This paper presents an efficient algorithm for the dynamics simulation and analysis of multi-flexible-body systems. This algorithm formulates and solves the nonlinear equations of motion for mechanical systems with interconnected flexible bodies subject to the limitations of modal superposition, and body substructuring, with arbitrarily large rotations and translations. The large rotations or translations are modelled as rigid body degrees of freedom associated with the interconnecting kinematic joint degrees of freedom. The elastic deformation of the component bodies is modelled through the use of modal coordinates and associated admissible shape functions. Apart from the approximation associated with the elastic deformations, this algorithm is exact, non-iterative, and applicable to generalized multi-flexible chain and tree topologies. In its basic form, the algorithm is both time and processor optimal in its treatment of the nb joint variables, providing O(log(nb)) turnaround time per temporal integration step, achieved with O(nb) processors. The actual cost associated with the parallel treatment of the nf flexible degrees of freedom depends on the specific parallel method chosen for dealing with the individual coefficient matrices which are associated locally with each flexible body.

1.
Hooker
,
W. W.
, and
Margulies
,
G.
, 1965, “
The Dynamical Attitude Equations for an n-Body Satellite
,”
J. Astronaut. Sci.
0021-9142,
7
(
4
), pp.
123
128
.
2.
Walker
,
M. W.
, and
Orin
,
D. E.
, 1982, “
Efficient Dynamic Computer Simulation of Robotic Mechanisms
,”
ASME J. Dyn. Syst., Meas., Control
0022-0434,
104
, pp.
205
211
.
3.
Armstrong
,
W. W.
, 1979, “
Recursive Solution to the Equations of Motion of an N-Link Manipulator
,” Fifth World Congress on the Theory of Machines and Mechanisms,
2
, pp.
1342
1346
.
4.
Featherstone
,
R.
, 1983, “
The Calculation of Robotic Dynamics Using Articulated Body Inertias
,”
Int. J. Robot. Res.
0278-3649,
2
(
1
), pp.
13
30
.
5.
Brandl
,
H.
,
Johanni
,
R.
, and
Otter
,
M.
, 1986, “
A Very Efficient Algorithm for the Simulation of Robots and Similar Multibody Systems Without Inversion of the Mass Matrix
,” IFAC/IFIP/IMACS Symposium, Vienna, Austria, pp.
95
100
.
6.
Bae
,
D. S.
, and
Haug
,
E. J.
, 1987, “
A Recursive Formation for Constrained Mechanical System Dynamics: Part I, Open Loop Systems
,”
Mech. Struct. Mach.
0890-5452,
15
(
3
), pp.
359
382
.
7.
Kurdila
,
A. J.
,
Menon
,
R. G.
, and
Sunkel
,
J. W.
, 1993, “
Nonrecursive Order N Formulation of Multibody Dynamics
,”
J. Guid. Control Dyn.
0731-5090,
16
(
5
) pp.
838
844
.
8.
Anderson
,
K. S.
, 1995, “
An Order-N Formulation for the Motion Simulation of General Multi-Rigid-Body Tree System
,”
Comput. Struct.
0045-7949,
46
(
3
), pp.
547
559
.
9.
Jain
,
A.
, 1991, “
Unified Formulation of Dynamics for Serial Rigid Multibody Systems
,”
J. Guid. Control Dyn.
0731-5090,
14
(
3
), pp.
531
542
.
10.
Kim
,
S. S.
, and
Haug
,
E. J.
, 1988, “
Recursive Formulation for Flexible Multibody Dynamics: Part I, Open-Loop Systems
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
71
(
3
) pp.
293
314
.
11.
Jain
,
A. K.
, and
Rodriguez
,
G.
, 1992, “
Recursive Flexible Multibody System Dynamics Using Spatial Operators
,”
J. Guid. Control Dyn.
0731-5090,
15
(
6
), pp.
1453
1466
.
12.
Anderson
,
K. S.
, 1995, “
Efficient Modelling of General Multibody Dynamic Systems with Flexible Components
,”
Computational Dynamics in Multibody Systems
,
Kluwer Academic Publishers
,
The Netherlands
, pp.
79
97
.
13.
Bauchau
,
O. A.
, 2003, “
Formulation of Modal-Based Elements in Nonlinear, Flexible Multibody Dynamics
,”
Int. J. Multiscale Comp. Eng.
1543-1649,
1
(
2–3
), pp.
161
180
.
14.
Banerjee
,
A. K.
, 1993, “
Block-Diagonal Equations for Multibody Elastodynamics with Geometric Stiffness and Constraints
,”
J. Guid. Control Dyn.
0731-5090,
16
(
6
), pp.
1092
1100
.
15.
Shabana
,
A. A.
, 1997, “
Flexible Multibody Dynamics: Review of Past and Recent Developments
,”
Multibody Syst. Dyn.
