Skip to main content
SpringerLink
Log in

Nonlinear dynamics and chaotic control of a flexible multibody system with uncertain joint clearance

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The nonlinear dynamics of a flexible multibody system with interval clearance size in a revolute joint is investigated in this work. The system is modeled by using a unified mesh of absolute nodal coordinate formulation (ANCF), that is, the flexible parts are meshed via the finite elements of the ANCF and the rigid parts are described via the ANCF reference nodes (ANCF-RNs). The kinetic models of all revolute joints are formulated by using ANCF reference node (ANCF-RN) coordinates. The influence of the Lund-Grenoble and the modified Coulomb’s friction models on the system dynamics is comparatively studied. The Chebyshev tensor product sampling method is used to generate the samples of the interval clearance size. With the purpose to maintain the continuous contact of the clearance joint, a modified extended delayed feedback control (EDFC) is used to stabilize the chaotic motion of the flexible multibody system. Finally, the dynamics of a planar slider–crank mechanism with interval clearance size in a revolute joint is studied, as a benchmark example, to check the effectiveness of the presented computation method and the modified EDFC.

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
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29

Similar content being viewed by others

References

  1. Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M.: Kinematics and Dynamics of Multibody Systems with Imperfect Joints: Models and Case Studies. Lecture Notes in Applied and Computational Mechanics, vol. 34. Springer, Berlin (2008)

    MATH  Google Scholar 

  2. Flores, P., Ambrósio, J.: Revolute joints with clearance in multibody systems. Comput. Struct. 82, 1359–1369 (2004)

    Article  Google Scholar 

  3. Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M.: Dynamics of multibody systems with spherical clearance joints. J. Comput. Nonlinear Dyn. 1(3), 240–247 (2006)

    Article  MATH  Google Scholar 

  4. Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M.: Spatial revolute joints with clearances for dynamic analysis of multibody systems. Proc. Inst. Mech. Eng. Part K J. Multi-body Dyn. 220(4), 257–271 (2006)

    Google Scholar 

  5. Flores, P., Leine, R., Glocker, C.: Modeling and analysis of planar rigid multibody systems with translational clearance joints based on the non-smooth dynamics approach. Multibody Syst. Dyn. 23, 165–190 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M., Koshy, C.S.: A study on dynamics of mechanical systems including joints with clearance and lubrication. Mech. Mach. Theory 41(3), 247–261 (2006)

    Article  MATH  Google Scholar 

  7. Flores, P., Ambróio, J., Claro, J.C.P., Lankarani, H.M., Koshy, C.S.: Lubricated revolute joints in rigid multibody system. Nonlinear Dyn. 56, 277–295 (2009)

    Article  MATH  Google Scholar 

  8. Flores, P., Lankarani, H.M.: Spatial rigid-multi-body systems with lubricated spherical clearance joints: modeling and simulation. Nonlinear Dyn. 60, 99–114 (2010)

    Article  MATH  Google Scholar 

  9. Ravn, P.: A continuous analysis method for planar multibody systems with joint clearance. Multibody Syst. Dyn. 2(1), 1–24 (1998)

    Article  MATH  Google Scholar 

  10. Dubowsky, S.: On predicting the dynamic effects of clearances in planar mechanisms. J. Eng. Ind. 96(1), 317–323 (1974)

    Article  Google Scholar 

  11. Erkaya, S., Uzmay, I.: Investigation on effect of joint clearance on dynamics of four-bar mechanism. Nonlinear Dyn. 58, 179–198 (2009)

    Article  MATH  Google Scholar 

  12. Erkaya, S., Uzmay, I.: Determining link parameters using genetic algorithm in mechanisms with joint clearance. Mech. Mach. Theory 44, 222–234 (2009)

    Article  MATH  Google Scholar 

  13. Erkaya, S.: Prediction of vibration characteristics of a planar mechanism having imperfect joints using neural network. J. Mech. Sci. Technol. 26(5), 1419–1430 (2012)

    Article  Google Scholar 

  14. Muvengei, O., Kihiu, J., Ikua, B.: Numerical study of parametric effects on the dynamic response of planar multibody systems with differently located frictionless revolute clearance joints. Mech. Mach. Theory 53, 30–49 (2012)

    Article  Google Scholar 

  15. Brutti, C., Coglitore, G., Valentini, P.P.: Modeling 3D revolute joint with clearance and contact stiffness. Nonlinear Dyn. 66(4), 531–548 (2011)

