Modeling and analysis of planar rigid multibody systems with translational clearance joints based on the non-smooth dynamics approach

The main purpose of this paper is to present and discuss a methodology for a dynamic modeling and analysis of rigid multibody systems with translational clearance joints. The methodology is based on the non-smooth dynamics approach, in which the interaction of the elements that constitute a translational clearance joint is modeled with multiple frictional unilateral constraints. In the following, the most fundamental issues of the non-smooth dynamics theory are revised. The dynamics of rigid multibody systems are stated as an equality of measures, which are formulated at the velocity-impulse level. The equations of motion are complemented with constitutive laws for the normal and tangential directions. In this work, the unilateral constraints are described by a set-valued force law of the type of Signorini’s condition, while the frictional contacts are characterized by a set-valued force law of the type of Coulomb’s law for dry friction. The resulting contact-impact problem is formulated and solved as a linear complementarity problem, which is embedded in the Moreau time-stepping method. Finally, the classical slider-crank mechanism is considered as a demonstrative application example and numerical results are presented. The results obtained show that the existence of clearance joints in the modeling of multibody systems influences their dynamics response.

[1]  D. T. Greenwood.  Principles of dynamics , 1965 .

[2]  G. Dantzig,et al.  COMPLEMENTARY PIVOT THEORY OF MATHEMATICAL PROGRAMMING , 1968 .

[3]  J. N Fawcett,et al.  Dynamics of the slider-crank mechanism with clearance in the sliding bearing , 1974 .

[4]  J. Moreau,et al.  Unilateral Contact and Dry Friction in Finite Freedom Dynamics , 1988 .

[5]  Parviz E. Nikravesh,et al.  Computer-aided analysis of mechanical systems , 1988 .

[6]  E. J. Haug,et al.  Computer aided kinematics and dynamics of mechanical systems. Vol. 1: basic methods , 1989 .

[7]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[8]  Jiri Rohn,et al.  A note on solvability of a class of linear complementarity problems , 1993, Math. Program..

[9]  S. Shaw,et al.  Chaotic and periodic dynamics of a slider-crank mechanism with slider clearance , 1994 .

[10]  Jeffrey C. Trinkle,et al.  Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with coulomb friction , 1996, Math. Program..

[11]  Friedrich Pfeiffer,et al.  Multibody Dynamics with Unilateral Contacts , 1996 .

[12]  D. Stewart,et al.  AN IMPLICIT TIME-STEPPING SCHEME FOR RIGID BODY DYNAMICS WITH INELASTIC COLLISIONS AND COULOMB FRICTION , 1996 .

[13]  D. Stewart,et al.  Time-stepping for three-dimensional rigid body dynamics , 1999 .

[14]  Michel Saint Jean,et al.  The non-smooth contact dynamics method , 1999 .

[15]  Panagiotis D. Panagiotopoulos,et al.  Inequality constraints with elastic impacts in deformable bodies. The convex case , 2000 .

[16]  C. Glocker.  On frictionless impact models in rigid-body systems , 2001, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[17]  Henk Nijmeijer,et al.  Periodic motion and bifurcations induced by the Painlevé paradox , 2002 .

[18]  C. Lamarque,et al.  Dynamics Investigation of Three Coupled Rods with a Horizontal Barrier , 2003 .

[19]  C. Glocker,et al.  A Set-Valued Force Law for Spatial Coulomb-Contensou Friction , 2003 .

[20]  D. Marghitu,et al.  Dynamic analysis of a planar rigid-link mechanism with rotating slider joint and clearance , 2003 .

[21]  Jorge Ambrósio,et al.  Revolute joints with clearance in multibody systems , 2004 .

[22]  J. Ambrósio,et al.  Dynamic Analysis for Planar Multibody Mechanical Systems with Lubricated Joints , 2004 .

[23]  H. Nijmeijer,et al.  Dynamics and Bifurcations ofNon - Smooth Mechanical Systems , 2006 .

[24]  C. Glocker,et al.  Formulation and Preparation for Numerical Evaluation of Linear Complementarity Systems in Dynamics , 2005 .

[25]  F. Pfeiffer,et al.  Simulation of Unilateral Constrained Systems with Many Bodies , 2005 .

[26]  F. Pfeiffer,et al.  Numerical aspects of non-smooth multibody dynamics , 2006 .

[27]  Ke Zhang,et al.  The FEM analysis and approximate model for cylindrical joints with clearances , 2007 .

[28]  C Chapman,et al.  Impulsive response of an automatic transmission system with multiple clearances: Formulation, simulation and experiment , 2007 .

[29]  N. Wouw,et al.  Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact , 2008 .

[30]  N. Wouw,et al.  Stability and Convergence of Mechanical Systems with Unilateral Constraints , 2008 .

[31]  Selçuk Erkaya,et al.  A neural–genetic (NN–GA) approach for optimising mechanisms having joints with clearance , 2008 .

[32]  P. Flores.  Kinematics and Dynamics of Multibody Systems with Imperfect Joints: Models and Case Studies , 2008 .

[33]  Shlomo Djerassi.  Collision with friction; Part B: Poisson’s and Stornge’s hypotheses , 2008 .

[34]  S. Djerassi.  Collision with friction; Part A: Newton’s hypothesis , 2009 .

[35]  Jorge Ambrósio,et al.  Translational Joints With Clearance in Rigid Multibody Systems , 2008 .

[36]  Christoph Glocker,et al.  Step size adjustment and extrapolation for time‐stepping schemes in non‐smooth dynamics , 2008 .

[37]  Yunqing Zhang,et al.  Dynamics of spatial flexible multibody systems with clearance and lubricated spherical joints , 2009 .

[38]  Alan Bowling,et al.  Energetically consistent simulation of simultaneous impacts and contacts in multibody systems with friction , 2009 .

[39]  Paulo Flores,et al.  Modeling and simulation of wear in revolute clearance joints in multibody systems , 2009 .

[40]  C. Glocker.  Set-Valued Force Laws: Dynamics of Non-Smooth Systems , 2012 .