[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Stability and Bifurcations Analysis for 2-DOF Vibroimpact System by Parameter Continuation Method. Part I: Loading Curve

Journal of Applied Nonlinear Dynamics 4(4) (2015) 357--370 | DOI:10.5890/JAND.2015.11.003

V.A. Bazhenov; P.P. Lizunov; O.S. Pogorelova; T.G. Postnikova; V.V. Otrashevskaia

Kyiv National University of Construction and Architecture, 31, Povitroflotskiy avenu, Kyiv, Ukraine

Download Full Text PDF

 

Abstract

Vibroimpact system dynamic behaviour is studied by numerical parametric continuation technique combined with shooting and Newton- Raphson’s methods. The technique is adapted to two-body twodegree- of-freedom vibroimpact system under periodic excitation. Impact is simulated by nonlinear contact interaction force based on Hertz’s contact theory. Stability or instability of obtained periodic solutions is determined by monodromy matrix eigenvalues based on Floquet’s theory. Analysis of dynamic behaviour for specific vibroimpact system was performed. The instability zones, different oscillatory regimes and bifurcation points were found. Poincare sections were also constructed.

References

  1. [1]  Babitsky, V.I. (1998), Theory of Vibro-impact Systems and Applications, Springer.
  2. [2]  Kobrinskii, A.E., & Kobrinskii, A.A. (1973), Vibroimpact Systems, Nauka, Moscow.
  3. [3]  Ragulskene, V.L. (1974), Vibro-impact Systems, Mintis, Vilna.
  4. [4]  Ibrahim, R.A. (2009), Vibro-impact Dynamics: Modeling, Mapping and Applications, (Vol. 43). Springer.
  5. [5]  Stronge, W.J. (2004), Impact Mechanics, Cambridge university press.
  6. [6]  Ivanov, A.P. (1997), Dynamics of Systems with Mechanical Impacts, International Education Program, Moscow, 336.
  7. [7]  Luo, A. C., & Guo, Y. (2012), Vibro-impact Dynamics, John Wiley & Sons.
  8. [8]  Ma, Y., Banerjee, S., Wiercigroch, M., & Pavlovskaia, E. (2008), The nature of the normal form map for soft impacting systems, International Journal of Non-Linear Mechanics, 43(6), 504-513.
  9. [9]  Ajibose, O., Wiercigroch, M., Pavlovskaia, E., & Akisanya, A. (2008, June), Influence of contact force models on the global and local dynamics of drifting impact oscillator, In Proceedings of the 8th World Congress on Computational Mechanics (WCCM＆08) and 5th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS＆08).
  10. [10]  Bishop, S.R. (1994), Impact oscillators. Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences, 347(1683), 347-351.
  11. [11]  Mikhlin, Y. V., Vakakis, A. F., & Salenger, G. (1998), Direct and inverse problems encountered in vibroimpact oscillations of a discrete system, Journal of sound and vibration, 216(2), 227-250.
  12. [12]  Foale, S., & Bishop, S. R. (1994), Bifurcations in impact oscillations, Nonlinear Dynamics, 6(3), 285-299.
  13. [13]  Toulemonde, C., & Gontier, C. (1998), Sticking motions of impact oscillators, European Journal of Mechanics- A/Solids, 17(2), 339-366.
  14. [14]  Blazejczyk-Okolewska, B., Czolczynski, K., & Kapitaniak, T. (2009), Dynamics of a two-degree-of-freedom cantilever beam with impacts, Chaos, Solitons & Fractals, 40(4), 1991-2006.
  15. [15]  Peterka, F. (1971), An investigation of the motion of impact dampers, paper I, II, III, Strojnicku Casopis, XXI, c, 5.
  16. [16]  Bichri, A., Belhaq,M., & Perret-Liaudet, J. (2011), Control of vibroimpact dynamics of a single-sided Hertzian contact forced oscillator, Nonlinear Dynamics, 63(1-2), 51-60.
  17. [17]  Allgower, E.L., & Georg, K. (2003), Introduction to numerical continuation methods (Vol. 45). SIAM.
  18. [18]  Nayfeh, A.H., & Balachandran, B. (1995), Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, Wiley.
  19. [19]  Seydel, R. (2010), Practical Bifurcation and Stability Analysis, Springer, New York.
  20. [20]  Shalashilin, V.I., & Kuznetsov, E. B. (2003), Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics, Springer.
  21. [21]  Grigolyuk, E.I., & Shalashilin, V.I. (1988), Problems of Nonlinear Deformation: Parameter Continuation Method in Nonlinear Problems of Solid Mechanics.
  22. [22]  Gouliaev, V.I., Bazhenov, V.A., & Gotsuliak, E. A. (1983), Stability of Periodical Processes in Non-Linear Mechanical Systems, Vyshcha Shkola, Kiev.
  23. [23]  Sracic, M.W., & Allen, M.S. (2011), Numerical continuation of periodic orbits for harmonically forced non linear systems. In Civil Engineering Topics, Volume 4 (pp. 51-69), Springer, New York.
  24. [24]  Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., & Golinval, J.C. (2009), Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques, Mechanical Systems and Signal Processing, 23(1), 195-216.
  25. [25]  Patel, B. P., Ibrahim, S. M., & Nath, Y. (2009), Periodic response of nonlinear dynamical system with large number of degrees of freedom, Sadhana, 34(6), 1033-1037.
