Dissipative Contact Force Models

  • Paulo FloresEmail author
  • Hamid M. Lankarani
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 226)


In this chapter, some of most relevant dissipative contact force models utilized in multibody dynamics are presented. In particular, attention is given to the constitutive laws proposed by Kelvin and Voigt, Hunt and Crossley, Lankarani and Nikravesh, and Flores et al. Additionally, other nonlinear contact force models are briefly presented. The contact force models are mostly developed based on the Hertzian theory, augmented with terms to accommodate the energy dissipation associated with the contact-impact process. The main differences and similarities among these continuous contact force models are discussed by utilizing an application example based on the externally collision between two spherical solids, and a comparative study of the various contact force models is presented. The variables utilized in the study are the contact force and the penetration between the contacting bodies.


Dissipative contact force models Kelvin-Voigt model Hunt and crossley model Lankarani and nikravesh model Flores et al. model 


  1. Alves J, Peixinho N, Silva MT, Flores P, Lankarani HM (2015) A comparative study of the viscoelastic constitutive models for frictionless contact interfaces in solids. Mech Mach Theory 85:172–188CrossRefGoogle Scholar
  2. Anagnostopoulos SA (1988) Pounding of buildings in series during earthquakes. Earthquake Eng Struct Dynam 16:443–456CrossRefGoogle Scholar
  3. Anderson RWG, Long AD, Serre T (2009) Phenomenological continuous contact-impact modelling for multibody simulations of pedestrian-vehicle contact interactions based on experimental data. Nonlinear Dyn 58:199–208CrossRefzbMATHGoogle Scholar
  4. Askari E, Flores P, Dabirrahmani D, Appleyard R (2014) Study of the friction-induced vibration and contact mechanics of artificial hip joints. Tribol Int 70:1–10CrossRefGoogle Scholar
  5. Atkinson KA (1989) An introduction to numerical analysis, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  6. Beer FB, Johnston ER (1997) Vector mechanics for engineers. Statics and DynamicsGoogle Scholar
  7. Bibalan PT, Featherstone R (2009) A study of soft contact models in simulink. In: Proceedings of the Australasian Conference on Robotics and Automation (ACRA), 2–4 Dec 2009, Sydney, Australia, 8 pGoogle Scholar
  8. Bordbar MH, Hyppänen T (2007) Modeling of binary collision between multisize viscoelastic spheres. J Numer Anal, Ind Appl Math 2(3–4):115–128zbMATHGoogle Scholar
  9. Brach RM (1991) Mechanical impact dynamics. Rigid body collisions. Wiley, New YorkGoogle Scholar
  10. Brilliantov NV, Spahn F, Hertzsch J-M, Pöschel T (1996) Model for collisions in granular gases. Phys Rev E 53:5382–5392CrossRefGoogle Scholar
  11. Dopico D, Luaces A, Gonzalez M, Cuadrado J (2009) Dealing with multiple contacts in a human-in-the-loop application. In: Arczewski K, Frączek J, Wojtyra M (eds) Proceedings of Multibody Dynamics 2009, ECCOMAS Thematic Conference, Warsaw, Poland, June 29–July 2, 17 pGoogle Scholar
  12. Dubowsky S, Deck JF, Costello H (1987) The dynamic modeling of flexible spatial machine systems with clearance connections. J Mech Transm Autom Des 109:87–94CrossRefGoogle Scholar
  13. Dubowsky S, Freudenstein F (1971) Dynamic analysis of mechanical systems with clearances, part 1: formulation of dynamic model. J Eng Ind 93:305–309CrossRefGoogle Scholar
  14. Dubowsky S, Gardner TN (1977) Design and analysis of multilink flexible mechanism with multiple clearance connections. J Eng Ind 99:88–96CrossRefGoogle Scholar
  15. Dubowsky S, Young SC (1975) An experimental and theoretical study of connection forces in high-speed mechanisms. J Eng Ind 97:1166–1174CrossRefGoogle Scholar
  16. Flores P, Ambrósio J (2010) On the contact detection for contact-impact analysis in multibody systems. Multibody Sys Dyn 24(1):103–122MathSciNetCrossRefzbMATHGoogle Scholar
  17. Flores P, Machado M, Silva MT, Martins JM (2011) On the continuous contact force models for soft materials in multibody dynamics. Multibody Sys Dyn 25:357–375CrossRefzbMATHGoogle Scholar
  18. Fox B, Jennings LS, Zomaya AY (2001) Numerical computation of differential-algebraic equations for non-linear dynamics of multibody systems involving contact forces. J Mech Des 123:272–281CrossRefGoogle Scholar
