Skip to main content
SpringerLink
Log in
SpringerLink
Log in

The SD Oscillator

  • Chapter
  • First Online:
A Smooth and Discontinuous Oscillator

Part of the book series: Springer Tracts in Mechanical Engineering ((STME))

  • 718 Accesses

Abstract

This chapter introduces briefly the smooth and discontinuous (SD) oscillator which is a simple mechanical model or a geometrical oscillator with both smooth and discontinuous dynamics depending on the value of a geometrical parameter . Like the traditional harmonic oscillator , this model is also a simple mass-spring system comprising a lumped mass linked by a pair of linear springs pinged to its rigid supports vibrating along the perpendicular bisector of the supports. The overview of the system with the definitions and the fundamental properties without detailed explanation, which will be expanded different topics discussed in the following chapters from Chaps. 3 to 6.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    This chapter involves a common work with Professors Marian Wiercigroch, Ekaterina Pavlovskaia, Celso Grebogi, J. Michael T. Thompson, Center for Applied Dynamics Research, Aberdeen University, King’s College, Aberdeen AB24 3UE, Scotland, UK.

References

  1. Woo, K. C., Pavlovskaia, E., Wiercigroch, M., & Rodger, A. A. (2003). Modelling of ground moling dynamics by an impact oscillator with a frictional slider. Meccanica, 38(1), 85–97.

    Article  MATH  Google Scholar 

  2. Karpenko, E. V., Pavlovskaia, E., & Wiercigroch, M. (2004). Nonlinear dynamic interactions of a jeffcott rotor with a preloaded snubber ring. Journal of Sound and Vibration, 276(1–2), 361–379.

    Google Scholar 

  3. Banerjee, S., & Chakrabarty, K. (1998). Nonlinear modeling and bifurcations in the boost converter. IEEE Transactions on Power Electronics, 13(2), 252–260.

    Article  Google Scholar 

  4. Banerjee, S., Ott, E., Yorke, J. A., & Yuan, G. H. (1997). Anomalous bifurcations in dc-dc converters: boderline collisions in piecewise smooth maps. In 28th annual IEEE power electronics specialists’ conference, PESC ’97 record, (Vol. 2), 22–27 Jun 1997, St. Louis, MO, pp. 1337–1344. IEEE.

    Google Scholar 

  5. Slotine, J. J., & Sastry, S. S. (1983). Tracking control of non-linear systems using sliding surfaces with application to robot manipulators. International Journal of Control, 38(2), 465–492.

    Article  MathSciNet  MATH  Google Scholar 

  6. Richard, P. Y., Cormerais, H., & Buisson, J. (2006). A generic design methodology for sliding mode control of switched systems. Nonlinear Analysis, Theory, Methods and Applications, 65(9), 1751–1772.

    Article  MathSciNet  MATH  Google Scholar 

  7. Luo, A. C. J., & Burkhardt, H. (1995). Intensity-based cooperative bidirectional stereo matching with simultaneous detection of discontinuities and occlusions. International Journal of Computer Vision, 15(3), 171–188.

    Article  Google Scholar 

  8. Nishita, T., Sederberg, T. W., & Kakimoto, M. (1990). Ray tracing trimmed rational surface patches. Computer Graphics, 24(4), 337–345.

    Article  Google Scholar 

  9. Kribs-Zaleta, C. M. (2004). To switch or taper off: The dynamics of saturation. Mathematical Biosciences, 192(2), 137–152.

    Article  MathSciNet  MATH  Google Scholar 

  10. Drulhe, S., Ferrari-Trecate, G., De Jong, H., & Viari, A. (2006). Reconstruction of switching thresholds in piecewise-affine models of genetic regulatory networks. In J. P. Hespanha & A. Tiwari (Eds.), Hybrid systems: computation and control (Vol. 3927, pp. 184–199)., Lecture notes in computer science Heidelberg: Springer.

    Chapter  Google Scholar 

  11. Filippov, A. F. (1988). Differential equations with discontinuous right-hand sides: Control systems (Vol. 18)., Mathematics and its applications The Netherlands: Springer. Originally published in Russian.

