New exact solutions for free vibrations of thin orthotropic rectangular plates

In this paper, a novel separation of variables is presented for solving the exact solutions for the free vibrations of thin orthotropic rectangular plates with all combinations of simply supported (S) and clamped (C) boundary conditions, and the correctness of the exact solutions are proved mathematically. The exact solutions for the three cases SSCC, SCCC, and CCCC are successfully obtained for the first time, although it was believed that they are unable to be obtained. The new exact solutions are further validated by extensive numerical comparisons with the solutions of FEM and those available in the literature.

[1]  Book Reviews : FLOW INDUCED VIBRATIONS S.S. Chen and M.D. Bernstein, Editors ASME, New York, New York, 1979, 152 pages , 1981 .

[2]  A. Kerr,et al.  NATURAL VIBRATION ANALYSIS OF CLAMPED RECTANGULAR ORTHOTROPIC PLATES , 1996 .

[3]  J. Reddy,et al.  Bending, stability and free vibration of thin orthotropic plates by simplified mixed finite elements , 1978 .

[4]  Robin S. Langley,et al.  Free vibration analysis of thin coplanar rectangular plate assemblies Part II: further results for generally orthotropic plates , 1996 .

[5]  W. Cleghorn,et al.  Generic free vibration of orthotropic rectangular plates with clamped and simply supported edges , 1993 .

[6]  H. Saunders,et al.  Literature Review : Books: THEORY AND ANALYSIS OF PLATES -- C LASSICA L AND NUMERICAL METHODS Rudolph Szilard Prentice Hall, Inc., Englewood Cliffs, N.J. (1974) , 1974 .

[7]  Y. Z. Chen,et al.  Evaluation of fundamental vibration frequency of an orthotropic bending plate by using an iterative approach , 1998 .

[8]  R. E. Rossi,et al.  Vibrations of a rectangular orthotropic plate with a free edge: a comparison of analytical and numerical results , 1998 .

[9]  Rama B. Bhat,et al.  Plate Characteristic Functions and Natural Frequencies of Vibration of Plates by Iterative Reduction of Partial Differential Equation , 1993 .

[10]  S. M. Dickinson,et al.  On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates , 1986 .

[11]  A. W. Leissa,et al.  Recent Research in Plate Vibrations. 1973 - 1976: Complicating Effects , 1978 .

[12]  A. N. Bercin.  Letter to the Editor: Free Vibration Solution for Clamped Orthotropic Plates Using the Kantorovich Method , 1996 .

[13]  M. Biancolini,et al.  Approximate solution for free vibrations of thin orthotropic rectangular plates , 2005 .

[14]  D. J. Gorman.  Accurate free vibration analysis of clamped orthotropic plates by the method of superposition , 1990 .

[15]  G. Oyibo,et al.  Generic buckling curves for specially orthotropic rectangular plates , 1983 .

[16]  V. W. Snyder,et al.  Free vibrations of rectangular orthotropic plates with a pair of parallel edges simply supported , 1990 .

[17]  R. Bhat.  Natural frequencies of rectangular plates using characteristic orthogonal polynomials in rayleigh-ritz method , 1986 .

[18]  D. J. Gorman.  Accurate Free Vibration Analysis of the Completely Free Orthotropic Rectangular Plate by the Method of Superposition , 1993 .

[19]  Santosh Kapuria,et al.  Levy-type piezothermoelastic solution for hybrid plate by using first-order shear deformation theory , 1997 .

[20]  N. S. Bardell,et al.  Free vibration analysis of thin coplanar rectangular plate assemblies — Part I: theory, and initial results for specially orthotropic plates , 1996 .

[21]  D. J. Gorman,et al.  Free Vibration Analysis of Rectangular Plates , 1982 .

[22]  L. M. Cook,et al.  Upper and lower bounds to the natural frequencies of vibration of clamped rectangular orthotropic plates , 1978 .

[23]  K. Bhaskar,et al.  Accurate and elegant free vibration and buckling studies of orthotropic rectangular plates using untruncated infinite series , 2008 .

[24]  Wei Ding,et al.  Accurate free vibration analysis of completely free symmetric cross-ply rectangular laminated plates , 2003 .

[25]  K. Bhaskar,et al.  Untruncated infinite series superposition method for accurate flexural analysis of isotropic/orthotropic rectangular plates with arbitrary edge conditions , 2008 .

[26]  Yang Xiang,et al.  Research on thick plate vibration: a literature survey , 1995 .

[27]  D. J. Gorman,et al.  Vibration Analysis of Plates by the Superposition Method , 1999 .

[28]  Serge Abrate,et al.  Free vibration, buckling, and static deflections of functionally graded plates , 2006 .

[29]  Wei Chen,et al.  3D free vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported , 2005 .

[30]  Arthur W. Leissa,et al.  Recent Research in Plate Vibrations: Classical Theory , 1977 .

[31]  Arthur W. Leissa,et al.  Vibration of Plates , 2021, Solid Acoustic Waves and Vibration.

[32]  A. W. Leissa,et al.  Literature Review: survey and analysis of the Shock and Vibration literature : Recent Studies in Plate Vibrations: 1981-85 Part II. Complicating Effects , 1987 .

[33]  R. L. Ramkumar,et al.  Free vibration solution for clamped orthotropic plates using Lagrangian multiplier technique , 1987 .

[34]  K. Bhaskar,et al.  SIMPLE AND EXACT SERIES SOLUTIONS FOR FLEXURE OF ORTHOTROPIC RECTANGULAR PLATES WITH ANY COMBINATION OF CLAMPED AND SIMPLY SUPPORTED EDGES , 2004 .

[35]  Rhys Jones,et al.  Application of the extended Kantorovich method to the vibration of clamped rectangular plates , 1976 .

[36]  T. Sakiyama,et al.  Free vibration analysis of orthotropic rectangular plates with variable thickness and general boundary conditions , 2005 .

[37]  Rama B. Bhat,et al.  Natural frequencies of orthotropic rectangular plates obtained by iterative reduction of the partial differential equation , 1996 .

[38]  Leonard Meirovitch,et al.  Elements Of Vibration Analysis , 1986 .

[39]  W. Ritz.  Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. , 1909 .

[40]  H. Al-khaiat.  Free vibration analysis of orthotropic plates by the initial value method , 1989 .

[41]  Free-Vibration Analysis of Point-Supported Orthotropic Plates , 1994 .

[42]  F. Holzweißig,et al.  A. W. Leissa, Vibration of Plates. (Nasa Sp‐160). VII + 353 S. m. Fig. Washington 1969. Office of Technology Utilization National Aeronautics and Space Administration. Preis brosch. $ 3.50 , 1971 .

[43]  James T. Stadter,et al.  Upper and Lower Bounds for the Frequencies of Rectangular Clamped Plates , 1967 .

[44]  S. Timoshenko,et al.  THEORY OF PLATES AND SHELLS , 1959 .

[45]  A. W. Leissa,et al.  Literature Review : Plate Vibration Research, 1976 - 1980: Complicating Effects , 1981 .

[46]  L. Rayleigh,et al.  The theory of sound , 1894 .