presence of all three in-plane stress components. Using this in-plane solution, vibration analysis of rectangular plates was carried out for various plate aspect ratios and the results were compared with corresponding values from finite element analysis.

Macromechanical characteristics deduced from tests on glass/epoxy composites with optical methods

AbstractExperimental results are required to verify the simulation results obtained by the homogenisation procedure of a biaxial weftknitted fabric reinforced composite with the Binary Model [1]. Tensile, bending and shear tests were performed to investigate and validate the elastic macro‐mechanical behaviour of the composite material. Beside other methods an optical whole‐field measuring system was used to estimate the deformation in all tests. Additionally, natural frequencies and the appropriate mode shapes of plates were used to get the elements of the compliance matrix. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates. Part I: iterative and explicit analytical solution for non-linear transverse vibrations

The effects of large vibration amplitudes on the first two axisymmetric mode shapes of clamped thin isotropic circular plates are examined. The theoretical model based on Hamilton's principle and spectral analysis developed previously by Benamar et al. for clamped–clamped beams and fully clamped rectangular plates is adapted to the case of circular plates using a basis of Bessel's functions. The model effectively reduces the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. Numerical results are given for the first and second axisymmetric non-linear mode shapes for a wide range of vibration amplitudes. For each value of the vibration amplitude considered, the corresponding contributions of the basic functions defining the non-linear transverse displacement function and the associated non-linear frequency are given. The non-linear frequencies associated to the fundamental non-linear mode shape predicted by the present model were compared with numerical results from the available published literature and a good agreement was found. The non-linear mode shapes exhibit higher bending stresses near to the clamped edge at large deflections, compared with those predicted by linear theory. In order to obtain explicit analytical solutions for the first two non-linear axisymmetric mode shapes of clamped circular plates, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri et al. for beams and rectangular plates, has been adapted to the case of clamped circular plates, leading to explicit expressions for the higher basic function contributions, which are shown to be in a good agreement with the iterative solutions, for maximum non-dimensional vibration amplitude values of 0.5 and 0.44 for the first and second axisymmetric non-linear mode shapes, respectively.

Vibration of Plates with Constrained V-Notches or Cracks

This paper reports the first known free vibration solutions for thin circular plates with V-notches having various edge conditions. The classical Ritz method is employed with 2 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 to account for the bending moment singularities at the sharp re-entrant corner of the V-notch. Extensive convergence studies summarized herein confirm that the corner functions substantially enhance the convergence and accuracy of nondimensional frequencies for circular plates having a free circumferential edge and various combinations edge conditions of the V-notch. Accurate (5 significant figures) frequencies are presented for clamped-free, clamped-hinged, and hinged-free notches for the spectra of notch angles, causing a re-entrant vertex corner of the radial edges. For very small notch angles, a clamped-free, clamped-hinged, or hinged-free radial crack ensues. One general observation is that, for the range of notch angles considered, there is a substantial increase in the first 6 frequencies as the notch depth increases. The frequency increase with increasing notch depth is more pronounced in the higher modes than the lower ones, and is quite substantial for segmental plates with notch angles equal to 180 deg. A large reduction in frequencies is also observed as the notch angle decreases at a constant notch depth. A new database of accurate frequencies and mode shapes for sectorial, semicircular and segmental plates is presented with which future solutions drawn from alternative numerical procedures and finite element and boundary element techniques may be compared. Normalized contours of the transverse vibratory displacement are shown for plates with various notch depths and notch angles of 5, 30, 60, 90, and 180 deg.

Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular and annular plates with nonlinear thickness variation along the radial direction. Unlike conventional plate theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u s , u z , and u θ in the radial, thickness, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the s and z directions. Potential (strain) and kinetic energies of the plates are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the plates. Numerical results are presented for completely free, annular and circular plates with uniform, linear, and quadratic variations in thickness. Comparisons are also made between results obtained from the present 3-D and previously published thin plate (2-D) data.

