The well known Euler-Lagrange equations of motion for constrained variational problems are derived using the principle of virtual work. These equations are used in the modelling of multibody systems and result in differential-algebraic equations of high index. Here they concern an N-link pendulum, a heavy aircraft towing truck and a heavy off-highway track vehicle. The differential-algebraic equation is cast as an ordinary differential equation through differentiation of the constraint equations. The resulting system is computed using the integration routine LSODAR, the Euler and fourth order Runge-Kutta methods. The difficulty to integrate this system is revealed to be the result of many highly oscillatory forces of large magnitude acting on many bodies simultaneously. Constraint compliance is analyzed for the three different integration methods and the drift of the constraint equations for the three different systems is shown to be influenced by nonlinear contact forces.

Issue Section:
Technical Papers
Keywords:
nonlinear dynamical systems, mechanical contact, variational techniques, Runge-Kutta methods, numerical analysis, Multibody Dynamics, Constrained Variational Systems, Mathematical Modelling, Differential-Algebraic Equations, Virtual Work, Track Vehicle, Nonlinear Contact Forces
Topics:
Algebra, Computation, Modeling, Multibody systems, Pendulums, Rollers, Trucks, Vehicles, Runge-Kutta methods, Aircraft, Wheels, Nonlinear dynamics
1.
Pereira, M. F. O. S., and Ambrosio, J. A. C., Eds., 1994, Computer-Aided Analysis of Rigid and Flexible Mechanical Systems, Kluwer Academic, NATO ASI Series, Vol. 268, Boston.
2.
Ryan, R. R., 1990, ADAMS-Multibody System Analysis Software, Multibody Systems Handbook, Springer-Verlag, Berlin.
3.
Smith, R. C., and Haug, E. J., 1990, DADS—Dynamic Analysis and Design System. Multibody Systems Handbook, Springer-Verlag, Berlin.
4.
Simeon, B., Fuhrer, C., and Rentrop, R., 1991, Differential-Agebraic Equations in Vehicle System Dynamics, Surveys on Mathematics for Industry, Springer-Verlag, Berlin.
5.
Brenan, K. E., Campbell, S. L., and Petzold, L. R., 1989, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Elsevier Science, New York, NY.
6.
Shabana, A. A., 1994, Computational Dynamics, Wiley, New York.
7.
Nakanishi
,
T.
, and
Shabana
,
A. A.
,
1994
, “
Contact Forces in the Nonlinear Dynamic Analysis of Tracked Vehicles
,”
Int. J. Numer. Comput. Methods Eng.
,
37
,
1251
–
1275
.
8.
Nakanishi
,
T.
, and
Shabana
,
A. A.
,
1994
, “
On the Numerical Solution of Tracked Vehicle Dynamic Equations
,”
Nonlinear Dyn.
,
6
, pp.
391
–
417
.
9.
Sarwar
,
M. K.
,
Nakanishi
,
T.
, and
Shabana
,
A. A.
,
1995
, “
Chain Link Deformation in the Nonlinear Dynamics of Tracked Vehicles
,”
J. Vib. Control
,
1
, pp.
201
–
224
.
10.
Petzold, L. R., and Hindmarsh, A. C., LSODAR, Computing and Mathematics Research Division, 1-316 Lawrence Livermore National Laboratory, Livermore, CA 94550.
11.
Ascher, U. M., and Petzold, L. R., 1992, Numerical Methods for Boundary Value Problems in Differential-Algebraic Equations, Recent Developments in Numerical Methods and Software for ODEs/DAEs/PDEs, G. D. Byrne and W. E. Schiesser, Eds., World Scientific, New Jersey.
12.
Gear
,
C. W.
, and
Petzold
,
L. R.
,
1984
, “
ODE Methods for the Solution of Differential/Algebraic Systems
,”
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
,
21
,
No. 4, August
No. 4, August
.
13.
Petzold
,
L. R.
,
1982
, “
Differential/Algebraic Equations are not ODEs
,”
SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput.
,
3
,
No. 3, September
No. 3, September
.
14.
Petzold, L. R., 1983, A Description of DASSL: A Differential/Algebraic System Solver, Scientific Computing, R. Stepleman et al., Eds., North-Holland, Amsterdam.
15.
Petzold, L. R., 1991, Methods and Software for Differential-Algebraic Systems, NATO ASI Series, Vol. F 69, Real Time Integration Methods for Mechanical System Simulation, E. J. Haug, Ed., Springer-Verlag, Berlin, Heidelberg.
16.
Petzold
,
L. R.
,
Ren
,
Y.
, and
Maly
,
T.
,
1997
, “
Regularization of Higher-Index Differential-Algebraic Equations with Rank-Deficient Constraints
,”
SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput.
,
18
, No.
3
, pp.
753
–
774
, May.
17.
Yen
,
J.
, and
Petzold
,
L. R.
,
1998
, “
An Efficient Newton-Type Iteration for the Numerical Solution of Highly Oscillatory Constrained Multibody Dynamic Systems
,”
SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput.
,
19
, No.
5
, pp.
1513
–
1534
, September.
18.
Byrne, G. D., and Schiesser, W. E., 1992, An Overview of Recent Developments in Numerical Methods and Software for ODEs/DAEs/PDEs, Recent Development in Numerical Methods and Software for ODEs/DAEs/PDEs, G. D. Byrne and W. E. Schiesser, Eds., World Scientific, New Jersey.
19.
Gear, C. W., 1991, An Introduction to Numerical Methods for ODEs and DAEs, NATO ASI Series, Vol. F 69. Real Time Integration Methods for Mechanical System Simulation, E. J. Haug, Ed., Springer-Verlag, Berlin, Heidelberg.
20.
Gladwell, G. M. L., 1980, Contact Problems in the Classical Theory of Elasticity, Sijthoff and Noordhoff, The Netherlands.
21.
Sears, F. W., Zemansky, M. W., and Young, H. D., 1987, University Physics, 7th Ed., Addison-Wesley.
Copyright © 2001
 by ASME
You do not currently have access to this content.