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where the functions F,G,f,g,f, and g are evaluated at epoch tj for a time interval of t* -tj. It is easily seen that det B^ = 1. It is the simplicity of B i3 that dictated the definition of the X variables.In many applications involving the perturbation matrix &ij, it may be possible to simplify the computation by working directly with the X variables. This is particularly true for the differential correction of orbits. By evaluating the corrections as a column matrix X 0 at the epoch to, the only value of Aj~l that is required is AQ~I. Furthermore, advantage can be taken of the sparseness of B^ to reduce considerably the required computation. Compared with the use of corrections to classical orbital elements, the X variables have the advantage of being valid for any two-body trajectory. Nomenclature a,b = dimensions of plate a rs = coefficient in the series expansion of deflection D = Eh*/12(l -z/ 2 ), flexural rigidity of the plate E = Young's modulus of the material of the plate h = plate thickness k = (p/i/Z>) 1/2 coa 2 /7r 2 , frequency parameter m,n,r,s = indices in the deflection series M = maximum value of index m N = maximum value of index n W(xi,yi) = deflection of the plate x,y = rectangular coordinates, see inset of Fig. 1 £i,2/i = oblique coordinates, see inset of Fig.

Natural frequencies and mode shapes of a wide range of skew membranes are obtained by the Rayleigh-Ritz method expressing the deflection as a double Fourier sine series in oblique coordinates. Since the eigenvalues of a polygonal simply supported plate are the squares of the eigenvalues of a polygonal membrane of the same geometry and the eigenfunctions are identical, the results of the present paper readily provide the vibration characteristics of simply supported skew plates as well. This detailed study revealed interesting features like the skew angle splitting the degenerate frequencies of rectangular membranes into distinct ones and the “frequency crossing” of some modes that belong to opposite symmetry groups of the skew membrane. With changing skew angle, the nodal patterns of some modes undergo a gradual change to other totally different patterns. Since, for some modes the diagonals happen to be the nodal lines, such modes can be interpreted as the appropriate modes of the corresponding triangular membrane and consequently, of triangular simply supported plates.

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