Pure Elastic Contact Force Models

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


The most important pure elastic constitutive laws commonly utilized to model and analyze contact-impact events in the context of multibody mechanical system dynamics are presented in this chapter. Additionally, the fundamental issues related to the generalized contact kinematics, developed under the framework of multibody system dynamics formulation, are briefly described. In this process, the main contact parameters are determined, namely the indentation or pseudo-penetration of the potential contacting points, and the normal contact velocity. Subsequently, the linear Hooke’s contact force model and the nonlinear Hertz’s law are presented together with a demonstrative example of application. Some other elastic contact force models are also briefly described.


Multibody dynamics Contact kinematics Elastic contact force models Hooke force model Hertz force 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. 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
  3. Atkinson KA (1989) An introduction to numerical analysis, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  4. Bei Y, Fregly BJ (2004) Multibody dynamic simulation of knee contact mechanics. Med Eng Phys 26:777–789CrossRefGoogle Scholar
  5. Brändlein J, Eschamann P, Hasbargen L (1998) Die Wälzlagerpraxis. Handbuch für die Berechnung und Gestaltung von Lagerungen. Vereinigte Fachverlage, GermanyGoogle Scholar
  6. Dietl P, Wensing J, van Nijen GC (2000) Rolling bearing damping for dynamic analysis of multi-body systems—experimental and theoretical results. Proc Inst Mech Eng Part K J Multibody Dyn 214(1):33–43CrossRefGoogle Scholar
  7. 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
  8. Flores P (2015) Concepts and Formulations for Spatial Multibody Dynamics. Springer, BerlinGoogle Scholar
  9. Flores P, Ambrósio J (2004) Revolute joints with clearance in multibody systems. Comput Struct 82:359–1369CrossRefGoogle Scholar
  10. Flores P, Ambrósio J (2010) On the contact detection for contact-impact analysis in multibody systems. Multibody Syst Dyn 24(1):103–122MathSciNetCrossRefzbMATHGoogle Scholar
  11. Flores P, Ambrósio J, Claro JCP, Lankarani HM (2006) Influence of the contact-impact force model on the dynamic response of multibody systems. Proc Inst Mech Eng Part K J Multibody Dyn 220(1):21–34Google Scholar
  12. Flores P, Ambrósio J, Claro JCP, Lankarani HM (2008) Translational joints with clearance in rigid multi-body systems. J Comput Nonlinear Dyn 3:0110071–10CrossRefGoogle Scholar
  13. Flores P, Ambrósio J, Claro JP (2004) Dynamic analysis for planar multibody mechanical systems with lubricated joints. Multibody Syst Dyn 12(1):47–74CrossRefzbMATHGoogle Scholar
  14. Gilardi G, Sharf I (2002) Literature survey of contact dynamics modeling. Mech Mach Theory 37:1213–1239MathSciNetCrossRefzbMATHGoogle Scholar
  15. Glocker C (2001) On frictionless impact models in rigid-body systems. Philos Trans Math Phys Eng Sci 359:2385–2404MathSciNetCrossRefzbMATHGoogle Scholar
  16. Glocker C (2004) Concepts for modeling impacts without friction. Acta Mech 168:1–19CrossRefzbMATHGoogle Scholar
  17. Goldsmith W (1960) Impact—the theory and physical behaviour of colliding solids. Edward Arnold Ltd, London, EnglandzbMATHGoogle Scholar
  18. Goodman LE, Keer LM (1965) The contact stress problem for an elastic sphere indenting an elastic cavity. Int J Solids Struct 1:407–415CrossRefGoogle Scholar
  19. Hertz H (1881) Über die Berührung fester elastischer Körper. J reine und angewandte Mathematik 92:156–171MathSciNetGoogle Scholar
  20. Hippmann G (2004) An algorithm for compliant contact between complexly shaped bodies. Multibody Syst Dyn 12:345–362MathSciNetCrossRefzbMATHGoogle Scholar
  21. Hunt KH, Crossley FRE (1975) Coefficient of restitution interpreted as damping in vibroimpact. J Appl Mech 7:440–445CrossRefGoogle Scholar
  22. Johnson KL (1961) Energy dissipation at spherical surfaces in contact transmitting oscillating forces. J Mech Eng Sci 3:362–368CrossRefGoogle Scholar
  23. Johnson KL (1982) One hundred years of Hertz contact. Proc Inst Mech Eng 196:363–378CrossRefGoogle Scholar
  24. Johnson KL (1999) Contact mechanics. Cambridge University Press, CambridgeGoogle Scholar
  25. 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
  26. Krempf P, Sabot J (1993) Identification of the damping in a Hertzian contact from experimental non-linear response curve. In: Proceedings of the IUTAM symposium on identification of mechanical systems. University of Wuppertal, GermanyGoogle Scholar
