VIBRATION MODES AND FREQUENCIES OF TIMOSHENKO BEAMS WITH ATTACHED RIGID BODIES

The equations of motion and boundary conditions for a free-fr ee Timos henko beam with rigid bodies attached at the endpoints are derived. Th e natural boundary conditions, for an end that has an attached rigid b ody, that include the effects of the body mass, first moment of mass, and moment of inertia are included. The frequency equation for a free- free Timoshenko beam with rigid bodies attached at its ends which incl udes all the effects mentioned above is presented and given in terms o f the fundamental frequency equations for Timoshenko beams that have n o attached rigid bodies. It is shown how any support/rigid-body condit ion may be easily obtained by inspection from the reported frequency e quation. The mode shapes and the orthogonality condition, which includ e the contribution of the rigid-body masses, first moments, and moment s of inertia, are also developed. Finally, the effect of the first mom ent of the attached rigid bodies is considered in an illustrative exam ple.