A solution procedure for studying the dynamic responses of a non-uniform Timoshenko beam with general time-dependent boundary conditions is developed by generalizing the method of Mindlin-Goodman and utilizing the exact solutions of non-uniform Timoshenko beam vibration given by Lee and Lin. A general change of dependent variable with shifting functions is introduced and the physical meanings of these shifting functions are further explored. The orthogonality condition for the eigenfunctions of a non-uniform Timoshenko beam with elastic boundary conditions is also derived. Several limiting cases and their corresponding procedures are revealed. Finally, the influence of the spring constant on the steady response of a beam subjected to a harmonic base excitation is investigated.
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