Smart Structures and Systems

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Free and transient responses of linear complex stiffness system by Hilbert transform and convolution integral

  • Bae, S.H. (School of Mechanical Engineering, Pusan National University) ;
  • Cho, J.R. (Department of Naval Architecture and Ocean Engineering, Hongik University) ;
  • Jeong, W.B. (School of Mechanical Engineering, Pusan National University)
  • Received : 2015.04.14
  • Accepted : 2016.02.14
  • Published : 2016.05.25
⟨ Previous Next ⟩

Abstract

This paper addresses the free and transient responses of a SDOF linear complex stiffness system by making use of the Hilbert transform and the convolution integral. Because the second-order differential equation of motion having the complex stiffness give rise to the conjugate complex eigen values, its time-domain analysis using the standard time integration scheme suffers from the numerical instability and divergence. In order to overcome this problem, the transient response of the linear complex stiffness system is obtained by the convolution integral of a green function which corresponds to the unit-impulse free vibration response of the complex system. The damped free vibration of the complex system is theoretically derived by making use of the state-space formulation and the Hilbert transform. The convolution integral is implemented by piecewise-linearly interpolating the external force and by superimposing the transient responses of discretized piecewise impulse forces. The numerical experiments are carried out to verify the proposed time-domain analysis method, and the correlation between the real and imaginary parts in the free and transient responses is also investigated.

