Transverse vibrations of a rotating uniform cantilever beam with tip mass as predicted by using beam characteristic orthogonal polynomials in the Rayleigh-Ritz method

Abstract

Natural frequencies and mode shapes of a rotating uniform cantilever beam with a tip mass are studied by using beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The set of orthogonal polynomials which satisfy the geometrical boundary conditions are generated by using the Gram-Schmidt process. The results are compared with those obtained by the Myklestad method, the extended Galerkin method and finite element methods. The variation of natural frequencies with the speed of rotation is plotted for several parameter combinations such as setting angle, tip mass, moment of inertia of tip mass, etc. Mode shapes at different rotational speeds are also plotted. Use of orthogonal polynomials for the deflection shapes enables the computation of higher natural frequencies of any order to be accomplished without facing any numerical difficulties, which is not the case when arbitrary polynomial expressions are used.