  26. [26]  Watson, L. T. (2001), Theory of globally convergent probability-one homotopies for nonlinear programming, SIAM Journal on Optimization, 11(3), 761-780.
  27. [27]  Ge, Y., Watson, L. T., Collins Jr, E. G., & Bernstein, D. S. (1994, December). Probability-one homotopy algorithms for full and reduced order H 2/H ꝏ controller synthesis, In Decision and Control, 1994., Proceedings of the 33rd IEEE Conference on (Vol. 3, pp. 2672-2677). IEEE.
  28. [28]  Watson, L. T., Billups, S. C., & Morgan, A. P. (1987), Algorithm 652: HOMPACK: A suite of codes for globally convergent homotopy algorithms, ACM Transactions on Mathematical Software (TOMS), 13(3), 281-310.
  29. [29]  Padmanabhan, C., & Singh, R. (1995), Analysis of periodically excited non-linear systems by a parametric continuation technique, Journal of Sound and Vibration, 184(1), 35-58.
  30. [30]  Padmanabhan, C., & Singh, R. (1995), Dynamics of a piecewise non-linear system subject to dual harmonic excitation using parametric continuation, Journal of Sound and Vibration, 184(5), 767-799.
  31. [31]  Byrtus, M. Dynamic Analysis of Reduced Order Large Rotating Vibro-Impact Systems.
  32. [32]  Rigaud, E., & Perret-Liaudet, J. (2003), Experiments and numerical results on non-linear vibrations of an impacting Hertzian contact. Part 1: harmonic excitation, Journal of Sound and Vibration, 265(2), 289-307.
  33. [33]  Perret-Liaudet, J., & Rigaud, E. (2006), Response of an impacting Hertzian contact to an order-2 subharmonic excitation: Theory and experiments, Journal of Sound and Vibration, 296(1), 319-333.
  34. [34]  Gontier, C., & Toulemonde, C. (1997), Approach to the periodic and chaotic behaviour of the impact oscillator by a continuation method, European Journal of Mechanics. A. Solids, 16(1), 141-163.
  35. [35]  Yoshitake, Y., Sueoka, A., Miyuki, T., Hamano, T., Kitayama, S., & Tamura, S. (2004), Development of shooting method for impact systems, JSME International Journal Series C, 47(3), 834-844.
  36. [36]  Bazhenov, V.A., Pogorelova, O.S., & Postnikova, T.G. (2011), The development of continuation after parameter method for vibroimpact systems provided the impact is simulated by contact interaction force, Strength of Materials and Theory of Structures, 87, 62-72.
  37. [37]  Bazhenov, V.A., Pogorelova, O.S., & Postnikova, T.G. (2012), Theoretical principles of dynamic behavior analysis for vibroimpact systems, Strength of Materials and Theory of Structures, 89, 39-49.(in Ukrainian)
  38. [38]  Bazhenov, V.A., Pogorelova, O.S., & Postnikova, T.G. (2013), Dynamic behaviour analysis of different types vibroimpact systems, LAP LAMBERT Academic Publ. GmbH and Co. KG Dudweileg, Germany.(in Russian)
  39. [39]  Bazhenov, V.A., Pogorelova, O.S., & Postnikova, T.G. (2013), Comparison of two impact simulation methods used for nonlinear vibroimpact systems with rigid and soft impacts, Journal of Nonlinear Dynamics, 2013.
  40. [40]  Bazhenov, V.A., Pogorelova, O.S., Postnikova, T.G., & Goncharenko, S.N. (2009), Comparative analysis of modeling methods for studying contact interaction in vibroimpact systems, Strength of Materials, 41(4), 392-398.
  41. [41]  Goldsmith, W. (1964), Impact, the Theory and Physical Behavior of Colliding Solids.
  42. [42]  Johnson, K. L. (1974), Contact Mechanics, 1985, Cambridge University Press, Cambridge.
  43. [43]  Machado, M., Moreira, P., Flores, P., & Lankarani, H. M. (2012), Compliant contact force models in multibody dynamics: Evolution of the Hertz contact theory, Mechanism and Machine Theory, 53, 99-121.
  44. [44]  Perret-Liaudet, J., & Rigaud, E. (2009), Non Linear Dynamic Behaviour of an One-Sided Hertzian Contact Excited by an External Gaussian White Noise Normal Excitation, In Vibro-Impact Dynamics of Ocean Systems and Related Problems (pp. 215-215), Springer Berlin Heidelberg.
  45. [45]  Holodniok,M., Klic, A., Kubicek, M.,& Marek, M. (1984), Methods of Analysis of Nonlinear Dynamical Models, Academia, Prague.
  46. [46]  Kubicek, M. (1976), Algorithm 502: dependence of solution of nonlinear systems on a parameter [c5], ACM Trans. Math. Softw., 2(1), 98-107.
  47. [47]  Lamarque, C.H., & Janin, O. (2000), Modal analysis of mechanical systems with impact non-linearities: limitations to a modal superposition, Journal of Sound and Vibration, 235(4), 567-609.
  48. [48]  Ivanov, A.P. (2012), Analysis of discontinuous bifurcations in nonsmooth dynamical systems, Regular and Chaotic Dynamics, 17(3-4), 293-306.
  49. [49]  Leine, R.I., Van Campen, D.H., & Van de Vrande, B.L. (2000), Bifurcations in nonlinear discontinuous systems, Nonlinear dynamics, 23(2), 105-164.

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates