Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Stability of negative stiffness viscoelastic systems


Authors: Yun-Che Wang and Roderic Lakes
Journal: Quart. Appl. Math. 63 (2005), 34-55
MSC (2000): Primary 74B10; Secondary 74C10, 74D05
DOI: https://doi.org/10.1090/S0033-569X-04-00938-6
Published electronically: December 17, 2004
MathSciNet review: 2126568
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Abstract | References | Similar Articles | Additional Information

Abstract: We analytically investigate the stability of a discrete viscoelastic system with negative stiffness elements both in the time and frequency domains. Parametric analysis was performed by tuning both the amount of negative stiffness in a standard linear solid and driving frequency. Stability conditions were derived from the analytical solutions of the differential governing equations and the Lyapunov stability theorem. High frequency response of the system is studied. Stability of singularities in the dissipation $\tan \delta$ is discussed. It was found that stable singular $\tan \delta$ is achievable. The system with extreme high stiffness analyzed here was metastable. We established an explicit link for the divergent rates of the metastable system between the solutions of differential governing equations in the time domain and the Lyapunov theorem.


References [Enhancements On Off] (What's this?)

  • 1. R. Bulatovic, On the Lyapunov stability of linear conservative gyroscopic systems, C. R. Acad. Sci. Paris, t. 324, Serie II b, p. 679-683 (1997).
  • 2. P. Gallina, About the stability of non-conservative undamped systems, J. Sound Vibration 262 (2003), no. 4, 977–988. MR 1971816, https://doi.org/10.1016/S0022-460X(02)01428-1
  • 3. R. S. Lakes, ``Extreme damping in composite materials with a negative stiffness phase", Physical Review Letters, 86, 2897-2900 (26 March 2001).
  • 4. R. S. Lakes, ``Extreme damping in compliant composites with a negative stiffness phase" Philosophical Magazine Letters, 81, 95-100 (2001).
  • 5. R. S. Lakes, T. Lee, A. Bersie, and Y. C. Wang, ``Extreme damping in composite materials with negative stiffness inclusions", Nature, 410, 565-567 (29 March 2001).
  • 6. R. S. Lakes and W. J. Drugan, Dramatically stiffer elastic composite materials due to a negative stiffness phase?, J. Mechanics and Physics of Solids, 50, 979-1009 (2002).
  • 7. Horst Leipholz, Stabilität elastischer Systeme, Verlag G. Braun, Karlsruhe, 1980 (German). Wissenschaft and Technik. [Science and technology]. MR 602696
    Horst Leipholz, Stability of elastic systems, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics of Elastic Stability, vol. 7, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980. Translated from the German. MR 595163
  • 8. L. Meirovitch, Methods of Analytical Dynamics, McGraw-Hill, New York, 1970.
  • 9. J. W. Morris Jr. and C. R. Krenn, The internal stability of an elastic solid, Philosophical Magazine A, 80, 12, 2827-2840 (2000).
  • 10. S. H. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books, Cambridge, Massachusetts, 1994.
  • 11. Y. C. Wang and R. S. Lakes, Extreme thermal expansion, piezoelectricity, and other coupled field properties in composites with a negative stiffness phase, J. Appl. Phys. 90, 6458-6465 (2001).
  • 12. Y. C. Wang and R. S. Lakes, Extreme stiffness systems due to negative stiffness elements, American Journal of Physics, 72, 40-50 (2004a).
  • 13. Y. C. Wang and R. S. Lakes, Stable extremely high-loss discrete viscoelastic systems due to negative stiffness elements, Appl. Phys. Lett. 84, 4451-4453 (2004b).
  • 14. Y. C. Wang and R. S. Lakes, Negative Stiffness Induced Extreme Viscoelastic Mechanical Properties: Stability and Dynamics, Philos. Mag., accepted (2004c).

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Additional Information

Yun-Che Wang
Affiliation: Department of Engineering Physics, Engineering Mechanics Program, University of Wisconsin-Madison, 147 Engineering Research Building, 1500 Engineering Drive, Madison, WI 53706-1687

Roderic Lakes
Affiliation: Department of Engineering Physics, Engineering Mechanics Program, Biomedical Engineering Department; Materials Science Program and Rheology Research Center, University of Wisconsin-Madison, 147 Engineering Research Building, 1500 Engineering Drive, Madison, WI 53706-1687
Email: lakes@engr.wisc.edu

DOI: https://doi.org/10.1090/S0033-569X-04-00938-6
Received by editor(s): December 15, 2003
Published electronically: December 17, 2004
Article copyright: © Copyright 2004 Brown University


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