A SERIES EXPANSION APPROACH TO INTERPRETING THE SPECTRA OF THE TIMOSHENKO BEAM

The Timoshenko beam model results in two fourth order partial differential equations in time and space. Consequently, solving the boundary value problem yields two independent sequences of natural frequencies and two corresponding sequences of mode shapes. A particular natural frequency and its corresponding mode shape describe one particular solution to the boundary value problem of the Timoshenko beam. From an eigenfunction expansion sense, all these possible solutions have to be considered in the complete series expansion of the solution. However, the question of whether these two independent sequences of natural frequencies, implies the existence of two distinct spectra of frequencies, has been a long standing topic of debate, and hitherto has not been resolved completely. The object of this paper is to provide answers to some of the issues raised by this debate. In this context, the complete solution in a series form to the Timoshenko beam is investigated, and it is shown for the first time that a particular mode shape of the solution is naturally expressed by an ordered pair of characteristic values, rather than a single characteristic value. This representation facilitates the progressive ordering of all the natural frequencies of the system and their respective mode shapes in a single set, and eliminates the remaining argument for the two spectra interpretation.