Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections

Optimizing the dynamic response of mechanical systems is often a necessary step during the early stages of product development cycle. This is a complex problem that requires to carry out the sensitivity analysis of the system dynamics equations if gradient-based optimization tools are used. These dynamics equations are often expressed as a highly nonlinear system of ordinary differential equations or differential-algebraic equations, if a dependent set of generalized coordinates with its corresponding kinematic constraints is used to describe the motion. Two main techniques are currently available to perform the sensitivity analysis of a multibody system, namely the direct differentiation and the adjoint variable methods. In this paper, we derive the equations that correspond to the direct sensitivity analysis of the index-3 augmented Lagrangian formulation with velocity and acceleration projections. Mechanical systems with both holonomic and nonholonomic constraints are considered. The evaluation of the system sensitivities requires the solution of a tangent linear model that corresponds to the Newton–Raphson iterative solution of the dynamics at configuration level, plus two additional nonlinear systems of equations for the velocity and acceleration projections. The method was validated in the sensitivity analysis of a set of examples, including a five-bar linkage with spring elements, which had been used in the literature as benchmark problem for similar multibody dynamics formulations, a point-mass system subjected to nonholonomic constraints, and a full-scale vehicle model.

[1]  E. Haug,et al.  Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems , 1982 .

[2]  J. J. Uicker,et al.  IMP (Integrated Mechanisms Program), A Computer-Aided Design Analysis System for Mechanisms and Linkage , 1972 .

[3]  R. Ledesma,et al.  Augmented lagrangian and mass-orthogonal projection methods for constrained multibody dynamics , 1996 .

[4]  Nathan M. Newmark,et al.  A Method of Computation for Structural Dynamics , 1959 .

[5]  M. Silva,et al.  A parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems , 2009 .

[6]  M. Géradin,et al.  Flexible Multibody Dynamics: A Finite Element Approach , 2001 .

[7]  Milton A. Chace,et al.  The Automatic Generation of a Mathematical Model for Machinery Systems , 1973 .

[8]  Adrian Sandu,et al.  Direct and Adjoint Sensitivity Analysis of Ordinary Differential Equation Multibody Formulations , 2014, Journal of Computational and Nonlinear Dynamics.

[9]  Javier García de Jalón,et al.  Kinematic and Dynamic Simulation of Multibody Systems , 1994 .

[10]  Juan C. García Orden Energy Considerations for the Stabilization of Constrained Mechanical Systems with Velocity Projection , 2010 .

[11]  Adrian Sandu,et al.  Dynamic Response Optimization of Complex Multibody Systems in a Penalty Formulation using Adjoint Sensitivity , 2014, ArXiv.

[12]  A Schaffer,et al.  Stabilized index-1 differential-algebraic formulations for sensitivity analysis of multi-body dynamics , 2006 .

[13]  József Kövecses,et al.  Use of penalty formulations in dynamic simulation and analysis of redundantly constrained multibody systems , 2013 .

[14]  B. Paul,et al.  Computer Analysis of Machines With Planar Motion: Part 1—Kinematics , 1970 .

[15]  Claus Führer,et al.  Numerical Methods in Multibody Dynamics , 2013 .

[16]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[17]  Javier Cuadrado,et al.  A comparison in terms of accuracy and efficiency between a MBS dynamic formulation with stress analysis and a non‐linear FEA code , 2001 .

[18]  Wim Desmet,et al.  Validation of a Real-Time Multibody Model for an X-by-Wire Vehicle Prototype Through Field Testing , 2015 .

[19]  E. Haug Design sensitivity analysis of dynamic systems , 1987 .

[20]  M. A. Serna,et al.  A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems , 1988 .

[21]  P. Nikravesh,et al.  Optimal Design of Mechanical Systems With Constraint Violation Stabilization Method , 1985 .

[22]  Daniel Dopico,et al.  On the Stabilizing Properties of Energy-Momentum Integrators and Coordinate Projections for Constrained Mechanical Systems , 2007 .

[23]  J. Baumgarte Stabilization of constraints and integrals of motion in dynamical systems , 1972 .

[24]  A. Avello,et al.  Optimization of multibody dynamics using object oriented programming and a mixed numerical-symbolic penalty formulation , 1997 .

[25]  C. W. Gear,et al.  Automatic integration of Euler-Lagrange equations with constraints , 1985 .

[26]  D. Bestle,et al.  Sensitivity analysis of constrained multibody systems , 1992, Archive of Applied Mechanics.

[27]  Javier García de Jalón,et al.  Kinematic and Dynamic Simulation of Multibody Systems: The Real Time Challenge , 1994 .

[28]  Daniel Dopico,et al.  Determination of Holonomic and Nonholonomic Constraint Reactions in an Index-3 Augmented Lagrangian Formulation With Velocity and Acceleration Projections , 2014 .

[29]  O. Bauchau,et al.  Review of Contemporary Approaches for Constraint Enforcement in Multibody Systems , 2008 .

[30]  Javier Cuadrado,et al.  Intelligent Simulation of Multibody Dynamics: Space-State and Descriptor Methods in Sequential and Parallel Computing Environments , 2000 .

[31]  Javier García de Jalón,et al.  Multibody Dynamics Optimization by Direct Differentiation Methods Using Object Oriented Programming , 1996 .

[32]  Daniel Dopico,et al.  An Efficient Unified Method for the Combined Simulation of Multibody and Hydraulic Dynamics: Comparison with Simplified and Co-Integration Approaches , 2011 .

[33]  Juan C. García Orden,et al.  Controllable velocity projection for constraint stabilization in multibody dynamics , 2012 .

[34]  Andrei Schaffer,et al.  Stability of the Adjoint Differential-Algebraic Equation of the Index-3 Multibody System Equation of Motion , 2005, SIAM J. Sci. Comput..

[35]  Alfonso Callejo Goena,et al.  Dynamic Response Optimization of Vehicles through Efficient Multibody Formulations and Automatic Differentiation Techniques , 2014 .

[36]  John McPhee,et al.  Symbolic Sensitivity Analysis of Multibody Systems , 2013 .

[37]  Werner Schiehlen,et al.  Multibody System Dynamics: Roots and Perspectives , 1997 .

[38]  L. Petzold,et al.  Adjoint sensitivity analysis for differential-algebraic equations: algorithms and software☆ , 2002 .

[39]  Javier García de Jalón,et al.  Sensitivity-Based, Multi-Objective Design of Vehicle Suspension Systems , 2015 .

[40]  K. Anderson,et al.  Analytical Fully-Recursive Sensitivity Analysis for Multibody Dynamic Chain Systems , 2002 .

[41]  Daniel Dopico,et al.  Dealing with multiple contacts in a human-in-the-loop application , 2011 .

[42]  Javier Cuadrado,et al.  Multibody Dynamics: Computational Methods and Applications , 2007 .

[43]  Thomas R. Kane,et al.  THEORY AND APPLICATIONS , 1984 .

[44]  Daniel Dopico,et al.  Behaviour of augmented Lagrangian and Hamiltonian methods for multibody dynamics in the proximity of singular configurations , 2016, Nonlinear Dynamics.

[45]  J. P. Dias,et al.  Sensitivity Analysis of Rigid-Flexible Multibody Systems , 1997 .

[46]  Benjamin Boudon,et al.  Design methodology of a complex CKC mechanical joint with a representation energetic tool multi-Bond graph: application to the helicopter , 2014 .

[47]  M. A. Serna,et al.  Dynamic analysis of plane mechanisms with lower pairs in basic coordinates , 1982 .