Free vibration analysis of arbitrarily shaped polygonal plates with simply supported edges using a sub-domain method

As an extension of the NDIF method developed by the authors, a practical analytical method for the free vibration analysis of a simply supported polygonal plate with arbitrary shape is proposed. Especially, the method is more effective for plates highly concave shapes because it employs a sub-domain method dividing the plate of interest with two sub-plates. The approximate solution of each sub-plate is assumed by linearly superposing plane waves propagated from edges of the sub-plate. Sub-system matrix equations for the two sub-plates are extracted by applying the simply supported boundary condition to the edges of each sub-plate (excepting the common interface of the two sub-plates). Finally, the sub-system matrix equations is merged into a single system matrix equation for the entire plate by considering the compatibility condition that the two sub-plates have the same displacement and slope at the common interface. The eigenvalues and mode shapes of the single plate are obtained from the determinant of a system matrix extracted from the entire system matrix equation. It is shown by several case studies that the proposed method has a good convergence characteristics and yields accurate eigenvalues and mode shapes, compared with another analytical method (NDIF method) and FEM (NASTRAN).

[1]  J. Z. Zhu,et al.  The finite element method , 1977 .

[2]  Z. Ding.  Vibration of an arbitrarily shaped membrane carrying elastically mounted masses , 1992 .

[3]  Young-Seok Kang,et al.  VIBRATION ANALYSIS OF ARBITRARILY SHAPED MEMBRANES USING NON-DIMENSIONAL DYNAMIC INFLUENCE FUNCTION , 1999 .

[4]  Inderjit Chopra,et al.  Vibration of simply-supported trapezoidal plates II. Unsymmetric trapezoids , 1971 .

[5]  Toshihiro Irie,et al.  Free vibration of regular polygonal plates with simply supported edges , 1981 .

[6]  Snehashish Chakraverty,et al.  Transverse vibration of simply supported elliptical and circular plates using boundary characteristic orthogonal polynomials in two variables , 1992 .

[7]  Arthur W. Leissa,et al.  The free vibration of rectangular plates , 1973 .

[8]  S. Durvasula,et al.  Vibration of skew plates , 1973 .

[9]  Free vibration analysis of an unsymmetric trapezoidal membrane , 2004 .

[10]  Sang-Wook Kang,et al.  Free Vibration Analysis of Arbitrarily Shaped Plates With Smoothly Varying Free Edges Using NDIF Method , 2008 .

[11]  Minoru Hamada.  Compressive or Shearing Buckling Load and Fundamental Frequency of a Rhomboidal Plate with All Edges Clamped , 1959 .

[12]  Shin-Hyoung Kang,et al.  APPLICATION OF FREE VIBRATION ANALYSIS OF MEMBRANES USING THE NON-DIMENSIONAL DYNAMIC INFLUENCE FUNCTION , 2000 .

[13]  S. Durvasula,et al.  Natural Frequencies and Modes of Skew Membranes , 1968 .

[14]  R. Blevins,et al.  Formulas for natural frequency and mode shape , 1984 .

[15]  S. Durvasula,et al.  Natural frequencies and modes of clamped skew plates. , 1969 .

[16]  L. Meirovitch.  Analytical Methods in Vibrations , 1967 .

[17]  M. G. Milsted,et al.  Use of trigonometric terms in the finite element method with application to vibrating membranes , 1974 .

[18]  H. D. Conway.  The Bending, Buckling, and Flexural Vibration of Simply Supported Polygonal Plates by Point-Matching , 1961 .

[19]  Sang-Wook Kang,et al.  Free vibration analysis of composite rectangular membranes with a bent interface , 2004 .

[20]  K. Nagaya.  Vibrations and Dynamic Response of Membranes With Arbitrary Shape , 1978 .

[21]  Jagannath Mazumdar,et al.  Transverse vibration of membranes of arbitrary shape by the method of constant-deflection contours , 1973 .

[22]  J. Mazumdar,et al.  Vibration analysis of plates of arbitrary shape—A new approach , 1979 .

[23]  Sang-Wook Kang,et al.  FREE VIBRATION ANALYSIS OF ARBITRARILY SHAPED PLATES WITH A MIXED BOUNDARY CONDITION USING NON-DIMENSIONAL DYNAMIC INFLUENCE FUNCTIONS , 2002 .

[24]  H. D. Conway,et al.  The free flexural vibrations of triangular, rhombic and parallelogram plates and some analogies , 1965 .

[25]  S. M. Dickinson.  THE BUCKLING AND FREQUENCY OF FLEXURAL VIBRATION OF RECTANGULAR ISOTROPIC AND ORTHOTROPIC PLATES USING RAYLEIGH'S METHOD , 1978 .

[26]  Shin-Hyoung Kang,et al.  FREE VIBRATION ANALYSIS OF ARBITRARILY SHAPED PLATES WITH CLAMPED EDGES USING WAVE-TYPE FUNCTIONS , 2001 .

[27]  C. Brebbia,et al.  Boundary Element Techniques , 1984 .

[28]  S. Durvasula.  Free vibration of simply supported parallelogrammic plates. , 1969 .

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

[30]  Carlos Alberto Brebbia,et al.  The Boundary Element Method for Engineers , 1978 .

[31]  Lee,et al.  Eigenmode analysis of arbitrarily shaped two-dimensional cavities by the method of point-matching , 2000, The Journal of the Acoustical Society of America.

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