Skip to main content
Log in

Dynamic modeling and analysis of wear in spatial hard-on-hard couple hip replacements using multibody systems methodologies

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

Abstract

Wear plays a key role in primary failure of artificial hip articulations. Thus, the main goal of this work is to investigate the influence of friction-induced vibration on the predicted wear of hard hip arthroplasties. This desideratum is reached by developing a three-dimensional multibody dynamic model for a hip prosthesis taking the spatial nature of the physiological loading and motion of the human body into account. The calculation of the intra-joint contact forces developed is based on a continuous contact force approach that accounts for the geometrical and materials properties of the contacting surfaces. In addition, the friction effects due to the contact between hip components are also taken into account. The vibration of the femoral head inside the cup associated with stick-slip friction, negative-sloping friction and dynamic variation in intra-joint contact force has been also incorporated in the present hip articulation model. The friction-induced vibration increases the sliding distance of the contact point between the head and cup surfaces by altering its micro- and macro-trajectories, and consequently affects the wear. In the present work, the Archard’s wear law is considered and embedded in the dynamic hip multibody model, which allows for the prediction of the wear developed in the hip joint. With the purpose of having more realistic wear simulation conditions, the geometries of the acetabular cup and femoral head are updated throughout the dynamic analysis. The main results obtained from computational simulations for ceramic-on-ceramic and metal-on-metal hip prostheses are compared and validated with those available in the best-published literature. Finally, from the study performed in the present work, it can be concluded that an important source of the high wear rates observed clinically may be due to friction-induced vibration.

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

Access this article

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Ibrahim, R.A.: Friction-induced vibration, chatter, squeal, and chaos. Part I: mechanics of contact and friction. Appl. Mech. Rev. 47(7), 209–226 (1994)

    Article  Google Scholar 

  2. Teoh, S.H., Chan, W.H., Thampuran, R.: An elasto-plastic finite element model for polyethylene wear in total hip arthroplasty. J. Biomech. 35(3), 323–330 (2002)

    Article  Google Scholar 

  3. Sfantos, G.K., Aliabadi, M.H.: Total hip arthroplasty wear simulation using the boundary element method. J. Biomech. 40, 378–389 (2007)

    Article  Google Scholar 

  4. Jourdan, F., Samida, A.: An implicit numerical method for wear modelling applied to a hip joint prosthesis problem. Comput. Methods Appl. Mech. Eng. 198, 2209–2217 (2009)

    Article  MATH  Google Scholar 

  5. Bevill, S.L., Bevill, G.R., Penmetsa, J.R., Petrella, A.J., Rullkoetter, P.J.: Finite element simulation of early creep and wear in total hip arthroplasty. J. Biomech. 38, 2365–2375 (2005)

    Article  Google Scholar 

  6. Kang, L., Galvin, A.L., Fisher, J., Jin, Z.: Enhanced computational prediction of polyethylene wear in hip joints by incorporating cross-shear and contact pressure in addition to load and sliding distance: effect of head diameter. J. Biomech. 42, 912–918 (2009)

    Article  Google Scholar 

  7. Matsoukas, G., Willing, R., Kim, Y.: Total hip wear assessment: a comparison between computational and in vitro wear assessment techniques using ISO14242 loading and kinematics. J. Biomech. Eng. 131, 1–11 (2009)

    Google Scholar 

  8. Mattei, L., Di Puccio, F., Ciulli, E.: A comparative study of wear laws for soft-on-hard hip implants using a mathematical wear model. Tribol. Int. 63, 66–77 (2013)

    Article  Google Scholar 

  9. Liu, F., Leslie, I., Williams, S., Fisher, J., Jin, Z.: Development of computational wear simulation of metal-on-metal hip resurfacing replacements. J. Biomech. 41, 686–694 (2008)

    Article  Google Scholar 

  10. Uddin, M.S., Zhang, L.C.: Predicting the wear of hard-on-hard hip joint prostheses. Wear 301, 192–200 (2013)

    Article  Google Scholar 

  11. Mattei, L., Di Puccio, F.: Wear simulation of metal-on-metal hip replacements with frictional contact. J. Tribol. 135(2), 021402, 12p (2013)

    Article  Google Scholar 

  12. Dowson, D., Jin, Z.: Metal-on-metal hip joint tribology. Proc. Inst. Mech. Eng. H J. Eng. Med. 220, 107–118 (2006)

  13. Essner, A., Sutton, K., Wang, A.: Hip simulator wear comparison of metal-on-metal, ceramic-on-ceramic and cross-linked UHMWPE bearings. Wear 259, 992–995 (2005)

