Direct integration of the equations of multibody dynamics using central differences and linearization

Abstract A methodology for integrating rigid body dynamics for the analysis of multibody systems is presented. The novelty lies in the fact that the equation system is solved directly by means of central differences as a second-order integration method. To obtain the best achievable convergence, the equilibrium is solved iteratively by the exact Newton method. Thus, it is possible to achieve the system solution directly without having to reduce the differential order. This decreases the number of unknowns. In return, it is necessary to linearize the equations. The rotation of each element is described by parameterization under a unit quaternion. In this paper the necessary developments for the modelization of the spherical and rotational joints are included. The constraints imposed by these joints, as well as the quaternion norm, are introduced into the model through a null space matrix. The reactions produced by these constraints are also eliminated from the system by using null space. Several examples are analyzed through the implementation of the methodology in Octave. The accuracy of the method is verified with results obtained from commercial software. The examples include benchmark problems.

[1]  Dan Negrut,et al.  The Newmark Integration Method for Simulation of Multibody Systems: Analytical Considerations , 2005 .

[2]  Dan Negrut,et al.  Implicit Runge-Kutta Integration of the Equations of Multibody Dynamics in Descriptor Form , 1999 .

[3]  Roy Featherstone,et al.  Rigid Body Dynamics Algorithms , 2007 .

[4]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

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

[6]  Michael F. Steigerwald BDF Methods for DAEs in Multi-body Dynamics: Shortcomings and Improvements , 1990 .

[7]  J. G. Jalón,et al.  Multibody dynamics with redundant constraints and singular mass matrix: existence, uniqueness, and determination of solutions for accelerations and constraint forces , 2013 .

[8]  M. Géradin,et al.  Numerical Integration of Second Order Differential—Algebraic Systems in Flexible Mechanism Dynamics , 1994 .

[9]  Louis Komzsik What Every Engineer Should Know About Computational Techniques of Finite Element Analysis , 2005 .

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

[11]  Thomas J. R. Hughes,et al.  Improved numerical dissipation for time integration algorithms in structural dynamics , 1977 .

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

[13]  D. Dopico,et al.  Penalty, Semi-Recursive and Hybrid Methods for MBS Real-Time Dynamics in the Context of Structural Integrators , 2004 .

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

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

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

[17]  Dan Negrut,et al.  An Implicit Runge–Kutta Method for Integration of Differential Algebraic Equations of Multibody Dynamics , 2003 .

[18]  Ahmed A. Shabana,et al.  Dynamics of Multibody Systems , 2020 .

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

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

[21]  D. Pogorelov,et al.  Differential–algebraic equations in multibody system modeling , 1998, Numerical Algorithms.

[22]  Youdan Kim,et al.  Introduction to Dynamics and Control of Flexible Structures , 1993 .

[23]  W. Blajer Augmented Lagrangian Formulation: Geometrical Interpretation and Application to Systems with Singularities and Redundancy , 2002 .

[24]  Daniel Dopico,et al.  Benchmarking of augmented Lagrangian and Hamiltonian formulations for multibody system dynamics , 2015 .

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

[26]  E. Bayo,et al.  Singularity-free augmented Lagrangian algorithms for constrained multibody dynamics , 1994, Nonlinear Dynamics.

[27]  J. Ambrósio,et al.  Stabilization Methods for the Integration of DAE in the Presence of Redundant Constraints , 2003 .

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

[29]  Domenico Guida,et al.  On the Computational Methods for Solving the Differential-Algebraic Equations of Motion of Multibody Systems , 2018 .

[30]  Arend L. Schwab,et al.  Dynamics of Multibody Systems , 2007 .

[31]  Dan Negrut,et al.  On an Implementation of the Hilber-Hughes-Taylor Method in the Context of Index 3 Differential-Algebraic Equations of Multibody Dynamics (DETC2005-85096) , 2007 .

[32]  Linda R. Petzold,et al.  The numerical solution of higher index differential/algebraic equations by implicit methods , 1989 .

[33]  U. Lugrís,et al.  A benchmarking system for MBS simulation software: Problem standardization and performance measurement , 2006 .

[34]  C. D. Mote,et al.  Optimization methods for engineering design , 1971 .

[35]  Linda,et al.  A Time Integration Algorithm forFlexible Mechanism Dynamics : TheDAE-Method , 1996 .

[36]  C. W. Gear,et al.  Simultaneous Numerical Solution of Differential-Algebraic Equations , 1971 .

[37]  Olivier A. Bauchau,et al.  Flexible multibody dynamics , 2010 .

[38]  C. Pappalardo A natural absolute coordinate formulation for the kinematic and dynamic analysis of rigid multibody systems , 2015 .

[39]  K. Bathe,et al.  Stability and accuracy analysis of direct integration methods , 1972 .