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Multibody System Dynamics (MSD) Multibody System Dynamics (MSD) Pierangelo Masarati Revision March 9, 2020 Copyright c 2019-2020 Pierangelo Masarati Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano Questo documento `esoggetto a copyright da parte degli autori e da parte del Dipartimento di Scienze e Tecnologie Aerospaziali del Politecnico di Milano. La sua riproduzione e diffusione `elibera purch´e avvenga per scopi consentiti dalle leggi vigenti in materia editoriale nello Stato italiano, e non avvenga per fini di lucro1. La riproduzione parziale per uso personale e l’uso per scopi istruzionali sono liberamente consentiti. E` condizione irrinunciabile che il documento non venga modificato al di fuori o contro il parere degli autori, che `evincolante, e che non venga privato di questo avviso di copyright. This document is subjected to copyright by the authors and the Department of Aerospace Science and Technology of Politecnico di Milano. Its reproduction and diffusion is free provided it complies with Italian copyright rules and not for profit2. The partial reproduction for personal use and the use for instructional purposes are allowed. The document cannot be modified without or against the consent of the authors, which is binding. This copyright notice cannot be removed. 1E` consentita la stampa su carta e la fotocopia da parte di terzi per fini didattici, a un costo ragionevole, senza alcuna alterazione dei contenuti o del formato. 2The reproduction “as is” in hard copy is allowed by third parties for teaching purposes, at a reasonable cost, without any modification of the contents nor of the format. i ii Contents 1 Introduction 1-1 1.1 Objectives...................................... ....... 1-1 1.2 Resources ....................................... ...... 1-1 1.2.1 Journals...................................... 1-1 1.2.2 Organizations................................. ...... 1-1 1.2.3 Events........................................ 1-2 1.2.4 FreeSoftware.................................. ..... 1-2 I Basics 1-3 2 Kinematics of Mechanical Systems 2-1 2.1 KinematicsofaPointinSpace . .......... 2-1 2.1.1 Position...................................... 2-1 2.1.2 Velocity ...................................... 2-1 2.1.3 Acceleration.................................. ...... 2-1 2.2 Rigid-BodyKinematicsinSpace . ........... 2-2 2.2.1 Orientation ................................... ..... 2-2 2.2.2 ParametrizationofRotations . .......... 2-4 2.2.3 RotationDifferentiation . ......... 2-4 2.2.4 ParametrizationofAngularVelocity . ............ 2-7 2.2.5 Parametrization of Other Rotation Derivatives . ............... 2-7 2.2.6 AngularAcceleration. ........ 2-7 2.3 RelativeMotion.................................. ........ 2-7 2.4 IncrementalOrientationApproach . ............. 2-8 2.4.1 Updated-Updated Orientation Approach . ............ 2-9 3 Dynamics of Mechanical Systems 3-1 3.1 DynamicsofaPointMass ............................ ........ 3-1 3.2 Invariants ...................................... ....... 3-3 3.2.1 KineticEnergy ................................. ..... 3-3 3.3 DynamicsofaRigidBody............................ ........ 3-4 3.3.1 EquilibriumofaContinuum. ........ 3-4 3.3.2 KineticEnergyofaContinuum . ........ 3-4 3.3.3 RigidityConstraint. ........ 3-5 3.3.4 InertiaForcesofaRigidBody . ........ 3-6 3.3.5 EquationsofMotionofaRigidBody. ......... 3-7 3.3.6 KineticEnergyofaRigidBody. ........3-10 3.4 DynamicsofaSystemofRigidBodies . ...........3-12 iii 4 Analytical Mechanics 4-1 4.1 VirtualWorkPrinciple. .......... 4-2 4.2 LagrangeEquationsoftheSecondKind . ............ 4-2 4.3 Kinematic Constraints, Lagrange Multipliers, and Lagrange Equations of the First Kind . 4-3 4.3.1 HolonomicConstraints. ........ 4-3 4.3.2 Non-HolonomicConstraints . ......... 4-4 4.4 Gauss’Principle ................................. ........ 4-5 II DynamicsofConstrainedMechanicalSystems 4-7 5 Integration of Ordinary Differential Equations 5-1 5.1 NumericalMethods................................ ........ 5-1 5.1.1 GeneralLinearMultistepMethods . .......... 5-1 5.1.2 SingleStepMethods ............................. ...... 5-2 5.1.3 Implicitvs.ExplicitMethods . .......... 5-2 5.2 Accuracy ........................................ ..... 5-2 5.3 Stability....................................... ....... 5-3 5.3.1 LinearStability............................... ....... 5-3 5.3.2 AlgorithmicStabilityvs.Accuracy . ............ 5-4 5.3.3 FirstDahlquistbarrier. ......... 5-6 5.3.4 SecondDahlquistbarrier . ........ 5-6 5.3.5 UsefulMethods................................. ..... 5-7 5.4 Intermediate Evaluations of the Problem . ............... 5-7 5.4.1 Stability..................................... 