Nonlinear Torsional Vibrations of Thin-Walled Beams of Open Section

1975 ◽  
Vol 42 (1) ◽  
pp. 240-242 ◽  
Author(s):  
C. Kameswara Rao
Keyword(s):  
Differential Equation ◽  
Large Amplitude ◽  
Torsional Vibrations ◽  
Simply Supported ◽  
Open Section ◽  
Governing Differential Equation ◽  
Thin Walled

An attempt has been made to derive and solve the governing differential equation of large amplitude torsional vibrations of simply supported doubly symmetric thin-walled beams of open section. Graphs indicating the influence of large amplitudes on nonlinear period of torsional vibrations for various nondimensional beam constants are presented.

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.

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.

The Torsional Vibration of Uniform Thin-Walled Beams of Open Section

1969 ◽  
Vol 73 (704) ◽  
pp. 672-674 ◽  
Author(s):  
J. B. Carr
Keyword(s):  
Torsional Vibration ◽  
Characteristic Functions ◽  
Shear Stresses ◽  
Simply Supported ◽  
Open Section ◽  
Torsional Resistance ◽  
Approximate Frequency ◽  
Thin Walled ◽  
Image Position ◽  
Fixed End

The pure torsional vibration of uniform thin-walled beams of open section, which is governed by the differential equation has been extensively analysed by Gere He derived the exact frequency equations for beams with a variety of end conditions. However, these equations are, in most cases, highly transcendental. This note uses an energy approach to obtain approximate frequency equations for the fixed-fixed and the fixed-simply-supported beams. A fixed end is one which allows no twist and no warping and a simply-supported end allows no twist but permits warping to take place freely. The approximating functions used are those corresponding to the exact solution of the problem if the torsional resistance caused by the St Venant system of shear stresses is zero. These functions are similar to the characteristic functions of simple beams in flexure.

On Asymptotic Analysis for Large Amplitude Nonlinear Free Vibration of Simply Supported Laminated Plates

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
S. K. Lai ◽  
C. W. Lim ◽  
Y. Xiang ◽  
W. Zhang
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Nonlinear Differential Equation ◽  
Large Amplitude ◽  
Analytical Approximation ◽  
Laminated Plates ◽  
Thin Plates ◽  
Nonlinear Free Vibration ◽  
Simply Supported ◽  
Analytical Approximations

An analytical approximation is developed for solving large amplitude nonlinear free vibration of simply supported laminated cross-ply composite thin plates. Applying Kirchhoff’s hypothesis and the nonlinear von Kármán plate theory, a one-dimensional nonlinear second-order ordinary differential equation with quadratic and cubic nonlinearities is formulated with the aid of an energy function. By imposing Newton’s method and harmonic balancing to the linearized governing equation, we establish the higher-order analytical approximations for solving the nonlinear differential equation with odd nonlinearity. Based on the nonlinear differential equation with odd and even nonlinearities, two new nonlinear differential equations with odd nonlinearity are introduced for constructing the analytical approximations to the nonlinear differential equation with general nonlinearity. The analytical approximations are mathematically formulated by combining piecewise approximate solutions from such two new nonlinear systems. The third-order analytical approximation with better accuracy is proposed here and compared with other numerical and approximate methods with respect to the exact solutions. In addition, the method presented herein is applicable to small as well as large amplitude vibrations of laminated plates. Several examples including large amplitude nonlinear free vibration of simply supported laminated cross-ply rectangular thin plates are illustrated and compared with other published results to demonstrate the applicability and effectiveness of the approach.

Free-Vibration Analysis of Rectangular Plates With Clamped-Simply Supported Edge Conditions by the Method of Superposition

1977 ◽  
Vol 44 (4) ◽  
pp. 743-749 ◽  
Author(s):  
D. J. Gorman
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Formidable Challenge ◽  
Edge Conditions ◽  
Governing Differential Equation

In this paper attention is focused on the free-vibration analysis of rectangular plates with combinations of clamped and simply supported edge conditions. Plates with at least two opposite edges simply supported are not considered as they have been analyzed in a separate paper. It is well known that the family of problems considered here have presented researchers with a formidable challenge over the years. This is because they are not directly amenable to Le´vy-type solutions. It has been pointed out in the literature that most of the existing solutions are approximate in that they either do not satisfy exactly the governing differential equation or the boundary conditions, or both. In a new approach taken by the author the method of superposition is exploited for handling these dynamic problems. It is found that solutions of any degree of exactitude are easily obtained. The governing differential equation is completely satisfied and the boundary conditions are satisfied to any degree of exactitude by merely increasing the number of terms in the series. Convergence is shown to be remarkably rapid and tabulated results are provided for a large range of parameters. The immediate applicability of the method to problems involving elastic restraint or inertia forces along the plate edges has been discussed in an earlier publication.

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