Heat Induced Vibration of a Rectangular Plate

1974 ◽  
Vol 96 (3) ◽  
pp. 1015-1021 ◽  
Author(s):  
N. D. Jadeja ◽  
Ta-Cheng Loo
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Rectangular Plate ◽  
Equation Of Motion ◽  
The Other ◽  
Double Trigonometric Series ◽  
Deflection Curve ◽  
Thermally Induced ◽  
Simply Supported ◽  
Governing Differential Equation

The purpose of this paper is to investigate thermally induced vibration of a rectangular plate with one edge fixed and other three edges simply supported. The plate was subjected to a sinusoidal heat input, which varied with respect to time, on one face while the other face of the plate was insulated. An approximate solution to the governing differential equation of motion of the plate was assumed in the form of a double trigonometric series which satisfied all the boundary conditions. Galerkin’s method was then used to obtain the deflection curve for the plate and corresponding stresses at various points in the plate. Certain interesting phenomena indicate the possibility of predicting early fatigue failure. Results are presented in graph forms and discussed.

Buckling and Vibration of Isosceles Triangular Plates having the Two Equal Edges Clamped and the Other Edge Simply–Supported

1955 ◽  
Vol 59 (530) ◽  
pp. 151-152 ◽  
Author(s):  
Hugh L. Cox ◽  
Bertram Klein
Keyword(s):  
Differential Equation ◽  
Axial Load ◽  
Unit Length ◽  
Approximate Solutions ◽  
The Other ◽  
Thin Plates ◽  
Critical Buckling Load ◽  
Simply Supported ◽  
Governing Differential Equation ◽  
Image Position

Approximate Solutions obtained by the method of collocation are presented for the lowest critical buckling load of an isosceles triangular plate loaded as shown in Fig. 1. Also, the fundamental frequency is given. The base of the triangle is simply supported and the other equal edges are clamped. The usual assumptions regarding the bending of thin plates are made. The governing differential equation for the plate loaded as shown in Fig. 1 is1where D is the plate stiffness, N is axial load per unit length, w is deflection, positive downward, and the quantities a and h are dimensions shown in Fig.1.

Vibrations of Rectangular Plates With Nonuniform Elastic Edge Supports

1980 ◽  
Vol 47 (4) ◽  
pp. 891-895 ◽  
Author(s):  
A. W. Leissa ◽  
P. A. A. Laura ◽  
R. H. Gutierrez
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Exact Solutions ◽  
Ritz Method ◽  
Equation Of Motion ◽  
The Other ◽  
Rectangular Plates ◽  
Governing Differential Equation ◽  
Vibration Problems ◽  
Elastic Edge

Two methods are introduced for the solution of free vibration problems of rectangular plates having nonuniform, elastic edge constraints, a class of problems having no previous solutions in the literature. One method uses exact solutions to the governing differential equation of motion, and the other is an extension of the Ritz method. Numerical results are presented for problems having parabolically varying rotational constraints.

scholarly journals Analytical solution to bending of stiffened and continuous antisymmetric laminates

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Liecheng Sun ◽  
Issam E. Harik
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Displacement Function ◽  
The Other ◽  
Order Partial Differential Equation ◽  
Eighth Order ◽  
Coupling Problem ◽  
Simply Supported ◽  
General Response

AbstractAnalytical Strip Method is presented for the analysis of the bending-extension coupling problem of stiffened and continuous antisymmetric thin laminates. A system of three equations of equilibrium, governing the general response of antisymmetric laminates, is reduced to a single eighth-order partial differential equation (PDE) in terms of a displacement function. The PDE is then solved in a single series form to determine the displacement response of antisymmetric cross-ply and angle-ply laminates. The solution is applicable to rectangular laminates with two opposite edges simply supported and the other edges being free, clamped, simply supported, isotropic beam supports, or point supports.

Effect of Nonhomogeneity on Vibration of Orthotropic Rectangular Plates of Varying Thickness Resting on Pasternak Foundation

2009 ◽  
Vol 131 (1) ◽  
Author(s):  
Roshan Lal ◽  
Dhanpati
Keyword(s):  
Differential Equation ◽  
Plate Theory ◽  
The Other ◽  
Thickness Variation ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Varying Thickness ◽  
Aspect Ratios ◽  
Displacement Mode

Free transverse vibrations of nonhomogeneous orthotropic rectangular plates of varying thickness with two opposite simply supported edges (y=0 and y=b) and resting on two-parameter foundation (Pasternak-type) have been studied on the basis of classical plate theory. The other two edges (x=0 and x=a) may be any combination of clamped and simply supported edge conditions. The nonhomogeneity of the plate material is assumed to arise due to the exponential variations in Young’s moduli and density along one direction. By expressing the displacement mode as a sine function of the variable between simply supported edges, the fourth order partial differential equation governing the motion of such plates of exponentially varying thickness in another direction gets reduced to an ordinary differential equation with variable coefficients. The resulting equation is then solved numerically by using the Chebyshev collocation technique for two different combinations of clamped and simply supported conditions at the other two edges. The lowest three frequencies have been computed to study the behavior of foundation parameters together with other plate parameters such as nonhomogeneity, density, and thickness variation on the frequencies of the plate with different aspect ratios. Normalized displacements are presented for a specified plate. A comparison of results with those obtained by other methods shows the computational efficiency of the present approach.

scholarly journals ANALYTICAL BENDING SOLUTION OF ALL CLAMPED ISOTROPIC RECTANGULAR PLATE ON WINKLER’S FOUNDATION USING CHARACTERISTIC ORTHOGONAL POLYNOMIAL

