Skip to main content
SpringerLink
Log in

Limit Trajectories in a Non-holonomic System of a Ball Moving Inside a Spherical Cavity

  • Original Paper
  • Published:
Journal of Vibration Engineering & Technologies Aims and scope Submit manuscript

Abstract

Purpose

The authors analyze the regular and distinctive patterns of the free motion of a ball type tuned mass damper.

Methods

The governing differential system modeling movement of a heavy ball rolling inside a spherical cavity is formulated and investigated; six degrees of freedom with three non-holonomic constraints and no slipping are assumed. Predominance of the Appell-Gibbs approach over the conventional Lagrangian procedure is pointed out when complicated non-holonomic systems are in question. General properties of the differential system in the normal form are discussed and possibilities of further investigation using semi-analytical methods are outlined. Simultaneously, a wide program of numerical simulation is presented concerning the homogeneous system with a number of initial condition settings and other parameter variants.

Results

Seven types of limit solutions (or limit trajectories) have been found and physically interpreted together with their neighborhood. The set of limit trajectories represents boundaries separating solution groups of a certain character. The shape and general character of regular solutions within particular domains delimited by these limit solutions were analyzed to facilitate a practical application of this theoretical background.

Conclusions

The paper represents a contribution to the theoretical background of the ball type tuned mass damper used in Civil Engineering. The analysis provides an insight to the possible character of a free motion of a ball type tuned mass damper under various configurations; this way it helps to analyze possible critical parameters of the system or interaction with the structure. Assumptions of further investigation are outlined.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Alsuwaiyan A, Shaw S (2002) Performance and dynamic stability of general-path centrifugal pendulum vibration absorbers. J Sound Vib 252(5):791–815. https://doi.org/10.1006/jsvi.2000.3534

    Article  Google Scholar 

  2. Arnold V (1978) Mathematical methods of classical mechanics. Springer, New York

    Book  Google Scholar 

  3. Aston P (1999) Bifurcations of the horizontally forced spherical pendulum. Comput Methods Appl Mech Eng 170(3):343–353. https://doi.org/10.1016/S0045-7825(98)00202-3

    Article  MathSciNet  MATH  Google Scholar 

  4. Bajodah AH, Hodges DH, Chen YH (2003) New form of Kane’s equations of motion for constrained systems. J Guid Control Dyn 26(1):79–88. https://doi.org/10.2514/2.5017

    Article  Google Scholar 

  5. Bajodah AH, Hodges DH, Chen YH (2004) Nonminimal generalized Kane’s impulse-momentum relations. J Guid Control Dyn 27(6):1088–1092. https://doi.org/10.2514/1.7157

    Article  Google Scholar 

  6. Baruh H (1998) Analytical dynamics. McGraw-Hill, New York

    Google Scholar 

  7. Benettin G, Galgani L, Giorgilli A, Strelcyn JM (1980) Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian sy stems; a method for computing all of them. part 2: Numer Appl Mecc 15(1):21–30

    MATH  Google Scholar 

  8. Bloch A, Marsden J, Zenkov D (2005) Nonholonomic dynamics. Not Am Math Soc 52:324–333

    MathSciNet  MATH  Google Scholar 

  9. Bogoljubov N, Mitropolskij Y (1961) Asymptotic methods in the theory of nonlinear oscillations, 2nd edn. Gordon & Breach Science Publishers Ltd., New York

    Google Scholar 

  10. Desloge E (1988) The Gibbs–Appell equation of motion. Am J Phys 56(9):841–846

    Article  MathSciNet  Google Scholar 

  11. Dieci L, Jolly M, Vleck EV (2011) Numerical techniques for approximating Lyapunov exponents and their implementation. J Comput Nonlinear Dyn 6(1):011003–7

    Article  Google Scholar 

  12. Emami M, Zohoor H, Sohrabpour S (2009) Solving high order nonholonomic systems using Gibbs-Appell method. In: Balan V (ed) Proceedings of The international conference differential geometry and dynamical systems. Geometry Balkan Press, Mangalia, pp 70–79

  13. Fenz DM, Constantinou MC (2006) Behaviour of the double concave friction pendulum bearing. Earthq Eng Struct Dyn 35(11):1403–1424. https://doi.org/10.1002/eqe.589

