Natural Frequencies Of Plate Research Articles

NATURAL FREQUENCIES OF ORTHOTROPIC RECTANGULAR PLATES OBTAINED BY ITERATIVE REDUCTION OF THE PARTIAL DIFFERENTIAL EQUATION

Natural frequencies of orthotropic rectangular plates are obtained by successive reduction of the plate partial differential equation and by solving the resulting ordinary differential equation exactly. The reduction of the partial differential equation is carried out by assuming an approximate solution satisfying the boundary conditions along one direction and employing a Galerkin averaging technique known as the Kantorovich method. The resulting ordinary differential equation along the other direction is solved exactly, which provides the natural frequencies of the plate. Natural frequencies of the orthotropic rectangular plates with various boundary conditions are obtained by using this approach, and presented along with the discussion of the convergence and accuracy of the results.

A non-destructive evaluation of the material properties of a composite laminated plate

A non-destructive method for the evaluation of material properties of a rectangular, anisotropic, homogeneous plate with four free edges is presented. The method consists of two steps. In the first step, a certain number of the plate's natural frequencies are experimentally measured. In the second step, the plate rigidities are varied in a theoretical model, so that the calculated natural frequencies match as close as possible the corresponding experimental values. Two such models are presented, based on the Classical Lamination Theory and on a Higher Order Shear Deformation Theory. High order Lagrange polynomials are used as deflection functions and the Rayleigh-Ritz procedure is employed to arrive at the solution. The identification of the plate rigidities is done by means of an iterative Bayesian parameter estimation method, where possible measurement errors or rigidities' uncertainties can be taken into account.

Natural frequencies of plates with arbitrary concentrated mass and stiffness modifications

Vibration analysis of plates has been an active research subject and numerous technical papers have been accumulated. Though analytical solutions have been found for plates with specified forms of mass or stiffness modifications, the problem of plates with arbitrary mass and stiffness modifications has not been addressed to date despite its important practical implications. In this paper, a new and effective method has been developed to predict exactly the natural frequencies and mode shapes of plates with arbitrary mass, stiffness and damping modifications. Plate receptances are derived based on mode superposition and are then used to calculate the receptances of the modified plate. Natural frequencies and mode shapes of the modified plate are identified by analyzing the receptance data based on the concept of modal analysis. Numerical results are given to demonstrate the practical applicability of the proposed method.

DYNAMICS AND CONTROL OF NON-LINEAR CIRCULAR PLATES WITH PIEZOELECTRIC ACTUATORS

Linear dynamics and distributed control of piezoelectric laminated continua have been intensively studied in recent years. In this paper, dynamics, electromechanical couplings, and control of piezoelectric laminated circular plates with an initial non-linear large deformation are investigated. It is assumed that the transverse non-linear components is much more prominent than the other two in-plane components—the von Karman type geometrical non-linear deformation. In addition, the piezoelectric layers are uniformly distributed on the top and bottom surfaces of the circular plate. Accordingly, the control effect is introduced via an equivalent control moment on the circumference. Dynamic equations and boundary conditions including elastic and piezoelectric couplings are formulated, and solutions are derived. Control of the plate's non-linear deflections and natural frequencies using high control voltages are studied, and their non-linear effects are evaluated.

Single-layer plate theories applied to the flexural vibration of completely free thick laminates

Two different refined single-layer plate theories are used to analyze the flexural vibration of thick symmetrically laminated rectangular plates with all edges free. Both theories account for parabolic distribution of the transverse shear strains through the thickness of the plate. The first theory includes the transverse normal effect and employs a three-dimensional constitutive law, whereas the second neglects this effect and employs plane-stress reduced stiffnesses. Numerical results are obtained by using the Ritz method. Characteristic orthogonal polynomials are used in the series representing the displacement functions. Convergence studies are shown to verify the accuracy of the approximate method. The accuracy of the results is further assessed by comparison with results from three-dimensional finite element analyses. Numerical results are presented for various plates including cross-ply and angle-ply laminated plates. The effect of transverse shear and normal deformation on the plate natural frequencies is illustrated by comparing the frequencies of the two higher order theories and the classical plate theory for decreasing length-to-thickness ratio.

