Analysis of vibration of clamped square plates by the Rayleigh-Ritz method with asymptotic solutions from a modified Bolotin method

Estimates of flexural frequencies of clamped square plates are initially obtained by the modified Bolotin's method. The mode shapes in “each direction” are then determined and the product functions of these mode shapes are used as admissible functions in the Rayleigh-Ritz method. The data for the first twenty eigenvalues in each of the three (four) symmetric groups obtained by the (i) Bolotin, (ii) Rayleigh and (iii) Rayleigh-Ritz methods are reported here. The Rayleigh estimates are found to be much closer to the true eigenvalues than the Bolotin estimates. The present product functions are found to be much superior to the conventional beam eigenmodes as admissible functions in the Rayleigh-Ritz method of analysis.

Stiff string approximations in Rayleigh-Ritz method for rotating beams

The governing differential equation of the rotating beam reduces to that of a stiff string when the centrifugal force is assumed as constant. The solution of the static homogeneous part of this equation is enhanced with a polynomial term and used in the Rayleighs method. Numerical experiments show better agreement with converged finite element solutions compared to polynomials. Using this as an estimate for the first mode shape, higher mode shape approximations are obtained using Gram-Schmidt orthogonalization. Estimates for the first five natural frequencies of uniform and tapered beams are obtained accurately using a very low order Rayleigh-Ritz approximation.

Macromechanical behaviour of composite laminates

An improved higher order transverse shear deformation theory is employed to arrive at modified constitutive relations which can be used in the flexural, buckling and vibration analysis of laminated plates and shells. The strain energy for such systems is then expressed in terms of the displacements and the rotations for ready reference and use. Numerical values of vibration frequencies are obtained using this formulation employing Ritz's method of analysis. The results are compared with those available in the literature to validate the analysis presented.

Some studies on free vibration of composite laminates

Free vibration analysis is carried out to study the vibration characteristics of composite laminates using the modified shear deformation, layered, composite plate theory and employing the Rayleigh-Ritz energy approach. The analysis is presented in a unified form so as to incorporate all different combinations of laminate boundary conditions and with full coverage with regard to the various design parameters of a laminated plate. A parametric study is made using a beam characteristic function as the admissible function for the numerical calculations. The numerical results presented here are for an example case of fully clamped boundary conditions and are compared with previously published results. The effect of parameters, such as the aspect ratio of plates, ply-angle, number of layers and also the thickness ratios of plies in laminates on the frequencies of the laminate, is systematically studied. It is found that for anti-symmetric angle-ply or cross-ply laminates unique numerical values of the thickness ratios exist which improve the vibration characteristics of such laminates. Numerical values of the non-dimensional frequencies and nodal patterns, using the thickness ratio distribution of the plies, are then obtained for clamped laminates, fabricated out of various commonly used composite materials, and are presented in the form of the design curves.

Natural frequencies of polar orthotropic annular plates

The classical Rayleigh-Ritz method with simple polynomials as admissible functions has been used for obtaining natural frequencies of transversely vibrating polar orthotropic annular plates. The method in conjunction with transformations introduced in the analysis has been found to be quite effective, particularly for large hole sizes. Estimates of natural frequencies corresponding to modes with one as well as two nodal diameters are obtained for the nine combinations of clamped, simply supported and free edge conditions and for various values of rigidity ratio and hole sizes. Based on the variation of eigenvalue parameter with rigidity ratio, the frequencies of these modes as well as those of axisymmetric modes have been expressed by means of simple formulae in terms of rigidity ratio and the frequencies of corresponding modes in the isotropic case. These formulae have been used in determining the fundamental frequencies of orthotropic plates.

Natural frequencies of circumferentially truncated sector plates with simply supported straight edges

The classical Rayleigh-Ritz method in conjunction with suitable co-ordinate transformations is found to be effective for accurate estimation of natural frequencies of circumferentially truncated circular sector plates with simply supported straight edges. Numerical results are obtained for all the nine combinations of clamped, simply supported and free boundary conditions at the circular edges and presented in the form of graphs. The analysis confirms an earlier observation that the plate behaves like a long rectangular strip as the width of the plate in the radial direction becomes small.

Vibration of generally orthotropic skew plates

Vibration problem of generally orthotropic plates with particular attention to plates of skew geometry is studied. The formulation is based on orthotropic plate theory with arbitrary orientation of the principal axes of orthotropy. The boundary conditions considered are combinations of simply supported, clamped, and free-edge conditions. Approximate solution for frequencies and modes is obtained by the Ritz method using products of appropriate beam characteristic functions as admissible functions. The variation of frequencies and modes with orientation of the axes of orthotropy is examined for different skew angles and boundary conditions. Features such as "crossings" and "quasi-degeneracies" of the frequency curves are found to occur with variation of the orientation of the axes of orthotropy for a given geometry of the skew plate. It is also found that for each combination of skew angle and side ratio, a particular orientation of the axes gives the highest value for the fundamental frequency of the plate.

Vibration of skew plates

The vibration problems of skew plates with different edge conditions involving simple support and clamping have been considered by using the variational method of Ritz, a double series of beam characteristic functions being employed appropriate to the combination of the edge conditions. Natural frequencies and modes of vibration have been obtained for different combinations of side ratio and skew angle. These detailed studies reveal several interesting features concerning the frequency curves and nodal patterns. The results presented should, in addition, be of considerable value and practical significance in design applications.

Natural frequencies of circumferentially truncated sector plates with simply supported straight edges

The classical Rayleigh-Ritz method in conjunction with suitable co-ordinate transformations is found to be effective for accurate estimation of natural frequencies of circumferentially truncated circular sector plates with simply supported straight edges. Numerical results are obtained for all the nine combinations of clamped, simply supported and free boundary conditions at the circular edges and presented in the form of graphs. The analysis confirms an earlier observation that the plate behaves like a long rectangular strip as the width of the plate in the radial direction becomes small.

On the use of a coordinate transformation for analysis of axisymmetric vibration of polar orthotropic annular plates

Estimates of natural frequencies corresponding to axisymmetric modes of flexural vibration of polar orthotropic annular plates have been obtained for various combinations of clamped, simply supported and free edge conditions. A coordinate transformation in the radial direction has been used to obtain effective solutions by the classical Rayleigh-Ritz method. The analysis with this transformation has been found to be advantageous in computations, particularly for large hole sizes, over direct analysis. Numerical results have been obtained for various values of hole sizes and rigidity ratio. The eigenvalue parameter has been found to vary more or less linearly with the rigidity ratio. A comparison with the results for isotropic plates has brought out some interesting features.

Stability Analysis of Composite Skew Plates Using a Dynamic Method

Stability analysis is carried out considering free lateral vibrations of simply supported composite skew plates that are subjected to both direct and shear in-plane forces. An oblique stress component representation is used, consistent with the skew-geometry of the plate. A double series, expressed in Chebyshev polynomials, is used here as the assumed deflection surface and Ritz method of solution is employed. Numerical results for different combinations of side ratios, skew angle, and in-plane loadings that act individually or in combination are obtained. In this method, the in-plane load parameter is varied until the fundamental frequency goes to zero. The value of the in-plane load then corresponds to a critical buckling load. Plots of frequency parameter versus in-plane loading are given for a few typical cases. Details of crossings and quasi degeneracies of these curves are presented.