    Article  Google Scholar 

  14. Galassi M, Davies J, Theiler J, Gough B, Jungman G, Booth M, Rossi F (2009) GNU scientific library: reference manual, 3rd edn. Network Theory Ltd. http://www.gnu.org/software/gsl/

  15. Hamel G (1978) Theoretische mechanik. Springer, Berlin

    MATH  Google Scholar 

  16. Haxton R, Barr A (1972) The autoparametric vibration absorber. J Eng Ind 94(1):119–125. https://doi.org/10.1115/1.3428100

    Article  Google Scholar 

  17. Kane T (1985) Dynamics, theory and applications. McGraw-Hill, New York

    Google Scholar 

  18. Lee W, Hsu C (1994) A global analysis of an harmonically excited spring-pendulum system with internal resonance. J Sound Vib 171(3):335–359

    Article  MathSciNet  Google Scholar 

  19. Leung A, Kuang J (2006) On the chaotic dynamics of a spherical pendulum with a harmonically vibrating suspension. Nonlinear Dyn 43:213–238. https://doi.org/10.1007/s11071-006-7426-8

    Article  MathSciNet  MATH  Google Scholar 

  20. Lewis A (1996) The geometry of the Gibbs–Appell equations and Gauss’s principle of least constraints. Rep Math Phys 38(1):11–28

    Article  MathSciNet  Google Scholar 

  21. Lichtenberg A, Lieberman M (1983) Regular and stochastic motion. Springer, New York

    Book  Google Scholar 

  22. Lurie A (2002) Analytical mechanics. Foundations of Engineering Mechanics. Springer, Berlin Heidelberg. https://doi.org/10.1007/978-3-540-45677-3

    Book  Google Scholar 

  23. Martins LA, Lara-Molina FA, Koroishi EH, Cavalini AA Jr (2019) Optimal design of a dynamic vibration absorber with uncertainties. J Vib Eng Technol. https://doi.org/10.1007/s42417-019-00084-6

    Article  Google Scholar 

  24. Miles J, Zhou QP (1993) Parametric excitation of a detuned spherical pendulum. J Sound Vib 164(2):237–250

    Article  MathSciNet  Google Scholar 

  25. Moser J (1973) Stable and random motions in dynamical systems. Princeton University Press, Princeton

    MATH  Google Scholar 

  26. Nabergoj R, Tondl A (1994) A simulation of parametric ship rolling: effects of hull bending and torsional elasticity. Nonlinear Dyn 6:265–284

    Article  Google Scholar 

  27. Neimark JI, Fufaev NA (1972) Dynamics of nonholonomic systems. Translations of mathematical monographs, vol 33. AMS, Providence, Rhode Island

    MATH  Google Scholar 

  28. Náprstek J, Fischer C (2009) Auto-parametric semi-trivial and post-critical response of a spherical pendulum damper. Comput Struct 87(19–20):1204–1215

    Article  Google Scholar 

  29. Náprstek J, Fischer C (2013) Dynamic response of a heavy ball rolling inside a spherical dish under external excitation. In: Zolotarev I (ed) Proc engineering mechanics 2013. IT AS CR, Prague Paper #46, pp 96–106

  30. Náprstek J, Fischer C (2016) Dynamic behavior and stability of a ball rolling inside a spherical surface under external excitation. In: Zingoni A (ed) Insights and innovations in structural engineering, mechanics and computation. Taylor & Francis, London, pp 214–219

  31. Náprstek J, Fischer C (2017) Non-holonomic dynamics of a ball moving inside a spherical cavity. Procedia Eng 199:613–618

    Article  Google Scholar 

  32. Náprstek J, Fischer C (2018) Appell-Gibbs approach in dynamics of non-holonomic systems. In: Reyhanoglu M (ed) Nonlinear systems—modeling, estimation, and stability, chap. 1. IntechOpen, London, pp 3–30. https://doi.org/10.5772/intechopen.76258

  33. Náprstek J, Fischer C (2018) Forced movement of a ball in spherical cavity under kinematic excitation. In: Fischer, C, Náprstek J (eds) Engineering mechanics 2018. ITAM, CAS, Praue, pp 573–576. https://doi.org/10.21495/91-8-573

