A lower bound for the smallest frequency determined by means of the Rayleigh-Ritz method

Abstract

The present paper deals with the derivation of a simple expression which constitutes a valid lower bound with respect to the Rayleigh-Ritz lowest frequency in the case of a vibrating mechanical system. The expression is obtained by diagonalizing the strain energy matrix currently constructed when using the classical Rayleigh-Ritz method. The approach is first employed in a situation where the strain energy matrix is a priori diagonal due to the type of coupled displacement fields present in the system, and it is shown that the calculated lower bounds are acceptable from a practical viewpoint. The methodology is then applied to the determination of lower bounds of natural frequencies in planar structures. It is then concluded that it seems both possible and convenient to construct such a limit leading to simple algebraic expressions which are effective from a designer's viewpoint, since the geometric and mechanical parameters which come into play in the dynamic behavior of the system can be easily varied as opposed to the case in which complicated transcendental equations are obtained.