Modeling of transversal dynamics of stepped beams in case of boundary-excitation

In this paper a dynamic model of a stepped beam excited by an electrodynamical actuator with non-negligible dynamics at one end is derived. As model ansatz the superposition of rigid-body and modal dynamics is chosen. This results in the assumption of separated rigid-body and modal dynamics. The modal part of the beam dynamics is modeled as a cantilever beam with inhomogeneous boundary conditions. The solution of the modal transversal displacement is developed in a fourier series with the eigenfunctions of the spatial eigenvalue problem (EVP) as basis functions. The EVP is solved for a stepped beam by defining spatial-transfer-matrices. It can be seen that the location of the discontinuity in the cross-sectional area is an appropriate degree of freedom for relocating a particular eigenfrequency in combination of amplifying the corresponding eigenmode. Finally, the entire system dynamics with two different stepped beams is measured and the results are compared with the theoretical considerations.