Acta Mechanica Solida Sinica

, Volume 25, Issue 1, pp 61–72

# Vibrations of Two Beams Elastically Coupled Together at an Arbitrary Angle

• Wen L. Li
• Murilo W. Bonilha
• Jie Xiao
Article

## Abstract

A general analytical method is developed for the vibrations of two beams coupled together at an arbitrary angle. The stiffness of a joint can take any value from zero to infinity to better model many real-world coupling conditions. Both flexural and longitudinal waves are included to account for the cross-coupling effects at the junctions. Each displacement component is here invariantly expressed, regardless of the coupling or boundary conditions, as a Fourier series supplemented by several closed-form functions to ensure the uniform convergence of the series expansions. Examples are presented to compare the current solution with finite element and experimental results.

## Key words

coupled beams beams free vibrations

## References

1. [1]
Hambric, S.A., Power flow and mechanical intensity calculations in structural finite element analysis. Journal of Vibration and Acoustics, 1990, 112: 542–549.
2. [2]
Gavric, L. and Pavic, G., A finite element method for computation of structural intensity by the normal mode approach. Journal of Sound and Vibration, 1993, 164: 29–43.
3. [3]
Szwerc, R.P., Burroughs, C.B., Hambric, S.A. and McDevitt, T.E., Power flow in coupled bending and longitudinal waves in beams. Journal of the Acoustical Society of America, 2000, 107: 3186–3195.
4. [4]
Mace, B.R. and Shorter, P., Energy flow models from finite element analysis. Journal of Sound and Vibration, 2000, 233: 369–389.
5. [5]
Rabbiolo, G., Bernhard, R.J. and Milner, F.A., Definition of a high-frequency threshold for plates and acoustical spaces. Journal of Sound and Vibration, 2004, 277: 647–667.
6. [6]
Bercin, A.N. and Langley, R.S., Application of the dynamic stiffness technique to the inplane vibrations of plate structures. Computers and Structures, 1996, 59: 869–875.
7. [7]
Langley, R.S., Analysis of power flow in beams and frameworks using the direct-dynamic stiffness method. Journal of Sound and Vibration, 1990, 136: 439–452.
8. [8]
Park, D.H., Hong, S.Y., Kil, F.G. and Jeon, J.J., Power flow models and analysis of in-plane waves in finite coupled think plates. Journal of Sound and Vibration, 2001, 244: 651–668.
9. [9]
Doyle, J.F., Wave Propagation in Structures. Springer, Berlin, 1989.
10. [10]
Ahmida, K.M. and Arruda, J.R.F., Spectral element-based prediction of active power flow in Timoshenko beams. International Journal of Solids and Structures, 2001, 38: 1669–1679.
11. [11]
Igawa, H., Komatru, K., Yamaguchi, I. and Kasai, T., Wave propagation analysis of frame structures using the spectral element method. Journal of Sound and Vibration, 2004, 277: 1071–1081.
12. [12]
Keane, A.J. and Price, W.G., A note on the power flowing between two conservatively coupled multi-modal sub-system. Journal of Sound and Vibration, 1991, 144: 185–196.
13. [13]
Keane, A.J., Energy flows between arbitrary configurations of conservatively coupled multi-modal elastic subsystems. Proceedings of the Royal Society of London A, 1992, 436: 537–568.
14. [14]
Beshara, M. and Keane, A.J., Vibrational power flows in beam networks with compliant and dissipative joints. Journal of Sound and Vibration, 1997, 203: 321–339.
15. [15]
Shankar, K. and Keane, A.J., Power flow predictions in a structure of rigidly joined beams using receptance theory. Journal of Sound and Vibration, 1995, 180: 867–890.
16. [16]
Farag, N.H. and Pan, J., Dynamic response and power flow in three-dimensional coupled beam structures. I. Analytical modeling. Journal of the Acoustical Society of America, 1997, 102: 315–325.
17. [17]
Keane, A.J., Energy flows between arbitrary configurations of conservatively coupled multi-modal elastic subsystems. Proceedings of the Royal Society of London A, 1992, 436: 537–568.
18. [18]
Li, W.L., Bonilha, M.W. and Xiao, J., Vibrations and power flows in a coupled beam system. Journal of Vibration and Acoustics, 2007, 129: 617–622.
19. [19]
Li, W.L., Free vibration of beams with general boundary conditions. Journal of Sound and Vibration, 2000, 237: 709–725.
20. [20]
Li, W.L., Comparison of Fourier sine and cosine series expansions for beams with arbitrary boundary conditions. Journal of Sound and Vibration, 2002, 255: 185–194.
21. [21]
ANSYS Release 6.1. ANSYS Structural Analysis Guide. ANSYS Inc, 2002.Google Scholar