Many multi-beam energy harvesters appearing in the literature require custom analytical or finite-element models to compute their eigensolutions and piezoelectric coupling effects. This paper discusses the use of the transfer matrix method to derive analytical solutions to beam structures with point-wise discontinuities or with lumped inertias between members or at the tip. In this method, transfer matrices are developed for the beam’s states (deflection, slope, shear force, and bending moment) analogously to the state transition matrix of a linear system. Euler-Bernoulli beam theory is used to derive transfer matrices for the uniform beam segments, and point transfer matrices are derived to handle discontinuities in the structure. The transfer matrix method is shown to be advantageous for analyzing complex structures because the size of the transfer matrix does not grow with increasing number of components in the structure. Furthermore, the same formulation can be used for a wide range of geometries, including arbitrary combinations of beam segments — single- or multi-layered — and lumped inertias. The eigensolution of the transfer matrix is shown to produce the natural frequencies and mode shapes for these structures. Subsequently, the electromechanical coupling effects are incorporated and the base excitation problem is considered. The electromechanical equations of motion are decoupled by mode and shown to be a generalization of existing analytical models. Parametric case studies are provided for beam structures with varying piezoelectric layer coverage.

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