Abstract
Based on one-dimensional theory of elastodynamics, the vibration and wave propagation in members of a framed structure are analyzed by the recently developed method of reverberation-ray matrix. Unidirectional traveling wave solutions for axial, torsional, and two flexural waves in six modes that are reverberated in structural members through repeated reflection and multiple scattering by joints of the structure are expressed in matrix form with two sets of unknown amplitude coefficients. From joint coupling equations and from compatibility conditions of displacements in dual coordinates of each member, the two sets of unknowns are determined in terms of a reverberation-ray matrix for given source excitations at joints. Free and forced wave motion in steady-state response can all be evaluated from the matrix solutions, and transient response is determined in one more step of Fourier inverse transform. The method is particularly effective to determine the early time transient response through Neumann series expansion of the inverse transformed solution with a special numerical algorithm.
Similar content being viewed by others
References
Howard S.M., Pao Y.H.: Analysis and experiments on stress waves in planar trusses. J. Eng. Mech. 124, 884–891 (1998)
Pao Y.H., Keh D.C., Howard S.M.: Dynamic response and wave propagation in plane trusses and frames. AIAA J. 37, 594–603 (1999)
Chen J.F., Pao Y.H.: Effects of causality and joint conditions on method of reverberation-ray matrix. AIAA J. 41, 1138–1142 (2003)
Boussinesq J.: Applications des potentiels á l’étude de l’équilibre et du mouvement des solides léastiques. Gauthier-Villars, Paris (1885)
Inglis C.E.: A Mathematical Treatise on Vibration in Railway Bridges. The University Press, Cambridge (1934)
Nowacki W.: Dynamics of Elastic Systems. Wiley, New York (1963)
Pestel E.C., Leckie F.A.: Matrix Methods in Elasto Mechanics. McGraw-Hill, New York (1963)
Clough R.W., Penzien J.: Dynamics of Structures, 2nd edn. McGraw Hill, New York (1993)
Doyle J.F.: Wave Propagation in Structures, 2nd edn. Springer, New York (1997)
Pao, Y.H., Nie, G.H., Keh, D.C.: Dynamic response and wave propagation in three-dimensional framed structures (2007, submitted for publication)
Pao Y.H., Su X.Y., Tian J.Y.: Reverberation matrix method for propagation of sound in a multilayered liquid. J. Sound Vib. 230, 743–760 (2000)
Su X.Y., Tian J.Y., Pao Y.H.: Application of the reverberation-ray matrix to the propagation of elastic waves in a layered solid. Int. J. Solids Struct. 39, 5447–5463 (2002)
Chen W.Q., Wang H.M., Bao R.H.: On calculating dispersion curves of waves in functionally graded elastic plate. Compos. Struct. 81, 233–242 (2007)
Pao Y.H., Chen W.Q., Su X.Y.: The reverberation-ray matrix and transfer matrix analyses of unidirectional wave motion. Wave Motion 44, 419–438 (2007)
Weaver W. Jr, Gere J.M.: Matrix Analysis of Framed Structures, 2nd edn. Van Nostrand, New York (1980)
Doyle J.F.: Static and Dynamic Analysis of Structures. Kluwer, Dordrecht (1991)
Mindlin R.D.: An Introduction to the Mathematical Theory of Vibrations of Elastic Plates. World Scientific, Singapore (2006)
Gilbert F., Backus G.E.: Propagator matrices in elastic wave and vibration problems. Geophysics 31, 326–332 (1966)
Nagem R.J., Williams J.H.: Dynamic analysis of large space structure using transfer matrices and joint coupling matrices. Mech. Struct. Mach. 17, 349–371 (1989)
Pao Y.H., Sun G.: Dynamic bending strains in planar trusses with pinned or rigid joints. J. Eng. Mech. 129, 324–332 (2003)
Chopra A.K.: Dynamics of Structures. Prentice-Hall, New Jersey (2001)
Miao, F.X., Sun, G., Pao, Y.H.: Vibration mode analysis of frames by the method of reverberation ray matrix. J. Struct. Eng. (2008, in press)
Guo, Y.Q.: The Method of Reverberation-Ray Matrix and its Application. Doctoral Dissertation (in Chinese), Zhejiang University, Hangzhou, China (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
In honor of Professor Franz Ziegler on his 70th birthday.
Rights and permissions
About this article
Cite this article
Pao, YH., Chen, WQ. Elastodynamic theory of framed structures and reverberation-ray matrix analysis. Acta Mech 204, 61–79 (2009). https://doi.org/10.1007/s00707-008-0012-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-008-0012-z