analysis of asymmetric vibration of non-uniform polar or tho tropic

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ANALYSIS OF ASYMMETRIC VIBRATION

OF NON-UNIFORM POLAR ORTHOTROPIC

ANNULAR PLATE USING ANSYS

Presented BySourabh SinhaME/CAAD/1001/09

Under the Guidance of Dr. Rampal SinghDepartment Of Applied MechanicsBirla Institute Of Technology, Ranchi



INTRODUCTION

The research work deals with the asymmetric freetransverse vibration of rectangular orthotropic plates of variable thickness along the x-axis when two opposite

edges parallel to x-axis are taken as simply-supportedand the other two edges are taken elastically restrained

against translation and rotation.

On the basis of classical plate theory , the fourth order

linear differential equation with variable coefficientsgoverning the motion would be derived using Newtonssecond law of motion which along with the boundaryconditions would be of great utility while analyzing thevibrational behavior of the plate using ANSYS.



LITERATURE SURVEY

There are lots of literature dealing with the freevibrations of elastic rectangular plates. Indeed varioussummaries identified atleast 500 references , about half

of which deal with the free vibration of thin , isotropic ,homogenous , uniform thickness rectangular plates ,mostly with classical boundary conditions.

W.Voigt [1] gave a classical solution to solve thevibrational problems of the rectangular plates.

W.Ritz [2] presented the well known Ritz method tosolve the general class vibrational problems.

Iguchi [3] obtained the solution for uniform thickerrectangular plates.



Hearmon [4] obtained the fundamental frequency of vibrations of rectangular wood and plywood plates byconsidering all sides simply-supported.

The first comprehensive collection of solutions forvibrational problems for rectangular plates was given byWarburton [5] by using Rayleigh-Ritz method with beamfunction for rectangular plates with combination of edges.

Gumeniute [6] suggested a finite difference approach toderive a formula for the fundamental frequency of asimply-supported rectangular plate having a linearvariable thickness.



Appl and Zorowski [7] studied the fundamentalfrequency of simply-supported rectangular plate with thethickness varying only in one direction.

Jain and Soni [8] studied free vibrations of rectangularplates of parabolically varying thickness.

Laura and Grossi [9] studied transverse vibrations of a

rectangular plate elastically restrained against rotation

along three edges and free on the fourth edge. Leissa et al [10] analyzed free vibrations of rectangular

plates having non-uniform , elastic edge constraints.



Leissa [11] presnted an excellent review of the literature (1976-1980) on vibrations analysis of the rectangular plates includingcomplicating effects.

Bhat et al [12] compared the natural frequencies of transversevibrations of rectangular plates of non-uniform thicknessobtained by different approaches namely-the Rayleighmethod with two different shape functions.

Bambill et al [13] determined the fundamental frequency of anorthotropic rectangular plate of linearly varying thickness and

with a free edge and other three edges subjected in allpossible supported combinations (clamped or simply-supported) by using the optimized Rayleigh-Ritz method withthe first terms of a pseudo-Fourier expansion and also byusing the finite element method.



THE PRESENT INVESTIGATION

The literature survey has shown that there is agrowing interest in the study of free transversevibration of orthotropic plates of variable

thickness due to economy , weight and strengthconsiderations in many engineering structureslike aerospace , naval , nuclear and other similarstructures.

The present research work is an attempt to study

the free asymmetric transverse vibration of non-uniform orthotropic plates of those geometricalconfigurations which are of versatile use indifferent engineering structures.



OBJECTIVE

The main objective of this thesis is to analyze the resultobtained by the work done with the results of thoseavailable in the literature.



METHODOLOGY ADOPTED

The tool or the software used for this thesis work isANSYS.

ANSYS finite element analysis enable engineers to:

Build computer models or transfer CAD models of structures , products , components , or systems.

Apply operatingloads or other design performanceconditions.

Study the physical responses , such as stress levels ,temperature distributions , or the impact of electromagnetic fields.

Optimize a design early in the development process toreduce production costs.



The ANSYS program has a comprehensive graphical userinterface that gives users easy , interactive access toprogram functions , commands , documentation , and

reference material . An intuitive menu system helps usernavigate through the ANSYS program.



REFERNCES

1. VOIGT, W. :Bemerkungen zu dem problem dertransversalen Schwingungen rechteckiger Platten,Nachr. Ges. Wiss., Gottingen, No.6, pp.225-230, 1893.

2. RITZ, W. : Theorie der trasnsversalen Schwingungeneiner quadratischen Platten mit freien Randern, Ann.Physik, Bd. 28, pp.737-786, 1909.

3. IGUCHI : Die eigenwert probleme fur die elastische

rechteckige platen, Hokkaido University Memory of theFaculty of Engineering, 305-372, 1938.

4. HEARMOM, R.F.S. : The fundamental frequency of vibration of rectangular wood and plywood plates,

Proceedings of the Physical Society, London,V

ol.58,pp.78-92, 1946.



REFERNCES

5. WARBURTON, G.B. : The vibration of rectangular plates,

Proceedings of the Institution of Mechanical Engineers, 168,371-381, 1954.

6. GUMENIUK,V.S. : Determination of the free vibrationfrequencies of variable thickness plates, Dopov, ANUkr SSR,

2, 130-133, 1955, (in Ukrainian).

7. APPL F.C. and BYERS N.R. : Fundamental frequency of simply

supported rectangular plates of linearly varying thickness.

Journal of Applied Mechanics, 32, 163-167, 1965.

8. JAIN, R.K. and SONI, S.R. : Free vibrations of rectangular plates

of parabolically varying thickness , Indian Journal of Pure andApplied Mathematics, 4, 267-277, 1973.



REFERNCES

9. LAURA, P.A.A. and GROSSI, R. : Transverse vibrations of a

rectangular plate elastically restrained against rotation alongthree edges and free edges and free on the fourth edge,

Journal of Sound andV

ibration, 59, 355-368, 1978. 10. LEISSA, A.W. and LAURA, P.A.A. and GUTIERREZ, R.H. :Vibrations of rectangular plates with non-uniform elastic edge

supports, Journal of Applied Mechanics,Vol. 47, 1980.

12. BHAT, R.B., LAURA, P.A.A., GUTIERREZ, R.G., CORTINEZ,

V.H. and SANZI, H.C. : Numerical experiments on thedetermination of natural frequencies of transverse vibration of rectangular plates of non-uniform thickness, Journal of Sound

andVibration,138(2), 205-219, 1990.



REFERNCES

13. BAMBILL, D.V., ROSSIT, C.A., LAURA, P.A.A. andROSSI, R.E. : Transverse vibration of an isotropicrectangular plate of linearly varying thickness and with a

free wedge, Journal of Sound andVibration, 235(3), 530-538, 2000.



THANKS

analysis of asymmetric vibration of non-uniform polar or tho tropic

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