Vibratory Bending of Damped Laminated Plates

1969 ◽  
Vol 91 (4) ◽  
pp. 1081-1090 ◽  
Author(s):  
R. A. Ditaranto ◽  
J. R. McGraw
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Natural Frequencies ◽  
Laminated Plates ◽  
Simultaneous Equations ◽  
Laminated Plate ◽  
Partial Differential ◽  
Simply Supported ◽  
First Case ◽  
Viscoelastic Layers

The natural frequencies and associated composite loss factor have been determined for a finite-length laminated plate having alternate elastic and viscoelastic layers. Partial differential equations in terms of the variables of the plate are derived and, with the loading equation for a freely vibrating plate, a set of simultaneous partial differential equations is formed. Of two solutions considered the first is general and the second satisfies the boundary condition for a simply supported plate. In both cases, the resulting algebraic simultaneous equations are complex since the shear modulus of the viscoelastic material is a complex expression. In the first case, the expressions could not be solved directly since the value of the eigenvalues depended upon the boundary conditions, whereas the eigenvalues for the simply supported plate could be easily chosen. The simply supported case is solved and the results plotted for specific dimensionless parameters.

Effect of Shear Deformations on the Bending of Rectangular Plates

1960 ◽  
Vol 27 (1) ◽  
pp. 54-58 ◽  
Author(s):  
V. L. Salerno ◽  
M. A. Goldberg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Plate Theory ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Classical Plate Theory ◽  
Simply Supported ◽  
Wide Range

The three partial differential equations derived by Dr. E. Reissner2, 3 have been reduced to a fourth-order partial differential equation resembling that of the classical plate theory and to a second-order differential equation for determining a stress function. The general solution for the two partial differential equations has been applied to a simply supported plate with a constant load p and to a plate with two opposite edges simply supported and the other two edges free. Numerical calculations have been made for the simply supported plate and the results compared with those of classical theory. The calculations for a wide range of parameters indicate that the deviation is small.

scholarly journals On Solving One-Dimensional Partial Differential Equations With Spatially Dependent Variables Using the Wavelet-Galerkin Method

2013 ◽  
Vol 80 (6) ◽  
Author(s):  
Simon Jones ◽  
Mathias Legrand
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Frequency Response ◽  
Natural Frequencies ◽  
Basis Functions ◽  
Mode Shapes ◽  
Response Functions ◽  
Partial Differential ◽  
Frequency Response Functions

The discrete orthogonal wavelet-Galerkin method is illustrated as an effective method for solving partial differential equations (PDE's) with spatially varying parameters on a bounded interval. Daubechies scaling functions provide a concise but adaptable set of basis functions and allow for implementation of varied loading and boundary conditions. These basis functions can also effectively describe C0 continuous parameter spatial dependence on bounded domains. Doing so allows the PDE to be discretized as a set of linear equations composed of known inner products which can be stored for efficient parametric analyses. Solution schemes for both free and forced PDE's are developed; natural frequencies, mode shapes, and frequency response functions for an Euler–Bernoulli beam with piecewise varying thickness are calculated. The wavelet-Galerkin approach is shown to converge to the first four natural frequencies at a rate greater than that of the linear finite element approach; mode shapes and frequency response functions converge similarly.

On the analytical approach for the bending/stretching of linearly elastic functionally graded rectangular plates with two opposite edges simply supported

2009 ◽  
Vol 223 (9) ◽  
pp. 2009-2016 ◽  
Author(s):  
A R Saidi ◽  
E Jomehzadeh
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Rectangular Plate ◽  
Plate Theory ◽  
Functionally Graded ◽  
Mindlin Plate ◽  
Transverse Displacement ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Simply Supported

In this article, a new analytical method for bending—stretching analysis of thick functionally graded (FG) rectangular plates is presented. Using this method, the governing equations of FG rectangular plates based on the first-order shear deformation or Mindlin plate theory are decoupled. Five coupled partial differential equations of the Mindlin FG plate are converted into two uncoupled partial differential equations in terms of transverse displacement and a new function. It is analytically shown that by introducing an equivalent flexural rigidity, the equations of FG rectangular plate become similar to those of the homogeneous isotropic plate. Solving these equations, the solutions are obtained for the FG rectangular plate with two opposite edges simply supported. A comparison of the present results with available solutions from previous studies is made and a good agreement can be seen. Also, the numerical results for stress and deflection of the FG rectangular plate with various boundary conditions are obtained.

Three-Dimensional Elasticity Study of Vibration of a Composite Shell Panel with Embedded Piezoelectric Sensors

2012 ◽  
Vol 186 ◽  
pp. 87-97
Author(s):  
Alireza R. Daneshmehr ◽  
Samaun Nili ◽  
A.R. Nateghi ◽  
Shirjan Hussaini
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Natural Frequencies ◽  
Composite Shell ◽  
Three Dimensional ◽  
Trigonometric Function ◽  
Partial Differential ◽  
Shell Panel ◽  
Three Dimensional Elasticity

In this paper, Free vibration analysis of a finite length composite shell panel with an embedded piezoelectric sensor, using three-dimensional elasticity solution, is presented. To this end, two different methods are applied to solve the governing equations of the problem. In the first method, the displacement field is derived using trigonometric function expansion in circumferential and longitudinal directions. Using the method of changing variables, the governing partial differential equations are reduced to ordinary differential equations. Then these equations are solved simultaneously with outer and inner boundary conditions to give the natural frequencies and shape modes of the shell panel. In the second method the highly coupled partial differential equations are reduced to ordinary differential equations by means of trigonometric function expansion in circumferential and axial directions and then the finite difference method is applied to evaluate the obtained differential equations in radial direction. Then, the natural frequencies of the multi-layered panel are calculated using the obtained ordinary differential equations. At last, some numerical examples are presented to compare the results obtained by these two different methods. Three layered laminated shell panel is assumed to be [0/90/P].

Sign in / Sign up

Export Citation Format

Share Document