    Article  Google Scholar 

  14. Flores, P.: Modeling and simulation of wear in revolute clearance joints in multibody systems. Mech. Mach. Theory 44, 1211–1222 (2009)

    Article  MATH  Google Scholar 

  15. Meng, H.C., Ludema, K.C.: Wear models and predictive equations: their form and content. Wear 181–183, 443–457 (1995)

    Article  Google Scholar 

  16. Mukras, S., Kim, N.H., Sawyer, W.G., Jackson, D.B., Bergquist, L.W.: Numerical integration schemes and parallel computation for wear prediction using finite element method. Wear 266, 822–831 (2009)

    Article  Google Scholar 

  17. Su, Y., Chen, W., Tong, Y., Xie, Y.: Wear prediction of clearance joint by integrating multi-body kinematics with finite-element method. Proc. Inst. Mech. Eng. J J. Eng. Tribol. 1(224), 815–823 (2010)

    Article  Google Scholar 

  18. Quental, C., Folgado, J., Ambrósio, J., Monteiro, J.: A multibody biomechanical model of the upper limb including the shoulder girdle. Multibody Syst. Dyn. 28(1–2), 83–108 (2012)

    Article  MathSciNet  Google Scholar 

  19. Hegadekatte, V., Huber, N., Kraft, O.: Finite element based simulation of dry sliding wear. Model. Simul. Mater. Sci. Eng. 13, 57–75 (2005)

    Article  Google Scholar 

  20. Flodin, A., Andersson, S.: A simplified model for wear prediction in helical gears. Wear 249, 285–292 (2001)

    Article  Google Scholar 

  21. Raimondi, M.T., Santambrogio, C., Pietrabissa, R., Raffelini, F., Molfetta, L.: Improved mathematical model of the wear of the cup articular surface in hip joint prostheses and comparison with retrieved components. Proc. Inst. Mech. Eng. H J. Eng. Med. 215(4), 377–391 (2001)

    Article  Google Scholar 

  22. Archard, J.F.: Contact and rubbing of flat surfaces. J. Appl. Phys. 24, 981–988 (1953)

    Article  Google Scholar 

  23. Barbour, P.S.M., Stone, M.H., Fisher, J.: A hip joint simulator study using simplified loading and motion cycles generating physiological wear paths and rates. Proc. Inst. Mech. Eng. H J. Eng. Med. 213(6), 455–467 (1999)

    Article  Google Scholar 

  24. Saikko, V., Calonius, O., Kernen, J.: Effect of slide track shape on the wear of ultra-high molecular weight polyethylene in a pin-on-disk wear simulation of total hip prosthesis. J. Biomed. Mater. Res. B Appl. Biomater. 69B(2), 141–148 (2004)

    Article  Google Scholar 

  25. Ramamurti, B., Bragdon, C.R., O’Connor, D.O., Lowenstein, J.D., Jasty, M., Estok, D.M., Harris, W.H.: Loci of movement of selected points on the femoral head during normal gait: three-dimensional computer simulation. J. Arthoplasty 11(7), 845–852 (1996)

    Article  Google Scholar 

  26. Saikko, V., Calonius, O.: Slide track analysis of the relative motion between femoral head and acetabular cup in walking and hip simulator. J. Biomech. 35(4), 455–464 (2002)

    Article  Google Scholar 

  27. Sariali, E., Stewart, T., Jin, Z., Fisher, J.: Three-dimensional modelling of in vitro hip kinematics under micro-separation regime for ceramic on ceramic total hip prosthesis: an analysis of vibration and noise. J. Biomech. 43, 326–333 (2010)

    Article  Google Scholar 

  28. Sawyer, W.G.: Wear predictions for a simple-cam including the coupled evolution of wear and load. J. Soc. Tribol. Lubr. Eng. 57, 31–36 (2001)

  29. Weiss, C., Hothan, A., Huber, G., Morlock, M., Hoffmann, N.: Friction-induced whirl vibration: root cause of squeaking in total hip arthroplasty. J. Biomech. 45, 297–303 (2012)

    Article  Google Scholar 

  30. Askari, E., Flores, P., Dabirrahmani, D., Appleyard, R.: Study of the friction-induced vibration and contact mechanics of artificial hip joints. Tribol. Int. 70, 1–10 (2014)

    Article  Google Scholar 

  31. Askari, E., Flores, P., Dabirrahmani, D., Appleyard, R.: Nonlinear vibration and dynamics of ceramic on ceramic artificial hip joints: a spatial multibody modelling. Nonlinear Dyn. 76, 1365–1377 (2014)

