finite element methods; impact; implicit; nonlinear dynamics; time integration; Non-smooth contact dynamics; friction
Abstract :
[en] This work studies the impact between spherical rigid bodies in the frame of nonsmooth contact dynamics considering friction effects. A new impact element formulation based on the classical instantaneous local Newton impact law is presented. The kinematics properties of the spheres are described by a rigid body formulation with translational and rotational degrees of freedom referred to an inertial frame. In addition, an extension of the nonsmooth generalized-alpha time integration scheme applied to collisions with multiple impacts including Coulomb's friction law is given. Six numerical examples are presented to evaluate the robustness and the performance of the proposed methodology.
Disciplines :
Mechanical engineering
Author, co-author :
Sánchez, Eliana; Centro de Investigación de Métodos Computacionales (CIMEC), Santa Fe, Argentina
Cosimo, Alejandro; Centro de Investigación de Métodos Computacionales (CIMEC), Santa Fe, Argentina
Bruls, Olivier ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire des Systèmes Multicorps et Mécatroniques
Cardona, Alberto; Centro de Investigación de Métodos Computacionales (CIMEC), Santa Fe, Argentina
Cavalieri, Federico J.; Centro de Investigación de Métodos Computacionales (CIMEC), Santa Fe, Argentina
Language :
English
Title :
Simulation of impacts between spherical rigid bodies with frictional effects
Publication date :
30 October 2024
Journal title :
International Journal for Numerical Methods in Engineering
CONICET - Consejo Nacional de Investigaciones Científicas y Técnicas
Funding text :
This work received financial support from Consejo Nacional de Investigaciones Cientificas y Técnicas (CONICET) PIP11220200101688CO and Universidad Tecnologica Nacional PID-UTN AMECAFE0008102TC which are gratefully acknowledged.
Nguyen NS, Brogliato B. Multiple Impacts in Dissipative Granular Chains. Vol 72. Springer Science & Business Media; 2013.
Flores P, Machado M, Silva MT, Martins JM. On the continuous contact force models for soft materials in multibody dynamics. Multibody Syst Dyn. 2011;25(3):357-375.
Hu G, Hu Z, Jian B, Liu L, Wan H. On the determination of the damping coefficient of non-linear spring-dashpot system to model hertz contact for simulation by discrete element method. J Comput. 2011;6:984-988.
Hunt KH, Crossley FRE. Coefficient of restitution interpreted as damping in vibroimpact. J Appl Mech. 1975;42(2):440-445.
Lankarani HM, Nikravesh PE. A contact force model with hysteresis damping for impact analysis of multibody systems. J Mech Des. 1990;112(3):369-376.
Zhang J, Li W, Zhao L, He G. A continuous contact force model for impact analysis in multibody dynamics. Mech Mach Theory. 2020;153:103946.
Zhao P, Liu J, Li Y, Chenchen W. A spring-damping contact force model considering normal friction for impact analysis. Nonlinear Dyn. 2021;105:1437-1457.
Rodrigues da Silva M, Marques F, Tavares da Silva M, Flores P. A compendium of contact force models inspired by Hunt and Crossley's cornerstone work. Mech Mach Theory. 2022;167:104501.
Moreau JJ. Unilateral contact and dry friction in finite freedom dynamics. Nonsmooth Mechanics and Applications. Springer; 1988:1-82.
Pfeiffer F, Glocker C. Multibody Dynamics with Unilateral Contacts. Wiley-VCH Verlag GmbH; 1996.
Brogliato B. Nonsmooth Mechanics. Springer International Publishing; 2016.
Acary V. Higher order event capturing time-stepping schemes for nonsmooth multibody systems with unilateral constraints and impact. Appl Numer Math. 2012;62:1259-1275.
Acary V. Projected event-capturing time-stepping schemes for nonsmooth mechanical systems with unilateral contact and Coulomb's friction. Comput Methods Appl Mech Eng. 2013;256:224-250.
Schindler T, Acary V. Timestepping schemes for nonsmooth dynamics based on discontinuous Galerkin methods: definition and outlook. Math Comput Simul. 2014;95:180-199.
Studer C. Numerics of Unilateral Contacts and Friction: Modeling and Numerical Time Integration in Non-Smooth Dynamics. Lecture Notes in Applied and Computational Mechanics. Vol 47. Springer; 2009.
