Transfer matrix method for the free and forced vibration analyses of multi-step Timoshenko beams coupled with rigid bodies on springs

• Published in: Applied Mathematical Modelling Vol. 87; pp. 152 - 170
• Main Authors: Chen, Gangli, Zeng, Xiaoyun, Liu, Xingbao, Rui, Xiaoting
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.11.2020; Elsevier BV
• Subjects: Dynamic characteristics; Eigenvectors; Forced vibration; Free vibration; Hybrid system; Hybrid systems; Linear equations; Mathematical models; Matrix methods; Mechanical systems; Mode superposition method; Orthogonality; Rigid structures; Robustness (mathematics); Springs (elastic); Timoshenko beam; Timoshenko beams; Transfer matrices; Transfer matrix method; Vibration; Transfer matrix method; Vibration; Orthogonality; Timoshenko beam; Hybrid system
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2020.05.023

•A novel method for the free and forced vibration analyses of the hybrid system is developed.•The free vibration characteristics are obtained by solving exact homogeneous linear equations.•The forced vibration analysis is conducted by introducing orthogonal augmented eigenvectors.•The present method is computationally efficient with high precision. Multi-step Timoshenko beams coupled with rigid bodies on springs can be regarded as a generalized model to investigate the dynamic characteristics of many structures and mechanical systems in engineering. This paper presents a novel transfer matrix method for the free and forced vibration analyses of the hybrid system. It is modeled as a chain system, where each beam and each rigid body with its supporting spring are dealt with one element, respectively. The transfer equation of each element is deduced based on separation of variables method. The system overall transfer equation is obtained by substituting an element transfer equation into another. Then, the free vibration characteristics are acquired by solving exact homogeneous linear equations. To compute the forced vibration response with modal superposition method, the body dynamic equations and augmented eigenvectors are established, and the orthogonality of augmented eigenvectors is mathematically proved. Without high-order global dynamic equation or approximate spatial discretization, the free and forced vibration analyses of the hybrid system are achieved efficiently and accurately in this study. As an analytical approach, the present method is easy, highly stylized, robust, powerful and general for the complex hybrid systems containing any number of Timoshenko beams and rigid bodies. Four numerical examples are implemented, and the results show that this method is computationally efficient with high precision.