Exact free vibration analysis of generalized multi-step Timoshenko beams coupled with spring-supported rigid bodies

This paper presents a linear version of the reduced multibody system transfer matrix method, specifically designed for the exact analysis of free vibrations in hybrid models composed of Timoshenko beams, rigid bodies, and springs. The method is flexible, designed to handle various boundary conditions and any combination of beams, rigid bodies, and springs. We treat each beam segment and spring-supported rigid body as independent elements. Thus, viewing the overall model as a chain system simplifies the analysis. The essence of this method is the recursive transfer of mechanical information between elements, which is contained in the reduced transfer equations. The reduced transfer equations for the spring-supported rigid bodies and Timoshenko beams are derived in detail. The accuracy, high precision, and higher-order modal analysis capabilities of this method are validated through numerical examples. Furthermore, the improvement of the numerical stability by the segmentation strategy is analyzed, and the orthogonality between the augmented eigenvectors is proved mathematically and numerically. The concise, structured and highly programmable greatly simplifies the process of handling complex hybrid systems containing any number of Timoshenko beams and rigid bodies.