    Article  Google Scholar 

  16. Orden, J.C.G.: Analysis of joint clearances in multibody systems. Multibody Syst. Dyn. 13(4), 401–420 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pezzuti, E., Stefanelli, R., Valentini, P.P., Vita, L.: Computer-aided simulation and testing of spatial linkages with joint mechanical errors. Int. J. Numer. Methods Eng. 65(11), 1735–1748 (2006)

    Article  MATH  Google Scholar 

  18. Valentini, P., Stefanelli, R., Vita, L.: Modelling hydrodynamic journal bearing in 3D multibody systems. In: Proceedings of ASME-IDETC/CIE 2005, Long Beach, California, USA, pp. 24–28 (2005)

  19. Tian, Q., Zhang, Y., Chen, L., Yang, J.: Simulation of planar flexible multibody systems with clearance and lubricated revolute joints. Nonlinear Dyn. 60, 489–511 (2010)

    Article  MATH  Google Scholar 

  20. Tian, Q., Liu, C., Machado, M., Flores, P.: A new model for dry and lubricated cylindrical joints with clearance in spatial flexible multibody system. Nonlinear Dyn. 64, 25–47 (2011)

    Article  MATH  Google Scholar 

  21. Tian, Q., Sun, Y., Liu, C., Hu, H., Flores, P.: Elastohydrodynamic lubricated cylindrical joints for rigid-flexible multibody dynamics. Comput. Struct. 114, 106–120 (2013)

    Article  Google Scholar 

  22. Shabana, A.A.: An Absolute Nodal Coordinates Formulation for the Large Rotation and Deformation Analysis of Flexible Bodies. Technical Report. No. MBS96-1-UIC, University of Illinois at Chicago (1996)

  23. Dufva, K., Shabana, A.A.: Analysis of thin plate structures using the absolute nodal coordinate formulation. Proc. Inst. Mech. Eng. Part K J. Multi-body Dyn. 219, 345–355 (2005)

    Google Scholar 

  24. Mikkola, A.M., Shabana, A.A., Sanchez-Rebollo, C., Jimenez-Octavio, J.R.: Comparison between ANCF and B-spline surfaces. Multibody Syst. Dyn. 30(2), 119–138 (2013)

    Article  MathSciNet  Google Scholar 

  25. Liu, C., Tian, Q., Hu, H.Y.: Dynamics of a large scale rigid-flexible multibody system composed of composite laminated plate. Multibody Syst. Dyn. 26(3), 283–305 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lan, P., Shabana, A.A.: Integration of B-spline geometry and ANCF finite element analysis. Nonlinear Dyn. 61, 193–206 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schiehlen, W.: Research trends in multibody system dynamics. Multibody Syst. Dyn. 18, 3–13 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shabana, A.A.: Flexible multibody dynamics review of past and recent developments. Multibody Syst. Dyn. 1, 189–222 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhu, J., Ting, K.L.: Uncertainty analysis of planar and spatial robots with joint clearances. Mech. Mach. Theory 35, 1239–1256 (2000)

    Article  Google Scholar 

  30. Parenti-Castelli, V., Venanzi, S.: Clearance influence analysis on mechanisms. Mech. Mach. Theory 40, 1316–1329 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wu, W., Rao, S.S.: Uncertainty analysis and allocation of joint tolerances in robot manipulators based on interval analysis. Reliab. Eng. Syst. Saf. 92, 54–64 (2007)

    Article  Google Scholar 

  32. Chen, G., Wang, H., Lin, Z.: A unified approach to the accuracy analysis of planar parallel manipulators both with input uncertainties and joint clearance. Mech. Mach. Theory 64, 1–17 (2013)

    Article  Google Scholar 

  33. Kumaraswamy, U., Shunmugam, M.S., Sujatha, S.: A unified framework for tolerance analysis of planar and spatial mechanisms using screw theory. Mech. Mach. Theory 69(6), 168–184 (2013)

    Article  Google Scholar 

  34. Li, X., Ding, X., Chirikjian, G.S.: Analysis of angular-error uncertainty in planar multiple-loop structures with joint clearances. Mech. Mach. Theory 91, 69–85 (2015)

    Article  Google Scholar 

  35. Muvengei, M., Kihiu, J., Ikua, B.: Effects of clearance size on the dynamic response of planar multi-body systems with differently located frictionless revolute clearance joints. J. Energy Technol. Policy 74, 954 (2012)

    Google Scholar 

  36. Flores, P.: Dynamic analysis of mechanical systems with imperfect kinematic joints. Dissertation, University of Minho, Guimarães (2004)

