Generalized Differential Quadrature Method for Vibration Analysis of Cantilever Trapezoidal FG Thick Plate

Document Type: Research Paper

Authors

Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran

Abstract

This paper presents a numerical solution for vibration analysis of a cantilever trapezoidal thick plate. The material of the plate is considered to be graded through the thickness from a metal surface to a ceramic one according to a power law function. Kinetic and strain energies are derived based on the Reissner-Mindlin theory for thick plates and using Hamilton's principle, the governing equations and boundary conditions are derived in the Cartesian coordinates. A transformation of coordinates is used to convert the equations and boundary conditions from the original coordinate into a new computational coordinates. Generalized differential quadrature method (GDQM) is selected as a strong method and natural frequencies and corresponding modes are derived. The accuracy and convergence of the proposed solution are confirmed using results presented by other authors. Finally, the effect of the power law index, angles and thickness of the plate on the natural frequencies are investigated.

Keywords

[1] Chopra I., Durvasula S., 1971, Vibration of simply supported trapezoidal plates I. symmetric trapezoids, Journal of Sound and Vibration 19: 379-392.
[2] Chopra I., Durvasula S., 1972, Vibration of simply supported trapezoidal plates II. un-symmetric trapezoids, Journal of Sound and Vibration 20: 125-134.
[3] Orris R.M., Petyt M., 1973, A finite element study of the vibration of trapezoidal plates, Journal of Sound and Vibration 27: 325-344.
[4] Srinivasan R.S., Babu B.J.C., 1983, Free vibration of cantilever quadrilal plates, Journal of the Acoustical Society of America 73: 851-855.
[5] Maruyama K., Ichinomiya O., Narita Y., 1983, Experimental study of the free vibration of clamped trapezoidal plates, Journal of Sound and Vibration 88: 523-534.
[6] Bert C.W., Malik M., 1996, Differential quadrature method for irregular domains and application to plate vibration, International Journal of Mechanical Sciences 38: 589-606.
[7] Xing Y., Liu B., 2009, High-accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain, International Journal for Numerical Methods in Engineering 80: 1718-1742.
[8] Shufrin I., Rabinovitch O., Eisenberger M., 2010, A semi-analytical approach for the geometrically nonlinear analysis of trapezoidal plates, International Journal of Mechanical Sciences 52: 1588-1596.
[9] Zhou L., Zheng W.X., 2008, Vibration of skew plates by the MLS-Ritz method, International Journal of Mechanical Sciences 50: 1133-1141.
[10] Zamani M., Fallah A., Aghdam M.M., 2012, Free vibration analysis of moderately thick trapezoidal symmetrically laminated plates with various combinations of boundary conditions, European Journal of Mechanics - A/Solids 36: 204-212.
[11] Hosseini-Hashemi Sh., Fadaee M., Atashipour S.R., 2011, A new exact analytical approach for free vibration of Reissner–Mindlin functionally graded rectangular plates, International Journal of Mechanical Sciences 53: 11-22.
[12] Shaban M., Alipour M.M., 2011, Semi-analytical solution for free vibration of thick functionally graded plates rested on elastic foundation with elastically restrained edge, Acta Mechanica Solida Sinica 24: 340-354.
[13] Hasani Baferani A., Saidi A.R., Ehteshami H., 2011, Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation, Composite Structure 93: 1842-1853.
[14] Zhu P., Liew K.M., 2011, Free vibration analysis of moderately thick functionally graded plates by local Kriging meshless method, Composite Structure 93: 2925-2944.
[15] Hosseini-Hashemi Sh., Salehipour H., Atashipour S.R, Sburlati R., 2013, On the exact in-plane and out-of-plane free vibration analysis of thick functionally graded rectangular plates: Explicit 3-D elasticity solutions, Composites Part B 46: 108-115.
[16] Jin G., Su Z., Shi Sh., Ye T., Gao S., 2014, Three-dimensional exact solution for the free vibration of arbitrarily thick functionally graded rectangular plates with general boundary conditions, Composite Structure 108: 565-577.
[17] Xia P., Long S.Y., Cui H.X., Li G.Y., 2009, The static and free vibration analysis of a nonhomogeneous moderately thick plate using the meshless local radial point interpolation method, Engineering Analysis with Boundary Elements 33: 770-777.
[18] Huang M., Ma X.O., Sakiyama T., Matuda M., Morita C., 2005, Free vibration analysis of plates using least-square-based on finite difference method, Journal of Sound and Vibration 288: 931-955.
[19] Nguyen-Xuan H., Liu G.R., Thai-Hoang C., 2010, An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner-Minslin, Computer Methods in Applied Mechanics and Engineering 199: 471-489.
[20] Leung A.Y.T., Zhu B., 2005, Transverse vibration of Mindlin Plates on two-parameter foundations by analytical trapezoidal p-elements, Journal of Engineering Mechanics 131: 1140-1145.
[21] Huang C.S., Leissa A.W., Chang M.J., 2005, Vibrations of skewed cantilevered triangular, trapezoidal and parallelogram Mindlin plates with considering corner stress singularities, International Journal for Numerical Methods in Engineering 62: 1789-1806.
[22] Abrate S., 2006, Free vibration, buckling, and static deflections of functionally graded plates, Composites Science and Technology 66: 2383-2394.
[23] Zhao X., Lee Y.Y., Liew K.M., 2009, Free vibration analysis of functionally graded plates using the element-free kp-Ritz method, Journal of Sound and Vibration 319: 918- 939.
[24] Eftekhari S.A., Jafari A.A., 2013, Modified mixed Ritz-DQ formulation for free vibration of thick rectangular and skew plates with general boundary conditions, Applied Mathematical Modelling 37: 7398-7426.
[25] Petrolito J., 2014, Vibration and stability analysis of thick orthotropic plates using hybrid-Trefftz elements, Applied Mathematical Modelling 38:5858-5869.
[26] Mindlin R.D., 1951, Influence of rotary inertia and shear on flexural motions of isotropic elastic plates, Journal of Applied Mechanics 18: 31-38.
[27] Liew K.M., Wang C.M., Xiang Y., Kitipornchai S., 1998, Vibration of Mindlin Plates, Elsevier.
[28] Kaneko T., 1975, On Timoshenko’s correction for shear in vibrating beams, Journal of Physics D: Applied Physics 8: 1928-1937.
[29] Bert C.W., Malik M., 1996, Differential quadrature method in computational mechanics: A review, Applied Mechanics Reviews 49: 1-28.