the finite strip method, Ramakrishnan and Kunukkasseril [4] studied the free vibration of annular sector plates, Cheung et al. [5] analyzed arbitrarily shaped plates by mapping the Cartesian coordinate system into the natural coordinate plane by means of serendipity shape functions using polynomial splines as displacement functions, Mizusawa [6] studied skew plates with different boundary constraints, Geannakakes [7] applied the semi-analytical finite strip method for the analysis of arbitrarily shaped plates using Hermitian polynomials as shape functions. In the present study, the problem of arbitrary-shaped vibrating Kirchhoff plates with general boundary conditions is analyzed using the Rayleigh-Ritz method [8]. The solution is expressed in terms of a linear combination of functions, which in the present study are selected as products of eigenfunctions of homogeneous uniform prismatic beams in flexural vibration [9]. General boundary conditions are introduced in the functional of the potential energy by additional terms, and both trigonometric and polynomial interpolations are implemented for mapping the shape of the plate in Cartesian coordinates into natural coordinates [5-7]. Flexural free vibration analysis of different shaped plates is then performed: skew, trapezoid and triangular plates, plates with parabolic edges, elliptic sector and annular plates. The proposed method can be directly applied also to variable thickness plates and nonhomogeneous plates, with variable density and stiffness. Purely elastic plates are considered; however the method may also be applied to the analysis of viscoelastic plates, as proposed by Catania et al. [10-11]. 2 Mapping technique An arbitrary-shaped plate in Cartesian coordinates x and y can be expressed by the mapping of a square plate defined in its natural coordinates [ and K [5], as shown in Fig. 1. The (generally non-conformal) mapping of the Cartesian system can be expressed as: 1 1 ( , ) ( , ), ( , ) ( , ) p p i i i i i i x xP y yP [ K [ K [ K [ K ¦ ¦ (1) where xi and yi (i = 1, 2, … p) are the coordinates of p points on the boundary of the plate, and Pi are interpolation functions. K [ (–1,1) (1,1) (–1,–1) (1,–1) 4 3 1 2 K [ (–1,1) (1,1) (–1,–1) (1,–1) 4 3 1 2 9 5 6 8 7 (0,1) (0,0) (–1, 0) (0,–1) (1,0) Fig. 1 Linear (left) and quadratic Lagrange (right) regions In order to evaluate the differential operators needed in plate analysis, the following relations can be written in the form: 1 1 1 1 , ( , ) p p i i i i i i p p i i i i i i P P x y x P P x y y [ [ [ [ K K K K ª º w w w ½ w ½ « » ° w ° ° ° w w w ° ° ° ° « » ® ¾ ® ¾ w « » w w w ° ° ° ° « » °w ° °w ° w w ¯ ¿ ¯ ¿ ¬ ¼ ¦ ¦ ¦ ¦ J J (2) and consequently: > @ 2 2 2 2 T 2 2 2 2 2 2 2 2 2 2 2 , x y x y [ [ [ K [ K K [ K K [ K ª º ½ ½ w w « » ° ° ° ° w w ½ w « » ° ° ° ° ° ° « » ° ° ° ° w ½ w w w w w w w ° ° ° ° ° ° « » ® ¾ ® ¾ ® ¾ ® ¾ w w w w w w w w w « » ¯ ¿ ° ° ° ° ° ° « » ° w ° ° ° °w ° w ¯ ¿ « » ° ° ° ° « » w w w w °¯ °¿ °¯ °¿ ¬ ¼ A B C C A AB (3) 78
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