  37. Rahmanian, S., Ghazavi, M.R.: Bifurcation in planar slider–crank mechanism with revolute clearance joint. Mech. Mach. Theory 91, 86–101 (2015)

    Article  Google Scholar 

  38. Olyaei, A.A., Ghazavi, M.R.: Stabilizing slider–crank mechanism with clearance joints. Mech. Mach. Theory 53, 17–29 (2012)

    Article  Google Scholar 

  39. Moore, R.E.: Interval Analysis. Prentice-Hall, Englewood Cliffs (1966)

    MATH  Google Scholar 

  40. Alefeld, G.: Interval analysis: theory and applications. J. Comput. Appl. Math. 121, 421–464 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  41. Jackson, K.R., Nedialkov, N.S.: Some recent advances in validated methods for IVPs for ODEs. Appl. Numer. Math. 42(1–3), 269–284 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  42. Makino, K., Berz, M.: Efficient control of the dependency problem based on Taylor model methods. Reliab. Comput. 5(1), 3–12 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wu, J.L., Zhao, Y., Chen, S.: An improved interval analysis method for uncertain structures. Struct. Eng. Mech. 20(6), 713–726 (2005)

    Article  Google Scholar 

  44. Lin, Y., Stadtherr, M.A.: Validated solutions of initial value problems for parametric ODEs. Appl. Numer. Math. 57(10), 1145–1162 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wu, J.L., Luo, Z., Zhang, Y.Q., Zhang, N., Chen, L.P.: Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions. Int. J. Numer. Methods Eng. 95(7), 608–630 (2013)

    Article  MathSciNet  Google Scholar 

  46. Wang, Z., Tian, Q., Hu, H.: Dynamics of spatial rigid-flexible multibody systems with uncertain interval parameters. Nonlinear Dyn. 84(2), 527–548 (2015)

    Article  MathSciNet  Google Scholar 

  47. Wu, J.L., Zhang, Y.Q., Chen, L.P., Luo, Z.: A Chebyshev interval method for nonlinear dynamic systems under uncertainty. Appl. Math. Model. 37(6), 4578–4591 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Wu, J.L., Luo, Z., Zhang, N., Zhang, Y.Q.: A new sampling scheme for developing metamodels with the zeros of Chebyshev polynomials. Eng. Optim. 47(9), 1264–1288 (2014)

    Article  MathSciNet  Google Scholar 

  49. Wu, J.L., Luo, Z., Zhang, N., Zhang, Y.Q.: A new interval uncertain optimization method for structures using Chebyshev surrogate models. Comput. Struct. 146, 185–196 (2015)

    Article  Google Scholar 

  50. Jalon, J., Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems. Springer, Berlin (2012)

    Google Scholar 

  51. Jalón, J.G.D.: Twenty-five years of natural coordinates. Multibody Syst. Dyn. 18(1), 15–33 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  52. García-Vallejo, D., Escalona, J.L., Mayo, J., Domínguez, J.: Describing rigid-flexible multibody systems using absolute coordinates. Nonlinear Dyn. 34(1–2), 75–94 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  53. Shabana, A.A.: Dynamics of Multibody Systems. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  54. Sopanen, J., Mikkola, A.: Description of elastic forces in absolute nodal coordinate formulation. Nonlinear Dyn. 34, 53–74 (2003)

    Article  MATH  Google Scholar 

  55. Gerstmayr, J., Matikainen, M.K., Mikkola, A.M.: A geometrically exact beam element based on the absolute nodal coordinate formulation. Multibody Syst. Dyn. 20, 359–384 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  56. Shen, Z., Li, P., Liu, C., Hu, G.: A finite element beam model including cross-section distortion in the absolute nodal coordinate formulation. Nonlinear Dyn. 77, 1019–1033 (2014)

    Article  Google Scholar 

  57. García-Vallejo, D., Mayo, J., Escalona, J.L., Domínguez, J.: Three-dimensional formulation of rigid-flexible multibody systems with flexible beam elements. Multibody Syst. Dyn. 20(1), 1–28 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  58. Shabana, A.A.: ANCF reference node for multibody system analysis. Proc. Inst. Mech. Eng. Part K J. Multi-body Dyn. 229(1), 109–112 (2014)

    Google Scholar 

  59. Shabana, A.A.: ANCF tire assembly model for multibody system applications. J. Comput. Nonlinear Dyn. 10(1), 024504–024504-4 (2015)

