Natural vibration of circular and annular thin plates by Hamiltonian approach

Abstract The present paper deals with the natural vibration of thin circular and annular plates using Hamiltonian approach. It is based on the conservation principle of mixed energy and is constructed in a new symplectic space. A set of Hamiltonian dual equations with derivatives with respect to the radial coordinate on one side of the equations and to the angular coordinate on the other side are obtained by using the variational principle of mixed energy. The separation of variables is employed to solve Hamiltonian dual equations of eigenvalue problem. Analytical frequency equations are obtained based on different cases of boundary conditions. The natural frequencies are the roots of the frequency equations and corresponding mode functions are in terms of the dual variables q1(r, θ). Three basic edge-constraint cases for circular plates and nine edge-constraint cases for annular plates are calculated and the results are compared well with existing ones.

[1]  Robin S. Langley,et al.  FREE VIBRATION OF THIN, ISOTROPIC, OPEN, CONICAL PANELS , 1998 .

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

[3]  Rama B. Bhat,et al.  VIBRATION OF RECTANGULAR PLATES USING PLATE CHARACTERISTIC FUNCTIONS AS SHAPE FUNCTIONS IN THE RAYLEIGH–RITZ METHOD , 1996 .

[4]  Yoshihiro Narita,et al.  Natural frequencies of free, orthotropic elliptical plates , 1985 .

[5]  K. Liew,et al.  Three-dimensional elasticity solutions for free vibrations of circular plates: A polynomials-Ritz analysis , 1999 .

[6]  M. Petyt,et al.  Natural frequencies for free vibration of nonhomogeneous elliptic and circular plates using two-dimensional orthogonal polynomials , 1997 .

[7]  K. Liew,et al.  Buckling And Vibration Of Annular Mindlin Plates With Internal Concentric Ring Supports Subject To In-Plane Radial Pressure , 1994 .

[8]  I. Stiharu,et al.  FREE VIBRATION OF ANNULAR ELLIPTIC PLATES USING BOUNDARY CHARACTERISTIC ORTHOGONAL POLYNOMIALS AS SHAPE FUNCTIONS IN THE RAYLEIGH–RITZ METHOD , 2001 .

[9]  T. Muhammad,et al.  Free in-plane vibration of isotropic non-rectangular plates , 2004 .

[10]  C. Lim,et al.  On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates , 2009 .

[11]  K. Liew,et al.  AXISYMMETRIC FREE VIBRATION OF THICK ANNULAR PLATES , 1999 .

[12]  Y. Xiang,et al.  On the missing modes when using the exact frequency relationship between Kirchhoff and Mindlin plates , 2005 .

[13]  Yoshihiro Narita,et al.  Natural frequencies of simply supported circular plates , 1980 .

[14]  V. Thevendran,et al.  Vibration Analysis of Annular Plates with Concentric Supports Using a Variant of Rayleigh-Ritz Method , 1993 .

[15]  A. Leung,et al.  Analytic stress intensity factors for finite elastic disk using symplectic expansion , 2009 .

[16]  G. D. Xistris,et al.  VIBRATION OF RECTANGULAR PLATES BY REDUCTION OF THE PLATE PARTIAL DIFFERENTIAL EQUATION INTO SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS , 1997 .

[17]  C. Y. Wang,et al.  Examination of the fundamental frequencies of annular plates with small core , 2005 .

[18]  Kenzo Sato,et al.  Free‐flexural vibrations of an elliptical plate with free edge , 1973 .

[19]  F. Ebrahimi,et al.  An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory , 2008 .

[20]  C. Wang,et al.  Buckling of annular plates elastically restrained against rotation along edges , 1996 .

[21]  Yang Xiang,et al.  Flexural vibration of shear deformable circular and annular plates on ring supports , 1993 .

[22]  Rama B. Bhat,et al.  CLOSED FORM APPROXIMATION OF VIBRATION MODES OF RECTANGULAR CANTILEVER PLATES BY THE VARIATIONAL REDUCTION METHOD , 1996 .

[23]  Li Yongqiang,et al.  Free vibration analysis of circular and annular sectorial thin plates using curve strip Fourier p-element , 2007 .

[24]  H. Carrington.  CXXXV. The frequencies of vibration of flat circular plates fixed at the circumference , 1925 .

[25]  Ding Zhou,et al.  Three-dimensional vibration analysis of circular and annular plates via the Chebyshev–Ritz method , 2003 .

[26]  S. M. Dickinson,et al.  On the free, transverse vibration of annular and circular, thin, sectorial plates subject to certain complicating effects , 1989 .

[27]  A. Leung,et al.  The mode III stress/electric intensity factors and singularities analysis for edge-cracked circular piezoelectric shafts , 2009 .

[28]  A. Leung,et al.  Hamiltonian Approach to Analytical Thermal Stress Intensity Factors—Part 1: Thermal Intensity Factor , 2010 .

[29]  Bo Liu,et al.  New exact solutions for free vibrations of rectangular thin plates by symplectic dual method , 2009 .