A General Analytical Solution for Free Vibration of Rectangular Plates Resting on Fixed Supports and With Attached Masses

1992 ◽  
Vol 114 (2) ◽  
pp. 239-245 ◽  
Author(s):  
R. K. Singal ◽  
D. J. Gorman
Keyword(s):  
Free Vibration ◽  
Analytical Procedure ◽  
Thin Plates ◽  
Mode Shapes ◽  
Experimental Program ◽  
Space Vehicles ◽  
Superposition Method ◽  
Rectangular Plates ◽  
Solar Panels ◽  
Good Agreement

A comprehensive analytical procedure based on the superposition method is described for establishing the free vibration frequencies and mode shapes of thin plates resting on rigid point supports and with attached masses. Effects of rotary inertia of the attached masses are incorporated into the analysis and are shown to be highly significant. Results of an extensive experimental program are reported and very good agreement is demonstrated between theory and experiment. The analytical procedure has application in numerous contemporary industrial problems, in particular, in the design of solar panels for space vehicles and in the field of electronic packaging.

Free Vibration of Perforated Plates

2009 ◽  
Vol 40 (5) ◽  
pp. 35-40 ◽  
Author(s):  
Sasank Sekhar Hota ◽  
Payodhar Padhi ◽  
Mrutyunjay Rout
Keyword(s):  
Free Vibration ◽  
Mesh Size ◽  
Thin Plates ◽  
Mode Shapes ◽  
Uniform Mesh ◽  
Rectangular Plates ◽  
Zero Energy ◽  
Perforated Plates ◽  
First Order ◽  
First Time

A subparametric triangular plate bending element of first order shear deformation has been combined for the first time with the approach of constraints that helps maintain uniform mesh size and shape even while dealing with cutouts of arbitrary shapes. This is a distinct improvement over the existing practices of cutout modeling. Further the formulation being based on the subparametric element offers the scope of achieving matching modes, which enables the model to remain free from problems of locking and spurious zero energy modes while solving problems of very thin plates. Benchmark examples on free vibration of rectangular plates with cutouts have been solved to test the accuracy of the model. The author's own problems are also presented on mode shapes.

Combinations for the Free Vibration Behaviors of Anisotropic Rectangular Plates Under General Edge Conditions

1999 ◽  
Author(s):  
Yoshihiro Narita
Keyword(s):  
Free Vibration ◽  
Structural Design ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Technical Information ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Models ◽  
Edge Conditions ◽  
Anisotropic Rectangular Plates

Abstract The free vibration behavior of rectangular plates provides important technical information in structural design, and the natural frequencies are primarily affected by the boundary conditions as well as aspect and thickness ratios. One of the three classical edge conditions, i.e., free, simple supported and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations, the present paper introduces the Polya counting theory in combinatorial mathematics, and formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three edge conditions and is used to numerically verify the numbers. In numerical experiments, the number of combinations in the free vibration behaviors is determined for some plate models by using the derived formulas, and are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the Ritz method.

Analysis of Clamped Rectangular Plates

1940 ◽  
Vol 7 (4) ◽  
pp. A139-A142
Author(s):  
Dana Young
Keyword(s):  
Middle Surface ◽  
Lateral Loading ◽  
Thin Plates ◽  
Superposition Method ◽  
Rectangular Plates ◽  
Lagrange’S Equation

Abstract This paper attempts to solve the problem of the bending action of rectangular plates clamped at all four edges and subjected to lateral loading. Analytical in nature, the author’s investigation is based upon the ordinary theory of bending of thin plates as treated in Lagrange’s equation of the middle surface. The superposition method is used and applied to a number of loadings not hitherto studied.

scholarly journals Green’s function approach to frequency analysis of thin circular plates

2016 ◽  
Vol 64 (1) ◽  
pp. 181-188
Author(s):  
K.K. Żur
Keyword(s):  
Power Series ◽  
Free Vibration ◽  
Influence Function ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Thin Plates ◽  
Circular Plates ◽  
Natural Modes ◽  
Good Agreement ◽  
Analytical Frequency

Abstract The free vibration analysis of homogeneous and isotropic circular thin plates by using the Green’s functions is considered. The formulae for construction of the influence function for all nodal diameters are presented in a closed form. The limited independent solutions of differential Euler equations were expanded in the Neumann power series using the method of successive approximation. This approach allows to obtain the analytical frequency equations as power series rapidly convergent to exact eigenvalues for different number of nodal diameters. The first ten dimensionless frequencies for eight different natural modes of circular plates are calculated. A part of obtained results have not been presented yet in open literature for thin circular plates. The results of investigation are in good agreement with selected results obtained by other methods presented in literature.