    Article  MathSciNet  Google Scholar 

  60. Flores, P., Lankarani, H.M.: Dynamic response of multibody systems with multiple clearance joints. J. Comput. Nonlinear Dyn. 7(3), 636–647 (2011)

    Google Scholar 

  61. Lankarani, H.M., Nikravesh, P.E.: A contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des. 112, 369–376 (1990)

    Article  Google Scholar 

  62. Ambrósio, J.A.C.: Impact of rigid and flexible multibody systems: deformation description and contact models. Nato Asi. 103, 57–81 (2003)

    MATH  Google Scholar 

  63. Muvengei, O., Kihiu, J., Ikua, B.: Dynamic analysis of planar multi-body systems with LuGre friction at differently located revolute clearance joints. Multibody Syst. Dyn. 28(4), 369–393 (2012)

    Article  MathSciNet  Google Scholar 

  64. Yamashita, H., Matsutani, Y., Sugiyama, H.: Longitudinal tire dynamics model for transient braking analysis: ANCF-LuGre tire model. J. Comput. Nonlinear Dyn. 10(3), 031003 (2014)

    Article  Google Scholar 

  65. Yu, C., Yu, S., Chong, C.: Dynamic analysis of a planar slider–crank mechanism with clearance for a high speed and heavy load press system. Mech. Mach. Theory 98, 81–100 (2016)

    Article  Google Scholar 

  66. Wang, Q., Tian, Q., Hu. H.Y.: Dynamic simulation of frictional multi-zone contacts of thin beams. Nonlinear Dyn. 83(4), 1919–1937 (2015)

  67. Liu, C., Tian, Q., Hu, H.Y.: New spatial curved beam and cylindrical shell elements of gradient deficient absolute nodal coordinate formulation. Nonlinear Dyn. 70, 1903–1918 (2012)

    Article  MathSciNet  Google Scholar 

  68. Arnold, M., Brüls, O.: Convergence of the generalized-\(\alpha \) scheme for constrained mechanical systems. Multibody Syst. Dyn. 18(2), 185–202 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  69. Jeffery, J.: Numerical Analysis and Scientific Computation. Addison Wesley, Cambridge (2004)

    Google Scholar 

  70. Shabana, A.A., Yakoub, R.Y.: Three dimensional absolute nodal coordinate formulation for beam elements: theory. J. Mech. Des. 123(4), 606–613 (2000)

    Article  Google Scholar 

  71. Pyragas, K.: Delayed feedback control of chaos. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 364, 2309–2334 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  72. Pyragas, K., Pyragas, V.: Using ergodicity of chaotic systems for improving the global properties of the delayed feedback control method. Phys. Rev. E. 80, 067201 (2009)

    Article  Google Scholar 

  73. Just, W., Reibold, E., Kacperski, K., Fronczak, P., Holyst, J.A., Benner, H.: Influence of stable Floquet exponents on time-delayed feedback control. Phys. Rev. E. 61, 5045–5056 (2000)

    Article  Google Scholar 

  74. Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A. 170, 421–428 (1992)

    Article  Google Scholar 

  75. Socolar, J.E.S., Sukow, D.W., Gauthier, D.J.: Stabilizing unstable periodic orbits in fast dynamical systems. Phys. Rev. E. 50, 3245–3248 (1994)

    Article  Google Scholar 

Download references

Acknowledgments

This research was supported in part by National Natural Science Foundation of China under Grants 11290151 and 11472042. The work was also supported by the Portuguese Foundation for Science and Technology with the reference project UID/EEA/04436/2013, by FEDER funds through the COMPETE 2020—Programa Operacional Competitividade e Internacionalização (POCI) with the reference project POCI-01-0145-FEDER-006941. Finally, the authors are much indebted to the anonymous reviewers for useful comments, recommendations and suggestions.

Author information

Authors and Affiliations

  1. MOE Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Zhe Wang, Qiang Tian & Haiyan Hu

  2. Department of Mechanical Engineering, University of Minho Campus de Azurem, 4804-533, Guimarães, Portugal

    Paulo Flores

Authors
  1. Zhe Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiang Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Haiyan Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Paulo Flores
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiang Tian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Z., Tian, Q., Hu, H. et al. Nonlinear dynamics and chaotic control of a flexible multibody system with uncertain joint clearance. Nonlinear Dyn 86, 1571–1597 (2016). https://doi.org/10.1007/s11071-016-2978-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2978-8

Keywords

Search

Navigation