On the dynamic response of beams with multiple geometric or material discontinuities

An analytic framework is developed for determining closed form expressions for the natural frequencies, mode shapes, and frequency response function for Euler–Bernoulli beams with any number of geometric or material discontinuities. The procedure uses a convenient matrix formulation to generalize the single discontinuity beam problem to beams with multiple step changes. Specifically, the multiple discontinuity beam problem is solved by analyzing the total structure as a series of distinct Euler–Bernoulli elements with continuity and compatibility enforced at separation locations. The method yields each respective section's eigenmode which may then be superpositioned to give the entire beam's mode shape and derivation of the frequency response function follows. Although the Euler–Bernoulli beam problem is demonstrated, any one-dimensional continuous structure is amenable to the prescribed analysis. Theoretical predictions are experimentally validated as well.