A Time Integration Algorithm forFlexible Mechanism Dynamics : TheDAE-Method

This paper introduces a new family of second-order methods for solving the index-2 differential-algebraic equations (DAEs) of motion for flexible mechanism dynamics. These methods, which extend the α-methods for ODEs of structural dynamics to DAEs, possess numerical dissipation that can be controlled by the user. Convergence and stability analysis is given and verifies that the DAE α-methods introduce no additional oscillations and preserve the stability of the underlying ODE system. Convergence of the Newton iteration, which can be a source of difficulty in solving nonlinear oscillatory systems with large stepsizes, is achieved via a coordinate-split modification to the Newton iteration. Numerical results illustrate the effectiveness of the new methods for simulation of flexible mechanisms.

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

[2]  Christian Lubich,et al.  Extrapolation integrators for constrained multibody systems , 1991, IMPACT Comput. Sci. Eng..

[3]  F. Potra,et al.  On the Numerical Solution of Euler-Lagrange Equations∗ , 1991 .

[4]  J. Yen,et al.  Numerical Solution of Nonlinear Oscillatory Multibody Dynamic Systems. , 1996 .

[5]  P. J. Pahl,et al.  Development of an implicit method with numerical dissipation from a generalized ingle-step algorithm for structural dynamics , 1988 .

[6]  E. Griepentrog,et al.  Differential-algebraic equations and their numerical treatment , 1986 .

[7]  Werner Schiehlen,et al.  Advanced Multibody System Dynamics , 1899 .

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

[9]  J. Yen.  Constrained equations of motion in multibody dynamics as ODEs on manifolds , 1993 .

[10]  W. Rheinboldt,et al.  On the numerical solution of the Euler-Lagrange equations , 1995 .

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

[12]  Manuel S. Pereira,et al.  Computer-Aided Analysis of Rigid and Flexible Mechanical Systems , 1994 .

[13]  B. M. Fulk.  MATH , 1992 .

[14]  Roy R. Craig,et al.  Structural Dynamics: An Introduction to Computer Methods , 1981 .

[15]  C. Lubich.  Integration of stiff mechanical systems by Runge-Kutta methods , 1993 .

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

[17]  G. B. Warburton.  Structural Dynamics—An Introduction to Computer Methods, Roy R. Craig. John Wiley & Sons, New York (1981), 527+XV, price £20.45 , 1982 .

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

[19]  Linda R. Petzold,et al.  An Efficient Newton-Type Iteration for the Numerical Solution of Highly Oscillatory Constrained Multibody Dynamic Systems , 1998, SIAM J. Sci. Comput..

[20]  J. Davenport.  Editor , 1960 .

[21]  T. E. Hull,et al.  Numerical solution of initial value problems , 1966 .

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

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

[24]  E. Haug.  Dynamics of articulated structures , 1989 .

[25]  C. Lubich.  Integration of stiff mechanical systems by Runge-Kutta methods , 1992 .

[26]  Roger Patient,et al.  Basic methods , 1984, Nature.

[27]  Bernd Simeon,et al.  Modelling a flexible slider crank mechanism by a mixed system of DAEs and PDEs , 1996 .

[28]  O. C. Zienkiewicz,et al.  An alpha modification of Newmark's method , 1980 .

[29]  B. Leimkuhler,et al.  Numerical solution of differential-algebraic equations for constrained mechanical motion , 1991 .

[30]  E. Haug,et al.  Dynamics of Articulated Structures. Part I. Theory , 1986 .

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

[32]  Linda R. Petzold,et al.  Numerical solution of highly oscillatory ordinary differential equations , 1997, Acta Numerica.