Effect of Non-Homogeneity on Thermally Induced Vibration of Orthotropic Visco-Elastic Rectangular Plate of Linearly Varying Thickness

ABSTRACT

The analysis presented here is to study the effect of non-homogeneity on thermally induced vibration of orthotropic visco-elastic rectangular plate of linearly varying thickness. Thermal vibrational behavior of non-homogeneous rectangular plates of variable thickness having clamped boundary conditions on all the four edges is studied. For non–homogeneity of the plate material, density is assumed to vary linearly in one direction. Using the method of separation of variables, the governing differential equation is solved. An approximate but quite convenient frequency equation is derived by using Rayleigh-Ritz technique with a two-term deflection function. Time period and deflection at different points for the first two modes of vibration are calculated for various values of temperature gradients, non- homogeneity constant, taper constant and aspect ratio. Comparison studies have been carried out with non-homogeneous visco-elastic rectangular plate to establish the accuracy and versatility.

The analysis presented here is to study the effect of non-homogeneity on thermally induced vibration of orthotropic visco-elastic rectangular plate of linearly varying thickness. Thermal vibrational behavior of non-homogeneous rectangular plates of variable thickness having clamped boundary conditions on all the four edges is studied. For non–homogeneity of the plate material, density is assumed to vary linearly in one direction. Using the method of separation of variables, the governing differential equation is solved. An approximate but quite convenient frequency equation is derived by using Rayleigh-Ritz technique with a two-term deflection function. Time period and deflection at different points for the first two modes of vibration are calculated for various values of temperature gradients, non- homogeneity constant, taper constant and aspect ratio. Comparison studies have been carried out with non-homogeneous visco-elastic rectangular plate to establish the accuracy and versatility.

KEYWORDS

Non-Homogeneous, Orthotropic, Visco-Elastic, Variable Thickness, Rectangular Plate, Vibration, Thermal Gradient

Non-Homogeneous, Orthotropic, Visco-Elastic, Variable Thickness, Rectangular Plate, Vibration, Thermal Gradient

Cite this paper

nullA. Gupta and P. Singhal, "Effect of Non-Homogeneity on Thermally Induced Vibration of Orthotropic Visco-Elastic Rectangular Plate of Linearly Varying Thickness,"*Applied Mathematics*, Vol. 1 No. 4, 2010, pp. 326-333. doi: 10.4236/am.2010.14043.

nullA. Gupta and P. Singhal, "Effect of Non-Homogeneity on Thermally Induced Vibration of Orthotropic Visco-Elastic Rectangular Plate of Linearly Varying Thickness,"

References

[1] Z. Sobotka, “Free Vibration of Visco-Elastic Orthotropic Rectangular Plates,” Acta Technica, Vol. 23, No. 6, 1978, pp. 678-705.

[2] A. K. Gupta and A. Khanna, “Vibration of Viscoelastic Rectangular Plate with Linearly Thickness Variations in Both Directions,” Journal of Sound and Vibration, Vol. 301, No. 3-5, 2007, pp. 450-457.

[3] A. W. Leissa, “Vibration of Plate,” NASA SP-60, 1969.

[4] S. R. Li and Y. H. Zhou, “Shooting Method for Non Linear Vibration and Thermal Buckling of Heated Orthotropic Circular Plates,” Journal of Sound and Vibration, Vol. 248, No. 2, 2001, pp. 379-386.

[5] D. V. Bambill, C. A. Rossil, P. A. A. Laura and R. E. Rossi, “Transverse Vibrations of an Orthotropic Rectangular Plate of Linearly Varying Thickness and with a Free Edge,” Journal of Sound and Vibration, Vol. 235, No. 3, 2000, pp. 530-538.

[6] J. S. Tomar and A. K. Gupta, “Effect of Thermal Gradient on Frequencies of Orthotropic Rectangular Plate Whose Thickness Varies in Two Directions,” Journal ofSound and Vibration, Vol. 98, No. 2, 1985, pp. 257- 262.

[7] J. S. Tomar and A. K. Gupta, “Thermal Effect on Frequencies of an Orthotropic Rectangular Plate of Linearly Varying Thickness,” Journal of Sound and Vibration, Vol. 90, No. 3, 1983, pp. 325-331.

[8] J. S. Tomar and A. K. Gupta, “Effect of Exponential Temperature Variation on Frequencies of an Orthotropic Rectangular Plate of Exponentially Varying Thickness,” Proceeding of the Workshop on Computer Application in Continum Mechanics, Roorkee, 11-13 March 1986, pp. 183-188.

[9] U. S. Gupta, R. Lal and S. Sharma, “Vibration Analysis of Non-Homogenous Circular Plate of Nonlinear Thickness Variation by Differential Quadrature Method,” Journal of Sound and Vibration, Vol. 298, No. 4-5, 2006, pp. 892-906.

[10] A. K. Gupta, T. Johri and R. P. Vats, “Thermal Effect on Vibration of Non-Homogeneous Orthotropic Rectangular Plate Having Bi-Directional Parabolically Varying Thickness,” Proceeding of International Conference in World Congress on Engineering and Computer Science 2007 (WCECS 2007), San Francisco, 24-26 October 2007, pp. 784-787.

[11] A. K. Gupta, A. Kumar and D. V. Gupta, “Vibration of Visco-Elastic Orthotropic Parallelogram Plate with Linearly Thickness Variation,” Proceeding of International Conference in World Congress on Engineering and Computer Science 2007 (WCECS 2007), San Francisco, 24-26 October 2007, pp. 800-803.

