Modern Mechanical Engineering, 2011, 1, 1-5

doi:10.4236/mme.2011.11001 Published Online August 2011 (http://www.SciRP.org/journal/mme)

Copyright © 2011 SciRes. MME

Analysis of Plate Vibration under Exponentially

Varying Thermal Condition

Tripti Johri, Indu Johri

Department of Applied Sciences & Humanities, J. P. Institute of Engineering & Technology, Meerut, India

E-mail: {tripti_johri1, indu_johri}@rediffmail.com

Received July 23, 2011; revised August 6, 2011; accepted August 15, 2011

Abstract

A mathematical model is developed to assist the design engineers by analyzing the vibration response of

non-homogeneous orthotropic rectangular plate under exponentially varying thermal condition. Plate thick-

ness varies parabolically in both directions. Using Rayleigh Ritz approach, frequency parameter and two

term deflection function is calculated for diverse values of taper constants. For the best comprehension of the

vibration analysis, results are depicted graphically.

Keywords: Vibration, Non-Homogeneous, Plate, Thickness, Deflection Function, Taper Constants

1. Introduction

Study of vibration responses of plate has always been a

principal concern for design engineers. These plates are

used in numerous industries for the construction of in-

numerable vital structures and devices, such as space

shuttles, rockets, air craft’s, shafts, plate heat exchanger

and many more. Satellite antenna booms are used in

space as electric field or gravity-gradient probes. Heated

by sunlight, a temperature gradient is built up across the

cross-section. The thermal strain produces bending and

torsion of the boom. Due to the twist the direction of the

sunlight changes with respect to the cross-section and

another temperature distribution is obtained which again

causes another deformation. Similarly, a serious problem

in mechanical design of heat exchanger is flow induced

vibration.

Certain structures are less vulnerable against vibration

impacts whereas certain other are more vulnerable. As

we all know that vibration effects are now cannot be ne-

glected, as our day to day life is affected by them; From

kitchen to exercise centre, vibration effect are experi-

enced. The only thing is we do not put an eye on them

deliberately. Juicer, Mixer, massager, WholeBody Vi-

bration Plate (for fitness), all encompasses vibration

effect. Those were few inevitable and positive aspects of

vibration. Controlled vibrations are utilized in health

industry, paper industry, structural engineering, and

aeronautical engineering and in many more industries.

But uncontrolled vibration causes devastation. Occur-

rences of Tsunami, earthquake, collapse of structures are

few such most common devastating effects of vibration.

Thus the study of vibration responses in advance is of

immense importance for sustainable and positive effects

of vibrations for the well being of humans.

Monograph written by Leissa [1] is ample source of

knowledge in the field of plate vibration. Leissa [2] pro-

vides abundant knowledge about the numerous compli-

cating effects that can be introduced in the analysis of

plate vibration. Tomar and Gupta [3] evaluated the ex-

ponential thermal gradient effect on the vibration of

orthotropic rectangular plate with variation in thickness.

Rahimi & Davoodinik [4] studied the thermal behavior

of functionally graded plates under the exponential and

hyperbolic temperature conditions. They concluded that

temperature distribution profile plays vital role in ther-

mal resultant distribution of stresses and strains for FGP.

Shang, Wang and Li [5] analyzed the deformation of

laminated plates under exponential distributions of tem-

perature through the thickness. The plate under consid-

eration is simply supported. Javaheri & Eslami [6] used

classical plate theory for the buckling analysis of func-

tionally graded plates under four different types of ther-

mal load.Gupta, Johri & Vats [7] studied thermally in-

duced vibrations of an orthotropic rectangular plate using

Rayleigh Ritz approach. Gupta, Johri & Vats [8] calcu-

lated deflection function and frequency parameter for a

rectangular plate under the effect of linear temperature

distribution where thickness of plate was varied para-

bolically in both directions.