Application of an augmented Lagrangian approach to multibody systems with equality motion constraints
The focus of this work is on dynamics of multibody systems subject to bilateral motion constraints. First, a new set of equations of motion is employed, expressed as a coupled system of strongly nonlinear second-order ordinary differential equations. After putting these equations in a weak form, the...
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| Published in | Nonlinear dynamics Vol. 99; no. 1; pp. 753 - 776 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.01.2020
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0924-090X 1573-269X |
| DOI | 10.1007/s11071-019-05059-6 |
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| Abstract | The focus of this work is on dynamics of multibody systems subject to bilateral motion constraints. First, a new set of equations of motion is employed, expressed as a coupled system of strongly nonlinear second-order ordinary differential equations. After putting these equations in a weak form, the position, velocity and momentum type quantities are assumed to be independent. This leads to a three-field set of equations of motion. Next, an alternative formulation is developed, based on optimization principles. It is shown that the equations of motion can eventually be cast in a form obtained by application of an augmented Lagrangian formulation, after introducing an appropriate set of penalty terms. This final set of equations is then used as a basis for developing a new time integration scheme. The validity and numerical efficiency of this scheme is verified by applying it to several example systems. In those examples, special emphasis is put on illustrating the advantages of the new method when applied to selected mechanical systems, involving redundant constraints or singular configurations. |
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| AbstractList | The focus of this work is on dynamics of multibody systems subject to bilateral motion constraints. First, a new set of equations of motion is employed, expressed as a coupled system of strongly nonlinear second-order ordinary differential equations. After putting these equations in a weak form, the position, velocity and momentum type quantities are assumed to be independent. This leads to a three-field set of equations of motion. Next, an alternative formulation is developed, based on optimization principles. It is shown that the equations of motion can eventually be cast in a form obtained by application of an augmented Lagrangian formulation, after introducing an appropriate set of penalty terms. This final set of equations is then used as a basis for developing a new time integration scheme. The validity and numerical efficiency of this scheme is verified by applying it to several example systems. In those examples, special emphasis is put on illustrating the advantages of the new method when applied to selected mechanical systems, involving redundant constraints or singular configurations. |
| Author | Natsiavas, S. Paraskevopoulos, E. Potosakis, N. |
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| Cites_doi | 10.1115/1.4027671 10.1007/s11071-016-2774-5 10.1137/0312021 10.1007/978-0-387-21792-5 10.1023/A:1019581227898 10.1007/978-94-011-6450-4 10.1016/j.ijnonlinmec.2015.07.007 10.1017/CBO9780511610523 10.1007/978-94-007-0335-3 10.1007/s11071-012-0727-1 10.1007/BF00047121 10.1115/1.3079826 10.1017/CBO9780511804441 10.1007/b97376 10.1007/s11071-012-0413-3 10.1007/s11044-016-9510-2 10.1007/s11071-014-1783-5 10.1017/S0962492904000212 10.1007/s11071-011-0224-y 10.1115/1.2389043 10.1016/j.ijnonlinmec.2009.01.006 10.1007/s11071-009-9579-8 10.1016/j.ijsolstr.2012.09.001 10.1007/BF01833296 10.1007/b98874 |
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| Keywords | Analytical dynamics Multibody dynamics Generalized Gauss principle Augmented Lagrangian formulation Bilateral constraints Weak form of equations of motion |