  19. Gharib M, Hurmuzlu Y (2012) A new contact force model for low coefficient of restitution impact. J Appl Mech 79(6):064506CrossRefGoogle Scholar
  20. Gilardi G, Sharf I (2002) Literature survey of contact dynamics modeling. Mech Mach Theory 37:1213–1239MathSciNetCrossRefzbMATHGoogle Scholar
  21. Goldsmith W (1960) Impact: The theory and physical behaviour of colliding solids. Edward Arnold Ltd, London, EnglandzbMATHGoogle Scholar
  22. Gonthier Y, McPhee J, Lange C, Piedboeuf J-C (2004) A regularized contact model with asymmetric damping and dwell-time dependent friction. Multibody Sys Dyn 11:209–233CrossRefzbMATHGoogle Scholar
  23. Greenwood DT (1965) Principles of dynamics. Prentice Hall, Englewood Cliffs, New JerseyGoogle Scholar
  24. Guess TM, Thiagarajan G, Kia M, Mishra M (2010) A subject specific multibody model of the knee with menisci. Med Eng Phys 32:505–515CrossRefGoogle Scholar
  25. Hartog JP (1985) Mechanical vibrations. Dover Publications, Inc., New YorkzbMATHGoogle Scholar
  26. Hegazy S, Rahnejat H, Hussain K (1999) Multi-Body dynamics in full-vehicle handling analysis under transient manoeuvre. Proc Inst Mech Eng Part K: J Multibody Dyn 213:19–31Google Scholar
  27. Herbert RG, McWhannell DC (1977) Shape and frequency composition of pulses from an impact pair. J Eng Ind 99:513–518CrossRefGoogle Scholar
  28. Hertz H (1881) Über die Berührung fester elastischer Körper. Journal reine und angewandte Mathematik 92:156–171MathSciNetGoogle Scholar
  29. Hossein B, Ebrahim A (2013) Dynamic properties of golden delicious and red delicious apple under normal contact force models. J Texture Stud 44(6):409–417CrossRefGoogle Scholar
  30. Hu S, Guo X (2015) A dissipative contact force model for impact analysis in multibody dynamics. Multibody Sys Dyn 35:131–151MathSciNetCrossRefGoogle Scholar
  31. Hunt KH, Crossley FRE (1975) Coefficient of restitution interpreted as damping in vibroimpact. J Appl Mech 7:440–445CrossRefGoogle Scholar
  32. Jankowski R (2006) Analytical expression between the impact damping ratio and the coefficient of restitution in the non-linear viscoelastic model of structural pounding. Earthq Eng Struct Dyn 35(4):517–524MathSciNetCrossRefGoogle Scholar
  33. Khulief YA, Shabana AA (1987) A continuous force model for the impact analysis of flexible multibody systems. Mech Mach Theory 22:213–224CrossRefGoogle Scholar
  34. Koshy CS, Flores P, Lankarani HM (2013) Study of the effect of contact force model on the dynamic response of mechanical systems with dry clearance joints: computational and experimental approaches. Nonlinear Dyn 73(1–2):325–338CrossRefGoogle Scholar
  35. Kuwabara G, Kono K (1987) Restitution coefficient in a collision between two spheres. Jpn J Appl Phys 26:1230–1233CrossRefGoogle Scholar
  36. Lankarani HM (1988) Canonical equations of motion and estimation of parameters in the analysis of impact problems. Ph.D. Dissertation, University of Arizona, Tucson, ArizonaGoogle Scholar
  37. Lankarani HM, Nikravesh PE (1990) A contact force model with hysteresis damping for impact analysis of multibody systems. J Mech Des 112:369–376CrossRefGoogle Scholar
  38. Lankarani HM, Nikravesh PE (1992) Hertz contact force model with permanent indentation in impact analysis of solids. ASME Advances in Design Automation, DE-vol 44–2, pp 377–312, ASME Design Technical Conferences, Scottsdale, AZ, USAGoogle Scholar
  39. Lankarani HM, Nikravesh PE (1994) Continuous contact force models for impact analysis in multibody systems. Nonlinear Dyn 5:193–207Google Scholar
  40. Lee HS, Yoon YS (1994) Impact analysis of flexible mechanical system using load-dependent ritz vectors. Finite Elem Anal Des 15:201–217CrossRefzbMATHGoogle Scholar
  41. Lee J, Herrmann HJ (1993) Angle of repose and angle of marginal stability: molecular dynamics of granular particles. J Phys A26:373–384Google Scholar
  42. Lee TW, Wang AC (1983) On the dynamics of intermittent-motion mechanisms, part 1: dynamic model and response. J Mech Trans Autom Des 105:534–540CrossRefGoogle Scholar
  43. Love AEH (1944) A treatise on the mathematical theory of elasticity. Dover Publications, New York, Fourth Edi-tionzbMATHGoogle Scholar
  44. Machado M, Moreira P, Flores P, Lankarani HM (2012) Compliant contact force models in multibody dynamics: evolution of the Hertz contact theory. Mech Mach Theory 53:99–121CrossRefGoogle Scholar
