Forced Vibrations of Damped Non-homogeneous Timoshenko Beams

This work is the next of a series on vibrations of non-homogeneous structures. It addresses the lateral harmonic forcing, with spatial dependencies, of a two-segment damped Timoshenko beam. In the series, frequency response functions (FRFs) were determined for segmented structures, such as rods and beams, using analytic and numerical approaches. These structures are composed of stacked cells, which are made of different materials and may have different geometric properties. The goal is the determination of frequency response functions (FRFs). Two approaches are employed. The first approach uses displacement differential equations for each segment, where boundary and interface continuity conditions are used to determine the constants involved in the solutions. Then the response, as a function of forcing frequency, can be obtained. This procedure is unwieldy, and determining particular integrals can become difficult for arbitrary spatial variations. The second approach uses logistic functions to model segment discontinuities. The result is a system of partial differential equations with variable coefficients. Numerical solutions are developed with the aid of MAPLE (R) software. For free/fixed boundary conditions, spatially constant force, and viscous damping, excellent agreement is found between the methods. The numerical approach is then used to obtain FRFs for cases including spatially varying load.