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| References | BoydSVandenbergheLConvex Optimization2004New YorkCambridge University Press10.1017/CBO9780511804441 RektorysKVariational Methods in Mathematics, Science and Engineering1977DordrechtD. Reidel Publishing Company10.1007/978-94-011-6450-4 Paraskevopoulos, E., Potosakis, N., Natsiavas, S.: Application of an augmented Lagrangian methodology to dynamics of multibody systems with equality constraints. In: 9th GRACM International Congress on Computational Mechanics, Chania, Greece (2018) PapalukopoulosCNatsiavasSDynamics of large scale mechanical models using multi-level substructuringASME J. Comput. Nonlinear Dyn.20072405110.1115/1.2389043 PapastavridisJGTensor Calculus and Analytical Dynamics1999Boca RatonCRC Press RockafellarRAugmented Lagrange multiplier functions and duality in nonconvex programmingSIAM J. Control19741226828538416310.1137/0312021 IFToMM T.C. for Multibody Dynamics, Library of Computational Benchmark Problems. http://www.iftomm-multibody.org/benchmark BlochAMNonholonomic Mechanics and Control2003New YorkSpringer10.1007/b97376 BertsekasDPConvex Optimization Theory2009BelmontAthena Scientific1242.90001 GreenwoodDTPrinciples of Dynamics1988Englewood CliffsPrentice-Hall Inc. ParaskevopoulosENatsiavasSOn application of Newton’s law to mechanical systems with motion constraintsNonlinear Dyn.201372455475304398110.1007/s11071-012-0727-1 DopicoDGonzalezFCuadradoJKövecsesJDetermination of holonomic and nonholonomic constraint reactions in an index-3 augmented Lagrangian formulation with velocity and acceleration projectionsASME J. Comput. Nonlinear Dyn.2014904100610.1115/1.4027671 BertsekasDPConstraint Optimization and Lagrange Multiplier Methods1982New YorkAcademic Press0572.90067 KurdilaAJJunkinsJLHsuSLyapunov stable penalty methods for imposing holonomic constraints in multibody system dynamicsNonlinear Dyn.1993451820708.70013 MotionSolve v14.0, User Guide, Altair Engineering Inc., Irvine, California, USA BauchauOAFlexible Multibody Dynamics2011LondonSpringer10.1007/978-94-007-0335-3 García OrdenJCEnergy considerations for the stabilization of constrained mechanical systems with velocity projectionNonlinear Dyn.2010604962261000810.1007/s11071-009-9579-8 García OrdenJCCondeSCControllable velocity projection for constraint stabilization in multibody dynamicsNonlinear Dyn.201268245257290417210.1007/s11071-011-0224-y NocedalJWrightSJNumerical Optimization1999New YorkSpringer10.1007/b98874 LeineRINijmeijerHDynamics and Bifurcations of Non-smooth Mechanical Systems2013BerlinSpringer1068.70003 PacejkaHBTyre and Vehicle Dynamics20123OxfordButterworth-Heinemann Potosakis, N., Paraskevopoulos, E., Natsiavas, S.: Numerical integration of a new set of equations of motion for a class of multibody systems using an augmented Lagrangian approach. In: 5th Joint International Conference on Multibody System Dynamics, Lisbon, Portugal (2018) GeradinMCardonaAFlexible Multibody Dynamics2001New YorkWiley0874.73072 BayoELedesmaRAugmented Lagrangian and mass-orthogonal projection methods for constrained multibody dynamicsNonlinear Dyn.19969113130139400410.1007/BF01833296 TheodosiouCNatsiavasSDynamics of finite element structural models with multiple unilateral constraintsInt. J. Non-linear Mech.20094437138210.1016/j.ijnonlinmec.2009.01.006 ŽenišekANonlinear Elliptic and Evolution Problems and their Finite Element Approximations1990LondonAcademic Press0731.65090 FloresPLeineRIGlockerChApplication of the nonsmooth dynamics approach to model and analysis of the contact-impact events in cam-follower systemsNonlinear Dyn.20126921172133294554510.1007/s11071-012-0413-3 BauchauOAEppleABottassoCLScaling of constraints and augmented Lagrangian formulations in multibody dynamics simulationsASME J. Comput. Nonlinear Dyn.2009402100710.1115/1.3079826 FrankelTThe Geometry of Physics: An Introduction1997New YorkCambridge University Press0888.58077 ParaskevopoulosENatsiavasSWeak formulation and first order form of the equations of motion for a class of constrained mechanical systemsInt. J. Non-linear Mech.20157720822210.1016/j.ijnonlinmec.2015.07.007 BlajerWAugmented Lagrangian formulation: geometrical interpretation and application to systems with singularities and redundancyMultibody Syst. Dyn.20028141159190509110.1023/A:1019581227898 MurrayRMLiZSastrySSA Mathematical Introduction to Robot Manipulation1994Boca RatonCRC Press0858.70001 ShabanaAADynamics of Multibody Systems20053New YorkCambridge University Press10.1017/CBO9780511610523 ParaskevopoulosENatsiavasSA new look into the kinematics and dynamics of finite rigid body rotations using Lie group theoryInt. J. Solids Struct.201350577210.1016/j.ijsolstr.2012.09.001 Neimark, J.