Free flexural vibration analysis of one-way stiffened plates by the free interface modal synthesis method

A numerical model is presented for predicting the natural frequencies of one-way stiffened plates with ribs having high ratios of flexural to shear rigidity. The model is based on the free interface modal synthesis method. Experimental validation using floors with wood I-joists and wood-based sheathing showed that the model has good numerical accuracy in the predictions of natural frequencies and mode shapes if analyses include shear deformation and rotatory inertia effects in ribs. Neglect of these effects can lead to large errors in the predicted natural frequencies for plates with ribs having high ratios of flexural to shear rigidity. Large errors can also be encountered in natural frequency prediction for plates with fairly low ratios of flexural to shear rigidity. This occurs with mode shapes that have multiple curvature along ribs if shear deformation and rotatory inertia effects are neglected. Key words: free flexural vibration, natural frequencies, ribbed plates, flexural rigidity, shear rigidity, modal synthesis.

Free vibration of rectangular isotropic plates with and without a concentrated mass

The natural frequencies of plates without and with a concentrated mass are analysed and presented. The effect of the various combinations of clamped (C) and simply-supported (S) conditions along the edges of the plates on natural frequencies is also presented. The Rayleigh-energy method is used in the theoretical formulation and single-term trigonometric functions are used to postulate the appropriate mode shapes. An extensive comparison of the results with well-known solutions is presented for plates carrying no concentrated mass. Comparison of the results for plates carrying a concentrated mass with published results is however limited. Lastly some experimental results for the case of the S-C-S-C plate carrying a concentrated mass are presented and discussed.

Transverse Vibration of Plate Structures With Elastically Restrained Edges by the Spline/Finite Strip Method

The newly developed spline finite strip method has been applied to determine the natural frequencies of plates and stiffened plates with edged elastically restrained against translation and rotation. The stiffener has been modelled so that it can be situated anywhere within the plate strip. The formulation has been generalized in such a manner that it can handle general shaped plates and stiffeners having arbitrary orientation and eccentricity. A consistent formulation has been adopted to incorporate the contribution of the edge restraints in the structural stiffness matrix. Numerical examples as available in the literature are solved and the comparison of the results indicates good accuracy of the method. New results in the form of natural frequencies corresponding to the higher modes particularly for plates/stiffened plates with nonclassical boundaries are presented.

Plate Characteristic Functions and Natural Frequencies of Vibration of Plates by Iterative Reduction of Partial Differential Equation

Natural frequency coefficients of rectangular plates and the corresponding plate characteristic functions are obtained by reduction of plate partial differential equation to an ordinary differential equation and solving it exactly. The reduction is carried out by assuming a deflection shape in one direction consistent with the boundary conditions and applying Galerkin’s averaging technique to eliminate the variable. The reduction method, commonly known as Kantorovich method, is applied sequentially on either directions of the plate and iterated until convergence is achieved for the natural frequency coefficients. The resulting plate characteristic functions are very good approximations to the normal modes of the plate. The results are tabulated for plates with combination of clamped, simply-supported, and free edge conditions.

A comprehensive analytical solution for free vibration of rectangular plates with classical edge coditions: Experimental verification

The problem of obtaining free vibration frequencies and mode shapes of rectangular plates resting on combinations of classical (i.e., clamped, simply supported, or free) edge supports is one that has been investigated for more than one hundred years. More recently, the superposition method has been developed for obtaining accurate analytical-type solutions for this family of problems. The object of this paper is to report on the results of numerous experimental tests carefully performed in order to verify the superposition method and associated computer software. Experimental and computed results are compared for a wide range of plate configurations. Very good agreement between theory and experiment has been obtained with regard to both plate natural frequencies and mode shapes. It is concluded that this computational procedure constitutes a powerful new tool for analysis of rectangular plate vibration problems.

