Causal Hysteretic Element
Publication: Journal of Engineering Mechanics
Volume 123, Issue 11
Abstract
In this paper the “causal hysteretic element” is constructed and analyzed. The dynamic stiffness of the proposed hysteretic model has the same imaginary part as the “ideal” hysteretic damper, but has the appropriate real part that makes the model causal. The proposed model is constructed by requiring that the real and imaginary parts of its transfer functions satisfy the Kramers-Kroning relations. This condition ensures that the corresponding time-response functions of the proposed model are zero at negative times. The causal hysteretic element is physically realizable at finite frequencies, but is not definable at ω= 0. The behavior of the proposed model is analyzed both in frequency and time domain. It is shown that the causal hysteretic element is the limiting case of a linear viscoelastic model with nearly frequency-independent dissipation. Finally, the response of a mass supported by the causal hysteretic element is discussed.
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References
1.
Abramowitz, M., and Stegun, I. A. (1970). Handbook of mathematical functions. Dover Publications, Inc., New York, N.Y.
2.
Biot, M. A. (1958). “Linear thermodynamics and the mechanics of solids.”Proc. 3rd U.S. Natl. Congr. Applied Mech., ASME, New York, N.Y., 1–18.
3.
Bird, B., Armstrong, R., and Hassager, O. (1987). Dynamic of polymeric liquids. John Wiley & Sons, Inc., New York, N.Y.
4.
Bishop, R. E. D.(1955). “The treatment of damping forces in vibration theory.”J. Royal Aeronautical Soc., 59(539), 738–742.
5.
Bishop, R. E. D., and Price, W. G. (1986). “A note on hysteretic damping of transient motions.”Random vibration-status and recent developments, I. Elishakoff and R. H. Lyon, eds., Elsevier, Amsterdam, The Netherlands, 39–45.
6.
Booij, H. C., and Thoone, P. J. M.(1982). “Generalization of Kramers-Kronig transforms and some approximations of relations between viscoelastic quantities.”Rheological Acta, 21, 15–24.
7.
Caughey, T. K. (1962). “Vibration of dynamic systems with linear hysteretic damping (linear theory).”Proc. 4th U.S. Natl. Congr. Appl. Mech., ASME, New York, N.Y., 87–97.
8.
Chopra, A. K. (1995). Dynamics of structures: theory and applications to earthquake engineering, Prentice-Hall, Inc., Englewood Cliffs, N.J.
9.
Clough, R. W., and Penzien, J. (1993). Dynamics of structures, 2nd Ed., McGraw-Hill, Inc., New York, N.Y.
10.
Crandall, S. H. (1963). “Dynamic response of systems with structural damping.”Air, space and instruments, Draper anniversary volume, S. Lees, ed., McGraw-Hill, Inc., New York, N.Y., 183–193.
11.
Crandall, S. H.(1970). “The role of damping in vibration theory.”J. Sound and Vibration, 11(1), 3–18.
12.
Crandall, S. H.(1991). “The hysteretic damping model in vibration theory.”Proc. Instn. Mech. Engrs., London, U.K., 205, 23–28.
13.
Dirac, P. A. M. (1958). The principles of quantum mechanics. Oxford University Press, Oxford, U.K.
14.
Inaudi, J. A., and Kelly, J. M.(1995). “Linear hysteretic damping and the Hilbert transform.”J. Engrg. Mech., ASCE, 121(5), 626–632.
15.
Inaudi, J. A., Zambrano, A., and Kelly, J. M. (1993). “On the analysis of linear structures with viscoelastic dampers.”Rep. No. UCB/EERC/93-09, Earthquake Engrg. Res. Ctr., Richmond, Calif.
16.
Lighthill, M. J. (1989). An introduction to Fourier analysis and generalized functions, Cambridge University Press, Cambridge, U.K.
17.
Makris, N. (1994a). “Constitutive models with complex parameters and the imaginary counterpart of records.”Proc. 1st World Conf. Struct. Control, International Association for Structural Control, Los Angeles, Calif., 1 WP3-63–WP3-72.
18.
Makris, N.(1994b). “The imaginary counterpart of recorded motions.”Earthquake Engrg. and Struct. Dynamics, 23, 265–273.
19.
Makris, N.(1997). “Dynamic stiffness, flexibility, impedance, mo- bility and the hidden delta function.”J. Engrg. Mech., ASCE, 123(11), 1202–1208.
20.
Makris, N., Inaudi, J. A., and Kelly, J. M.(1996). “Macroscopic models with complex coefficients and causality.”J. Engrg. Mech., ASCE, 122(6), 566–573.
21.
Myklestad, N. O.(1952). “The concept of complex damping.”J. Appl. Mech., ASME, 19(3), 284–286.
22.
Neumark, S. (1957). “Concept of complex stiffness applied to problems of oscillations with viscous and hysteretic damping.”Aero. Res. Council Rep. No. 3269, 1–34.
23.
Papoulis, A. (1987). The Fourier integral and its applications, McGraw-Hill, Inc., New York, N.Y.
24.
Reid, T. J.(1956). “Free vibration and hysteretic damping.”J. Royal Aeronautical Soc., 60, 283.
25.
Theodorsen, T., and Garrick, I. E. (1940). “Mechanism of flutter, a theoretical and experimental investigation of the flutter problem.”NACA TR Rep. No. 685, Natl. Advisory Com. for Aeronautics, Washington, D.C.
Information & Authors
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Published In
Journal of Engineering Mechanics
Volume 123 • Issue 11 • November 1997
Pages: 1209 - 1214
Copyright
Copyright © 1997 American Society of Civil Engineers.
History
Published online: Nov 1, 1997
Published in print: Nov 1997
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