  34. Náprstek J, Fischer C (2019) Stable and unstable solutions in auto-parametric resonance zone of a non-holonomic system. Nonlinear Dyn (accepted). https://doi.org/10.1007/s11071-019-04948-0

    Article  MATH  Google Scholar 

  35. Náprstek J, Fischer C, Pirner M, Fischer O (2011) Non-linear dynamic behaviour of a ball vibration absorber. In: Papadrakakis M, Fragiadakis M, Plevris V (eds) Proc. COMPDYN 2011, pp. CD ROM, paper 180. ECCOMAS—NTU Athens, Kerkyra, Corfu

  36. Náprstek J, Pirner M (2002) Non-linear behaviour and dynamic stability of a vibration spherical absorber. In: Smyth A (ed) Proc. 15th ASCE Engineering Mechanics Division Conference, pp. CD ROM, paper 150. Columbia Univ., New York

  37. Ott E (1993) Chaos in dynamical systems. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  38. Palacios A, Gross LM, Rockwood AP (1996) Dynamics and chaos: the spherical pendulum. Comput Graphics Forum 15(4):263–270. https://doi.org/10.1111/1467-8659.1540263

    Article  Google Scholar 

  39. Pars L (1972) A treatise on analytical dynamics, 2nd edn. Ox Bow Press, Connecticut

    MATH  Google Scholar 

  40. Pirner M (1994) Dissipation of kinetic energy of large-span bridges. Acta Tech CSAV 39:407–418

    Google Scholar 

  41. Pirner M, Fischer O (2000) The development of a ball vibration absorber for the use on towers. J Int Assoc Shell Spat Struct 41(2):91–99

    Google Scholar 

  42. Ren Y, Beards C (1994) A new receptance-based perturbative multi-harmonic balance method for the calculation of the steady state response of non-linear systems. J Sound Vib 172(5):593–604

    Article  Google Scholar 

  43. Schuster H (1988) Deterministic chaos, 2nd edn. VCH Publishers, Weinheim

    MATH  Google Scholar 

  44. Shaw SW, Schmitz PM, Haddow AG (2006) Tautochronic vibration absorbers for rotating systems. J Comput Nonlinear Dyn 1(4):283–293. https://doi.org/10.1115/1.2338652

    Article  Google Scholar 

  45. Soltakhanov S, Yushkov M, Zegzhda S (2009) Mechanics of non-holonomic systems. Springer, Berlin, Heidelberg

    Book  Google Scholar 

  46. Takei H, Shimazaki Y (2010) Vibration control effects of tuned cradle damped mass damper. J Appl Mech 13:587–594

    Google Scholar 

  47. Tondl A (1997) To the analysis of autoparametric systems. Z Angew Math Mech 77(6):407–418

    Article  MathSciNet  Google Scholar 

  48. Tritton DJ, Grooves M (1999) Lyapunov exponents for the Miles’ spherical pendulum equations. Phys D 126(1–2):83–98. https://doi.org/10.1016/S0167-2789(98)00263-2

    Article  MathSciNet  MATH  Google Scholar 

  49. Udwadia F, Kalaba R (1998) The explicit Gibbs–Appell equation and generalized inverse forms. Q Appl Math 56(2):277–288

    Article  MathSciNet  Google Scholar 

  50. Yong-Fen Q (1990) Gibbs-Appell’s equations of variable mass nonlinear nonholonomic mechanical systems. Appl Math Mech 11(10):973–983. https://doi.org/10.1007/BF02115681

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The kind support of the Czech Science Foundation project No. 17-26353J and the institutional support No. RVO 68378297 are gratefully acknowledged. Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum provided under the program “Projects of Large Research, Development, and Innovations Infrastructures” (CESNET LM2015042) is greatly appreciated.

Author information

Authors and Affiliations

  1. Institute of Theoretical and Applied Mechanics of the CAS, Prosecká 76, 190 00, Prague 9, Czech Republic

    Jiří Náprstek & Cyril Fischer

Authors
  1. Jiří Náprstek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cyril Fischer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiří Náprstek.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Náprstek, J., Fischer, C. Limit Trajectories in a Non-holonomic System of a Ball Moving Inside a Spherical Cavity. J. Vib. Eng. Technol. 8, 269–284 (2020). https://doi.org/10.1007/s42417-019-00132-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42417-019-00132-1

Keywords

Search

Navigation