[en] Mechanical systems are usually subjected not only to bilateral constraints but also to unilateral constraints. Inspired by the generalized-alpha time integration method for smooth flexible multibody dynamics, this paper presents a nonsmooth generalized-alpha method, which allows a consistent treatment of the nonsmooth phenomena induced by unilateral constraints and an accurate description of the structural vibrations during free motions. Both the algorithm and the implementation are illustrated in detail. Numerical example tests are given in the scope of both rigid and flexible body models, taking account for both linear and nonlinear systems, and comprising both unilateral and bilateral constraints. The extended nonsmooth generalized-alpha method is verified through comparison to the traditional Moreau-Jean method and the fully implicit Newmark method. Results show that the nonsmooth generalized-alpha method benefits from the accuracy and stability property of the classical generalized-alpha method with controllable numerical damping. In particular, when it comes to the analysis of flexible systems, the nonsmooth generalized-alpha method shows much better accuracy property than the other two methods.
Disciplines :
Mechanical engineering
Author, co-author :
Chen, Qiong-zhong
Acary, Vincent
Virlez, Geoffrey ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Ingénierie des véhicules terrestres
Bruls, Olivier ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire des Systèmes Multicorps et Mécatroniques
Language :
English
Title :
A nonsmooth generalized-alpha scheme for flexible multibody systems with unilateral constraints
Publication date :
November 2013
Journal title :
International Journal for Numerical Methods in Engineering
Hilber H, Hughes T, Taylor R. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering and Structural Dynamics 1977; 5(3):283-292.
Chung J, Hulbert G. A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized- α method. ASME Journal of Applied Mechanics 1993; 60(2):371-375.
Newmark NM. A method for computation for structural dyanmics. ASCE Journal of the Engineering Mechanics Division 1959; 85(3):67-94.
Géradin M, Cardona A. Flexible Multibody Dynamics: A Finite Element Approach. John Wiley & Sons: New York, 2001.
Cardona A, Géradin M. Time integration of the equations of motion in mechanism analysis. Computers and Structures 1989; 33(3):801-820.
Arnold M, Brüls O. Convergence of the generalized- α scheme for constrained mechanical systems. Multibody System Dynamics 2007; 18(2):185-202.
Lankarani H, Nikravesh P. Continuous contact force models for impact analysis in multibody analysis. Nonlinear Dynamics 1994; 5(2):193-207.
Pfeiffer F. On non-smooth dynamics. Meccanica 2008; 43(5):533-554.
Acary V, Brogliato B. Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics, Lecture Notes in Applied and Computational Mechanics, vol.35, Springer Verlag: Berlin Heidelberg, 2008.
Acary V. Higher order event capturing time-stepping schemes for nonsmooth multibody systems with unilateral constraints and impacts. Applied Numerical Mathematics 2012; 62(10):1259-1275.
Acary V. Numerical methods for the time integration of nonsmooth mechanical systems. Part II. Nonsmooth mechanical systems with impacts. published in 6th Workshop SDS2010 Structural Dynamical System: Computational Aspects, Monopoli, Italy, June 8-11, 2010.
Paoli L, Schatzman M. A numerical scheme for impact problems I: The one-dimensional case. SIAM Journal of Numerical Analysis 2002; 40(2):702-733.
Paoli L, Schatzman M. A numerical scheme for impact problems II: The multi-dimensional case. SIAM Journal of Numerical Analysis 2002; 40(2):734-768.
Moreau JJ. Unilateral contact and dry friction in finite freedom dynamics. In J.J. Moreau and P.D. Panagiotopoulos, Editors, Nonsmooth Mechanics and Applications, CISM Courses and Lectures, Wien - New York: Springer Verlag 1988; 302:1-82.
Jean M, Moreau JJ. Unilaterality and dry friction in the dynamics of rigid bodies collections. In Proc. of Contact Mech. Int. Symp., Vol.1, Curnier A (ed.). Presses Polytechniques et Universitaires Romandes: Lausanne, Switzerland, 1992; 31-48.
Jean M. The non-smooth contact dynamics method. Computer Methods in Applied Mechanics and Engineering 1999; 177(3-4):235-257.
Acary V. Projected event-capturing time-stepping schemes for nonsmooth mechanical systems with unilateral contact and Coulomb's friction. Computer Methods in Applied Mechanics and Engineering 2013; 256(10):224-250.
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 Verlag: Berlin Heidelberg, 2009.
