Matrix formulations for solving the configuration-dependent eigenvalue problem of a two-link flexible manipulator

This paper introduces two analytical matrix descriptions of the differential eigenvalue problem concerning each admissible posture of a two-link flexible manipulator. In the hypothesis of the Euler-Bernoulli beam theory, analytical natural frequencies and mode shapes are derived through both formulations, which are obtained according to an axial and transverse dynamic characterization of the manipulator links. As is demonstrated by several simulations, such formulations give identical analytical results, though having very different matrix structures. Moreover, additional simulations through a finite element package confirm the analytical predictions of both methods presented herein.