ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 15 (2020), 545 -- 560
This work is licensed under a Creative Commons Attribution 4.0 International License.
MODELLING THE DURABILITY OF MULTIBODY TOTAL HIP JOINT PROSTHESIS
Virgil Florescu, Laurentiu Rece, Aurel Gherghina and Adriana Tudorache
Abstract. The Total Hip Prosthesis (THP) is one of the biggest successes of the 20th century in the field of orthopaedic biomechanical engineering. The loss of stability THP are scheduled failure. The percentage of good functioning after 10 years of operation for classically THP is 94\According to the data from the Swedish Prosthesis Register, the working time of a prosthesis reaches an average of 15 years. It is well known that the premature failure of the THP implant, in which there is frictional slip between the acetabular cup and the femoral head, also depends on the surgical accuracy which is required to provide functional angles. One solution is to promote hip prosthesis with multibody rolling. Predicting their function and the analytical determination of the lastingness of multibody prostheses is a challenge. The purpose of this paper is to provide a reliable solution to make a prediction on the durability of a THP multibody implant with the use of computing resources available to any user. This may also be the basis for subsequent risk analyzes of implant failure depending on the physical aspects of the patient and the features of the prosthesis.
2020 Mathematics Subject Classification: 70G10, 70H03, 92-05.92C50
Keywords: total hip joint prosthesis sustainability, durability determination, sparse matrices
References
M. Abadie, Dynamic simulation of rigid bodies: modelling of frictional contacts. In: Brogliato, B. (ed.) Impacts in Mechanical Systems, 61--144, Analysis and Modelling. Lecture Notes in Physics, 551 Springer Berlin, 2000. MR1843042. Zbl 1004.70010.
V. Acary and B. Brogliato, Numerical Methods for Nonsmooth Dynamical Systems Lecture Notes in Applied and Computational Mechanics, 35, Springer Berlin, 2008. Zbl 1173.74001.
S. Adly, F. Nacry and L. Thibault, Preservation of prox-regularity of sets and application to constrained optimization. SIAM J. Optim. 26(1)(2016), 448--473. MR3458154. Zbl 1333.49031.
M. Anitescu, J.F. Cremer and F.A. Potra, On the existence of solutions to complementarity formulations of contact problems with friction. In: Ferris, M.C., Pang, J.S. (eds.) Complementarity and Variational Problems. State of the Art, (1997), 12--21. SIAM, Philadelphia. MR1445069. Zbl 0887.70009.
M. Anitescu and F.A. Potra, Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems, Nonlinear Dyn. 24 (1997), 405--437. MR1474672. Zbl 0899.70005.
H. Audren and A. Kheddar, 3-D robust stability polyhedron in multicontact, IEEE Trans. Robot. 34(2) (2018), 388--403 .
D. Baraff, Issues in computing contact forces for non-penetrating rigid bodies, Algorithmica 10 (2--4) (1993), 292--352. MR1231367. Zbl 0777.70006.
D.S. Bernstein, Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory, Princeton University Press, Princeton, 2005. Zbl 1075.15001.
A. Blumentals, B. Brogliato and F. Bertails-Descoubes, The contact problem in Lagrangian systems subject to bilateral and unilateral constraints, with or without sliding Coulomb's friction: a tutorial, Multibody Syst. Dyn. 38 (2016), 43--76. MR3523669. Zbl 1372.70044.
B. Brogliato, Inertial couplings between unilateral and bilateral holonomic constraints in frictionless Lagrangian systems, Multibody Syst. Dyn. 29 (2013), 289--325. MR3027721. Zbl 1271.70032.
L. Capitanu and V. Florescu, An Overview of the Researches on Hip Prosthesis with Rolling Friction, LAP LAMBERT Academic Publishing, 2015.
S. Caron, Q. C. Pham and Y. Nakamura, ZMP support areas for multicontact mobility under frictional constraints, IEEE Trans. Robot. 33(1)(2017), 67--80.
D. Choudhury, R. Horn and S. Pierce, Quasi-positive definite operators and matrices, Linear Algebra Appl. 99 (1988), 161--176. MR0925155. Zbl 0654.15022.
H. Dahlstr, Evaluation of a metal-on-metal total hip arthroplasty system-Ph. D Thesis, Stockholm (2017).
j. Fraczek and M. Wojtyra, On the unique solvability of a direct dynamics problem for mechanisms with redundant constraints and Coulomb friction, Mech. Mach. Theory 46 (2011), 312--334. Zbl 1336.70020.
