Harmonic Forcing of Damped Non-homogeneous Elastic Rods

This work is one of an ongoing series of investigations on the motions of non-homogeneous structures. In the series, natural frequencies, mode shapes and frequency response functions (FRFs) were determined for undamped segmented rods and beams, using analytic and numerical approaches. These structures are composed of stacked cells, which may have distinct geometric and material properties. Here, the steady state response, due to harmonic forcing, of a segmented damped rod is investigated. The objective is the determination of FRFs for the system. Two methods are employed. The first uses the 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 may become unpractical for arbitrary spatial forcing functions. The second approach uses logistic functions to model the segment discontinuities. This leads to a single partial differential equation with variable coefficients, which is solved numerically using MAPLE® software. For free-fixed boundary conditions and spatially constant force good agreement is found between the methods. The continuously varying functions approach is then used to obtain the response for a spatially varying force.