  27. Lankarani HM (1988) Canonical equations of motion and estimation of parameters in the analysis of impact problems. PhD Dissertation, University of Arizona, Tucson, Arizona, USAGoogle Scholar
  28. 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
  29. Liu C, Zhang K, Yang L (2005) The compliance contact model of cylindrical joints with clearances. Acta Mech Sin 21:451–458MathSciNetCrossRefzbMATHGoogle Scholar
  30. Liu C, Zhang K, Yang L (2006) Normal force-displacement relationship of spherical joints with clearances. J Comput Nonlinear Dyn 1:160–167CrossRefGoogle Scholar
  31. Liu C, Zhang K, Yang R (2007) The FEM analysis and approximate model for cylindrical joints with clearances. Mech Mach Theory 42:183–197CrossRefzbMATHGoogle Scholar
  32. Lopes DS, Silva MT, Ambrósio JA, Flores P (2010) A mathematical framework for rigid contact detection between quadric and superquadric surfaces. Multibody Syst Dyn 24(3):255–280MathSciNetCrossRefzbMATHGoogle Scholar
  33. Luo L, Nahon M (2011) Development and validation of geometry-based compliant contact models. J Comput Nonlinear Dyn 6:0110041–11CrossRefGoogle Scholar
  34. Machado M, Flores P, Ambrósio J (2014) A lookup-table-based approach for spatial analysis of contact problems. J Comput Nonlinear Dyn 9(4):041010CrossRefGoogle Scholar
  35. Machado M, Flores P, Ambrósio J, Completo A (2011) Influence of the contact model on the dynamic response of the human knee joint. Proc Inst Mech Eng Part K J Multibody Dyn 225(4):344–358Google Scholar
  36. Machado M, Flores P, Claro JCP, Ambrósio J, Silva M, Completo A, Lankarani HM (2010) Development of a planar multi-body model of the human knee joint. Nonlinear Dyn 60:459–478CrossRefzbMATHGoogle Scholar
  37. 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
  38. Mukras S, Mauntler A, Kim NH, Schmitz TL, Sawyer WG (2010) Evaluation of contact force and elastic foundation models for wear analysis of multibody systems. In: Proceedings of the ASME 2010 international design engineering technology conferences. Montreal, Quebec, Canada, Paper No: DETC2010-28750, 15–18 AugGoogle Scholar
  39. Nijen G (1997) On the overrolling of local imperfections in rolling bearings. Ph.D. Dissertation, University of Twente, The NetherlandsGoogle Scholar
  40. Nikravesh PE (1988) Computer-aided analysis of mechanical systems. Prentice Hall, Englewood Cliffs, New JerseyGoogle Scholar
  41. Pereira CM, Ramalho AL, Jmbrósio JA (2011) A critical overview of internal and external cylinder contact force models. Nonlinear Dyn 63:681–697CrossRefGoogle Scholar
  42. Pérez-González A, Fenollosa-Esteve C, Sancho-Bru JL, Sánchez-Marín FT, Vergara M, Rodríguez-Cervantes PJ (2008) A modified elastic foundation contact model for application in 3D models of the prosthetic knee. Med Eng Phys 30(3):387–398CrossRefGoogle Scholar
  43. Pombo J, Ambrósio J (2008) Application of a wheel-rail contact model to railway dynamics in small radius curved tracks. Multibody Syst Dyn 19(1–2):91–114CrossRefzbMATHGoogle Scholar
  44. Ravn P (1998) A continuous analysis method for planar multibody systems with joint clearance. Multibody Syst Dyn 2:1–24CrossRefzbMATHGoogle Scholar
  45. Sabot J, Krempf P, Janolin C (1998) Nonlinear vibrations of a sphere-plane contact excited by a normal load. J Sound Vib 214:359–375CrossRefGoogle Scholar
  46. Shigley JE, Mischke CR (1989) Mechanical engineering design. McGraw-Hill, New YorkGoogle Scholar
  47. 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
  48. Tian Q, Liu C, Machado M, Flores P (2011) A new model for dry and lubricated cylindrical joints with clearance in spatial flexible multibody systems. Nonlinear Dyn 64:25–47CrossRefzbMATHGoogle Scholar
  49. Tian Q, Zhang Y, Chen L, Flores P (2009) Dynamics of spatial flexible multibody systems with clearance and lubricated spherical joints. Comput Struct 87(13–14):913–929CrossRefGoogle Scholar
  50. Timoshenko SP, Goodier JN (1970) Theory of elasticity. McGraw Hill, New YorkzbMATHGoogle Scholar
  51. Yang DCH, Sun ZS (1985) A rotary model for spur gear dynamics. J Mech Transmissions Autom Des 107:529–535CrossRefGoogle Scholar
  52. Zhu SH, Zwiebel S, Bernhardt G (1999) Theoretical formula for calculating damping in the impact of two bodies in a multibody system. Proc Inst Mech Eng, Part C J Mech Eng Sci 213:211–216CrossRefGoogle 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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