D. Gamble, P. K. Jaiswal, I. Lutz and K. D. Johnston, The Use of Ceramics in Total Hip Arthroplasty, Orthopedics and Rheumatology Open Access Journal 4 (3) (2017) DOI: 10.19080/OROAJ.2017.04.555636.
J.B. Gilbert, C. Moler and R. Schreiber, Sparse matrices in MATLAB: Design and Implementation, Tech. Report (1991), Xerox, Palo Alto Research Center.
C. Glocker and F. Pfeiffer, Dynamical systems with unilateral contacts, Nonlinear Dyn. 3 (4) (1992), 245--259.
J.G. de Jalón and M.D. Gutteriez-Lopez, Multibody dynamics with redundant constraints and singular mass matrix: existence, uniqueness, and determinatio determination of solutions for accelerations and constraint forces, Multibody Syst. Dyn. 30 (3) (2013), 311--341. MR3102982. Zbl 1274.70010.
J.G. de Jalón, J. Unda and A. Avello, Natural coordinates for the computer analysis of multibody systems, Comput. Methods Appl. Mech. Eng. 56 (1986), 309–327. Zbl 0577.70004.
A. Laulusa and O.A. Bauchau, Review of classical approaches for constraint enforcement in multibody systems, J. Comput. Nonlinear Dyn. 3(1) (2008), 011004 .
R.I. Leine and N. van de Wouw, Stability and convergence of mechanical systems with unilateral constraints, Lecture Notes in Applied and Computational Mechanics 6 Springer Berlin, 2008. Zbl 1143.70001.
P. Lötstedt, Mechanical systems of rigid bodies subject to unilateral constraints, SIAM J. Appl. Math. 42(2) (1982), 281--296. MR0650224. Zbl 0489.70016.
D. Negrut, R. Serban and A. Tasora, Posing multibody dynamics with friction and contact as a differential complementarity problem, J. Comput. Nonlinear Dyn. 13 (2018), 014503.
J.S. Pang and J.C. Trinkle, Complementarity formulation and existence of solutions of dynamic rigid-body contact problems with Coulomb friction. Math. Program. 73(2) (1996), 199--226. MR1392162. Zbl 0854.70008.
A. Peiret, J. Kovecses and J. M. Font-Llagunes, Analysis of friction coupling and the Painleve paradox in multibody systems, Multibody System Dynamics, 45 (2019), 361-378. MR3910146. Zbl 1408.70006.
A. Peters, A. J.H. Veldhuijzen, M. Tijink, R. W. Poolman and R. M.H.A. Huis In T Veld - Patient restrictions following total hip arthroplasty: A national survey, Acta Orthop. Belg., 83 (2017), 45-52.
D. Rus, L. Capitanu and L. L. Badita, A qualitative correlation between friction coefficient and steel surface wear in linear dry sliding contact to polymers with SGF, Friction, 2(1) (2014), 47-57.
S.-M. Song and X. Gao, The mobility equation and the solvability of joint forces/torques in dynamic analysis, ASME Journal of Mechanical Design, 114 (1992), 257-262.
J.C. Trinkle, J.S. Pang, S. Sudarsky and G. Lo, On dynamic multi-rigid-body contact problems with Coulomb friction, J. Appl. Math. Mech./Z. Angew. Math. Mech., 77(4) (1997), 267--279. MR1449130. Zbl 0908.70008.
M. Wojtyra, Joint reactions in rigid body mechanisms with dependent constraints, Mech. Mach. Theory 44 (2009), 2265--2278. Zbl 1247.70026.
Virgil Florescu - correspondent author
Department of Technology Mechanics, Technical University of Civil Engineering,
69 Plevnei route Bucharest, Romania.
e-mail: florescuvirgil@yahoo.com
Laurentiu Rece
Department of Technology Mechanics, Technical University of Civil Engineering,
69 Plevnei route Bucharest, Romania.
e-mail: rece@utcb.ro
Aurel Gherghina
Ministerul Apărării, Romania.
e-mail: stefan@lew.ro
Adriana Tudorache
Faculty of Engineering, Constantin Brancusi University from Targu-Jiu,
Republicii Street no.1, Targu-Jiu, Romania.
e-mail: adriana_ty2006@yahoo.com