Keywords

References

  1. Bae, S.H., Cho, J.R. and Jeong, W.B. (2014a), "A discrete convolutional Hilbert transform with the consistent imaginary initial conditions for the time-domain analysis of five-layered viscoelastic sandwich beam", Comput. Method. Appl. M., 268, 245-263. https://doi.org/10.1016/j.cma.2013.09.010
  2. Bae, S.H., Cho, J.R. and Jeong, W.B. (2014b), "Time-duration extended Hilbert transform superposition for the reliable impact response analysis of five-layered damped sandwich beams", Finite Elem. Anal. Des., 90, 41-49. https://doi.org/10.1016/j.finel.2014.06.007
  3. Bae, S.H., Jeong, W.B. and Cho, J.R. (2014c), "Transient response of complex stiffness system using a green function from the Hilbert transform and the steady space technic", Proc. Inter.noise, 1-10.
  4. Chen, L.Y., Chen, J.T., Chen, C.H. and Hong, H.K. (1994), "Free vibrations of a SDOF system with hysteretic damping", Mech. Res. Commun., 21, 599-604. https://doi.org/10.1016/0093-6413(94)90023-X
  5. Chen, J.T. and You, D.W. (1997), "Hysteretic damping revisited", Adv. Eng. Softw., 28, 165-171. https://doi.org/10.1016/S0965-9978(96)00052-X
  6. Cho, J.R., Lee, H.W., Jeong, W.B., Jeong, K.M. and Kim, K.W. (2013), "Numerical estimation of rolling resistance and temperature distribution of 3-D periodic patterned tire", Int. J. Solids Struct., 50, 86-96. https://doi.org/10.1016/j.ijsolstr.2012.09.004
  7. Crandall, S.H. (1995), "A new hysteretic damping model?", Mech. Res. Commun., 22, 201-202. https://doi.org/10.1016/S0093-6413(99)80001-9
  8. Fink, M. (1992), "Time reversal of ultrasonic fields, I. Basic principles", IEEE T. Ultrason. Ferr., 39(5), 555-566. https://doi.org/10.1109/58.156174
  9. Genta, G. and Amati, N. (2010), "Hysteretic damping in rotordynamics: An equivalent formulation", J. Sound Vib., 329(22), 4772-4784. https://doi.org/10.1016/j.jsv.2010.04.036
  10. Hajianmaleki, M. and Qatu, M.S. (2013), "Vibrations of straight and curved composite beams: A review", Comput. Struct., 100, 218-232. https://doi.org/10.1016/j.compstruct.2013.01.001
  11. Inaudi, J. and Makris, N. (1996), "Time-domain analysis of linear hysteretic damping", Earthq. Eng. Struct. D., 25, 529-545. https://doi.org/10.1002/(SICI)1096-9845(199606)25:6<529::AID-EQE549>3.0.CO;2-P
  12. Johansson, M. (1999), The Hilbert Transform, Master Thesis, Vaxjo University.
  13. Li, Z., Qiao, G., Sun, Z., Zhao, H. and Guo, R. (2012), "Short baseline positioning with an improved time reversal technique in a multi-path channel", J. Marine Sci. Appl., 11(2), 251-257. https://doi.org/10.1007/s11804-012-1130-5
  14. Luo, H., Fang, X. and Ertas, B. (2009), "Hilbert transform and its engineering applications", AIAA J., 47(4), 923-932. https://doi.org/10.2514/1.37649
  15. Mead, D.J. and Markus, S. (1969), "The forced vibrations of a three-layer, damped sandwich beam with arbitrary boundary conditions", J. Sound Vib., 10(2), 163-175. https://doi.org/10.1016/0022-460X(69)90193-X
  16. Meirovitch, M. (1986), Elements of Vibration Analysis, McGraw-Hill.
  17. Mohammadi, F. and Sedaghati, R. (2012), "Linear and nonlinear vibration analysis of sandwich cylindrical shell with constrained viscoelastic core layer", Int. J. Mech. Sci., 54(1), 156-171. https://doi.org/10.1016/j.ijmecsci.2011.10.006
  18. Nguyen, H., Andersen, T. and Pedersen, G.F. (2005), "The potential use of time reversal techniques in multiple element antenna systems", IEEE Commun. Lett., 9(1), 40-42. https://doi.org/10.1109/LCOMM.2005.01011
  19. Rao, S.S. (1995), Mechanical Vibrations, 3rd eds, Singapore.
  20. Padois, T., Prax, C., Valeau, V. and Marx, D. (2012), "Experimental localization of an acoustic sound source in a wind-tunnel flow by using a numerical time-reversal technique", Acoust. Soc. Am., 132(4), 2397. https://doi.org/10.1121/1.4747015
  21. Sainsbury, M.G. and Masti, R.S. (2007), "Vibration damping of cylindrical shells using strain-energy-based distribution of an add-on viscoelastic treatment", Finite Elem. Anal. Des., 43, 175-192. https://doi.org/10.1016/j.finel.2006.09.003
  22. Salehi, M., Bakhtiari-Nejad, F. and Besharati, A. (2008), "Time-domain analysis of sandwich shells with passive constrained viscoelastic layers", Scientia Iranica, 15(5), 637-43.
  23. Wang, Z.C., Geng, D., Ren, W.X., Chen, G.D. and Zhang, G.F. (2015), "Damage detection of nonlinear structures with analytical mode decomposition and Hilbert transform", Smart Struct. Syst., 15(1), 1-13. https://doi.org/10.12989/sss.2015.15.1.001
  24. Won, S.G., Bae, S.H., Cho, J.R., Bae, S.R. and Jeong, W.B. (2013), "Three-layered damped beam element for forced vibration analysis of symmetric sandwich structures with a viscoelastic core", Finite Elem. Anal. Des., 68, 39-51. https://doi.org/10.1016/j.finel.2013.01.004
  25. Zhu, H., Hu, Y. and Pi, Y. (2014), "Transverse hysteretic damping characteristics of a serpentine belt: Modeling and experimental investigation", J. Sound Vib., 333(25), 7019-7035. https://doi.org/10.1016/j.jsv.2014.06.020

Cited by

  1. Transient response analysis of tapered FRP poles with flexible joints by an efficient one-dimensional FE model vol.59, pp.2, 2016, https://doi.org/10.12989/sem.2016.59.2.243
  2. Transient response of vibration systems with viscous-hysteretic mixed damping using Hilbert transform and effective eigenvalues vol.20, pp.3, 2016, https://doi.org/10.12989/sss.2017.20.3.263
  3. Development and Supervision of Robo-Advisors under Digital Financial Inclusion in Complex Systems vol.2021, pp.None, 2016, https://doi.org/10.1155/2021/6666089
  4. Using a Tuned-Inerto-Viscous-Hysteretic-Damper (TIVhD) for vibration suppression in multi-storey building structures vol.708, pp.1, 2016, https://doi.org/10.1088/1755-1315/708/1/012012