Effect of Foundation and Non-Homogeneity on the Vibrations of Polar Orthotropic Parabolically Tapered Circular Plates
Affiliation(s)
1 Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, India. 2 Department of Mathematics, Gurukul Kangri University, Haridwar, India. 3 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India.
1 Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, India. 2 Department of Mathematics, Gurukul Kangri University, Haridwar, India. 3 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India.
ABSTRACT
The effect of Pasternak foundation and non-homogenity on the axisymmetric vibrations of polar orthotropic parabolically varying tapered circular plates has been analyzed on the basis of classical plate theory. Ritz method has been used to find the numerical solution of the specified problem. The efficiency of the Ritz method depends on the choice of basis function based upon deflection of polar orthotropic plates. The effects of different plate parameters viz. elastic foundation, non-homogeneity, taper parameter and that of orthotropy on fundamental, second and third mode of vibration have been studied for clamped and simply-supported boundary conditions. Mode shapes for specified plates have been drawn for both the boundary conditions. Convergence and comparison studies have been carried out for specified plates.
The effect of Pasternak foundation and non-homogenity on the axisymmetric vibrations of polar orthotropic parabolically varying tapered circular plates has been analyzed on the basis of classical plate theory. Ritz method has been used to find the numerical solution of the specified problem. The efficiency of the Ritz method depends on the choice of basis function based upon deflection of polar orthotropic plates. The effects of different plate parameters viz. elastic foundation, non-homogeneity, taper parameter and that of orthotropy on fundamental, second and third mode of vibration have been studied for clamped and simply-supported boundary conditions. Mode shapes for specified plates have been drawn for both the boundary conditions. Convergence and comparison studies have been carried out for specified plates.
KEYWORDS
Pasternak Foundation, Parabolically Varying Thickness, Circular Plate, Polar Orthotropy, Non-Homogenity
Pasternak Foundation, Parabolically Varying Thickness, Circular Plate, Polar Orthotropy, Non-Homogenity
Cite this paper
Srivastava, S. , Sharma, S. and Lal, R. (2015) Effect of Foundation and Non-Homogeneity on the Vibrations of Polar Orthotropic Parabolically Tapered Circular Plates. Applied Mathematics, 6, 1563-1573. doi: 10.4236/am.2015.69139.
Srivastava, S. , Sharma, S. and Lal, R. (2015) Effect of Foundation and Non-Homogeneity on the Vibrations of Polar Orthotropic Parabolically Tapered Circular Plates. Applied Mathematics, 6, 1563-1573. doi: 10.4236/am.2015.69139.
References
[1] Ramaiah, G.K. and Kumar, V. (1973) Natural Frequencies of Polar Orthotropic Annular Plates. Journal of Sound and Vibration, 26, 517-531.
http://dx.doi.org/10.1016/S0022-460X(73)80217-2 [2] Dyka, C.T. and Carney, J.F. (1979) Vibration and Stability of Spanning Polar Orthotropic Annular Plates Reinforced with Edge Beams. Journal of Sound and Vibration, 6, 223-231.
http://dx.doi.org/10.1016/0022-460X(79)90647-3 [3] Gorman, D.G. (1982) Natural Frequencies of Polar Orthotropic Uniform Annular Plates. Journal of Sound and Vibration, 80, 145-154.
http://dx.doi.org/10.1016/0022-460X(82)90397-2 [4] Gorman, D.G. (1983) Natural Frequencies of Polar Orthotropic Variable Thickness Annular Plates. Journal of Sound and Vibration, 86, 47-60.
http://dx.doi.org/10.1016/0022-460X(83)90942-2 [5] Gunaratnam, D.G. and Bhattacharya, A.P. (1989) Transverse Vibration and Stability of Polar Orthotropic Circular Plates: High Level Relationship. Journal of Sound and Vibration, 132, 383-392.
http://dx.doi.org/10.1016/0022-460X(83)90942-2 [6] Gupta, U.S., Lal, R. and Jain, S.K. (1991) Buckling and Vibrations of Polar Orthotropic Circular Plates of Linearly Varying Thickness Resting on an Elastic Foundation. Journal of Sound and Vibration, 147, 423-434.
http://dx.doi.org/10.1016/0022-460X(91)90491-2 [7] Gupta, U.S., Lal, R. and Jain, S.K. (1993) Vibration and Buckling of Parabolically Tapered Polar Orthotropic Plates on Elastic Foundation. Indian Journal of Pure and Applied Mathematics, 24, 607-631. [8] Gupta, U.S., Lal, R. and Sagar, R. (1994) Effect of Elastic Foundation on Axisymmetric Vibrations of Polar Orthotropic Mindlin Circular Plates. Indian Journal of Pure and Applied Mathematics, 25, 1317-1326. [9] Gupta, U.S., Jain, S.K. and Jain, D. (1995) Method of Collocation by Derivatives in the Study of Axisymmeric Vibration of Circular Plates. Computer and Structures, 57, 841-845.