2017 ◽  
Vol 36 (3) ◽  
pp. 724-728
Author(s):  
EI Ogunjiofor ◽  
CU Nwoji
Keyword(s):  
Differential Equation ◽  
Rectangular Plate ◽  
Elastic Foundation ◽  
Winkler Foundation ◽  
Rectangular Plates ◽  
Deflection Coefficient ◽  
Percentage Difference ◽  
Governing Differential Equation ◽  
Isotropic Rectangular Plate

The analytical bending solution of all clamped rectangular plate on Winkler foundation using characteristic orthogonal polynomials (COPs) was studied. This was achieved by partially integrating the governing differential equation of rectangular plate on elastic foundation four times with respect to its independents x and y axis. The foundation was assumed to be homogeneous, elastic and isotropic. The governing differential equation was non-dimensionalised to make it consistent. The deflection polynomials functions were formulated. Thereafter, the Galerkin’s works method was applied to the governing differential equation of the plate on Winkler foundation to obtain the deflection coefficient, . Numerical example was presented at the end to compare the results obtained by this method and those from earlier studies. The percentage difference obtained for central deflection of all clamped rectangular plate loaded with UDL using the method and earlier research works  for K = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are: 0.000042, 0.000052, 0.000043, -0.000011, -0.000068, 0.000001, 0.000001, 0.000001, -0.000001, 0.000000, 0.000001. 0.000033, 0.000035, 0.000033, -0.000018, -0.000072, -0.000003, -0.000003, -0.000003, 0.000002, -0.000002, -0.000001. The result showed that an easy to use and understandable model was developed for determination of deflections of all clamped rectangular plates on Winkler’s elastic foundation using principle of COPs. http://dx.doi.org/10.4314/njt.v36i3.9

scholarly journals Analysis on free vibration and critical buckling load of a FGM porous rectangular plate

2021 ◽  
Vol 39 (2) ◽  
pp. 317-325
Author(s):  
Zhaochun Teng ◽  
Pengfei Xi
Keyword(s):  
Differential Equation ◽  
Elastic Modulus ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Natural Frequencies ◽  
Gradient Index ◽  
Rectangular Plates ◽  
Buckling Loads ◽  
Governing Differential Equation

The properties of functionally gradient materials (FGM) are closely related to porosity, which has effect on FGM's elastic modulus, Poisson's ratio, density, etc. Based on the classical theory of thin plates and Hamilton principle, the mathematical model of free vibration and buckling of FGM porous rectangular plates with compression on four sides is established. Then the dimensionless form of the governing differential equation is also obtained. The dimensionless governing differential equation and its boundary conditions are transformed by differential transformation method (DTM). After iterative convergence, the dimensionless natural frequencies and critical buckling loads of the FGM porous rectangular plate are obtained. The problem is reduced to the free vibration of FGM rectangular plate with zero porosity and compared with its exact solution. It is found that DTM gives high accuracy result. The validity of the method is verified in solving the free vibration and buckling problems of the porous FGM rectangular plates with compression on four sides. The results show that the elastic modulus of FGM porous rectangular plate decreases with the increase of gradient index and porosity. Furthermore, the effects of gradient index and porosity on dimensionless natural frequencies and critical buckling loads are further analyzed under different boundary conditions with constant aspect ratio, and the effects of aspect ratio and load on dimensionless natural frequencies under different boundary conditions.

Stability of Rectangular Plates Under Shear and Bending Forces

1936 ◽  
Vol 3 (4) ◽  
pp. A131-A135 ◽  
Author(s):  
Stewart Way
Keyword(s):  
Critical Load ◽  
Rectangular Plate ◽  
Bending Moment ◽  
The Other ◽  
Rectangular Plates ◽  
Bending Stress ◽  
Simply Supported ◽  
Tension And Compression ◽  
The Way

Abstract The author first discusses the problem of a plane, simply supported rectangular plate loaded by shearing forces in the plane of the plate on all four edges. There are two stiffeners attached one third and two thirds of the way along the plate. The critical load is calculated for various stiffener rigidities. Also, the rigidity necessary to keep the stiffeners straight when the plate buckles is found. This stiffener rigidity is found to be slightly larger than that necessary for a plate with one stiffener and the same panel dimensions as the plate with two stiffeners. The second problem discussed by the author is that of a plane, simply supported rectangular plate loaded by uniformly distributed edge shearing forces in the plane of the plate and linearly distributed tension and compression in the plane of the plate at the ends. The end forces vary from tension hσo, at one corner to—hσo, at the other corner, so that their resultant is a bending moment. The presence of the edge shearing forces is found to diminish the critical bending stress in this case. Calculations are made for various magnitudes of bending and shearing forces for plates of various proportions.

Boundary Value Problem Method to Investigate Superharmonic Resonance of MEMS Resonators

2017 ◽  
Author(s):  
Julio Beatriz ◽  
Martin Botello ◽  
Christian Reyes ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
The Other ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Amplitude Frequency Response ◽  
Problem Method ◽  
Mems Resonators

This paper deals with two different methods to analyze the amplitude frequency response of an electrostatically actuated micro resonator. The methods used in this paper are the method of multiple scales, which is an analytical method with one mode of vibration. The other method is based on system of odes which is derived using the partial differential equation of motion, as well as the boundary conditions. This system is then solved using a built in matlab function known as BVP4C. Results are then shown comparing the two methods, under a variety of parameters, including the influence of damping, voltage, and fringe.

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