    Article  MathSciNet  Google Scholar 

  32. Nikravesh, P.E.: Computer-Aided Analysis of Mechanical Systems. Prentice Hall, Englewood Cliffs (1988)

    Google Scholar 

  33. 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 

  34. Bergmann, G., Deuretzbacher, G., Heller, M., Graichen, F., Rohlmann, A., Strauss, J., Duda, G.N.: Hip contact forces and gait patterns from routine activities. J. Biomech. 34(7), 859–871 (2001)

    Article  Google Scholar 

  35. Ribeiro, A., Rasmussen, J., Flores, P., Silva, L.F.: Modeling of the condyle elements within a biomechanical knee model. Multibody Syst. Dyn. 28, 181–197 (2012)

    Article  MathSciNet  Google Scholar 

  36. Lopes, P.S.T.: Geometric and structural analysis of the locking mechanism between liner and acetabular cup. MSc Dissertation in Biomedical Engineering, University of Minho, Guimarães, Portugal (2007)

  37. Stops, A., Wilcox, R., Jin, Z.: Computational modelling of the natural hip: a review of finite element and multibody simulations. Comput. Methods Biomech. Biomed. Eng. 15(9), 963–979 (2012)

    Article  Google Scholar 

  38. Mattei, L., Di Puccio, F., Piccigallo, B., Ciulli, E.: Lubrication and wear modeling of artificial hip joints: a review. Tribol. Int. 44, 532–549 (2011)

    Article  Google Scholar 

  39. Blajer, W., Czaplicki, A., Dziewiecki, K., Mazur, Z.: Influence of selected modeling and computational issues on muscle force estimates. Multibody Syst. Dyn. 24, 473–492 (2010)

    Article  MATH  Google Scholar 

  40. Ambrosio, J.: Rigid and flexible multibody dynamics tools for the simulation of systems subjected to con-tact and impact conditions. Eur. J. Solids A/Solids 19, S23–44 (2000)

    Google Scholar 

  41. Cappozzo, A., Gazzani, F.: Joint kinematic assessment during physical exercise. In: Berme, N., Cappozzo, A. (eds.) Biomechanics of Human Movement: Applications in Rehabilitation, Sports and Ergonomics, pp. 263–274. Bertec Corp., Worthington, Ohio (1990)

  42. Flores, P., Lankarani, H.M.: Dynamic response of multibody systems with multiple clearance joints. ASME J. Comput. Nonlinear Dyn. 7(3), 031003, 13p (2012)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  45. Quental, C., Folgado, J., Ambrásio, J., Monteiro, J.: Critical analysis of musculoskeletal modelling complexity in multibody biomechanical models of the upper limb. Comput. Method Biomech. Biomed. Eng. 18(7), 749–759 (2015)

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

    Article  MATH  Google Scholar 

  47. 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 

  48. Ahmed, S., Lankarani, H.M., Pereira, M.F.O.S.: Frictional impact analysis in open loop multibody mechanical system. J. Mech. Des. 121, 119–127 (1999)

    Article  Google Scholar 

  49. Flores, P., Ambrósio, J.: On the contact detection for contact-impact analysis in multibody systems. Multibody Syst. Dyn. 24(1), 103–122 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  51. Silva, P., Silva, M.T., Martins, J.: Evaluation of the contact forces developed in the lower limb/orthosis interface for comfort design. Multibody Syst. Dyn. 24, 367–388 (2010)

    Article  MATH  Google Scholar 

  52. Lopes, D.S., Silva, M.T., Ambrósio, J.A., Flores, P.: A mathematical framework for contact detection between quadric and superquadric surfaces. Multibody Syst. Dyn. 24(3), 255–280 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  53. Machado, M., Flores, P., Ambrósio, J., Completo, A.: Influence of the contact model on the dynamic response of the human knee joint. Proc. Inst. Mech. Eng. K J. Multi-body Dyn. 225(4), 344–358 (2011)

    Google Scholar 

  54. Flores, P., Leine, R., Glocker, C.: Application of the nonsmooth dynamics approach to model and analysis of the contact-impact events in cam-follower systems. Nonlinear Dyn. 69, 2117–2133 (2012)

    Article  MathSciNet  Google Scholar 

  55. Koshy, C.S., Flores, P., Lankarani, H.M.: 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), 325–338 (2013)

    Article  Google Scholar 

  56. Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn. Springer, Berlin (1993)

    MATH  Google Scholar 

  57. 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 

  58. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Dover Publications, New York (1944)

