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

- Journal title : Smart Structures and Systems
- Volume 17, Issue 5, 2016, pp.753-771
- Publisher : Techno-Press
- DOI : 10.12989/sss.2016.17.5.753

Title & Authors

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

Bae, S.H.; Cho, J.R.; Jeong, W.B.;

Bae, S.H.; Cho, J.R.; Jeong, W.B.;

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

linear complex stiffness system;free and transient responses;time domain analysis;Hilbert transform;state-space formulation;convolution integral;

Language

English

Cited by

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.

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.

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.

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.

8.

Fink, M. (1992), "Time reversal of ultrasonic fields, I. Basic principles", IEEE T. Ultrason. Ferr., 39(5), 555-566.

9.

Genta, G. and Amati, N. (2010), "Hysteretic damping in rotordynamics: An equivalent formulation", J. Sound Vib., 329(22), 4772-4784.

10.

Hajianmaleki, M. and Qatu, M.S. (2013), "Vibrations of straight and curved composite beams: A review", Comput. Struct., 100, 218-232.

11.

Inaudi, J. and Makris, N. (1996), "Time-domain analysis of linear hysteretic damping", Earthq. Eng. Struct. D., 25, 529-545.

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.

14.

Luo, H., Fang, X. and Ertas, B. (2009), "Hilbert transform and its engineering applications", AIAA J., 47(4), 923-932.

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.

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.

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.

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.

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.

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.