Free Vibration of Timoshenko–Ehrenfest Beams and Frameworks Using Frequency-Dependent Mass and Stiffness Matrices

The frequency-dependent mass and stiffness matrices of a Timoshenko–Ehrenfest beam are developed through extensive application of symbolic computation. Explicit algebraic expressions for the frequency dependent shape functions and each of the independent elements of the frequency-dependent mass and stiffness matrices are presented concisely. The ensuing frequency-dependent mass and stiffness matrices of the Timoshenko–Ehrenfest beam are applied with particular reference to the Wittrick–Williams algorithm to investigate the free vibration characteristics of an individual Timoshenko–Ehrenfest beam and a framework. The results are discussed with significant conclusions drawn. The proposed method retains the exactness of results like the dynamic stiffness method, but importantly, it opens the possibility of including damping in the analysis