  45. Marhefka DW, Orin DE (1999) A compliant contact model with nonlinear damping for simulation of robotic systems. IEEE Trans Syst Man Cybern Part A Syst Hum 29(6):566–572CrossRefGoogle Scholar
  46. Maw N, Barber JR, Fawcett JN (1975) The oblique impact of elastic spheres. Wear, 101–114Google Scholar
  47. Minamoto H, Kawamura S (2011) Moderately high speed impact of two identical spheres. Int J Impact Eng 38:123–129CrossRefGoogle Scholar
  48. Moreira P, Flores P, Silva M (2012) A biomechanical multibody foot model for forward dynamic analysis. 2012 IEEE 2nd Portuguese Meeting in Bioengineering, ENGENG 2012, Coimbra, Portugal, 23–25 Feb 2012. Article Number 6331396Google Scholar
  49. Oden JT, Martins JAM (1984) Models and computational methods for dynamic friction phenomena. In: Proceedings: Fenomech III in: computer methods in applied mechanics and engineering, North Holland, AmsterdamGoogle Scholar
  50. Pamies-Vila R, Font-Llagunes JM, Lugris U, Cuadrado J (2014) Parameter identification method for a three-dimensional foot-ground contact model. Mech Mach Theory 75:107–116CrossRefGoogle Scholar
  51. Papetti S, Avanzini F, Rocchesso D (2011) Numerical methods for a nonlinear impact model: A comparative study with closed-form corrections. IEEE Trans Audio, Speech Lang 19:2146–2158CrossRefGoogle Scholar
  52. Pereira C, Ramalho A, Ambrósio J (2014) Applicability domain of internal cylindrical contact force models. Mech Mach Theory 78:141–157CrossRefGoogle Scholar
  53. Piskounov N (1990) Cálculo diferencial e integral. Edições Lopes da Silva, Porto, PortugalGoogle Scholar
  54. Ristow GH (1992) Simulating granular flow with molecular dynamics. J Phys I France 2:649–662CrossRefGoogle Scholar
  55. Rogers RJ, Andrews GC (1977) Dynamic simulation of planar mechanical systems with lubricated bearing clearances using vector-network methods. J Eng Ind 99:131–137CrossRefGoogle Scholar
  56. Schwab AL, Meijaard JP, Meijers P (2002) A comparison of revolute joint clearance model in the dynamic analysis of rigid and elastic mechanical systems. Mech Mach Theory 37:895–913CrossRefzbMATHGoogle Scholar
  57. Schwager T, Pöschel T (1998) Coefficient of normal restitution of viscous particles and cooling rate of granular gases. Phys Rev E 57(1):650–654CrossRefGoogle Scholar
  58. Shäfer J, Dippel S, Wolf ED (1996) Force Schemes in Simulations of Granular Materials. J Phys I Fr 6:5–20CrossRefGoogle Scholar
  59. Shivaswamy S (1997) Modeling contact forces and energy dissipation during impact in multibody mechanical systems. Ph.D. Dissertation, Wichita State University, Wichita, Kansas, USAGoogle Scholar
  60. Shivaswamy S, Lankarani HM (1997) Impact analysis of plate using quasi-static approach. J Mech Des 119:376–381CrossRefGoogle Scholar
  61. Silva PC, Silva MT, Martins JM (2010) Evaluation of the contact forces developed in the lower limb/orthosis interface for comfort design. Multibody Sys Dyn 24:367–388CrossRefzbMATHGoogle Scholar
  62. Steidel RF (1989) An introduction to mechanical vibrations, 3rd edn. Wiley, New YorkzbMATHGoogle Scholar
  63. Stronge WJ (2000) Impact mechanics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  64. Tsuji Y, Tanaka T, Ishida T (1992) Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technol 71(3):239–250CrossRefGoogle Scholar
  65. Wasfy TM, Noor AK (2003) Computational strategies for flexible multibody systems. Appl Mech Rev 56:553–613CrossRefGoogle Scholar
  66. Zhang X, Vu-Quoc L (2002) Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions. Int J Impact Eng 27:317–341CrossRefGoogle Scholar
  67. Zhang Y, Sharf I (2004) Compliant force modeling for impact analysis. In: Proceedings of the 2004 ASME international design technical conferences, Salt Lake City, Paper No. DETC2004-57220Google Scholar
  68. Zhang Y, Sharf I (2009) Validation of nonlinear viscoelastic contact force models for low speed impact. J Appl Mech 76(051002):12pGoogle Scholar
  69. Zhiying Q, Qishao L (2006) Analysis of impact process based on restitution coefficien. J Dyn Control 4:294–298Google Scholar
  70. Zukas JA, Nicholas T, Greszczuk LB, Curran DR (1982) Impact dynamics. Wiley, New YorkGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of MinhoGuimaraesPortugal
  2. 2.Department of Mechanical EngineeringWichita State UniversityWichitaUSA

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