Research Papers

Nonlinear Analysis of Shape Memory Devices With Duffing and Quadratic Oscillators

[+] Author and Article Information
Shantanu Rajendra Gaikwad

Mechanical and Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Sangareddy 502285, Telangana, India

Ashok Kumar Pandey

Mechanical and Aerospace Engineering,
Indian Institute of Technology Hyderabad,
Sangareddy 502285, Telangana, India
e-mail: ashok@iith.ac.in

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 15, 2016; final manuscript received September 6, 2017; published online October 9, 2017. Assoc. Editor: Mohammad Younis.

J. Comput. Nonlinear Dynam 13(1), 011003 (Oct 09, 2017) (8 pages) Paper No: CND-16-1562; doi: 10.1115/1.4037923 History: Received November 15, 2016; Revised September 06, 2017

In this paper, we investigate the linear and nonlinear response of shape memory alloy (SMA)-based Duffing and quadratic oscillator under large deflection conditions. In this study, we first present thermomechanical constitutive modeling of SMA with a single degree-of-freedom system. Subsequently, we solve equation to obtain linear frequency and nonlinear frequency response using the method of harmonic balance and validate it with numerical solution as well as averaging method under the isothermal condition. However, for nonisothermal condition, we analyze the influence of cubic and quadratic nonlinearity on nonlinear response based on method of harmonic balance. Analysis of results leads to various ways of controlling the nature and extent of nonlinear response of SMA-based oscillators. Such findings can be effectively used to control external vibration of different systems.

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Ge, G. , 2014, “ Response of a Shape Memory Alloy Beam Model Under Narrow Band Noise Excitation,” Math. Probl. Eng., 2014, p. 985467.
Otsuka, K. , and Wayman, C. M. , 1999, Shape Memory Materials, Cambridge University Press, Cambridge, UK. [PubMed] [PubMed]
Cveticanin, L. , 2004, “ Vibrations of the Nonlinear Oscillator With Quadratic Nonlinearity,” Physica A, 341, pp. 123–135. [CrossRef]
Vestroni, F. , and Capecchi, D. , 1999, “ Coupling and Resonance Phenomena in Dynamic Systems With Hysteresis,” IUTAM Symposium on New Applications of Nonlinear and Chaotic Dynamics in Mechanics, Ithaca, NY, July 27–Aug. 1, pp. 203–212.
Saadat, S. , Salichs, J. , Noori, M. , Hou, Z. , Davoodi, H. , Bar-On, I. , Suzuki, Y. , and Masuda, A. , 2002, “ An Overview of Vibration and Seismic Applications of NiTi Shape Memory Alloy,” Smart Mater. Struct., 11(2), p. 218. [CrossRef]
Dimitris, C. L. , 2008, Shape Memory Alloys: Modeling and Engineering Applications, Springer-Verlag, New York. [PubMed] [PubMed]
Bernardini, D. , and Vestroni, F. , 2003, “ Non-Isothermal Oscillations of Pseudoelastic Devices,” Int. J. Non-Linear Mech., 38(9), pp. 1297–1313. [CrossRef]
Lacarbonara, W. , Bernardini, D. , and Vestroni, F. , 2004, “ Nonlinear Thermomechanical Oscillations of Shape-Memory Devices,” Int. J. Solids Struct., 41(5), pp. 1209–1234. [CrossRef]
Bernardini, D. , and Rega, G. , 2010, “ The Influence of Model Parameters and of the Thermomechanical Coupling on the Behavior of Shape Memory Devices,” Int. J. Non-Linear Mech., 45(10), pp. 933–946. [CrossRef]
Bernardini, D. , 2001, “ On the Macroscopic Free Energy Functions for Shape Memory Alloys,” J. Mech. Phys. Solids, 49(4), pp. 813–837. [CrossRef]
Ivshin, Y. , and Pence, T. J. , 1994, “ A Thermomechanical Model for a One Variant Shape Memory Material,” J. Intell. Mater. Syst. Struct., 5(4), pp. 455–473. [CrossRef]
Moussa, M. O. , Moumni, Z. , Doar, O. , Touz, C. , and Zaki, W. , 2012, “ Non-Linear Dynamic Thermomechanical Behaviour of Shape Memory Alloys,” J. Intell. Mater. Syst. Struct., 23(14), pp. 1593–1611. [CrossRef]
Salichs, J. , Hou, Z. , and Noori, M. , 2001, “ Vibration Suppression of Structures Using Passive Shape Memory Alloy Energy Dissipation Devices,” J. Intell. Mater. Syst. Struct., 12(10), pp. 671–680. [CrossRef]
Thomson, P. , Balas, G. J. , and Leo, P. H. , 1995, “ The Use of Shape Memory Alloys for Passive Structural Damping,” Smart Mater. Struct., 4(1), p. 36. [CrossRef]
Clark, P. W. , Aiken, I . D. , Kelly, J. M. , Higashino, M. , and Krumme, R. , 1995, “ Experimental and Analytical Studies of Shape-Memory Alloy Dampers for Structural Control,” Smart Structures and Materials '95, San Diego, CA, Feb. 26–Mar. 3, Paper No. 2445.
Urabe, M. , 1965, “ Galerkin's Procedure for Nonlinear Periodic Systems,” Arch. Ration. Mech. Anal., 20(2), pp. 120–152. [CrossRef]
Nayfeh, A. H. , and Balachandran, B. , 1995, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods (Wiley Series in Nonlinear Sciences), Wiley, New York. [CrossRef]
Oberaigner, E. R. , Tanaka, K. , and Fischer, F. D. , 1996, “ Investigation of the Damping Behavior of a Vibrating Shape Memory Alloy Rod Using a Micromechanical Model,” Smart Mater. Struct., 5(4), p. 456. [CrossRef]
Raniecki, B. , Lexcellent, C. , and Tanaka, K. , 1992, “ Thermodynamic Models of Pseudoelastic Behaviour of Shape Memory Alloys,” Arch. Mech., 44(3), pp. 261–284.
Kovacic, I. , and Brennan, M. J. , 2011, The Duffing Equation: Nonlinear Oscillators and Their Behaviour, Wiley, Chichester, UK. [CrossRef]
Nayfeh, A. H. , and Sanchez, N. E. , 1989, “ Bifurcations in a Forced Softening Duffing Oscillator,” Int. J. Non-Linear Mech., 24(6), pp. 483–493. [CrossRef]
Hu, H. , 2006, “ Solution of a Quadratic Nonlinear Oscillator by the Method of Harmonic Balance,” J. Sound Vib., 293(1), pp. 462–468. [CrossRef]
Krylov, N. , and Bogoliubov, N. , 1949, Introduction to Non-Linear Mechanics, Princeton University Press, London.
Stoer, J. , and Bulirsch, R. , 2013, Introduction to Numerical Analysis, Vol. 12, Springer Science & Business Media, New York.