An exact solution for free vibration of thin functionally graded rectangular plates

2010 ◽  
Vol 225 (3) ◽  
pp. 526-536 ◽  
Author(s):  
A Hasani Baferani ◽  
A R Saidi ◽  
E Jomehzadeh
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Vibration Characteristics ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Functionally Graded Plates ◽  
Different Boundary Conditions

The aim of this article is to find an exact analytical solution for free vibration characteristics of thin functionally graded rectangular plates with different boundary conditions. The governing equations of motion are obtained based on the classical plate theory. Using an analytical method, three partial differential equations of motion are reformulated into two new decoupled equations. Based on the Navier solution, a closed-form solution is presented for natural frequencies of functionally graded simply supported rectangular plates. Then, considering Levy-type solution, natural frequencies of functionally graded plates are presented for various boundary conditions. Three mode shapes of a functionally graded rectangular plate are also presented for different boundary conditions. In addition, the effects of aspect ratio, thickness—length ratio, power law index, and boundary conditions on the vibration characteristics of functionally graded rectangular plates are discussed in details. Finally, it has been shown that the effects of in-plane displacements on natural frequencies of functionally graded plates under different boundary conditions have been studied.

scholarly journals A Highly Accurate and Efficient Analytical Approach to Bridge Deck Free Vibration Analysis

2000 ◽  
Vol 7 (6) ◽  
pp. 399-412 ◽  
Author(s):  
D.J. Gorman ◽  
L. Garibaldi
Keyword(s):  
Free Vibration ◽  
Bridge Deck ◽  
Bridge Decks ◽  
Free Vibration Analysis ◽  
Free Edge ◽  
Mode Shapes ◽  
Superposition Method ◽  
Type Solution ◽  
Edge Type ◽  
Edge Conditions

The superposition method is employed to obtain an accurate analytical type solution for the free vibration frequencies and mode shapes of multi-span bridge decks. Free edge conditions are imposed on the long edges running in the direction of the deck. Inter-span support is of the simple (knife-edge) type. The analysis is valid regardless of the number of spans or their individual lengths. Exact agreement is found when computed results are compared with known eigenvalues for bridge decks with all spans of equal length. Mode shapes and eigenvalues are presented for typical bridge decks of three and four span lengths. In each case torsional and non-torsional modes are studied.

Theoretical Modeling and Experimental Validation of Fluid-Loaded Micro Rectangular Plates

2009 ◽  
Author(s):  
Zhangming Wu ◽  
Mike T. Wright ◽  
Xianghong Ma
Keyword(s):  
Dynamic Testing ◽  
Mode Shapes ◽  
Experimental Investigations ◽  
Rectangular Plates ◽  
Micro Scale ◽  
Isotropic Plates ◽  
Testing Facility ◽  
Excitation Dynamic ◽  
Theoretical Simulations ◽  
Good Agreement

This paper presents a theoretical model on the vibration analysis of micro scale fluid-loaded rectangular isotropic plates, based on the Lamb’s assumption of fluid-structure interaction and the Rayleigh-Ritz energy method. An analytical solution for this model is proposed, which can be applied to most cases of boundary conditions. The dynamical experimental data of a series of microfabricated silicon plates are obtained using a base-excitation dynamic testing facility. The natural frequencies and mode shapes in the experimental results are in good agreement with the theoretical simulations for the lower order modes. The presented theoretical and experimental investigations on the vibration characteristics of the micro scale plates are of particular interest in the design of microplate based biosensing devices.

On the symplectic superposition method for new analytic free vibration solutions of side-cracked rectangular thin plates

2020 ◽  
Vol 489 ◽  
pp. 115695
Author(s):  
Zhaoyang Hu ◽  
Yushi Yang ◽  
Chao Zhou ◽  
Xinran Zheng ◽  
Rui Li
Keyword(s):  
Free Vibration ◽  
Thin Plates ◽  
Superposition Method
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