Free Vibration Analysis of Rotating Blades With Uniform Tapers

Abstract

A spectral finite element method (SFEM) is proposed to develop a low-degree-of-freedom model for dynamic analysis of rotating tapered beams. The method exploits semi-analytical progressive wave solutions of the governing partial differential equations. Only one single spectral finite element is needed to obtain any modal frequency or mode shape, which is as accurate or better than other approaches reported in the literature for straight or uniformly tapered beams. The minimum number of such spectral finite elements corresponds to the number of substructures, that is, beam sections with different uniform tapers, in a rotating beam to capture the complete system dynamic characteristics. The element assembly procedure is accomplished in the same fashion as the conventional finite element approach. Results are for a number of examples such as a straight beam and beams with uniform taper or compound tapers. Overall, for a rotating blade system, our SFEM provides highly accurate predictions for any modal frequency using a single element or very few elements corresponding to the number of uniform taper changes in the blade system. Nomenclature EI (x) = beam bending flexural stiffness EI 0 = reference beam bending flexural stiffness L = beam length M(x) = beam bending moment m(x) = beam mass per unit length m0 = reference beam mass per unit length R =o ffset length between beam and rotating hub T (x) = beam axial force due to centrifugal stiffening V (x) = beam shear force W(x) = beam bending mode shape function w(x, t) = beam transverse displacement α = beam mass per unit length constant βi = beam bending flexural stiffness constant, i = 1, 4 η = nondimensional axial force µ = nondimensional natural frequency � = beam rotation speed ω =e xcitation frequency