Brüls O, Acary V, Cardona A. Simultaneous enforcement of constraints at position and velocity levels in the nonsmooth generalized-α$$ \alpha $$ scheme. Comput Methods Appl Mech Eng. 2014;281:131-161.
Brüls O, Acary V, Cardona A. On the Constraints Formulation in the Nonsmooth Generalized-Alpha Method. Springer International Publishing; 2018:335-374.
Chen Q-Z, Acary V, Virlez G, Brüls O. A nonsmooth generalized-α$$ \alpha $$ scheme for flexible multibody systems with unilateral constraints. Int J Numer Methods Eng. 2013;96:487-511.
Capobianco G, Harsch J, Eugster SR, Leine RI. A nonsmooth generalized-alpha method for mechanical systems with frictional contact. Int J Numer Methods Eng. 2021;122(22):6497-6526.
Tasora A, Anitescu M. A complementarity-based rolling friction model for rigid contacts. Meccanica. 2013;48(7):1643-1659.
Huang J, Vicente da Silva M, Krabbenhoft K. Three-dimensional granular contact dynamics with rolling resistance. Comput Geotech. 2013;49:289-298.
Acary V, Bourrier F. Coulomb friction with rolling resistance as a cone complementarity problem. Eur J Mech A Solids. 2021;85:104046.
De Saxcé G, Feng ZQ. New inequality and functional for contact with friction: the implicit standard material approach. Mech Struct Mach. 1991;19(3):301-325.
De Saxcé G, Feng Z-Q. The bipotential method: a constructive approach to design the complete contact law with friction and improved numerical algorithms. Math Comput Model. 1998;28(4-8):225-245.
Sánchez E, Cosimo A, Brüls O, Cardona A, Cavalieri FJ. Non-smooth numerical solution for Coulomb friction, rolling and spinning resistance of spheres applied to flexible multibody system dynamics. Multibody Syst Dyn. 2023;59(1):1-35.
Darboux G. Étude géométrique sur les percussions et le choc des corps. Bull Sci Math Astron. 1880;4(1):126-160.
Liu C, Zhao Z, Brogliato B. Frictionless multiple impacts in multibody systems. I. Theoretical framework. Proc R Soc A Math Phys Eng Sci. 2008;464(2100):3193-3211.
Liu C, Zhao Z, Brogliato B. Frictionless multiple impacts in multibody systems. II. Numerical algorithm and simulation results. Proc R Soc A Math Phys Eng Sci. 2009;465(2101):1-23.
Glocker C. An Introduction to Impacts. Springer; 2006:45-101.
Poisson SD. Traité de Mécanique. 2nd ed. Bachelier; 1833.
Gharib M, Celik A, Hurmuzlu Y. Shock absorption using linear particle chains with multiple impacts. ASME J Appl Mech. 2011;78:031005.
Cuneyt Y, Gharib M, Hurmuzlu Y. Solving frictionless rocking block problems with multiple impacts. Proc R Soc A Math Phys Eng Sci. 2009;465:3323-3339.
Carretero-González R, Khatri D, Porter MA, Kevrekidis PG, Daraio C. Dissipative solitary waves in granular crystals. Phys Rev Lett. 2009;102(2):024102.
Job S, Melo F, Sokolow A, Sen S. How hertzian solitary waves interact with boundaries in a 1d granular medium. Phys Rev Lett. 2005;94(17):178002.
Yigit AS, Ulsoy AG, Scott RA. Spring-dashpot models for the dynamics of a radially rotating beam with impact. J Sound Vib. 1990;142(3):515-525.
Newton I. Philosophiae Naturalis Principia Mathematica. Royal Society of London; 1686.
Glocker C, Aeberhard U. The geometry of Newton's cradle. Advances in Mechanics and Mathematics. Springer; 2006:185-194.
Frémond M. Rigid bodies collisions. Phys Lett A. 1995;204(1):33-41.
Han I, Gilmore BJ. Multi-body impact motion with friction analysis, simulation, and experimental validation. J Mech Des. 1993;115(3):412.
Cosimo A, Cavalieri FJ, Cardona A, Brüls O. On the adaptation of local impact laws for multiple impact problems. Nonlinear Dyn. 2020;102(4):1997-2016.