[12] A. K. Gupta and L. Kumar, “Thermal Effect on Vibration of Non-Homogenous Visco-Elastic Rectangular Plate of Linear Varying Thickness,” Meccanica, Vol. 43, No. 1, 2008, pp. 47-54.

[13] A. K. Gupta, A. Khanna and D. V. Gupta, “Free Vibration of Clamped Visco-Elastic Rectangular Plate Having Bi-Directional Exponentially Thickness Variations,” Journal of Theotrocial and Applied Mechanics, Vol. 47, No. 2, 2009, pp. 457-471.

[14] A. K. Gupta, N. Aggarwal, D. V. Gupta, S. Kumar and P. Sharma, “Study of Non-Homogeneity on Free Vibration of Orthotropic Visco-Elastic Rectangular Plate of Parabolic Varying Thickness,” Advanced Studies of Theory Physics, Vol. 4, No. 10, 2010, pp. 467-486.

[15] K. Bhasker and B. Kaushik, “Simple and Exact Series Solutions for Flexure of Orthotropic Rectangular Plates with Any Combination of Clamped and Simply Supported Edges,” Composite Structure, Vol. 63, No. 1, 2004, pp. 63-81.

[1] Z. Sobotka, “Free Vibration of Visco-Elastic Orthotropic Rectangular Plates,” Acta Technica, Vol. 23, No. 6, 1978, pp. 678-705.

[2] A. K. Gupta and A. Khanna, “Vibration of Viscoelastic Rectangular Plate with Linearly Thickness Variations in Both Directions,” Journal of Sound and Vibration, Vol. 301, No. 3-5, 2007, pp. 450-457.

[3] A. W. Leissa, “Vibration of Plate,” NASA SP-60, 1969.

[4] S. R. Li and Y. H. Zhou, “Shooting Method for Non Linear Vibration and Thermal Buckling of Heated Orthotropic Circular Plates,” Journal of Sound and Vibration, Vol. 248, No. 2, 2001, pp. 379-386.

[5] D. V. Bambill, C. A. Rossil, P. A. A. Laura and R. E. Rossi, “Transverse Vibrations of an Orthotropic Rectangular Plate of Linearly Varying Thickness and with a Free Edge,” Journal of Sound and Vibration, Vol. 235, No. 3, 2000, pp. 530-538.

[6] J. S. Tomar and A. K. Gupta, “Effect of Thermal Gradient on Frequencies of Orthotropic Rectangular Plate Whose Thickness Varies in Two Directions,” Journal ofSound and Vibration, Vol. 98, No. 2, 1985, pp. 257- 262.

[7] J. S. Tomar and A. K. Gupta, “Thermal Effect on Frequencies of an Orthotropic Rectangular Plate of Linearly Varying Thickness,” Journal of Sound and Vibration, Vol. 90, No. 3, 1983, pp. 325-331.

[8] J. S. Tomar and A. K. Gupta, “Effect of Exponential Temperature Variation on Frequencies of an Orthotropic Rectangular Plate of Exponentially Varying Thickness,” Proceeding of the Workshop on Computer Application in Continum Mechanics, Roorkee, 11-13 March 1986, pp. 183-188.

[9] U. S. Gupta, R. Lal and S. Sharma, “Vibration Analysis of Non-Homogenous Circular Plate of Nonlinear Thickness Variation by Differential Quadrature Method,” Journal of Sound and Vibration, Vol. 298, No. 4-5, 2006, pp. 892-906.

[10] A. K. Gupta, T. Johri and R. P. Vats, “Thermal Effect on Vibration of Non-Homogeneous Orthotropic Rectangular Plate Having Bi-Directional Parabolically Varying Thickness,” Proceeding of International Conference in World Congress on Engineering and Computer Science 2007 (WCECS 2007), San Francisco, 24-26 October 2007, pp. 784-787.

[11] A. K. Gupta, A. Kumar and D. V. Gupta, “Vibration of Visco-Elastic Orthotropic Parallelogram Plate with Linearly Thickness Variation,” Proceeding of International Conference in World Congress on Engineering and Computer Science 2007 (WCECS 2007), San Francisco, 24-26 October 2007, pp. 800-803.

[12] A. K. Gupta and L. Kumar, “Thermal Effect on Vibration of Non-Homogenous Visco-Elastic Rectangular Plate of Linear Varying Thickness,” Meccanica, Vol. 43, No. 1, 2008, pp. 47-54.

[13] A. K. Gupta, A. Khanna and D. V. Gupta, “Free Vibration of Clamped Visco-Elastic Rectangular Plate Having Bi-Directional Exponentially Thickness Variations,” Journal of Theotrocial and Applied Mechanics, Vol. 47, No. 2, 2009, pp. 457-471.

[14] A. K. Gupta, N. Aggarwal, D. V. Gupta, S. Kumar and P. Sharma, “Study of Non-Homogeneity on Free Vibration of Orthotropic Visco-Elastic Rectangular Plate of Parabolic Varying Thickness,” Advanced Studies of Theory Physics, Vol. 4, No. 10, 2010, pp. 467-486.

[15] K. Bhasker and B. Kaushik, “Simple and Exact Series Solutions for Flexure of Orthotropic Rectangular Plates with Any Combination of Clamped and Simply Supported Edges,” Composite Structure, Vol. 63, No. 1, 2004, pp. 63-81.