Harmonic Forcing of a Two-Segment Euler-Bernoulli Beam

This study is on the forced motions of non-homogeneous elastic beams. Euler-Bernoulli theory is employed and applied to a two-segment configuration subject to harmonic forcing. The objective is to determine the frequency response function for the system. Two different solution strategies are used. In the first, analytic solutions are derived for the differential equations for each segment. The constants involved are determined using boundary and interface continuity conditions. The response, at a given location, can be obtained as a function of forcing frequency (FRF). The procedure is unwieldy. Moreover, determining particular integrals can be difficult for arbitrary spatial variations. An alternative method is developed wherein material and geometric discontinuities are modeled by continuously varying functions (here logistic functions). This results in a single differential equation with variable coefficients, which is solved numerically, for specific parameter values, using MAPLE®. The numerical solutions are compared to the baseline analytical approach for constant spatial dependencies. For validation purposes an assumed-modes solution is also developed. For a free-fixed boundary conditions example good agreement between the numerical methods and the analytical approach is found, lending assurance to the continuous variation model. Fixed-fixed boundary conditions are also treated and again good agreement is found