Skip to main content

Advertisement

SpringerLink
Log in

A dissipative contact force model for impact analysis in multibody dynamics

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

Abstract

In this work, a new dissipative contact force model, based on the foundation of Hertz contact law, is presented for impact analysis in multibody dynamics. A hysteresis damping force is introduced in the model for capturing the energy loss during the contact process. An approximate function, representing the relationship between the deformation velocity and deformation, is used to calculate the energy loss due to the damping force. The difference between the compression phase and restitution phase during the contact process is taken into account in the energy loss calculation. For illustration, four different contact force models are applied in a numerical example to compare their behaviors. The results are presented in the form of dynamic simulations in a multibody system, which allow comparison of the differences and similarities among the four contact models. They show the validity of our model for soft or hard contact problems.

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

Similar content being viewed by others

Abbreviations

i,j :

solid sphere

\(v_{i}^{ ( - )},v_{j}^{ ( - )}\) :

initial velocity

\(v_{i}^{ ( + )},v_{j}^{ ( + )}\) :

separation velocity

t (−) :

time of initial contact

t (+) :

time of separation

t (m) :

time of maximum deformation

R i ,R j :

radius of solid sphere

F N :

normal contact force

δ :

deformation or indentation

K :

generalized stiffness

n :

Hertz’s contact force exponent

T (−),T (+) :

kinetic energies of the two spheres before and after impact

E :

Young’s modulus

λ :

Poisson’s ratio

\(\dot{\delta}^{ ( + )}\) :

relative separation velocity

ΔE :

energy loss

e :

coefficient of restitution

m :

equivalent mass

\(\dot{\delta}^{ ( - )}\) :

initial deformation velocity

D :

hysteresis damping coefficient

c :

hysteresis damping factor

δ m :

maximum deformation

\(\dot{\delta}\) :

relative normal velocity

ΔE c ,ΔE r :

energy loss for compression or restitution

References

  1. Greenwood, D.T.: Principles of Dynamics. Prentice-Hall, Englewood Cliffs (1965)

    Google Scholar 

  2. Dubowsky, S., Deck, J.F., Costello, H.: The dynamic modeling of flexible spatial machine systems with clearance connections. J. Mech. Transm.—T. ASME 109(1), 87–94 (1987)

    Article  Google Scholar 

  3. Shabana, A.A.: Dynamics of Multibody Systems. Wiley, New York (1989)

    MATH  Google Scholar 

  4. Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts. Wiley, New York (1996)

    Book  MATH  Google Scholar 

  5. Bai, Z.F., Zhao, Y.: Dynamic behaviour analysis of planar mechanical systems with clearance in revolute joints using a new hybrid contact force model. Int. J. Mech. Sci. (2011). doi:10.1016/j.ijmecsci.2011.10.009

  6. Tian, Q., Zhang, Y., Chen, L., Flores, P.: Dynamics of spatial flexible multibody systems with clearance and lubricated spherical joints. Comput. Struct. 87(13–14), 913–929 (2009)

    Article  Google Scholar 

  7. Mukras, S., Mauntler, N.A., Kim, N.H., Schmitz, T.L., Sawyer, W.G.: Modeling a slider–crank mechanism with joint wear. SAE Int. J. Passeng. 2(1), 600 (2009)

    Article  Google Scholar 

  8. Erkaya, S., Uzmay, İ.: Optimization of transmission angle for slider–crank mechanism with joint clearances. Struct. Multidiscip. Optim. 37(5), 493–508 (2009)

    Article  Google Scholar 

  9. Wojtkowski, M., Pecen, J., Horabik, J., Molenda, M.: Rapeseed impact against a flat surface: Physical testing and DEM simulation with two contact models. Powder Technol. 198(1), 61–68 (2010)

    Article  Google Scholar 

  10. Machado, M., Flores, P., Claro, J., Ambrósio, J., Silva, M., Completo, A., Lankarani, H.: Development of a planar multibody model of the human knee joint. Nonlinear Dyn. 60(3), 459–478 (2010)

    Article  MATH  Google Scholar 

  11. Flores, P., Lankarani, H.: Spatial rigid-multibody systems with lubricated spherical clearance joints: modeling and simulation. Nonlinear Dyn. 60(1), 99–114 (2010)

    Article  MATH  Google Scholar 

  12. Zhao, Y., Bai, Z.F.: Dynamics analysis of space robot manipulator with joint clearance. Acta Astronaut. 68(7–8), 1147–1155 (2011)

    Article  Google Scholar 

  13. Flores, P.: A parametric study on the dynamic response of planar multibody systems with multiple clearance joints. Nonlinear Dyn. (2010). doi:10.1007/s1171-010-9676-71-010-8

  14. 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(2), 165–190 (2010)

    Article  MathSciNet  Google Scholar 

  15. Flores, P., Machado, M., Silva, M., Martins, J.: On the continuous contact force models for soft materials in multibody dynamics. Multibody Syst. Dyn. 25(3), 357–375 (2011)

    Article  MATH  Google Scholar 

  16. Tasora, A., Anitescu, M., Negrini, S., Negrut, D.: A compliant visco-plastic particle contact model based on differential variational inequalities. Int. J. Non-Linear Mech. 53, 2–12 (2013). doi:10.1016/j.ijnonlinmec.2013.01.010