Moreau JJ. Some numerical methods in multibody dynamics: application to granular materials. Eur J Mech A Solids. 1994;13:93-114.
Moreau JJ. Numerical aspects of the sweeping process. Comput Methods Appl Mech Eng. 1999;177(3-4):329-349.
Stewart DE, Trinkle JC. An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and Coulomb friction. Int J Numer Methods Eng. 1996;39(15):2673-2691.
Brogliato B, Zhang H, Liu C. Analysis of a generalized kinematic impact law for multibody-multicontact systems, with application to the planar rocking block and chains of balls. Multibody Syst Dyn. 2012;27(3):351-382.
Zhang H, Brogliato B, Liu C. Dynamics of planar rocking-blocks with Coulomb friction and unilateral constraints: comparisons between experimental and numerical data. Multibody Syst Dyn. 2013;32:1-25.
Zhang H, Brogliato B, Liu C. Study of the planar rocking-block dynamics with Coulomb friction: critical kinetic angles. ASME J Comput Nonlinear Dyn. 2013;8(3):021002.
Bowling A, Flickinger D, Harmeyer S. Energetically consistent simulation of simultaneous impacts and contacts in multibody systems with friction. Multibody Syst Dyn. 2009;22(1):27-45.
Flickinger DM, Bowling A. Simultaneous oblique impacts and contacts in multibody systems with friction. Multibody Syst Dyn. 2010;23:249-261.
Glocker C. Energetic consistency conditions for standard impacts. Multibody Syst Dyn. 2013;29(1):77-117.
Leine RI, van de Wouw N. Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact. Nonlinear Dyn. 2008;51(4):551-583.
Glocker C, Pfeiffer F. Multiple impacts with friction in rigid multibody systems. Nonlinear Dyn. 1995;7(4):471-497.
Leine RI, van Campen DH, Glocker C. Nonlinear dynamics and modeling of various wooden toys with impact and friction. J Vib Control. 2003;9:25-78.
Pfeiffer F. Multibody systems with unilateral constraints. J Appl Math Mech. 2001;65(4):665-670.
Pfeiffer F. Energy considerations for frictional impacts. Arch Appl Mech. 2010;80(1):47-56.
Géradin M, Cardona A. Flexible Multibody Dynamics: A Finite Element Approach. Wiley; 2001.
Cosimo A, Galvez J, Cavalieri FJ, Cardona A, Brüls O. A robust nonsmooth generalized-α$$ \alpha $$ scheme for flexible systems with impacts. Multibody Syst Dyn. 2020;48(2):127-149.
Alart P, Curnier A. A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput Methods Appl Mech Eng. 1991;92(3):353-375.
Galvez J, Cavalieri FJ, Cosimo A, Brüls O, Cardona A. A nonsmooth frictional contact formulation for multibody system dynamics. Int J Numer Methods Eng. 2020;121(16):3584-3609.
Cardona A, Klapka I, Géradin M. Design of a new finite element programming environment. Eng Comput. 1994;11:365-381.
Leine RI, van de Wouw N, eds. Stability and Convergence of Mechanical Systems with Unilateral Constraints. Springer; 2008.
Bosten A, Brüls O. On the evaluation of the tangential slip increment in quasi-static beam-to-beam contact problems. ECCOMAS Thematic Conference and IACM Special Interest Conference Highly Flexible Slender Structures (HFSS 2023). University of Rijeka; 2023.
Alciatore DG. The Illustrated Principles of Pool and Billiards. 1st ed. Sterling; 2017.
Wallace RE, Schroeder MC. Analysis of billiard ball collisions in two dimensions. Am J Phys. 1988;56(9):815-819.
Jewett B. Seeking Truth of Beliefs. Tech Talk, Billiards Digest. Technical Report. 1995.
Alciatore DG. Throw—Part II: Follow and Draw Effects, Dr. Dave's Illustrated Principles. Billiards Digest; 2006;29:10.
Alciatore DG. Throw—Part III: Follow and Draw Effects, Dr. Dave's Illustrated Principles. Billiards Digest; 2006;29:11.
Corral E, Gismeros R, Marques F, Flores P, Gómez Garcia MJ, Castejón C. Dynamic modeling and analysis of pool balls interaction. Multibody Dynamics 2019: Proceedings of the 9th ECCOMAS Thematic Conference on Multibody Dynamics. Springer; 2020:79-86.