Grahic Jump Location
Fig. 1

(a) Phase transformation of a SMA under loading and temperature effect, (b) stress–strain–temperature hysteresis loop of a SMA, and (c) hysteresis loop at different loading rates

Grahic Jump Location
Fig. 2

(a) A lumped model of SMA-based oscillator including cubic and quadratic nonlinear stiffness and (b) a schematic representation of SMA with internal temperature and surrounding temperature

Grahic Jump Location
Fig. 3

(a) Frequency response of SMA under isothermal condition with γ=0.2 and β0=0.0 showing the bifurcation points A at Ω=0.87 and B at Ω=0.84. (b) Variation of nondimensional pseudoelastic force f=(x−sgn(x)λξ) of SMA versus displacement, (c) time history of displacement, (d) FFT of displacement signals, (e) phase diagram, and (f) Poincare map of signals under isothermal condition for points A and B.

Grahic Jump Location
Fig. 4

(a) Validation of solutions based on harmonic balance method and method of averaging under isothermal case with the numerical solution when β0=0. Here, OA, AB, BC, CD, DE, and EF indicate different portions of response curve; AB is unstable portion and rest of the portions are stable. (b) Comparison of the frequency response curves of SMA-based cubic oscillator obtained from harmonic balance method and method of averaging under the isothermal condition when (γ=0.2) and β0 varies from −0.1 to 1.1. (c) Unstable (asterisk) and stable portions (circle) are shown for frequency response curves at different values of β0.

Grahic Jump Location
Fig. 5

(a) Validation of solutions based on harmonic balance method with the numerical solution under nonisothermal condition when β0=0. (b) Comparison of displacement-based frequency response curves of Duffing oscillator with and without SMA at different values of nonlinear constant β0. Variation of (c) displacement and (d) temperature-based frequency response curves for different values of β0.

Grahic Jump Location
Fig. 6

(a) Displacement frequency response curves and (b) temperature frequency response curves of cubic and quadratic oscillator with SMA




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