5-10 5.5 Integration of Problems in Second-Order Differential Form..................5-10 5.6 Integration of Implicit and Nonlinear Problems . .................5-10 5.6.1 Predictor-Corrector . ........5-11 5.6.2 LinearMultistepMethods . ........5-11 5.6.3 LinearMultistageMethods . ........5-12 6 Kinematic Constraints 6-1 6.1 LowerPairsandComplexConstraints . ............ 6-1 6.2 OperationswithConstraintEquations . .............. 6-3 6.3 PositionConstraints . .......... 6-4 6.3.1 SphericalHinge................................ ...... 6-4 6.3.2 InlineConstraint .............................. ....... 6-7 6.3.3 InplaneConstraint . ....... 6-8 6.3.4 DistanceConstraint . ....... 6-9 6.4 OrientationConstraints . ...........6-11 6.4.1 ParallelismandOrthogonality . ..........6-11 6.4.2 RevoluteConstraint . .6-14 6.4.3 CardanJoint ................................... 6-15 7 Minimal Coordinate Set 7-1 7.1 CoordinateReduction ............................. ......... 7-1 7.1.1 CoordinatePartitioning . ......... 7-2 7.1.2 Coordinate Reduction and Orthogonality . ............ 7-4 7.1.3 OtherProjectionMethods. ........ 7-7 7.1.4 Compliance with Holonomic Constraints . ............ 7-7 7.2 RecursiveFormulations . .......... 7-8 iv 8 Redundant Coordinate Set 8-1 8.1 Differential-AlgebraicEquations. ............... 8-1 8.2 Singular Perturbations Theory Interpretation . .................. 8-2 8.2.1 SlowSubsystem ................................. 8-4 8.2.2 FastSubsystem................................. ..... 8-4 8.3 ConceptofIndex .................................. ....... 8-4 8.3.1 Index1DAE ..................................... 8-5 8.3.2 Index2DAE ..................................... 8-6 8.3.3 Index3DAE ..................................... 8-6 8.4 IndexReduction .................................. ....... 8-7 8.4.1 Constraint Stabilization by Index Reduction . .............. 8-7 8.4.2 Baumgarte’sMethod. ...... 8-8 8.5 Initialization .................................. ......... 8-9 9 Unconstrained Coordinate Set 9-1 9.1 Gauss’Principle ................................. ........ 9-1 9.2 AugmentedLagrangian ............................. ........ 9-1 9.3 ForceProjectionMethod. .......... 9-2 9.3.1 StabilizedConstrainedDynamics . ........... 9-2 9.3.2 Stabilized Constrained Dynamics in ODE Form . ............ 9-3 9.3.3 Elimination of Drift at the Position Level . ............. 9-4 III Dynamics of Flexible Systems 9-7 10 Beam 10-1 11 Flexible Element 11-1 IV AdvancedProblemsandImplementationAspects 11-3 V Appendices 11-7 A Rotation Parametrizations A-1 A.1 ParametrizationofRotation. ............A-1 A.1.1 EulerAngles ................................... A-1 A.1.2 EulerVector ................................... A-2 A.1.3 Cayley-Gibbs-Rodrigues Parameters . ............A-3 A.1.4 Euler-Rodrigues Parameters and Quaternions . ..............A-4 A.2 ParametrizationofAngularVelocity . ..............A-5 A.2.1 EulerAngles ................................... A-5 A.2.2 EulerVector ................................... A-5 A.2.3 Cayley-Gibbs-Rodrigues Parameters . ............A-5 A.2.4 Euler-Rodrigues Parameters and Quaternions . ..............A-6 B Lagrange Equations of the Second Kind and Virtual Work Principle B-1 B.1 ContributionoftheInertiaForces. ..............B-1 B.2 ContributionoftheInertiaMoments . .............B-2 v vi List of Figures 3.1 Sir Isaac Newton (Woolsthorpe-by-Colsterworth, December 25, 1642–London, March 20, 1726)[fromWikipedia]. ....... 3-2 3.2 Jean Baptiste le Rond d’Alembert (Paris 1717–Paris 1783)[fromWikipedia]. 3-3 3.3 Leonhard Euler (Basel, April 15, 1707–Saint Petersburg, September 18, 1783) [from Wikipedia]. 3-5 4.1 Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia or Giuseppe Luigi Lagrangia or Lagrange (Turin, January 25, 1736 — Paris April 10, 1813) [from Wikipedia]. 4-1 6.1 Sketch of the lower kinematic pairs (adapted from [1]). ................... 6-2 6.2 Sketch of the geometry of a generic pair formulated according to Bauchau [2] (adapted from[1]). .......................................... 6-2 6.3 SketchofaCardano’sjoint. ...........6-16 vii viii List of Tables A.1 Admissible rotation sequences for (Tait-Bryan/Cardano)-Eulerangles . A-1 ix x List of Examples 5.1 Trapezoidrule ................................... ....... 5-8 5.2 Gauss-Lobattomethod. ......... 5-8 5.3 Heun’smethod .................................... ...... 5-9 8.1 Slowandfastmechanicalsubsystems . ............. 8-2 xi xii List of Exercises 3.1 Dynamicsofabar—puretranslation . ........... 3-7 3.2 Internalforcesinabar—puretranslation. ............... 3-8 3.3 Dynamicsofabar—rotation. ......... 3-8 5.1 Accuracyoftwo-stepBDF. ......... 5-2 5.2 StabilityofCrank-Nicolsonmethod. .............. 5-6 6.1 Sphericalhingewithnooffsets . ........... 6-4 6.2 Derivative of spherical hinge with no offsets . ................ 6-4 6.3 Linearization of
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