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 probl em yields two independent sequences of natural frequencies and two correspo nding sequences of mode shapes. A particular natural frequency and its corr esponding 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 expa nsion of the solution. However, the question of whether these two independe nt sequences of natural frequencies, implies the existence of two distinct spectra of frequencies, has been a long standing topic of debate, and hithe rto has not been resolved completely. The object of this paper is to provid e 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 sol ution is naturally expressed by an ordered pair of characteristic values, r ather than a single characteristic value. This representation facilitates t he progressive ordering of all the natural frequencies of the system and th eir respective mode shapes in a single set, and eliminates the remaining ar gument for the two spectra interpretation. (C) 2001 Academic Press.