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Abstract

This paper aims at presenting a method to determine the “exact” natural frequencies and mode shapes of a hybrid beam composed of multiple elastic beam segments and multiple rigid bodies with each rigid body connected with two adjacent elastic beam segments. Furthermore, each rigid body has its own mass and rotary inertia, and is supported by a translational spring and/or a rotational spring. First, based on the equations of the continuity of deformations and the equilibrium of moments and forces for each of the intermediate rigid bodies and boundary conditions, the coefficient matrices of the entire hybrid beam are derived. The overall coefficient matrix for the entire hybrid beam is obtained using the numerical assembly technique. The exact natural frequencies are determined by equating the determinant of the last overall coefficient matrix to zero. With respect to each of the natural frequencies, one may obtain the associated integration constants from the simultaneous equations. Finally, substituting these integration constants into the displacement functions for all the elastic beam segments and replacing the space occupied by each of the rigid bodies by a straight line, one determines each of the corresponding mode shapes of the hybrid beam. Finally, the influence of materials for the elastic beam segments on natural frequencies and mode shapes of the hybrid beam is studied.

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