Tuning of the natural frequency spectrum of a circular plate by in-plate stress

Initial stresses are often introduced deliberately into centrally clamped, peripherally free circular plates to increase the plate natural frequencies. Increased natural frequencies in the critical modes extend the range of stable rotation speed. Current axisymmetric initial stressing schemes increase the natural frequencies of the zero nodal circle, two and higher nodal diameter vibration modes, but the zero and one nodal diameter natural frequencies decrease. For sufficiently large initial stress, the zero nodal diameter natural frequency vanishes and the plate experiences divergence buckling, thereby limiting the initial stress that can be induced. In this paper a new initial stressing method is presented whereby the natural frequencies of all zero nodal circle modes increase simultaneously, thus avoiding buckling of the plate through excessive initial stress. The stress field is induced by applying uniform pressure to the boundaries of eccentric interior holes. Through proper location of the holes, the critical natural frequencies increase monotonically with increased pressure, and the magnitudes of the frequency shifts increase with the introduction of additional stressed holes. The initial stressing schemes can be directed towards optimal increases in specific vibration modes. Calculations predict natural frequency increases between 35% and 90% depending on the vibration mode. Experimental measurements are consistent with the model predictions.

Vibration of Orthotropic Rectangular Plates With Variable Thickness

The problem vibration of rectangular orthotropic plates with variable thickness and mixed boundary conditions are solved by a modified energy method. A general expression is written for the deflection of the plate without aiming at any particular combination of boundary conditions. Boundary conditions are satisfied approximately by adjusting a set of so-called fixity factors. A computer program has been developed to solve for natural frequencies of plates with variable thicknesses and having different orthotropic properties.

Vibration and stability of elasticity supported circular plates under conservative and non-conservative loads

This paper deals with the vibration and stability of circular plates elastically restrained against translation and rotation along concentric intermediate circles as well as at the outer edge. The plate is subjected to a horizontal or tangential radial load at the periphery. Both axisymmetric and non-axisymmetric vibrations are considered. It is assumed that the plate is thin, elastic, and isotropic. The solutions are obtained in terms of Bessel functions for both the complete and annular plate sections. The effect of the support stiffnesses on the plate natural frequency and the divergence and flutter instability loads are studied in detail.

Vibrations of Twisted Cantilevered Plates—Experimental Investigation

The experimental portion of a joint government/industry/university research study on the vibrational characteristics of twisted cantilevered plates is presented. The overall purpose of the research study was to assess the capabilities and limitations of existing analytical methods in predicting the vibratory characteristics of twisted plates. Thirty cantilevered plates were precision machined at the Air Force’s Aero Propulsion Laboratory. These plates, having five different degrees of twist, two thicknesses, and three aspect ratios representative of turbine engine blade geometries, were tested for their vibration mode shapes and frequencies. The resulting nondimensional frequencies and selected mode shapes are presented as a function of plate tip twist. The trends of the plate natural frequencies as a function of the governing geometric parameters are discussed. The effect of support compliance on the plate natural frequency and its impact on numerically modeling twisted plates is also presented.

Real Time Vibration Control of Rotating Circular Plates by Temperature Control and System Identification

A system for real-time control of the transverse vibration of a rotating circular plate, based on a thermal stressing technique and dynamic system identification, is presented in this paper. In this method the plate natural frequency spectrum is modified through the purposeful introduction of thermal membrane stresses. The critical speed is maximized. In effect vibration is controlled through real-time control of the plate design. Evaluations with computer simulation and experimental measurements on a thin circular plate verify the system capability to control transverse vibration in a changing thermal environment.

Dynamic behaviour of isotropic plates under combined acoustic excitation and static in-plane compression

A study is described which was carried out to investigate the dynamic response of an aircraft-type aluminium alloy plate, with fully clamped boundaries, subjected to combined acoustic excitation and uniaxial in-plane compression. Experiments were conducted in an acoustic tunnel with sound pressure levels up to 150 dB. The non-linear characteristics of the plate have been examined under sinusoidal and broad band random excitation and a study made of the statistical and spectral properties of the response. Theoretical and experimental estimates of the plate natural frequencies and mode shapes are described.

Improved natural frequencies for plates with a free edge

A simple correction formula is developed for the square of natural frequencies of plates with a free edge. The theory is applied to torsional vibrations of a long plate strip and to vibrations of a circular plate with a free edge and a number of nodal diameters. The paper concludes with the remark that the theory may be extended mutatis mutandis to vibrations of shells with a free edge.

A natural frequency analogy between spherically curved panels and flat plates

By way of a similitude analysis of spherically curved panels, an analogy has been established between the natural frequencies of such panels and the natural frequencies of plates that are similar in boundary shape and type. Also, a general similitude relationship for spherically curved panels has been derived.