Thermal Effect on Free Vibration of Non-Homogeneous Orthotropic Visco-Elastic Rectangular Plate of Parabolically Varying Thickness

ABSTRACT

A simple model presented here is to study the thermal effect on vibration of non-homogeneous orthotropic visco-elastic rectangular plate of parabolically varying thickness having clamped boundary conditions on all the four edges. For non-homogeneity of the plate material, density is assumed to vary linearly in one direction. Using the separation of variables method, the governing differential equation has been solved for vibration of non-homogeneous orthotropic viscoelastic rectangular plate. An approximate frequency equation is derived by using Rayleigh-Ritz technique with a two-term deflection function. Results are calculated for time period and deflection at different points, for the first two modes of vibration, for various values of temperature gradients, non-homogeneity constant, taper constant and aspect ratio and shown by graphs.

A simple model presented here is to study the thermal effect on vibration of non-homogeneous orthotropic visco-elastic rectangular plate of parabolically varying thickness having clamped boundary conditions on all the four edges. For non-homogeneity of the plate material, density is assumed to vary linearly in one direction. Using the separation of variables method, the governing differential equation has been solved for vibration of non-homogeneous orthotropic viscoelastic rectangular plate. An approximate frequency equation is derived by using Rayleigh-Ritz technique with a two-term deflection function. Results are calculated for time period and deflection at different points, for the first two modes of vibration, for various values of temperature gradients, non-homogeneity constant, taper constant and aspect ratio and shown by graphs.

KEYWORDS

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

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

Cite this paper

nullA. Gupta and P. Singhal, "Thermal Effect on Free Vibration of Non-Homogeneous Orthotropic Visco-Elastic Rectangular Plate of Parabolically Varying Thickness,"*Applied Mathematics*, Vol. 1 No. 6, 2010, pp. 456-463. doi: 10.4236/am.2010.16060.

nullA. Gupta and P. Singhal, "Thermal Effect on Free Vibration of Non-Homogeneous Orthotropic Visco-Elastic Rectangular Plate of Parabolically Varying Thickness,"

References

[1] A. W. Leissa, NASA SP-60, Vibration of Plate, 1969.

[2] A. W. Leissa, “Recent Studies in Plate Vibration 1981-1985 Part II, Complicating Effects,” Shock and Vibration Dig., Vol. 19, No. 3, 1987, pp. 10-24.

[3] 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, San- Francisco, October 2007, pp. 784-787.

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

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

[6] 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, October 1983, pp. 325-331.

[7] 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, March 1986, pp. 183- 188.

[8] R. Lal, “Transverse Vibrations of Orthotropic Non-Uniform Rectangular Plate with Continuously Varying Density,” Indian Journal of Pure and Applied Mathematics, Vol. 34, No. 4, 2003, pp. 587-606.

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

[10] A. K. Gupta and H. Kaur, “Study of the Effect of Thermal Gradient on Free Vibration of Clamped Visco-Elastic Rectangular Plate with Linearly Thickness Variation in Both Directions,” Meccanica, Vol. 43, No. 4, 2008, pp. 449- 458.

[11] A. K. Gupta, A. Kumar and Y. K. Gupta, “Vibration of Visco-Elastic Parallelogram Plate with Parabolic Thickness Variation,” Applied Mathematics, Vol. 1, No. 2, 2010, pp. 128-136.

[12] A. K. 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.

[1] A. W. Leissa, NASA SP-60, Vibration of Plate, 1969.

[2] A. W. Leissa, “Recent Studies in Plate Vibration 1981-1985 Part II, Complicating Effects,” Shock and Vibration Dig., Vol. 19, No. 3, 1987, pp. 10-24.

[3] 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, San- Francisco, October 2007, pp. 784-787.

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

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

[6] 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, October 1983, pp. 325-331.

[7] 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, March 1986, pp. 183- 188.

[8] R. Lal, “Transverse Vibrations of Orthotropic Non-Uniform Rectangular Plate with Continuously Varying Density,” Indian Journal of Pure and Applied Mathematics, Vol. 34, No. 4, 2003, pp. 587-606.

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

[10] A. K. Gupta and H. Kaur, “Study of the Effect of Thermal Gradient on Free Vibration of Clamped Visco-Elastic Rectangular Plate with Linearly Thickness Variation in Both Directions,” Meccanica, Vol. 43, No. 4, 2008, pp. 449- 458.

[11] A. K. Gupta, A. Kumar and Y. K. Gupta, “Vibration of Visco-Elastic Parallelogram Plate with Parabolic Thickness Variation,” Applied Mathematics, Vol. 1, No. 2, 2010, pp. 128-136.

[12] A. K. 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.