Chaos in Autoparametric Three Degree of Freedom System with SMA Spring
Wybrane pełne teksty z tego czasopisma
French-Polish Seminar of Mechanics (18 ; 31.05.2010 ; Warsaw, Poland)
In this paper is studied a three degree of freedom autoparametric system with two pendulums connected by shape memory alloys (SMA) spring in the neighborhood internal and external resonance. The system consists of the body of mass mi which is hung on a spring and a damper, and two connected by SMA spring pendulums of the length l₁ and l₂ and masses m₂ and m₃ mounted to the body of mass m₁. It is assumed, that the motion of the pendulums are damped by resistive forces. Shape memory alloys have ability to change their material properties. A polynomial constitutive model is assumed to describe the behavior of the SMA spring (it was assumed that the uniaxial stress σ is a fifth-degree polynomial of the strain). The equations of motion have been solved numerically and there were studied pseudoelastic effects associated with martensitic phase transformations. It was assumed that SMA presents two stable phases: austenite and martensite. Solutions for the system response are presented for specific values of the parameters of system. It was shown that in this type system one mode of vibrations may excite or damp another mode, and that except different kinds of periodic vibrations there may also appear chaotic vibrations. For the identification of the responses of the system various techniques, including chaos techniques such as bifurcation diagrams and time histories, power spectral densities (FFT), Poincare maps and exponents of Lyapunov maybe use.
Bibliogr. 26 poz., wykr.
- Baker, G. L., Gollub, J. P., 1998, Chaotic dynamics: an introduction, PWN, Warsaw (in polish).
- Cross, W. B., Kariotis, A. H, Stimler, F. J., 1970, Nitinol Characterization Study, Goodyear Aerospace Corporation Report, No Ger 14188 (NASA CR-1433), Akron, Ohio.
- Den Hartog, J. P., 1956, Mechanical Vibrations, McGraw-Hill Book Company, Inc.
- Drazin P. G., 1992, Nonlinear systems, Cambridge University Press.
- Falk, F., 1980, Model free energy, mechanics and thermodynamics of shape memory alloys, Acta metal. 28, 1773-1780.
- Jackson, C. M., Wagner, H. J., Wasilewski, R. J., 1972, 55-Nitinol-The Alloy with a Memory: Its Phisical Matallurgy, Properties and Applications, NASA-SP-5110, Washington, D.C.
- Liang, C., Rogers, C. A., 1990, One - Dimensional Thermodynamical Constitutive Relations for Shape Memory Materials, J. of Intell. Mater. Syts. And Struct., 1, 207-234.
- Liang, C., Rogers, C. A., 1991, One - Design of Shape Memory Alloys Coils and their Aplications in Vibrations Control, Recent Avdvances in Active Control of Sound and Vibrations, Rogers C. A. ed., Technomic Publishing, Lancaster Basel, 177-198.
- Muller, I., Xu, H., 1991, On the pseudo-elastic hysteresis, Acta Metall. Mater., 39 (3), 263-271.
- Moon, F. C., 1987, Chaotic Vibrations, John Wiley & Sons, Inc.
- Piccirillo, V., Balthazar, J. M., Pontes Jr., B. R., Feliks, J. L. P., 2007, On nonlinear response of a non-ideal vibrating system with shape memory alloy (SMA), 9th Conference on Dynamical Systems - Theory and Applications, Proceedings, Edit. J. Awrejcewicz, P. Olejnik, J. Mrozowski, Lodz, Dec. 17-20, Poland, 783-790.
- Piccirillo, V., Balthazar, J. M., Pontes Jr., B. R., Feliks, J. L. P., 2008, On Sommerfeld Effect in a Non-Ideal Vibrating System with SMA Using an Averaging Method, Euromech 498 Colloquium, Book of Abstracts, Lublin University of Technology, Poland, 267-271.
- Pietrzakowski, M., 1998, Dynamics of thermally activated shape memory alloy reinforced laminated beams, Journal of Tcheoretical and Applied Mechanics, 4, 36, 879-893.
- Pietrzakowski, M., 2000, Natural frequency modification of thermally activated composite plates, Meccanica, 1, 313-320.
- Rogers, C. A., Liang, C., Jia, J., 1989, Behavior of Shape Memory Alloy Reinforced Composite Plates - Parts I and II, Proceedings of the 30th Structures, Structural Dynamics and Material Conference, Mobile, AL, 2011-2017
- Sado, D., 1997, The energy transfer in nonlinearly coupled two-degree-of-freedom systems, Publishing House of the Warsaw University of Technology, Mechanika, 166, (in polish).
- Sado, D., 2004, The Dynamics of a Coupled Three Degree of Freedom Mechanical System, Mechanics and Mechanical Engineering, 7, 1, 29-39.
- Sado, D., 2006, The periodic and chaotic vibration of dynamical system with elastic pendulum, Proceedings of ESDA 2006, 8th Biennial ASME Conference on Engineering Systems Design and Analysis, Torino, Italy, ESDA2006-95636, 1-8.
- Sado, D., 2007, Nonlinear oscillations of an autoparametrical system with two coupled pendulums, Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information I Engineering Conference, IDETC/CIE 2007, Las Vegas, Nevada, USA, DETC2007-34699, 1-6.
- Sado, D., Gajos, K., 2006, Effect of damping on the periodic and chaotic vibration of system with double pendulum, Proceedings of ESDA2006, 8th Biennial ASME Conference on Engineering Systems Design and Analysis, Torino, Italy, ESDA2006-95637, 1-7.
- Sado, D., Pietrzakowski, M., 2010, Dynamics of thermally activated shape memory alloy in autoparametric systems with two pendulums, International Journal of Non-Linear Mechanics, 45, 859-865.
- Sado, D., Gajos, K., 2008, The dynamics of an autoparametrical system with two coupled on Engineering Systems Design and Analysis, ESDA 08, Haifa, Israel, ESDA 2008-59202, 1-6.
- Savi, M. A., Pacheco, P. M. C. L., Braga, A. M. B., 2002, Chaos in shape memory two-bar truss, Int. Journal of Non-Linear Mechanics, 37, 1387-1395.
- Szemplińska-Stupnicka, W., 2002, Chaos bifurkacje ifraktale wokół nas, Publishing House of the Warsaw University of Technology, Warsaw (in polish).
- Vyas, A., Bajaj, A. K., 2001, Dynamics of autoparametric vibration absorber using multiple pendulums, Journal of Sound and Vibration, 246 (1), 115-135.
- Vyas, A., Bajaj, A. K., 2006, Global dynamics of an autoparametric system with multiple pendulums, Journal of Computational and Nonlinear Dynamics, 1, 35-46.