Independent Variable Formulations

In Chapter 5 the dependent variable formulations for simulation of multibody systems were described. The derivation of equations of motion in terms of dependent states, though conceptually simple and easy to handle, leads to large-dimension governing DAEs, which results in computationally inefficient algorithms, burdened further with the constraint violation problem. An old and legitimate approach to the dynamics formulation is therefore to use a minimal number of independent state variables for a unique representation of motion by means of pure reduced-dimension ODEs. The numerical integration of the ODEs is usually by far more efficient compared to the integration of the governing DAEs, and the numerical solution is released from the problem of constraint violation. On the other hand, the minimal-form ODE formulations require often more modeling effort and skill. Effective and computer-oriented modeling procedures of this type are thus highly desirable.

[1]  W. Blajer.  Elimination of Constraint Violation and Accuracy Aspects in Numerical Simulation of Multibody Systems , 2002 .

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

[3]  W. Blajer.  A Geometric Unification of Constrained System Dynamics , 1997 .

[4]  T. Kane,et al.  ON THE DERIVATION OF EQUATIONS OF MOTION , 1965 .

[5]  W. Blajer.  An effective solver for absolute variable formulation of multibody dynamics , 1995 .

[6]  C. D. Meyer,et al.  Generalized inverses of linear transformations , 1979 .

[7]  N. K. Mani,et al.  Application of Singular Value Decomposition for Analysis of Mechanical System Dynamics , 1985 .

[8]  R. Rohrer,et al.  Automated Design of Biasing Circuits , 1971 .

[9]  Modelling and simulation of a sphere rolling on the inside of a rough vertical cylinder , 2006 .

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

[11]  H. Essén.  On the Geometry of Nonholonomic Dynamics , 1994 .

[12]  C. W. Gear,et al.  Differential-algebraic equations index transformations , 1988 .

[13]  J. C. Simo,et al.  Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum , 1991 .

[14]  Parviz E. Nikravesh,et al.  An adaptive constraint violation stabilization method for dynamic analysis of mechanical systems , 1985 .

[15]  Thomas R. Kane,et al.  Formulation of Equations of Motion for Complex Spacecraft , 1980 .

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

[17]  Roderic C. Deyo,et al.  Real-Time Integration Methods for Mechanical System Simulation , 1991 .

[18]  Werner Schiehlen,et al.  Research trends in multibody system dynamics , 2007 .

[19]  T. Kane.  Dynamics of Nonholonomic Systems , 1961 .

[20]  P. Nikravesh.  Initial condition correction in multibody dynamics , 2007 .

[21]  W. Blajer.  A geometrical interpretation and uniform matrix formulation of multibody system dynamics , 2001 .

[22]  R. M. Howe,et al.  Stability and Accuracy Analysis of Baumgartes Constraint Violation Stabilization Method , 1995 .

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

[24]  K Yagasaki,et al.  Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems; For Engineers, Physicists, and Mathematicians , 2002 .

[25]  Daniel Dopico,et al.  On the optimal scaling of index three DAEs in multibody dynamics , 2008 .

[26]  Dae-Sung Bae,et al.  A Stabilization Method for Kinematic and Kinetic Constraint Equations , 1990 .

[27]  Willi Kortiim.  A book review of :“Multibody Systems Handbook”W. SCHIEHLEN (ed.” Springer-Verlag. 1990 , 1990 .

[28]  R. M. Howe,et al.  Geometric Elimination of Constraint Violations in Numerical Simulation of Lagrangian Equations , 1994 .

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

[30]  P. Nikravesh.  Systematic reduction of multibody equations of motion to a minimal set , 1990 .

[31]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

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

[33]  E. Fuehrer C. Eich,et al.  Numerical Methods in Multibody Dynamies , 1992 .

[34]  John M. Hansen,et al.  Unconditionally energy stable implicit time integration: application to multibody system analysis and design , 2000 .

[35]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[36]  C. W. Gear,et al.  ODE METHODS FOR THE SOLUTION OF DIFFERENTIAL/ALGEBRAIC SYSTEMS , 1984 .

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

[38]  S. S. Kim,et al.  A General and Efficient Method for Dynamic Analysis of Mechanical Systems Using Velocity Transformations , 1986 .

[39]  P. Likins,et al.  Singular Value Decomposition for Constrained Dynamical Systems , 1985 .

[40]  W. Rheinboldt,et al.  Theoretical and numerical analysis of differential-algebraic equations , 2002 .

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

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

[43]  W. Blajer,et al.  A unified approach to the modelling of holonomic and nonholonomic mechanical systems , 1996 .

[44]  W. Schiehlen,et al.  A projective criterion to the coordinate partitioning method for multibody dynamics , 1994, Archive of Applied Mechanics.

[45]  Parviz E. Nikravesh,et al.  Systematic construction of equations of motion for rigid‐flexible multibody systems containing open and closed kinematic loops , 1991 .

[46]  George M. Lance,et al.  A Differentiable Null Space Method for Constrained Dynamic Analysis , 1987 .

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

[48]  I. Neĭmark,et al.  Dynamics of Nonholonomic Systems , 1972 .

[49]  E. A. Desloge,et al.  The Gibbs–Appell equations of motion , 1988 .

[50]  Charles William Gear.  An Introduction to Numerical Methods for ODEs and DAEs , 1990 .

[51]  Iain S. Duff,et al.  Computing the structural index , 1986 .

[52]  M. Lesser.  A geometrical interpretation of Kane’s Equations , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[53]  S. Lin,et al.  Stabilization Method for Numerical Integration of Multibody Mechanical Systems , 1998 .

[54]  Georg-P. Ostermeyer.  On Baumgarte Stabilization for Differential Algebraic Equations , 1990 .

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

[56]  C. W. Gear,et al.  The index of general nonlinear DAEs , 1995 .