Dynamic Stiffness Formulation Using Timoshenko Theory for Free Vibration of Rotating Beams

Dynamic stiffness matrices of rotating tapered Timoshenko beams are formulated and subsequently used to investigate their free vibration characteristics. The range of cross-sections considered covers a majority of the practical cases for which the area of cross-section and the second moment of area can respectively, vary linearly and cubically or alternatively, the corresponding variations can be of second (square) and fourth order. First, the governing differential equations of motion of the rotating tapered beam in free flap flexural vibration are derived for the general case using Hamilton’s principle, and by allowing for the effects of shear deformation, rotatory inertia, centrifugal stiffening, an arbitrary outboard force and a hub radius term. For harmonic oscillation the differential equations are solved for flexural displacement and section rotation using the Frobenius method of series solution. The expressions for shear force and bending moment at any crosssection of the beam are also obtained in explicit analytical form. Then the dynamic stiffness matrix is developed, by relating the amplitudes of forces and moments to those of the displacements and rotations at the ends of the harmonically vibrating tapered Timoshenko beam. Next, the Wittrick-Williams algorithm is used as a solution technique to the resulting dynamic stiffness matrix to compute the natural frequencies and mode shapes of some illustrative examples. The effects of shear deformation, rotatory inertia, rotational speed, taper ratio and hub radius on the natural frequencies and mode shapes are investigated. Results are discussed, and compared with published ones, wherever possible. The error that may incur as a result of using the Bernoulli-Euler theory is estimated for different rotational speeds and slenderness ratios of the beam. Finally some concluding remarks are made.