Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

The eigensolutions of wave propagation for repetitive structures

  • Zhong, Wanxie (Research Institute of Engineering Mechanics, Dalian Univ. of Tech.) ;
  • Williams, F.W. (Division of Structural Engineering, School of Engineering, Univ. of Wales College of Cardiff)
  • Published : 1993.10.25
⟨ Previous Next ⟩

Abstract

The eigen-equation of a wave traveling over repetitive structure is derived directly form the stiffness matrix formulation, in a form which can be used for the case of the cross stiffness submatrix $K_{ab}$ being singular. The weighted adjoint symplectic orthonormality relation is proved first. Then the general method of solution is derived, which can be used either to find all the eigensolutions, or to find the main eigensolutions for large scale problems.

Keywords

References

  1. Howson, W.P., Banerjee, J.R., and Williams, F.W. (1983), "Concise equations and program for exact eigensolutions of plane frames including member shear," Adv. Eng. Software, 5, 137-141. https://doi.org/10.1016/0141-1195(83)90108-0
  2. Lin, Wen-Wei (1987), "A new method for computing the closed loop eigenvalue of a discrete-time algebraic Riccati equation," Linear Algeb. and Its Appl. 96, 157-180. https://doi.org/10.1016/0024-3795(87)90342-9
  3. Mead, D.J. (1973), "A general theory of harmonic wave propagation in linear periodic systems with multiple coupling," Journ. Sound and Vib. 27, 235-260. https://doi.org/10.1016/0022-460X(73)90064-3
  4. Mead, D.J. (1975), "Wave propagation and natural modes in periodic systems;I. mono-coupled systems. II. multi-coupled systems, with and without damping," Journ. Sound and Vib., 40, 1-18, 19-39. https://doi.org/10.1016/S0022-460X(75)80227-6
  5. Miller, D.W., Hall, S.R. and Von Flotow, A.H. (1990), "Optimal control of power flow at structural junctions," Journ. Sound and Vib., 140, 475-497. https://doi.org/10.1016/0022-460X(90)90762-O
  6. Miller, D.W., and Von Flotow, A.H. (1989), "A travelling wave approach to power flow in structural networks." Journ. Sound and Vib. 128, 145-162. https://doi.org/10.1016/0022-460X(89)90686-X
  7. Signorelli, J. and Von Flotow, A.H. (1988), "Wave propagation, power flow, and resonance in a truss beam," Journ. Sound and Vib., 126, 127-144. https://doi.org/10.1016/0022-460X(88)90403-8
  8. Van Loan, C.F. (1984), "A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix," Linear Algeb. and Appl., 61, 233-251. https://doi.org/10.1016/0024-3795(84)90034-X
  9. Von Flotow, A.H. (1986), "Disturbance propagation in structural networks," Journ. Sound and Vib. 106, 433-450. https://doi.org/10.1016/0022-460X(86)90190-2
  10. Yong, Y., Lin, Y.K. (1989), "Propagation of decaying waves in periodic and piecewise periodic structures of finite length." Journ. Sound and Vib. 129, 99-118. https://doi.org/10.1016/0022-460X(89)90538-5
  11. Yong, Y. and Lin, Y.K. (1990), "Wave propagation for truss type structural networks." Proc. AIAA/ASME/ ASCE/AHS/ASC 31st Structures, Structural Dynamics and Materials Conference, Calif., Apr. 1990, paper AlAA-90-1082-CP, pp.2026-2035.
  12. Zhong, Wanxie (1992), "The inverse iteration method for the eigenproblem of large symplectic matrices," To publish in Comput. Struct. Mech. and Appl. (in Chinese)
  13. Zhong, Wanxie and Cheng, Gengdong (1991), "On regularization of singular control and stiffness shifting method," Proc. Asia-Pacific Conference on Comput. Mechanics, 373-378. Hong-Kong, Dec. 1991.
  14. Zhong, Wanxie, Lin, Jiahao and Qiu, Chunhang (1992), "Computational structural mechanics and optimal control," Int. J. Num. Meth. in Eng., 33, 197-211. https://doi.org/10.1002/nme.1620330113
  15. Zhong, Wanxie and Williams, F.W. (1991), "On the localization of the vibration mode of a sub-structural chain-type structure," Proc. of Mechanical Engineering Institution, Part C, 205, 281-288. https://doi.org/10.1243/PIME_PROC_1991_205_120_02
  16. Zhong, Wanxie and Williams, F.W. (1992), "Wave Problems for Repetitive Structures and Symplectic Mathematics," Proc. of Mechanical Engineering Institution, part C, 206, 371-379.
  17. Zhong, Wanxie and Yang, Zaishi (1992), "Partial differential equations and Hamiltonian system." Computational Mechanics in Structural Engineering, pp.32-48, Elsevier.
  18. Zhong, Wanxie;Zhong, Xiangxiang (1990), "Computational structural mechanics optimal control and semi-analytical method for PDE," Computers and Structures, 37(6), 993-1004. https://doi.org/10.1016/0045-7949(90)90011-P
  19. Zhong, Wanxie and Zhong, Xiangxiang (1991), "On the adjoint symplectic inverse substitution method for main eigensolutions ofa large Hamiltonian matrix." Journ. of Systems Eng., 1, 41-50.
  20. Zhong, Wanxie and Zhong, Xiangxiang (1992a), "Elliptic partial differential equation and optimal control," Numerical Methods for PDE. 8, 149-169. https://doi.org/10.1002/num.1690080206
  21. Zhong, Wanxie and Zhong, Xiangxiang (1992b), "On the computation of anti-symmetric matrices." Proc. of EPMESC-4th, 2(late papers 2), pp.1309-16, Dalian, PRC, July 30-Aug. 2nd, 1992.
  22. Zhong, Wanxie and Zhong, Xiangxiang (1993), "Method of separation of variables and Hamiltonian system," Numerical Methods for PDE, 9, 63-75. https://doi.org/10.1002/num.1690090107

Cited by

  1. Research on a systematic methodology for theory of elasticity vol.24, pp.7, 2003, https://doi.org/10.1007/BF02437818
  2. Localization investigation of stationary random wave transmission along damped ordered substructural chains vol.211, pp.3, 1997, https://doi.org/10.1243/0954406971521791
  3. Vertical random vibration analysis of adjacent building induced by highway traffic load vol.8, pp.8, 2016, https://doi.org/10.1177/1687814016659181
  4. Symplectic random vibration analysis of a vehicle moving on an infinitely long periodic track vol.329, pp.21, 2010, https://doi.org/10.1016/j.jsv.2010.05.004
  5. Symplectic analysis for periodical electro-magnetic waveguides vol.267, pp.2, 2003, https://doi.org/10.1016/S0022-460X(02)01451-7
  6. Symplectic analysis for elastic wave propagation in two-dimensional cellular structures vol.26, pp.5, 2010, https://doi.org/10.1007/s10409-010-0373-0
  7. Calculation model of the vehicle CRTS II coupling system with symplectic method 2018, https://doi.org/10.1177/0954409717703470
  8. Symplectic analysis of vertical random vibration for coupled vehicle–track systems vol.317, pp.1-2, 2008, https://doi.org/10.1016/j.jsv.2008.03.004
  9. Symplectic Approach on the Wave Propagation Problem for Periodic Structures with Uncertainty vol.32, pp.3, 2019, https://doi.org/10.1007/s10338-019-00084-9
  10. Extending Zhong-Williams scheme to solve repeated-root wave modes vol.519, pp.None, 1993, https://doi.org/10.1016/j.jsv.2021.116584