1384-5640,
1
, pp.
189
222
.
16.
Quinn
,
M.
, 1994,
Parallel Computing, Theory and Practice
,
McGraw-Hill
,
New York
.
17.
Fijany
,
A.
,
Shraf
,
I.
,
D’Eleuterio
,
G. M. T.
, 1995, “
Parallel O(log N) computation of manipulator forward dynamics
,”
IEEE Trans. Rob. Autom.
1042-296X,
11
(
3
), pp.
389
400
.
18.
Featherstone
,
R.
, 1999, “
A Divide-and-Conquer Articulated-Body Algorithm for Parallel O(log (n) ) Calculation of Rigid Body Dynamics. Part 1: Basic Algorithm
,”
Int. J. Robot. Res.
0278-3649,
18
(
3
), pp.
867
875
.
19.
Featherstone
,
R.
, 1999, “
A Divide-and-Conquer Articulated-Body Algorithm for Parallel O(log (n) ) Calculation of Rigid Body Dynamics. Part 2: Trees, Loops and Accuracy
,”
Int. J. Robot. Res.
0278-3649,
18
(
3
), pp.
876
892
.
20.
Shabana
,
A. A.
, 1996, “
Resonance Conditions and Deformable Body Coordinate Systems
,”
J. Sound Vib.
0022-460X,
192
(
1
), pp.
389
398
.
21.
Schwertassek
,
R.
,
Wallrapp
,
O.
, and
Shabana
,
A.
, 1999, “
Flexible Multibody Simulation and Choice of Shape Functions
,”
Nonlinear Dyn.
0924-090X,
20
, pp.
361
380
.
22.
Schwertassek
,
R.
,
Dombrowski
,
S.
, and
Wallrapp
,
O.
, 1999, “
Modal Representation of Stree in Flexible Multibody Simulations
,”
Nonlinear Dyn.
0924-090X,
20
, pp.
381
399
.
23.
Shabana
,
A. A.
, 1993, “
Substructuring in Flexible Multibody Dynamics
,” in
Proceedings of the Computer Aided Analysis of Rigid and Flexible Mechanical Systems: NATO-Advanced Study Institute
, Vol.
1
, pp.
377
396
.
24.
Roberson
,
R. E.
, and
Schwertassek
,
R.
, 1988,
Dynamics of Multibody Systems
,
Springer-Verlag
,
Berlin
.
25.
Kane
,
T. R.
, and
Levinson
,
D. A.
, 1985,
Dynamics: Theory and Application
,
McGraw-Hill
,
New York
.
26.
Mukherjee
,
R.
, and
Anderson
,
K. S.
, 2006, “
Orthogonal Complement Based Divide-and-Conquer Algorithm for Constrained Multibody Systems
,” Nonlinear Dynamics (in press).
27.
Kim
,
S. S.
, and
VanderPloeg
,
M. J.
, 1986, “
Generalized and Efficient Method for Dynamic Analysis of Mechanical Systems Using Velocity Transforms
,”
ASME J. Mech., Transm., Autom. Des.
0738-0666,
108
(
2
), pp.
176
182
.
28.
Nikravesh
,
P. E.
, 1990, “
Systematic Reduction of Multibody Equations to a Minimal Set
,”
Int. J. Non-Linear Mech.
0020-7462,
25
(
2–3
), pp.
143
151
.
29.
Ambrosia
,
J. A. C.
, 2001, “
Complex Flexible Multibody Systems with Application to Vehicle Dynamics
,”
Multibody Syst. Dyn.
1384-5640,
6
(
2
), pp.
163
182
.
30.
Botz
,
M.
, and
Hagedorn
,
P.
, 1997, “
Dynamic Simulation of multibody systems including planar elastic beams using Autolev
,”
Eng. Comput.
0264-4401,
14
(
4
), pp.
456
470
.
31.
Claus
,
H.
, 2001, “
A Deformation Approach to Stress Distribution in Flexible Multibody Systems
,”
Multibody Syst. Dyn.
1384-5640,
6
, pp.
143
161
.
32.
Seo
,
S.
, and
Yoo
,
H.
, 2002, “
Dynamic Analysis of Flexible Beams Undergoing Overall Motion Employing Linear Strain Measures
,”
AIAA J.
0001-1452,
40
(
2
), pp.
319
326
.
33.
Yoo
,
H.
,
Ryan
,
R.
, and
Scott
,
R.
, 1995, “
Dynamics of Flexible Beams Undergoing Overall Motion
,”
J. Sound Vib.
0022-460X,
181
(
2
), pp.
261
278
.
34.
Yoo
,
H.
, and
Shin
,
S.
, 1998, “
Vibration Analysis of Rotating Cantilever Beams
,”
J. Sound Vib.
0022-460X,
212
(
5
), pp.
807
828
.
This content is only available via PDF.
You do not currently have access to this content.