Transfer Matrix Method for the Determination of the Free Vibration of Two Elastically Coupled Beams

Abstract:

The feasibility of using the transfer matrix method (TMM) to compute the free vibration characteristics of a system composed of continues and discrete elements vibrating in a plane is explored theoretically. In the approach to the problem, a general analytical method based on TMM is developed for the vibrations of two uniform Euler-Bernoulli beams coupled by a spring. The components of the transfer matrix are all functions of the systems natural frequency. The overall transfer equation only involves boundary state vectors, whereas the state vectors at all other connection points do not appear. The state vectors at the boundary are composed of displacements, rotation angles, bending moments and shear forces, which are partly known and partly unknown. Moreover, the overall transfer matrix is independent of the degrees of the freedom. A non-trivial solution of the final overall transfer equation requires the coefficient matrix to be singular. This paper reduces the zero search of its determinate to a minimization problem and demonstrates a simple, robust algorithm being much more efficient than direct enumeration. A numerical result is presented to demonstrate the proposal method.

• Periodical: Applied Mechanics and Materials (Volume 372)
• Pages: 301-304
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.372.301
• Online since: August 2013
• Authors: Laith K. Abbas, Dieter Bestle, Xiao Ting Rui
• Keywords: Coupled Beams, Free Vibration, Recursive Eigenvalue Search, Transfer Matrix Method (TMM)

References

[1] X. Rui, L. Yun, Y. Lu, B. He, G. Wang, Transfer Matrix Method of Multibody System and its Application, Science Press, Beijing, 2008 (in Chinese).

[2] S. Kukla, J. Przybylski, L. Tomski, Longitudinal vibrations of rods coupled by translational springs, Journal of Sound and Vibration, 185 (1995) 717–722.

DOI: 10.1006/jsvi.1995.0411

[3] H. Ping Lin, S. Chang Chang, Free vibrations of two rods connected by multi-spring–mass systems, Journal of Sound and Vibration, 330 (2011) 2509-2519.

DOI: 10.1016/j.jsv.2010.12.008

[4] D. Bestle, L. K. Abbas and X. Rui, Recursive eigenvalue search algorithm for transfer matrix method of flexible multibody systems: submitted to Multibody System Dynamics (2013).

DOI: 10.1007/s11044-013-9399-y

[5] L. K. Abbas, J. Li, X. Rui, Transfer matrix method for the determination of the natural vibration characteristics of realistic thrusting launch vehicle—Part I, Mathematical Problems in Engineering, volume 2013, Article ID 764673, 16 pages, (2013).

DOI: 10.1155/2013/764673