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems. Translations of Mathematical Monographs, American Mathematical Society 33, Providence, RI (1972) BrenanKECampbellSLPetzholdLRNumerical Solution of Initial-Value Problems in Differential-Algebraic Equations1989New YorkNorth-Holland Adams, M.S.C.: User Guide. MSC Software Corporation, California, USA (2016) MarsdenJERatiuTSIntroduction to Mechanics and Symmetry19992New YorkSpringer10.1007/978-0-387-21792-5 NatsiavasSParaskevopoulosEA set of ordinary differential equations of motion for constrained mechanical systemsNonlinear Dyn.2015791911193810.1007/s11071-014-1783-5 BenziMicheleGolubGene H.LiesenJörgNumerical solution of saddle point problemsActa Numerica2005141137216834210.1017/S0962492904000212 GonzalezFDopicoDPastorinoRCuadradoJBehaviour of augmented Lagrangian and Hamiltonian methods for multibody dynamics in the proximity of singular configurationsNonlinear Dyn.20168514911508352013510.1007/s11071-016-2774-5 MashayekhiMJKövecsesJA comparative study between the augmented Lagrangian method and the complementarity approach for modeling the contact problemMultibody Syst. Dyn.201740327345366924610.1007/s11044-016-9510-2 DP Bertsekas (5059_CR22) 2009 JC García Orden (5059_CR33) 2010; 60 E Bayo (5059_CR23) 1996; 9 RM Murray (5059_CR2) 1994 DP Bertsekas (5059_CR13) 1982 KE Brenan (5059_CR7) 1989 HB Pacejka (5059_CR37) 2012 JG Papastavridis (5059_CR17) 1999 AM Bloch (5059_CR3) 2003 E Paraskevopoulos (5059_CR29) 2013; 50 F Gonzalez (5059_CR27) 2016; 85 T Frankel (5059_CR18) 1997 Michele Benzi (5059_CR28) 2005; 14 S Natsiavas (5059_CR9) 2015; 79 P Flores (5059_CR41) 2012; 69 E Paraskevopoulos (5059_CR12) 2015; 77 JC García Orden (5059_CR34) 2012; 68 MJ Mashayekhi (5059_CR42) 2017; 40 OA Bauchau (5059_CR6) 2011 K Rektorys (5059_CR10) 1977 OA Bauchau (5059_CR25) 2009; 4 5059_CR30 R Rockafellar (5059_CR40) 1974; 12 5059_CR31 C Papalukopoulos (5059_CR38) 2007; 2 A Ženišek (5059_CR11) 1990 S Boyd (5059_CR15) 2004 AA Shabana (5059_CR4) 2005 M Geradin (5059_CR5) 2001 W Blajer (5059_CR24) 2002; 8 DT Greenwood (5059_CR1) 1988 J Nocedal (5059_CR14) 1999 JE Marsden (5059_CR20) 1999 E Paraskevopoulos (5059_CR8) 2013; 72 5059_CR16 RI Leine (5059_CR21) 2013 D Dopico (5059_CR26) 2014; 9 AJ Kurdila (5059_CR32) 1993; 4 C Theodosiou (5059_CR39) 2009; 44 5059_CR19 5059_CR35 5059_CR36 |
| References_xml | – reference: BrenanKECampbellSLPetzholdLRNumerical Solution of Initial-Value Problems in Differential-Algebraic Equations1989New YorkNorth-Holland – reference: BertsekasDPConstraint Optimization and Lagrange Multiplier Methods1982New YorkAcademic Press0572.90067 – reference: LeineRINijmeijerHDynamics and Bifurcations of Non-smooth Mechanical Systems2013BerlinSpringer1068.70003 – reference: BlochAMNonholonomic Mechanics and Control2003New YorkSpringer10.1007/b97376 – reference: MarsdenJERatiuTSIntroduction to Mechanics and Symmetry19992New YorkSpringer10.1007/978-0-387-21792-5 – reference: BauchauOAEppleABottassoCLScaling of constraints and augmented Lagrangian formulations in multibody dynamics simulationsASME J. Comput. Nonlinear Dyn.2009402100710.1115/1.3079826 – reference: GreenwoodDTPrinciples of Dynamics1988Englewood CliffsPrentice-Hall Inc. – reference: IFToMM T.C. for Multibody Dynamics, Library of Computational Benchmark Problems. http://www.iftomm-multibody.org/benchmark – reference: GonzalezFDopicoDPastorinoRCuadradoJBehaviour of augmented Lagrangian and Hamiltonian methods for multibody dynamics in the proximity of singular configurationsNonlinear Dyn.20168514911508352013510.1007/s11071-016-2774-5 – reference: BauchauOAFlexible Multibody Dynamics2011LondonSpringer10.1007/978-94-007-0335-3 – reference: ŽenišekANonlinear Elliptic and Evolution Problems and their Finite Element Approximations1990LondonAcademic Press0731.65090 – reference: BayoELedesmaRAugmented Lagrangian and