    Article  Google Scholar 

  17. Bai, Z.F., Zhao, Y.: A hybrid contact force model of revolute joint with clearance for planar mechanical systems. Int. J. Non-Linear Mech. 48, 15–36 (2013). doi:10.1016/j.ijnonlinmec.2012.07.003

    Article  Google Scholar 

  18. Lin, Y., Haftka, R.T., Queipo, N.V., Fregly, B.J.: Surrogate articular contact models for computationally efficient multibody dynamic simulations. Med. Eng. Phys. 32(6), 584–594 (2010). doi:10.1016/j.medengphy.2010.02.008

    Article  Google Scholar 

  19. Anitescu, M., Potra, F.A., Stewart, D.E.: Time-stepping for three-dimensional rigid body dynamics. Comput. Methods Appl. Math. 177(3–4), 183–197 (1999). doi:10.1016/S0045-7825(98)00380-6

    MATH  MathSciNet  Google Scholar 

  20. Pang, J., Stewart, D.: Differential variational inequalities. Math. Program. 113(2), 345–424 (2008). doi:10.1007/s10107-006-0052-x

    Article  MATH  MathSciNet  Google Scholar 

  21. Tasora, A., Negrut, D., Anitescu, M.: Large-scale parallel multi-body dynamics with frictional contact on the graphical processing unit. Proc. Inst. Mech. Eng., Proc., Part K, J. Multi-Body Dyn. 222(4), 315–326 (2008)

    Google Scholar 

  22. Lankarani, H.M., Nikravesh, P.E.: A contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des. 112(3), 369–376 (1990)

    Article  Google Scholar 

  23. Flores, P.: Compliant contact force approach for forward dynamic modeling and analysis of biomechanical systems. Proc. IUTAM 2, 58–67 (2011)

    Article  Google Scholar 

  24. Machado, M., Moreira, P., Flores, P., Lankarani, H.M.: Compliant contact force models in multibody dynamics: Evolution of the Hertz contact theory. Mech. Mach. Theory 53, 99–121 (2012)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  26. Lankarani, H.M., Nikravesh, P.E.: Continuous contact force models for impact analysis in multibody systems. Nonlinear Dyn. 5(2), 193–207 (1994)

    Google Scholar 

  27. Hunt, K.H., Crossley, F.R.E.: Coefficient of restitution interpreted as damping in vibroimpact. J. Appl. Mech. 42(2), 440–445 (1975)

    Article  Google Scholar 

  28. Herbert, R.G., Mcwhannell, D.C.: Shape and frequency composition of pulses from an impact pair. J. Eng. Ind. 99(3), 513–518 (1977)

    Article  Google Scholar 

  29. Lee, T.W., Wang, A.C.: On the dynamics of intermittent-motion mechanisms. Part 1: Dynamic model and response. J. Mech. Transm.—T. ASME 105(3), 534–540 (1983)

    Article  Google Scholar 

  30. Gonthier, Y., Mcphee, J., Lange, C., Piedbœuf, J.: A regularized contact model with asymmetric damping and dwell-time dependent friction. Multibody Syst. Dyn. 11(3), 209–233 (2004). doi:10.1023/B:MUBO.0000029392.21648.bc

    Article  MATH  Google Scholar 

  31. Zhiying, Q., Qishao, L.: Analysis of impact process based on restitution coefficient. J. Dyn. Control 4, 294–298 (2006)

    Google Scholar 

  32. Jackson, R., Green, I., Marghitu, D.: Predicting the coefficient of restitution of impacting elastic-perfectly plastic spheres. Nonlinear Dyn. 60(3), 217–229 (2010). doi:10.1007/s11071-009-9591-z

    Article  MATH  Google Scholar 

  33. Seifried, R., Schiehlen, W., Eberhard, P.: The role of the coefficient of restitution on impact problems in multi-body dynamics. Proc. Inst. Mech. Eng., Proc., Part K, J. Multi-Body Dyn. 224(3), 279–306 (2010)

    Article  Google Scholar 

  34. Braccesi, C., Landi, L.: A general elastic–plastic approach to impact analysis for stress state limit evaluation in ball screw bearings return system. Int. J. Impact Eng. 34(7), 1272–1285 (2007)

    Article  Google Scholar 

  35. Zhang, X., Vu-Quoc, L.: Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions. Int. J. Impact Eng. D 27(3), 317–341 (2002)

    Article  Google Scholar 

  36. Ye, K., Zhu, H.: A note on the Hertz contact model with nonlinear damping for pounding simulation. Earthq. Eng. Struct. Dyn. 38(9), 1135–1142 (2008)

    Article  Google Scholar 

Download references

Acknowledgements

We would like to thank the financial supports form the Natural Science Foundation of China.

Author information

Authors and Affiliations

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian, 116024, China

    Shiwu Hu & Xinglin Guo

Authors
  1. Shiwu Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinglin Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinglin Guo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, S., Guo, X. A dissipative contact force model for impact analysis in multibody dynamics. Multibody Syst Dyn 35, 131–151 (2015). https://doi.org/10.1007/s11044-015-9453-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11044-015-9453-z

Keywords

Search

Navigation