Free vibration analysis of corner-supported rectangular plates with symmetrically distributed edge beams

Analytical type solutions are obtained for the free vibration frequencies and mode shapes of thin corner-supported rectangular plates with symmetrically distributed reinforcing beams, or strips, attached to the plate edges. The method of superposition is employed. Equations governing reactions at plate–beam interfaces are developed in dimensionless form. The approach is comprehensive in that both lateral and rotational stiffness, and inertia, of the beam are incorporated into the analysis. For illustrative purposes computed eigenvalues and mode shapes are presented for two plate–beam systems of realistic geometries. It is shown that the method is easily extended to cover the case where the edge beams do not have a symmetrical distribution. This appears to be the first comprehensive analytical study of this problem of industrial interest.

３．計算力学 平板の振動モードの縮退と連成挙動について

This paper presents the degenerate modes and modal coupling in vibrating rectangular plates with the same boundary conditions along four edges.The spline element method is applied to analyze frequencies and mode shapes of orthotropic rectangular plates with tapered thickness.Some of degenerate mode pairs are split in frequency in vibration of isotropic square plates with the four edges clamped, free or simply supported, respectively. And nondegenerate modes such as X mode or ring mode can occur in tapered square plates and orthotropic rectangular plates having certain width/thickness ratio and aspect ratio.

Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method

This paper describes a method for free vibration analysis of rectangular plates with any thicknesses, which range from thin, moderately thick to very thick plates. It utilises admissible functions comprising the Chebyshev polynomials multiplied by a boundary function. The analysis is based on a linear, small-strain, three-dimensional elasticity theory. The proposed technique yields very accurate natural frequencies and mode shapes of rectangular plates with arbitrary boundary conditions. A very simple and general programme has been compiled for the purpose. For a plate with geometric symmetry, the vibration modes can be classified into symmetric and antisymmetric ones in that direction. In such a case, the computational cost can be greatly reduced while maintaining the same level of accuracy. Convergence studies and comparison have been carried out taking square plates with four simply-supported edges as examples. It is shown that the present method enables rapid convergence, stable numerical operation and very high computational accuracy. Parametric investigations on the vibration behaviour of rectangular plates with four clamped edges have also been performed in detail, with respect to different thickness-side ratios, aspect ratios and Poisson’s ratios. These results may serve as benchmark solutions for validating approximate two-dimensional theories and new computational techniques in future.

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES: PART II—FIRST AND SECOND NON-LINEAR MODE SHAPES OF FULLY CLAMPED RECTANGULAR PLATES

In a previous series of papers, a semi-analytical model based on Hamilton's principle and spectral analysis has been developed for geometrically non-linear free vibrations occurring at large displacement amplitudes of clamped–clamped beams and fully clamped rectangular homogeneous and composite plates. In Part I of this series of papers, concerned with geometrically non-linear free and forced vibrations of various beams, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written in the modal basis, in the neighbourhood of each resonance has been developed. Simple explicit formulae, ready and easy to use for analytical or engineering purposes have been derived, which allows direct calculation of the basic function contributions to the first three non-linear mode shapes of the beams considered. Also, various possible truncations of the series expansion defining the first non-linear mode shape have been considered and compared with the complete solution, which showed that an increasing number of basic functions has to be used, corresponding to increasingly sized intervals of vibration amplitudes; starting from use of only one function, i.e., the first linear mode shape, corresponding to very small amplitudes, for which the linear theory is still valid, and ending by the complete series, involving six functions, corresponding to maximum vibration amplitudes at the beam middle point up to once the beam thickness. For higher amplitudes, a complementary second formulation has been developed, leading to reproduction of the known results via the solution of reduced linear systems of five equations and five unknowns. The purpose of this paper is to extend and adapt the approach described above to the geometrically non-linear free vibration of fully clamped rectangular plates in order to allow direct and easy calculation of the first, second and higher non-linear fully clamped rectangular plate mode shapes, with their associated non-linear frequencies and non-linear bending stress patterns. Also, numerical results corresponding to the first and second non-linear modes shapes of fully clamped rectangular plates with an aspect ratio α=0·6 are presented. Data concerning the higher non-linear modes, the aspect ratio effect, and the forced vibration case will be presented later.