mass-orthogonal projection methods for constrained multibody dynamicsNonlinear Dyn.19969113130139400410.1007/BF01833296 – reference: FloresPLeineRIGlockerChApplication of the nonsmooth dynamics approach to model and analysis of the contact-impact events in cam-follower systemsNonlinear Dyn.20126921172133294554510.1007/s11071-012-0413-3 – reference: MurrayRMLiZSastrySSA Mathematical Introduction to Robot Manipulation1994Boca RatonCRC Press0858.70001 – reference: BlajerWAugmented Lagrangian formulation: geometrical interpretation and application to systems with singularities and redundancyMultibody Syst. Dyn.20028141159190509110.1023/A:1019581227898 – reference: PapalukopoulosCNatsiavasSDynamics of large scale mechanical models using multi-level substructuringASME J. Comput. Nonlinear Dyn.20072405110.1115/1.2389043 – reference: BoydSVandenbergheLConvex Optimization2004New YorkCambridge University Press10.1017/CBO9780511804441 – reference: PacejkaHBTyre and Vehicle Dynamics20123OxfordButterworth-Heinemann – reference: DopicoDGonzalezFCuadradoJKövecsesJDetermination of holonomic and nonholonomic constraint reactions in an index-3 augmented Lagrangian formulation with velocity and acceleration projectionsASME J. Comput. Nonlinear Dyn.2014904100610.1115/1.4027671 – reference: RektorysKVariational Methods in Mathematics, Science and Engineering1977DordrechtD. Reidel Publishing Company10.1007/978-94-011-6450-4 – reference: Adams, M.S.C.: User Guide. MSC Software Corporation, California, USA (2016) – reference: MashayekhiMJKövecsesJA comparative study between the augmented Lagrangian method and the complementarity approach for modeling the contact problemMultibody Syst. Dyn.201740327345366924610.1007/s11044-016-9510-2 – reference: ShabanaAADynamics of Multibody Systems20053New YorkCambridge University Press10.1017/CBO9780511610523 – reference: García OrdenJCCondeSCControllable velocity projection for constraint stabilization in multibody dynamicsNonlinear Dyn.201268245257290417210.1007/s11071-011-0224-y – reference: ParaskevopoulosENatsiavasSOn application of Newton’s law to mechanical systems with motion constraintsNonlinear Dyn.201372455475304398110.1007/s11071-012-0727-1 – reference: BenziMicheleGolubGene H.LiesenJörgNumerical solution of saddle point problemsActa Numerica2005141137216834210.1017/S0962492904000212 – reference: PapastavridisJGTensor Calculus and Analytical Dynamics1999Boca RatonCRC Press – reference: Neimark, J.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems. Translations of Mathematical Monographs, American Mathematical Society 33, Providence, RI (1972) – reference: MotionSolve v14.0, User Guide, Altair Engineering Inc., Irvine, California, USA – reference: TheodosiouCNatsiavasSDynamics of finite element structural models with multiple unilateral constraintsInt. J. Non-linear Mech.20094437138210.1016/j.ijnonlinmec.2009.01.006 – reference: NocedalJWrightSJNumerical Optimization1999New YorkSpringer10.1007/b98874 – reference: NatsiavasSParaskevopoulosEA set of ordinary differential equations of motion for constrained mechanical systemsNonlinear Dyn.2015791911193810.1007/s11071-014-1783-5 – reference: FrankelTThe Geometry of Physics: An Introduction1997New YorkCambridge University Press0888.58077 – reference: KurdilaAJJunkinsJLHsuSLyapunov stable penalty methods for imposing holonomic constraints in multibody system dynamicsNonlinear Dyn.1993451820708.70013 – reference: Paraskevopoulos, E., Potosakis, N., Natsiavas, S.: Application of an augmented Lagrangian methodology to dynamics of multibody systems with equality constraints. 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| SubjectTerms | Automotive Engineering Classical Mechanics Control Differential equations Dynamical Systems Engineering Equations of motion Mathematical analysis Mechanical Engineering Mechanical systems Multibody systems Nonlinear equations Optimization Ordinary differential equations Original Paper Time integration Vibration |
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| Title | Application of an augmented Lagrangian approach to multibody systems with equality motion constraints |
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