DYNAMIC STIFFNESS FORMULATION AND FREE VIBRATION ANALYSIS OF CENTRIFUGALLY STIFFENED TIMOSHENKO BEAMS

The dynamic stiffness matrix of a centrifugally stiffened Timoshenko beam has been developed and used to carry out a free vibration analysis. The governing differential equations of motion of the beam in free vibration are derived using Hamiltonʹs principle and include the effect of an arbitrary hub radius. For harmonic oscillation the derivation leads to two different (but of similar form) fourth-order ordinary differential equations with variable coefficients that govern the amplitudes of bending displacement and bending rotation respectively. An outboard force at the end of the beam is taken into account which makes possible the free vibration analysis of rotating non-uniform or tapered Timoshenko beams. Using the Frobenius method of series solution and imposing boundary conditions, the dynamic stiffness matrix, which relates amplitudes of harmonically varying forces with the amplitudes of harmonically varying displacements at the ends of the element, is formulated. Applying the Wittrick–Williams algorithm to the resulting dynamic stiffness matrix the natural frequencies of a few carefully chosen illustrative examples are obtained. The results are compared with those available in the literature.