http://dx.doi.org/10.1016/0045-7949(95)00085-U [10] Ansari, A.H. (2000) Vibration of Plates of Variable Thickness. Ph.D. Thesis, University of Roorkee, Roorkee. [11] Gupta, A.P. and Bhardwaj, N. (2005) Free Vibration of Polar Orthotropic Circular Plates of Quadratically Varying Thickness Resting on Elastic Foundation. Applied Mathematical Modelling, 29, 137-157.
http://dx.doi.org/10.1016/j.apm.2004.07.010 [12] Gupta, U.S., Ansari, A.H. and Sharma, S. (2006) Buckling and Vibration of Polar Orthotropic Circular Plate Resting on Winkler Foundation. Journal of Sound and Vibration, 297, 457-476.
http://dx.doi.org/10.1016/j.jsv.2006.01.073 [13] Gupta, U.S., Lal, R. and Sharma, S. (2006) Thermal Effect on Axisymmetric Vibrations of Non-Uniform Polar Orthotropic Circular Plates with Elastically Restrained Edge. Proceedings of the 2nd International Congress on Computational Mechanics and Simulation (ICCMS-06), IIT Guwahati, 8-10 December 2006. [14] Gupta, U.S., Lal, R. and Sharma, S. (2006) Vibration Analysis of Non-Homogeneous Circular Plate of Non-Linear Thickness Variation by Differential Quadrature Method. Journal of Sound and Vibration, 298, 892-906.
http://dx.doi.org/10.1016/j.jsv.2006.05.030 [15] Gupta, U.S., Lal, R. and Sharma, S. (2007) Vibration of Non-Homogeneous Circular Mindlin Plates with Variable Thickness. Journal of Sound and Vibration, 302, 1-17.
http://dx.doi.org/10.1016/j.jsv.2006.07.005 [16] Sharma, S., Gupta, U.S. and Lal, R. (2010) Effect of Pasternak Foundation on Axisymmetric Vibration of Polar Orthotropic Annular Plates of Varying Thickness. Journal of Vibration and Acoustics, 132, Article ID: 041001. [17] Sharma, S., Srivastava, S. and Lal, R. (2011) Free Vibration Analysis of Circular Plate of Variable Thickness Resting on Pasternak Foundation. Journal of International Academy of Physical Sciences, 15, 1-13. [18] Sharma, S., Lal, R. and Srivastava, S. (2012) Effect of Pasternak Foundation on Axisymmetric Vibration of Polar Orthotropic Non-Homogeneous Circular Plate of Variable Thickness. International Journal of Computational Mathematics and Numerical Simulation, 5, 151-163. [19] Bahmyari, E. and Khedmati, M.R. (2013) Vibration Analysis of Non-Homogeneous Moderately Thick Plates with Point Supports Resting on Pasternak Elastic Foundation Using Element Free Galerkin Method. Engineering Analysis with Boundary Elements, 37, 1212-1238.
http://dx.doi.org/10.1016/j.enganabound.2013.05.003 [20] Sayyad, A.S. and Ghugal, Y.M. (2015) On the Free Vibration Analysis of Laminated Composite and Sandwich Plates: A Review of Recent Literature with Some Numerical Results. Composite Structures, 129, 177-201.
http://dx.doi.org/10.1016/j.compstruct.2015.04.007 [21] Eftekhari, S.A. and Jafari, A.A. (2013) A Simple and Accurate Ritz Formulation for Free Vibration of Thick Rectangular and Skew Plates with General Boundary Conditions. Acta Mechanica, 224, 193-209.
http://dx.doi.org/10.1007/s00707-012-0737-6 [22] Leissa, A.W. (1969) Vibration of Plates. NASA Report No. SP-160. [23] Sharma, S. (2006) Free Vibration Studies on Non-Homogeneous Circular and Annular Plates. Ph.D. Thesis, I.I.T. Roorkee, Roorkee.
[1] Ramaiah, G.K. and Kumar, V. (1973) Natural Frequencies of Polar Orthotropic Annular Plates. Journal of Sound and Vibration, 26, 517-531.
http://dx.doi.org/10.1016/S0022-460X(73)80217-2 [2] Dyka, C.T. and Carney, J.F. (1979) Vibration and Stability of Spanning Polar Orthotropic Annular Plates Reinforced with Edge Beams. Journal of Sound and Vibration, 6, 223-231.
http://dx.doi.org/10.1016/0022-460X(79)90647-3 [3] Gorman, D.G. (1982) Natural Frequencies of Polar Orthotropic Uniform Annular Plates. Journal of Sound and Vibration, 80, 145-154.
http://dx.doi.org/10.1016/0022-460X(82)90397-2 [4] Gorman, D.G. (1983) Natural Frequencies of Polar Orthotropic Variable Thickness Annular Plates. Journal of Sound and Vibration, 86, 47-60.
http://dx.doi.org/10.1016/0022-460X(83)90942-2 [5] Gunaratnam, D.G. and Bhattacharya, A.P. (1989) Transverse Vibration and Stability of Polar Orthotropic Circular Plates: High Level Relationship. Journal of Sound and Vibration, 132, 383-392.