Studer C, Leine RI, Glocker C. Step size adjustment and extrapolation for time stepping schemes in non-smooth dynamics. International Journal for Numerical Methods in Engineering 2008; 76(11):1747-1781.
Schindler T, Acary V. Timestepping schemes for nonsmooth dynamics based on discontinuous Galerkin methods: definition and outlook. Mathematics and Computers in Simulation 2013. In press.
Krause R, Walloth M. Presentation and comparison of selected algorithms for dynamic contact based on the Newmark scheme. Applied Numerical Mathematics 2012; 62(10):1393-1410.
Doyen D, Ern A, Piperno S. Time-integration schemes for the finite element dynamic Signorini problem. SIAM Journal on Scientific Computing 2011; 33(1):223-249.
Simo JC, Tarnow N. The discrete energy-momentum method. Conservering algorithms for nonlinear elastodynamics. Zangnew Math Phys 1992; 43(5):757-793.
Laursen T, Chawla V. Design of energy conserving algorithms for frictionless dynamic contact problems. International Journal for Numerical Methods in Engineering 1997; 40(5):863-886.
Chawla V, Laursen T. Energy consistent algorithms for frictional contact problem. International Journal for Numerical Methods in Engineering 1998; 42(5):799-827.
Laursen T, Love G. Improved implicit integrators for transient impact problems-geometric admissibility within the conserving framework. International Journal for Numerical Methods in Engineering 2002; 53(2):245-274.
Laursen T, Love G. Improved implicit integrators for transient impact problems: Dynamic frictional dissipation within an admissible conserving framework. Computer Methods in Applied Mechanics and Engineering 2003; 192(19):2223-2248.
Hauret P, LeTallec P. Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact. Computer Methods in Applied Mechanics and Engineering 2006; 195(37-40):4890-4916.
ZolghadrJahromi H, Izzuddin B. Energy conserving algorithms for dynamic contact analysis using Newmark methods. Computers and Structures 2013; 118:74-89.
Vola D, Pratt E, Jean M, Raous M. Consistent time discretization for dynamical frictional contact problems and complementarity techniques. European Journal of Computational Mechanics / Revue européenne des éléments finis 1998; 7(1-2-3):149-162.
Dumont Y, Paoli L. Vibrations of a beam between obstacles. Convergence of a fully discretized approximation. ESAIM, Mathematical Models and Numerical Analysis 2006; 40(4):705-734.
Schatzman M, Bercovier M. Numerical approximation of a wave equation with unilateral constraints. Mathematics of Computations 1989; 53(187):55-79.
Carpenter NJ, Taylor RL, Katona MG. Lagrange constraints for transient finite element surface contact. International Journal for Numerical Methods in Engineering 1991; 32(1):103-128.
Khenous H, Laborde P, Renard Y. On the discretization of contact problems in elastodynamics. In Analysis and simulation of contact problems, Vol.27, Wriggers P, et al. (ed.), Lecture Notes in Applied and Computational Mechanics. Springer Verlag: Berlin Heidelberg, 2006; 31-38.
Khenous HB, Laborde P, Renard Y. Mass redistribution method for finite element contact problems in elastodynamics. European Journal of Mechanics, A, Solids 2008; 27(5):918-932.
Deuflhard P, Krause R, Ertel S. A contact-stabilized Newmark method for dynamical contact problems. International Journal for Numerical Methods in Engineering 2007; 73(9):1274-1290.
Eich E. Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM Journal on Numerical Analysis 1993; 30(5):1467-1482.
Alart P, Curnier A. A mixed formulation for frictional contact problems prone to Newton like solution method. Computer Methods in Applied Mechanics and Engineering 1991; 92(3):353-375.
Christensen P, Klarbring A, Pang J, Stromberg N. Formulation and comparison of algorithms for frictional contact problems. International Journal for Numerical Methods in Engineering 1998; 42(1):145-172.
Cottle R, Pang J, Stone R. The Linear Complementarity Problem. Academic Press: Boston, MA, 1992.
Klarbring A. A mathematical programming approach to three-dimensional contact problems with friction. Computer Methods in Applied Mechanics and Engineering 1986; 58(2):175-200.
Chen Q, Acary V, Virlez G, Brüls O. A Newmark-type integrator for flexible systems considering nonsmooth unilateral constraints. Proceedings of the 2nd Joint International Conference on Multibody System Dynamics, Stuttgart, Germany, 2012.
Flores P, Leine R, Glocker C. Modeling and analysis of planar rigid multibody systems with translational clearance joints based on the non-smooth dynamics approach. Multibody System Dynamics 2010; 23(2):165-190.
