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PROCEEDINGS OF THE THIRTEENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING
Edited by: B.H.V. Topping and Y. Tsompanakis
Free Vibration of a Rotating Tapered Rayleigh Beam: A Dynamic Stiffness Method of Solution
J.R. Banerjee1 and D.R. Jackson2
1School of Engineering and Mathematical Sciences, City University London, United Kingdom
J.R. Banerjee, D.R. Jackson, "Free Vibration of a Rotating Tapered Rayleigh Beam: A Dynamic Stiffness Method of Solution", in B.H.V. Topping, Y. Tsompanakis, (Editors), "Proceedings of the Thirteenth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 76, 2011. doi:10.4203/ccp.96.76
Keywords: rotating tapered beam, free vibration, dynamic stiffness method, Frobenius method, Wittrick-Williams algorithm.
The dynamic stiffness method for free vibration analysis of a rotating tapered Rayleigh beam is developed in this paper to investigate its free vibration characteristics. The type of taper considered is such that the depth and, or the width of the beam cross-section vary linearly along the length of the beam so that the area of cross-section follows a linear or square variation whereas the corresponding second moment of area follows a cubic or fourth order variation, respectively. Such variations cover most of the practical cross-sections. First, the governing differential equation of motion of the beam in free bending vibration is derived by applying Hamilton's principle. The effects of centrifugal stiffening, an outboard force, an arbitrary hub radius and importantly, the rotatory inertia (so as to make it a Rayleigh beam) are taken into account. For harmonic oscillation the differential equation is solved by using the Frobenius method of series solution. The expressions for bending displacement, bending rotation, shear force and bending moment at any cross-section of the beam are then obtained from the series solution. Next, the dynamic stiffness matrix is formulated by applying the boundary conditions to relate the amplitudes of forces and moments to those of the displacements and rotations at the ends of the vibrating beam. Natural frequencies and mode shapes of some illustrative examples are computed using the ensuing dynamic stiffness matrix through the application of the Wittrick-Williams algorithm as solution technique. The results showing the effects of slenderness ratio, rotational speed and taper ratio on natural frequencies are discussed. This is followed by some concluding remarks.
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