Abstract:

The Rayleigh--Ritz method for continuous systems uses an N-term series with unspecified coefficients as the trial function for the Rayleigh ratio. Extremizing that ratio leads to N approximate natural frequencies (omega)[inf j][sup (N)], j=1,...,N. The upper bound theorem states that the true values obtained by solving the field equations are such that (omega)[inf j](less than or equal to)(omega)[inf j][sup (N)]. The separation theorem states that adding a single term to the aforementioned series yields new estimates (omega)[inf j][sup (N+1)], j=1,...,N+1, such that the previous (omega)[inf j][sup (N)] fall in the intervals between the new values. Taken together, these theorems constitute a proof that the lower eigensolutions obtained from the Rayleigh--Ritz method should converge to the true eigensolution. However, both theorems assume infinite precision arithmetic. This paper uses the simple case of a cantilever beam to examine the behavior of the Rayleigh--Ritz method with increasing series length. Natural frequencies derived from different classes of kinematically admissible basis functions, drawn from monomials, trigonometric functions, and Bessel functions are examined relative to the upper bound and separation theorems. Most selections fail to yield properly behaved solutions if the series length is extended beyond 10--12 terms. In some cases the eigenvalue solver fails to find real eigenvalues, or to find the correct number of eigenvalues. Some of the basis function sets permit formulating the Rayleigh ratio functional analytically, rather than by numerical integration, but the results are the same. The failure of the method is shown to stem from ill-conditioning that arises as a consequence of similarity in the appearance of higher-order basis functions.