    MATH  Google Scholar 

  59. Goldsmith, W.: Impact. The Theory and Physical Behaviour of Colliding Solids. Edward Arnold Ltd, London (1960)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  61. Hetzler, H., Schwarzer, D., Seemann, W.: Analytical investigation of steady-state stability and Hopf-bifurcations occurring in sliding friction oscillators with application to low-frequency disc brake noise. Commun. Nonlinear Sci. Numer. Simul. 12, 83–99 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  62. Kanga, J., Krousgrilla, C.M., Sadeghi, F.: Oscillation pattern of stick-slip vibrations. Int. J. Non-Linear Mech. 44, 820–828 (2009)

    Article  Google Scholar 

  63. Bengisu, M.T., Akay, A.: Stability of friction-induced vibrations in multi-degree-of-freedom systems. J. Sound Vib. 171, 557–570 (1994)

    Article  MATH  Google Scholar 

  64. Hertz, H.: Über die Berührung fester elastischer Körper. J. Reine Angew. Math. 92, 156–171 (1881)

    MathSciNet  Google Scholar 

  65. Craig, J.J.: Introduction to Robotics: Mechanics and Control, 2nd edn. Addison-Wesley Longman, Reading (1989)

  66. Dorlot, J.M.: Long-term effects of alumina components in total hip prostheses. Clin. Orthop. Rel. Res. 282, 47–52 (1992)

    Google Scholar 

  67. Mittelmeier, H., Heisel, J.: Sixteen years’ experience with ceramic hip pros-theses. Clin. Orthop. Rel. Res. 282, 64–72 (1992)

    Google Scholar 

  68. Affatato, S., Traina, F., De Fine, M., Carmignato, S., Toni, A.: Alumina-on-alumina hip implants: a wear study of retrieved components. J. Bone Joint Surg. 94–B, 37–42 (2012)

    Article  Google Scholar 

  69. Stewart, T., Nevelos, J., Tipper, J., Insley, G., Streicher, R., Ingham, E., Fisher, J.: Long term simulator studies of alumina ceramic/ceramic hip joints with swing phase micro-separation; analysis of wear and wear debris generation. In: Combined Orthopaedic Research Societies Meeting, Rhodes, Greece (2001)

  70. Walter, W.L., Kurtz, S.M., Esposito, C., Hozack, W., Holley, K.G., Garino, J.P., Tuke, M.A.: Retrieval analysis of squeaking alumina ceramic-on-ceramic bearings. J. Bone Joint Surg. 93–B(2), 1597–1601 (2011)

    Article  Google Scholar 

  71. Harun, M.N., Wang, F.C., Jin, Z.M., Fisher, J.: Long-term contact-coupled wear prediction for metal-on-metal total hip joint replacement. J. Eng. Tribol. 223, 993–1001 (2009)

    Google Scholar 

  72. Medley, J.B., Chan, F.W., Krygier, J.J., Bobyn, J.D.: Comparison of alloys and designs in a hip simulator study of metal on metal implants. Clin. Orthop. Rel. Res. 329, 148–149 (1996)

    Article  Google Scholar 

  73. Chan, F.W., Bobyn, J.D., Medley, J.B., Krygier, J.J., Tanzer, M.: The Otto Aufranc Award. Wear and lubrication of metal-on-metal hip implants. Clin. Orthop. Rel. Res. 369, 10–24 (1999)

    Article  Google Scholar 

  74. Liu, F., Jin, Z.M., Grigoris, P., Hirt, F., Rieker, C.: Contact mechanics of metal-on-metal hip implants employing a metallic cup with a Uhmwpe backing. Proc. Inst. Mech. Eng. H J. Eng. Med. 217(3), 207–213 (2003)

    Article  Google Scholar 

  75. Udofia, I.J., Yew, A., Jin, Z.M.: Contact mechanics analysis of metal-on-metal hip resurfacing prostheses. Proc. Inst. Mech. Eng. H J. Eng. Med. 218(5), 293–305 (2004)

    Article  Google Scholar 

  76. Askari, E., Flores, P., Dabirrahmani, D., Appleyard, R.: A computational analysis of squeaking hip prostheses. ASME J. Comput. Nonlinear Dyn. 10(2) (2015)

  77. Goldsmith, A.A., Dowson, D., Isaac, G.H., Lancaster, J.G.: A comparative joint simulator study of the wear of metal-on-metal and alternative material combinations in hip replacements. Proc. Inst. Mech. Eng. H J. Eng. Med. 214(1), 39–47 (2000)

