Vibration of Clamped Non-homogeneous Square Plate with non- uniform varying Thickness and Thermal Effect

Ashish Kumar Sharma¹*, Manoj Kumar Dhiman², Vijyeta Verma², Vandana²

¹Department of Mathematics, IEC University, Baddi (Solan), H.P., India

²Research Scholar Department of Mathematics, IEC University, Baddi (Solan), H.P., India

*Corresponding Author E-mail: ashishk482@gmail.com,m.manojdhiman@gmail.com, v.verma286@gmail.com,vandnamehta1991@gmail.com

ABSTRACT:

Plates with the variable thickness have great importance in the modern technology. A simple model is presented to study the effect of thermal gradient on vibration of non-homogenous square plate with varying thickness in x and y directions. Rayleigh-Ritz technique has been applied to obtain corresponding equation of frequency. The natural frequency for the first two modes of vibration is calculated for different values of thermal gradient, taper constant and non-homogenous constant at various points. All the results are presented in graphical and tabular form.

KEYWORDS: visco-elastic, vibration, square plate, thermal gradient, taper constant, frequency.

1. INTRODUCTION:

In the engineering we cannot move without considering the effect of vibration because all machines and engineering structures experiences vibrations. Plates of uniform and non-uniform thickness are widely used in aeronautical field, naval structure, submarine, nuclear reactor etc. The requirement to know the effect of temperature on visco-elastic plates of variable thickness has become vital with the advancement of technology. In literature there are many researchers who studied the vibration behavior of homogeneous or non-homogeneous plates with variable thickness, with or without consideration of temperature effects. An extensive review on linear vibration of plates has been given by Leissa [1] in his monograph and a series of review articles [2] Tomar and Gupta [3] studied the effect of taper constants in two directions on elastic plates, but not on visco-elastic plates. Bhatnagar and Gupta [4] studied the effect of thermal gradient on vibration of a visco-elastic circular plate of variable thickness. Gupta and Khanna [5] studied the effect of linearly varying thickness in both directions on vibration of a visco-elastic rectangular plate. Gupta and Khanna [6] studied the Vibration of clamped visco-elastic rectangular plate with parabolic thickness variations. Gupta and Khanna [7] analyzed free vibration of clamped rectangular plate with bi-direction exponentially thickness variations. Khanna and Sharma [8] have been studied on Vibration Analysis of Visco-Elastic Square Plate of Variable Thickness with Thermal Gradient. Khanna & Sharma [9] studied natural vibration of visco-elastic plate of varying thickness with thermal effect. Khanna & Sharma [10] studied Analysis of free vibrations of visco-elastic square plate of variable thickness with temperature effect. Khanna & Sharma [11] analyzed a computational prediction on vibration of square plate by varying thickness with bi-dimensional thermal effect. Khanna & Sharma [12] presented the study of free vibration of visco-elastic square plate of bi-parabolic variable thickness with two dimensional temperature variation i.e. linearly in x-direction but parabolic in y-direction. In this paper the analysis of study of thermal effect on the vibration of visco-elastic square plate whose thickness and thermal effect vary in x and y directions. Also, it is supposed that the plate is clamped on all the four edges. Due to temperature deviation, we suppose that non homogeneity occurs in modulus of elasticity. Frequency for the first two modes of vibration is calculated for various numerical values of thermal gradient, taper constant and non-homogenous constant. Findings are presented in graphical and tabular form.

2. ANALYSIS OF THE MODEL:

2.1 Description of the model:

A thin, isotropic and non-uniform four sided clamped square plate having length ‘a’ to be assumed as shown in figures 1 (a) and (b) below:

Figure 1.(a) c-c-c clamped square plate Figure 1.(b) Square plate with length ‘a’

5. CONCLUSION:

Authors compare the results of present paper with [8] the authors concluded that values of frequencies in present paper decrease for both the modes of vibrations for the corresponding values of taper parameter as thermal gradient increases. Therefore, engineers and technocrats are advised to go through the findings of the present paper before finalizing any design of a machine or mechanical equipments so that they can produce more effective and reliable structures.

REFERENCES:

1. Leissa A.W., 1969, Vibration of plate, NASA SP-160

2. Leissa A.W., 1987, Recent studies in plate vibration 1981-1985. Part II, complicating effects, the Shock and Vibration Dig., 19, 10-24.

3. Tomar J.S., Gupta A.K., 1985, Effect of thermal gradient on frequencies of an orthotropic rectangular plate whose thickness varies in two directions, J.Sound and Vibration, 98, 257-262.

4. Bhatanagar N.S., Gupta A.K., 1988, Vibration analysis of visco-elastic circular plate subjected to thermal gradient, Modelling, Simulation and Control, B, AMSE, 15, 17-31.

5. Gupta A.K., Khanna A., 2007, Vibration of visco-elastic rectangular plate with linearly thickness variations in both directions, J. Sound and Vibration, 301, 450-457.

6. Gupta A.K., Khanna A., 2008, Vibration of clamped visco-elastic rectangular plate with parabolic thickness variations, Journal shock and vibration, 15, 713-723.

7. Khanna A., Sharma A.K., 2011, Vibration Analysis of Visco-Elastic Square Plate of Variable Thickness with Thermal Gradient, International Journal of Engineering and Applied Sciences, Turkey, 3(4),1-6.

8. Khanna A., Sharma A.K., 2013, Natural Vibration of Visco-Elastic Plate of Varying Thickness with Thermal Effect, Journal of Applied Science and Engineering, 16(2), 135-140.

9. Khanna A., Sharma A.K., 2011, Analysis of free vibrations of visco-elastic square plate of variable thickness with temperature effect, International Journal of Applied Engineering Research, 2(2), 312-317.

10. Khanna A., Sharma A.K., 2012, A computational prediction on vibration of square plate by varying thickness with bi-dimensional thermal effect, International journal of emerging in engineering and development, 2(3), 191-196.

11. Khanna A., Sharma A.K., 2011, Study of Free Vibration of Visco-Elastic Square Plate of Variable Thickness with Thermal Effect, Innovative System Design and Engineering, U.S.A.2(4).

Received on 22.11.2016 Modified on 24.11.2016

DOI: 10.5958/2349-2988.2017.00011.0

Research J. Science and Tech. 2017; 9(1):71-75.