Free vibration of rotating tapered beams using the dynamic stiffness method

Free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method

Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method

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

Free Vibration Analysis of Rotating Blades With Uniform Tapers

In this study, problems related to the thermoelastic modeling and behavior of thin-walled beams made of functionally graded materials (FGMs) are addressed. In this context, two structural systems are considered: (i) rotating turbomachinery blades, and (ii) thin-walled beam structures spinning about their longitudinal axis. In all these cases, the structures operate in a high temperature environment. Under the spinning speed, gyroscopic forces are generated. Although conservative in nature, under their influence, as in nonconservative systems, instabilities by flutter and divergence can occur. In this context, the implications on their vibration and instability of conservative and gyroscopic forces considered in conjunction with a temperature field that yields the degradation of elastic properties are investigated. A continuously graded variation in composition of the ceramic and metal phases across the beam wall thickness in terms of a simple power law distribution is implemented. Results highlig...

Thin-Walled Beams Made of Functionally Graded Materials and Operating in a High Temperature Environment: Vibration and Stability

The free vibration of rotating beams is analyzed by means of a finite-element method of variable order. This method entails displacement functions that are a complete power series of a variable number of terms. The terms are arranged so that the generalized coordinates are composed of displacements and slopes at the element extremities and, additionally, displacements at certain points within the element. The displacement is assumed to be analytic within an element and thus can be approximated to any degree of accuracy desired by a complete power series. Numerical results are presented for uniform beams with zero and nonzero hub radii, tapered beams, and a nonuniform beam with discontinuities. Since the present method reduces to a conventional beam finite-element method for a cubic displacement function, the results are compared and found to be superior to the conventional results in terms of accuracy for a given number of degrees of freedom. Indeed, essentially exact eigenvalues and eigenvectors are obtained with this technique, which is far more rapidly convergent than other approaches in the literature.

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Free-Vibration Analysis of Rotating Beams by a Variable-Order Finite-Element Method

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Michael Yang Rutkowski

Papers

Free-Vibration Analysis of Rotating Beams by a Variable-Order Finite-Element Method

Free-vibration analysis of rotating beams by a variable-order finite-element method