    Article  Google Scholar 

  78. Sieber, H.P., Rieker, C.B., Kottig, P.: Analysis of 118 Second-generation metal-on-metal retrieved hip implants. J. Bone Joint Surg. 81–B(1), 46–50 (1999)

    Article  Google Scholar 

  79. Affatato, S., Taddei, P., Carmignato, S., Modena, E., Toni, A.: Severe damage of alumina-on-alumina hip implants: wear assessments at a microscopic level. J. Eur. Ceram. Soc. 32(14), 3647–3657 (2012)

    Article  Google Scholar 

  80. Reinisch, G., Judmann, K.P., Lhotka, C., Lintner, F., Zweymuller, K.A.: Retrieval study of un-cemented metal-on-metal hip prostheses revised for early loosening. Biomaterials 24(6), 1081–1091 (2003)

    Article  Google Scholar 

  81. Al-Hajjar, M., Fisher, J., Tipper, J.L., Williams, S., Jennings, L.M.: Wear of 36-mm BIOLOX(R) delta ceramic-on-ceramic bearing in total hip replacements under edge loading conditions. Proc. Inst. Mech. Eng. H J. Eng. Med. 227(5), 535–542 (2013)

    Article  Google Scholar 

  82. Fisher, J., Al-Hajjar, M., Williams, S., Tipper, J., Ingham, E., Jennings, L.: Simulation and measurement of wear in metal-on-metal bearings in vitro-understanding the reasons for increased wear. Orthop. Trauma 26(4), 253–258 (2012)

    Article  Google Scholar 

  83. Nevelos, J.E., Ingham, E., Doyle, C., Nevelos, A.B., Fisher, J.: The influence of acetabular cup angle on the wear of “BIOLOX Forte” alumina ceramic bearing couples in a hip joint simulator. J. Mater. Sci. Mater. Med. 12, 141–144 (2001)

    Article  Google Scholar 

  84. Hatton, A., Nevelos, J.E., Nevelos, A.A., Banks, R.E., Fisher, J., Ingham, E.: Alumina-alumina artificial hip joints. Part I: a histological analysis and characterisation of wear debris by laser capture microdissection of tissues retrieved at revision. Biomaterials 23(16), 3429–3440 (2002)

    Article  Google Scholar 

  85. Al-Hajjar, M., Jennings, L.M., Begand, S., Oberbach, T., Delfosse, D., Fisher, J.: Wear of novel ceramic-on-ceramic bearings under adverse and clinically relevant hip simulator conditions. J. Biomed. Mater. Res. B Appl. Biomater. 101(8), 1456–1462 (2013)

    Article  Google Scholar 

  86. Fialho, J.C., Fernandes, P.R., Eca, L., Folgado, J.: Computational hip joint simulator for wear and heat generation. J. Biomech. 40(11), 2358–2366 (2007)

    Article  Google Scholar 

  87. Liu, F., Fisher, J., Jin, Z.M.: Effect of motion inputs on the wear prediction of artificial hip joints. Tribol. Int. 63, 105–114 (2013)

    Article  Google Scholar 

  88. Scholes, S.C., Unsworth, A., Goldsmith, A.A.J.: A frictional study of total hip joint replacements. Phys. Med. Biol. 45, 3721–3735 (2000)

    Article  Google Scholar 

  89. Hall, R.M., Unsworth, A.: Friction in hip prostheses. Biomaterials 18, 1017–1026 (1997)

    Article  Google Scholar 

  90. Brockett, C., Williams, S., Jin, Z.M., Isaac, G., Fisher, J.: Friction of total hip replacements with different bearings and loading conditions. J. Biomed. Mater. Res. B Appl. Biomater. 81B(2), 508–515 (2007)

    Article  Google Scholar 

  91. Essner, A., Sutton, K., Wang, A.: Hip simulator wear comparison of metal-on-metal, ceramic-on-ceramic and crosslinked UHMWPE bearings. Wear 259(7–12), 992–995 (2005)

    Article  Google Scholar 

Download references

Acknowledgments

The first author gratefully acknowledges Macquarie University for his International Macquarie University Research Excellence Scholarship (iMQRES)—No. 2010017. The second author would like to thank the Portuguese Foundation for Science and Technology (FCT) through the project UID/EEA/04436/2013.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ehsan Askari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Askari, E., Flores, P., Dabirrahmani, D. et al. Dynamic modeling and analysis of wear in spatial hard-on-hard couple hip replacements using multibody systems methodologies. Nonlinear Dyn 82, 1039–1058 (2015). https://doi.org/10.1007/s11071-015-2216-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-2216-9

Keywords

Navigation