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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 99

Modelling Frequency Adjustment Effects Using Shape Memory Alloy Oscillators

L.X. Wang1 and R.V.N. Melnik2

1MCI, Faculty of Science and Engineering, University of Southern Denmark, Sonderborg, Denmark
2Mathematical Modelling and Computational Sciences, Wilfrid Laurier University, Waterloo ON, Canada

Full Bibliographic Reference for this paper
L.X. Wang, R.V.N. Melnik, "Modelling Frequency Adjustment Effects Using Shape Memory Alloy Oscillators", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 99, 2006. doi:10.4203/ccp.83.99
Keywords: vibration tuning, frequency adjustment, shape memory alloys, hysteresis.

Summary
Mechanical vibrations constitute a common phenomenon in applications of many engineering structures and systems. The result of such vibrations is usually desired to be tuned according to a specific application. In many application developments, a supplementary oscillator with a suitable frequency is attached to the primary system, by which the primary vibration can be tuned. For the cases where the primary vibration frequency is uncertain, it is always beneficial to make a supplementary vibration frequency continuously variable in a rather wide range. For construction of adaptive vibration absorbers, one kind of promising candidate material is shape memory alloys (SMA), because its stiffness can be varied by changing its temperature, due to its thermo-mechanical coupling properties [1,2,3].

In the present paper, we model the performance of a shape memory alloy oscillator used as a vibration absorber, which consists of a shape memory alloy rod and an end-mass. The vibration characteristic of the oscillator is continuously adjustable by changing its temperature, due to the thermo-mechanical coupling property of the shape memory alloy rod.

In order to capture the thermo-mechanical coupling property in the material, as well as hysteresis induced by phase transformations at low temperature, the mechanical behaviour of the SMA rod is first modelled using the modified Ginzburg-Landau theory by the following partial differential equation:

    (19)

where is the displacement of the rod, is its temperature, and strain; and are all material specific constants. The above model is shown be able to capture the thermo-mechanical coupling property and hysteresis induced by phase transformations, but its numerical analysis is not trivial [4].

For the convenience of the analysis, the above model is simplified. By taking into account the nature of the absorber, it can be approximated by the following model. representing a force-deformation relation for a nonlinear spring:

    (20)

where and are physical constants of the SMA rod, and are displacements of the two ends of the SMA rod which give the measure of the rod deformation when its length is fixed. Using the above model, the governing equations of the entire system, including the primary system and the attached absorber, can be formulated as follows:
(21)
where , and are the mass, spring stiffness, and friction coefficient in the primary system, respectively; is the mass in the absorber, and is the excitation.

The performance of the SMA oscillator used as a vibration absorber is then simulated numerically using the developed model. It is demonstrated that the SMA oscillator can be adjusted to match various vibration frequencies by changing its temperature. It is shown that at high temperature, the SMA oscillator has a similar behaviour with linear vibration absorbers and the vibration of the primary system can be suppressed to a rather small value. The frequency response of the SMA oscillator also indicates a certain similarity between the (nonlinear) SMA absorber and linear ones. It is also demonstrated that at low temperature, the SMA absorber can be used as a general vibration damper, because it is able to dissipate energy due to the hysteresis. However, the frequency response of the absorber exhibits in this case a much more complicated behavior compared to its linear counterpart.

References
1
S.M.T. Hashemia, and S.E. Khademb, Modeling and analysis of the vibration behavior of a shape memory alloy beam, Inter. Jour. Mech. Sci. 48 (2006) 44-52. doi:10.1016/j.ijmecsci.2005.09.011
2
S. Saadat, J. Salichs, M. Noori, Z. Hou, H. Davoodi, I. Bar-on, Y. Suzuki, and A. Masuda, An overview of vibration and seismic applications of NiTi shape memory alloy, Smart Materials and Structures, 11 (2002) 218-229. doi:10.1088/0964-1726/11/2/305
3
K. A. Williams, G. T. C. Chiu, and R. J. Bernhard, Dynamic modelling of a shape memory alloy adaptive tuned vibration absorber. Journal of Sound and Vibration 280 (2005) 211-234. doi:10.1016/j.jsv.2003.12.040
4
F. Falk, P. Konopka, Three-dimensional Landau theory describing the martensitic phase transformation of shape memory alloys. J.Phys.:Condens.Matter 2 (1990) 61-77. doi:10.1088/0953-8984/2/1/005

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