Free In-Plane Vibration of Rectangular Plates

BasedontheKantorovich ‐Krylovvariationalmethod (Kantorovich,L.V.,andKrylov,V.I., ApproximateMethods of Higher Analysis , Noordhoff International, Groningen, The Netherlands, 1964, pp. 241 ‐357), we analytically solve for modal frequency and displacement mode shapes of in-plane vibration of rectangular plates with free and clamped boundary conditions. Free and clamped boundary conditions do not allow closed-form solutions from partial differential equations. We develop analytical expressions of plate mode shapes consisting of a linear combination of progressive waves with unknown wave amplitudes. We then use these to calculate the modal frequencies and the wave amplitudes with analytical mode shape expressions. An iteration scheme is presented to calculate efe ciently the natural frequency and corresponding mode shape functions based on the Kantorovich ‐ Krylov method. Three cone gurations are considered: a plate with four edges clamped, a plate with three edges clamped and one edge free, and a plate with two parallel edges clamped and the other two edges free. The e rst six natural frequencies predicted by our approach are validated using NASTRAN and other analyses from the literature. Improvements in accuracy of the e rst six predicted natural frequencies were achieved using the Kantorovich ‐Krylov method when compared against other results in the literature. a = length of plate in x direction b = length of plate in y direction E = Young’ s modulus of plate material h = plate thickness k1;2 = two wave numbers in the solution of mode shape functions m;n = number of half-wavelengths for mode shapes

A Generalized kantorovich method and its application to free in-plane plate vibration problem

In this paper, we develop a generalized Kantorovich method to construct a closed from solution for two coupled partial differential equations (PDEs) with mixed boundary conditions. One of the applications of this method is the study of i-plane vibration of rectangular plates. We obtain analytical expressions of plate mode shapes comprising a linear combination of progressive waves with unknow wave amplitudes. Using these mode shapes, we iteratively calculate the modal frequencies and wave amplitudes. We consider three cases: (1)a Plate with four edges clamped;(2)a plate with three edges clamped and one edge free; and(3)a plate with two parallel edges clamped and the remaining two edges free. Improvements in accuracy of the first six predicted natural frequencies were achieved by using our method, when compared with prior results from the literature.

An Exact Approach For Free Vibration Analysis of Multi-Step Nonuniform Shear Plates

In this paper multi-story frame buildings are treated as shear plates with variably distributed mass and stiffness for free vibration analysis. An analytical model of a shear plate is proposed and an exact approach for determining natural frequencies and mode shapes of such shear plates is presented. The function for describing the distribution of mass of a shear plate along the plate height is an arbitrary one and the distribution of shear stiffness is expressed as a functional relation with the distribution of mass and vice versa. The exact solutions of one-step shear plates are obtained first for seven cases. Then the derived exact solutions of one-step shear plates are used to establish the frequency equations of multi-step shear plates by using the transfer matrix method. The numerical example shows that the natural frequencies and mode shapes of a shear plate calculated by the proposed methods are in good agreement with the measured data and those determined by the Ritz method and the finite element method, verifying the accuracy and applicability of the proposed methods.

F-1017 内部を数点で支持された周辺固定円板の固有振動解析(G10-1 連続体の解析)(G10 機械力学・計測制御部門一般講演)

The natural frequencies and the mode shapes of circular plate with clamped edge supported inside several points are determined. A partial integro-differential equation is solved using eigenfunctions and eigenfrequencies of circular plate with clamped edge. If the reaction forces at the supported points for free vibration are considered as harmonically excitation forces with unknown amplitudes, a periodic solution of forced vibration of circular plate with clamped edge is obtained. Using the conditions that the deflection must be zero at the supported points, simultaneous equations with respect to the unknown amplitudes of forces are obtained. The frequency equation is obtained from a necessary and sufficient condition for existence of the non-trivial solutions.