http://dx.doi.org/10.1016/0022-460X(83)90942-2 [6] Gupta, U.S., Lal, R. and Jain, S.K. (1991) Buckling and Vibrations of Polar Orthotropic Circular Plates of Linearly Varying Thickness Resting on an Elastic Foundation. Journal of Sound and Vibration, 147, 423-434.
http://dx.doi.org/10.1016/0022-460X(91)90491-2 [7] Gupta, U.S., Lal, R. and Jain, S.K. (1993) Vibration and Buckling of Parabolically Tapered Polar Orthotropic Plates on Elastic Foundation. Indian Journal of Pure and Applied Mathematics, 24, 607-631. [8] Gupta, U.S., Lal, R. and Sagar, R. (1994) Effect of Elastic Foundation on Axisymmetric Vibrations of Polar Orthotropic Mindlin Circular Plates. Indian Journal of Pure and Applied Mathematics, 25, 1317-1326. [9] Gupta, U.S., Jain, S.K. and Jain, D. (1995) Method of Collocation by Derivatives in the Study of Axisymmeric Vibration of Circular Plates. Computer and Structures, 57, 841-845.
http://dx.doi.org/10.1016/0045-7949(95)00085-U [10] Ansari, A.H. (2000) Vibration of Plates of Variable Thickness. Ph.D. Thesis, University of Roorkee, Roorkee. [11] Gupta, A.P. and Bhardwaj, N. (2005) Free Vibration of Polar Orthotropic Circular Plates of Quadratically Varying Thickness Resting on Elastic Foundation. Applied Mathematical Modelling, 29, 137-157.
http://dx.doi.org/10.1016/j.apm.2004.07.010 [12] Gupta, U.S., Ansari, A.H. and Sharma, S. (2006) Buckling and Vibration of Polar Orthotropic Circular Plate Resting on Winkler Foundation. Journal of Sound and Vibration, 297, 457-476.
http://dx.doi.org/10.1016/j.jsv.2006.01.073 [13] Gupta, U.S., Lal, R. and Sharma, S. (2006) Thermal Effect on Axisymmetric Vibrations of Non-Uniform Polar Orthotropic Circular Plates with Elastically Restrained Edge. Proceedings of the 2nd International Congress on Computational Mechanics and Simulation (ICCMS-06), IIT Guwahati, 8-10 December 2006. [14] Gupta, U.S., Lal, R. and Sharma, S. (2006) Vibration Analysis of Non-Homogeneous Circular Plate of Non-Linear Thickness Variation by Differential Quadrature Method. Journal of Sound and Vibration, 298, 892-906.
http://dx.doi.org/10.1016/j.jsv.2006.05.030 [15] Gupta, U.S., Lal, R. and Sharma, S. (2007) Vibration of Non-Homogeneous Circular Mindlin Plates with Variable Thickness. Journal of Sound and Vibration, 302, 1-17.
http://dx.doi.org/10.1016/j.jsv.2006.07.005 [16] Sharma, S., Gupta, U.S. and Lal, R. (2010) Effect of Pasternak Foundation on Axisymmetric Vibration of Polar Orthotropic Annular Plates of Varying Thickness. Journal of Vibration and Acoustics, 132, Article ID: 041001. [17] Sharma, S., Srivastava, S. and Lal, R. (2011) Free Vibration Analysis of Circular Plate of Variable Thickness Resting on Pasternak Foundation. Journal of International Academy of Physical Sciences, 15, 1-13. [18] Sharma, S., Lal, R. and Srivastava, S. (2012) Effect of Pasternak Foundation on Axisymmetric Vibration of Polar Orthotropic Non-Homogeneous Circular Plate of Variable Thickness. International Journal of Computational Mathematics and Numerical Simulation, 5, 151-163. [19] Bahmyari, E. and Khedmati, M.R. (2013) Vibration Analysis of Non-Homogeneous Moderately Thick Plates with Point Supports Resting on Pasternak Elastic Foundation Using Element Free Galerkin Method. Engineering Analysis with Boundary Elements, 37, 1212-1238.
http://dx.doi.org/10.1016/j.enganabound.2013.05.003 [20] Sayyad, A.S. and Ghugal, Y.M. (2015) On the Free Vibration Analysis of Laminated Composite and Sandwich Plates: A Review of Recent Literature with Some Numerical Results. Composite Structures, 129, 177-201.
http://dx.doi.org/10.1016/j.compstruct.2015.04.007 [21] Eftekhari, S.A. and Jafari, A.A. (2013) A Simple and Accurate Ritz Formulation for Free Vibration of Thick Rectangular and Skew Plates with General Boundary Conditions. Acta Mechanica, 224, 193-209.
http://dx.doi.org/10.1007/s00707-012-0737-6 [22] Leissa, A.W. (1969) Vibration of Plates. NASA Report No. SP-160. [23] Sharma, S. (2006) Free Vibration Studies on Non-Homogeneous Circular and Annular Plates. Ph.D. Thesis, I.I.T. Roorkee, Roorkee.