    Book  MATH  Google Scholar 

  12. Feigin, M. I. (1966). Resonance behaviour of a dynamical system with collisions. Journal of Applied Mathematics and Mechanics, 30(5), 1118–1123.

    Article  Google Scholar 

  13. Kunze, M. (2000). Non-smooth dynamical systems. Berlin: Springer.

    Book  MATH  Google Scholar 

  14. Peterka, F. (1981). Introduction to vibration of mechanical systems with internal impacts. Prague: Academia.

    Google Scholar 

  15. Shaw, S. W., & Holmes, P. J. (1983). A periodically forced piecewise linear oscillator. Journal of Sound and Vibration, 90(1), 129–155.

    Article  MathSciNet  MATH  Google Scholar 

  16. Nordmark, A. B. (1991). Non-periodic motion caused by grazing incidence in an impact oscillator. Journal of Sound and Vibration, 145(2), 279–297.

    Article  Google Scholar 

  17. Thompson, J. M. T., & Hunt, G. W. (1973). A general theory of elastic stability. London: Wiley.

    MATH  Google Scholar 

  18. Savi, M. A., & Pacheco, P. M. C. L. (2003). Transient chaos in an elasto-plastic beam with hardening. Journal of Brazilian Society of Mechanical Sciences and Engineering, 25(2), 189–193.

    Article  Google Scholar 

  19. Maor, E. (2007). The Pythagorean theorem: A 4,000-year history. Princeton: Princeton University Press.

    MATH  Google Scholar 

  20. Drake., E. T. (1996). Restless Genius: Robert Hooke and his earthly thoughts. Oxford: Oxford University Press.

    Google Scholar 

  21. Lewin, W. (1999). Hooke’s Law, simple harmonic oscillator., MIT course 8.01: Classical mechanics, lecture 10. (ogg) (videotape) Cambridge: MIT OCW.

    Google Scholar 

  22. Guckenheimer, J., & Holmes, P. (1983). Nonlinear oscillation, dynamical system and bifurcation of vector fields. New York: Springer.

    Book  MATH  Google Scholar 

  23. Lai, Y. C., & Lerner, D. (1998). Effective scaling regime for computing the correlation dimension from chaotic time series. Physica D-Nonlinear Phenomena, 115, 1–18.

    Article  MATH  Google Scholar 

  24. Tel, T. (1986). Characteristic exponents of chaotic repellers as eigenvalues. Physics Letters A, 119(2), 65–68.

    Article  MathSciNet  MATH  Google Scholar 

  25. Grebogi, C., Ott, E., & Yorke, J. A. (1983). Crises, sudden changes in chaotic attractors, and transient chaos. Physica D-Nonlinear Phenomena, 7(1–3), 181–200.

    Article  MathSciNet  MATH  Google Scholar 

  26. Pavlovskaia, E., & Wiercigroch, M. (2004). Analytical drift reconstruction for visco-elastic impact oscillators operating in periodic and chaotic regimes. Chaos, Solitons and Fractals, 19(1), 151–161.

    Google Scholar 

  27. Stefanski, A., & Kapitaniak, T. (2003). Estimation of the dominant lyapunov exponent of non-smooth systems on the basis of maps synchronization. Chaos Solitons and Fractals, 15(2), 233–244.

    Article  MathSciNet  MATH  Google Scholar 

  28. Thompson, J. M. T., & Ghaffari, R. (1983). Chaotic dynamics of an impact oscillator. Physical Review A, 27(3), 1741–1743.

    Article  MathSciNet  Google Scholar 

  29. Thompson, J. M. T., & Stewart, H. B. (2002). Nonlinear dynamics and chaos (2nd ed.). Chichester: Wiley and Sons.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Astronautics, Harbin Institute of Technology, Harbin, China

    Qingjie Cao

  2. Laboratoire de Mécanique et d’Acoustique, CNRS, Marseille Cedex13, France

    Alain Léger

Authors
  1. Qingjie Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alain Léger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qingjie Cao .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Cao, Q., Léger, A. (2017). The SD Oscillator . In: A Smooth and Discontinuous Oscillator. Springer Tracts in Mechanical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53094-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-53094-8_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53092-4

  • Online ISBN: 978-3-662-53094-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation