Extension of Rayleigh–Ritz method for eigenvalue problems with discontinuous boundary conditions applied to vibration of rectangular plates
Introduction
Computation of eigenvalues has a vast literature, much of it devoted to vibration of mechanical structures such as beams and plates. Computation of frequencies and modes has been pursued by analytical methods, variational methods principally Rayleigh–Ritz (R–R), and numerical discretization methods such as finite elements. Here we develop an extension of the R–R method to deal with discontinuous boundary conditions (DBC) where the discontinuity is within an edge. The R–R method starts by selecting a set of orthogonal functions, also named coordinate functions. These are usually products of trigonometric functions and/or orthogonal polynomials that satisfy the boundary conditions (BC). A trial function is then formed as a sum of these coordinate functions multiplied by undetermined coefficients. Introducing the trial function into the energy equation yields a quadratic form in the undetermined coefficients. The eigenvalues of the matrix of this quadratic form provide approximations to the eigenvalues of the boundary value problem. With suitable choice of the coordinate functions good approximation can be achieved.
This paper addresses the construction of coordinate functions for rectangular plates subject to DBC (Fig. 1b) contrasted with the ordinary BC (Fig. 1a). Geometry 1(a) can be treated by the established R–R method. In this standard arrangement the BC are constant along each edge but can vary among the edges. In geometry 1(b), the BC are discontinuous within a single edge. DBC such as 1(b) cannot be treated by the customary R–R method because the trial functions do not satisfy the discontinuous BC within edges. In all geometries, material properties of the rectangular plate may vary with position. We briefly review the three types of approaches used to treat vibration problems.
Much of the early work on vibration applied the classical separation of variables to continuous BC resulting in a nonlinear algebraic equation (characteristic equation) for the eigenvalues that is solved numerically [1]. Gorman [2], [3], [4] introduced a general superposition method involving expansions in suitable trigonometric and hyperbolic functions satisfying the differential equation. These expansions are supplemented by bending moment distributions on some edges, as needed to satisfy the boundary conditions. The resulting algebraic equations in the expansion coefficients and the bending moment distributions are solved numerically. Good results were obtained for different plates including those with free edges (e.g. cantilever) which at that time presented difficulties. Gorman later extended his method to plates with DBC [5]. He treated discontinuities between clamped and simply supported (S-S) segments by again introducing bending moment distributions to fully satisfy the BC at the edge of the discontinuity. The method was demonstrated for plates containing clamped and S-S edges and only for the first mode eigenvalue. It appears that the method can be extended to include free edges and higher mode eigenvalues at the cost of considerable increase in the analytical and computational effort. Narita et al. developed a series expansion to treat DBC in circular [6], and rectangular plates [7], [8]. The DBC were also expanded in a series resulting in complicated equations for frequencies and modes to be solved numerically.
Variational methods have been most commonly used in vibration analysis. Most of the early work employed the R–R method and was concerned with rectangular plates under a variety of BC. Leissa [1] reviewed the early literature going back to Ritz and obtained exact solutions for six combinations of BC amenable to an analytical treatment, and numerical solutions for 21 additional BC combinations. His coordinate functions were products of beam functions, i.e. functions that satisfy the beam BC. Bhat [9] introduced the more convenient orthogonal polynomials known as “boundary characteristic polynomials” (BCP), as coordinate functions and obtained results in good agreement with Leissa׳s exact and approximate solutions. The R–R method using BCP was subsequently applied by several researchers for a variety of plates, including those with variable properties, but not including DBC. The R–R method has the advantage of simplicity and the ability to handle spatially varying plate properties without any change in the coordinate functions that depend only on the boundary conditions. In the analytical methods the base functions satisfy the differential equations and are not suitable for plates with variable properties.
A different variational method was developed by Filipich and Rosales [10], [11], [12]. They constructed extremizing sequences the members of which are sums of coordinate functions. The BC do not have to be satisfied by each coordinate function individually but are satisfied by the sequence in the limit. This general method was applied among other problems to inhomogeneous rectangular plates consisting of sections of different densities separated by curved boundaries. As with the standard R–R method the continuous nature of the coordinate functions cannot satisfy BC discontinuities within edges.
A variational method in conjunction with the “penalty method” was used by Ilanko, and Monterrubio [13], [14], [15] that also follows the R–R approach. However, some of the essential BC and other constraints such as fixed points or line supports are handled by adding penalty terms to the energy function to be minimized. Calculations are repeated for increasing values of the penalty coefficients till convergence is reached. The method was successful in some problems with continuous BC along an edge. However, applying it to DBC resulted in frequencies increasing monotonically with penalty coefficient without convergence (see Example 3 in Section 3).
Numerically more intensive discretization methods are capable of handling DBC as well as standard BC. These methods include finite elements, differential quadratures, and boundary elements (boundary integral) leading to integral equations. Wei et al. [16] reviewed numerical methods applied to problems with DBC and developed a discrete singular convolution method for these problems. In the present paper we compare our results with those from a finely discretized finite element method.
The present method relies on the R–R method that is familiar and simpler than the series expansions in [8]. The key aspect here is the introduction of new coordinate functions consisting of sums of products of orthogonal polynomials capable of satisfying exactly DBC such as those in Fig. 1(b). These coordinate functions are still approximate because they involve a small discontinuity in the function and its derivatives at the interface between regions I and II (Fig. 1b). The present method can also be applied to two-dimensional problems of heat conduction, diffusion, and flow of slightly compressible fluids in porous media. In this work, the method is applied to the problem of plate vibration.
Section snippets
Analysis
The analysis is developed with reference to the BC arrangement shown in Fig. 1(b). Product functions are defined for region I and region II separately:where pi(x), qi(x), uj(y), and are polynomials and/or trigonometric functions orthonormal in respectively and satisfying BC1 only, BC4 only, BC2+BC5, and BC3+BC5, respectively. The specific BC need not be specified at this point.
Product functions
Example 1
The geometry is specified in Fig. 2(a) with . The pi functions were obtained by applying Gramm–Schmidt (G–S) ortho-normalization in the interval on the following set of polynomials satisfying the clamped BC at .The qi functions were obtained by applying G–S orthonormalization in the interval on the following set obeying the clamped BC at . The ui functions were obtained by applying G–S in the interval on the set
Conclusion
Two-part coordinate functions are defined that satisfy discontinuous boundary conditions and can be used in the R–R method for eigenvalue estimation. The two parts are matched approximately at the interface of their respective regions by minimizing the mean square error of functions and their x-derivatives. Several of the coordinate functions have zero eigenvalues µ producing zero matching error. However, additional functions are needed for use in the R–R method. These correspond to small but
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