Hamilton’s equations can be used to define the dynamics of a tree configured flexible multibody system. Their states are the generalized coordinates and momenta $(p,q)$. Numerical solution of these equations requires the time derivatives of the states be defined. Hamilton’s equations have the benefit that the time derivative of the system momenta are easy to compute. However, the generalized velocities $q̇$ need be solved from the system momenta as defined by $p=J(q)q̇$ to support the computation of $ṗ$ and the propagation of $q$. Because of the size of $J$, the determination of $q̇$ by linear equation solution schemes requires order $([N+∑i=1Nni]3)$ arithmetic operations, where $N$ is the number of bodies and $ni$ is the number of mode shape functions used to model the $ith$ body deformations. It has been shown that $q̇$ can be solved recursively from the momentum equations for rigid multibody systems (Naudet, Lefeber, and Terze, 2003, “Forward Dynamics of Open-Loop Multibody Mechanisms Using An Efficient Recursive Algorithm Based On Canonical Momenta,” Multibody Syst. Dyn., 10, pp. 45–59). This paper extends that result to flexible multibody systems. The overall arithmetic operations to solve for $q̇$ in this case is proportional to $N$ if the effort to solve for the flexible coordinate rates for each body is weighted the same as that for the joint rate. However, each time the flexible coordinates rate of a body is solved an order $(ni3)$ operations is incurred. Thus, the total computational effort for flexible multibody systems includes an additional order $(∑i=1Nni3)$ operations.

1.
Vereshchagin
,
A. F.
, 1974, “
Computer Simulation of the Dynamics of Complicated Mechanisms
,”
Eng. Cybern.
0013-788X,
12
(
6
), pp.
65
70
.
2.
Armstrong
,
A. A.
, 1979, “
Recursive Solution to the Equations of Motion of an N-Link Manipulator
,”
Proceedings of the Fifth World Congress on Theory of Machines and Mechanisms, ASME
.
3.
Bae
,
D. S.
, and
Haug
,
E. J.
, 1987, “
A Recursive Formulation for Constrained Mechanical System Dynamics: Part II, Closed Loop Systems
,”
Mech. Struct. Mach.
0890-5452,
15
, pp.
481
506
.
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.
Rodriguez
,
G.
, 1987, “
Kalman Filtering, Smoothing and Recursive Robot Arm Forward and Inverse Dynamics
,”
IEEE Journal of Robotics and Automation
0882-4967,
3
(
6
), pp.
624
639
.
6.
Rosenthal
,
D.
, 1990, “
An Order n Formulation for Robotic Systems
,”
J. Astronaut. Sci.
0021-9142,
38
(
4
), pp.
511
529
.
7.
Anderson
,
K. S.
, 1992, “
An Order N Formulation for Motion Simulation of General Constrained Multi-Rigid-Body Systems
,”
Comput. Struct.
0045-7949,
43
(
3m
), pp.
565
579
.
8.
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
.
9.
Goldstein
,
H.
, 1950,
Classical Mechanics
,
,
.
10.
Pars
,
L. A.
, 1965,
A Treatise on Analytical Dynamics
,
Heinemann
,
London
.
11.
Russell
,
W. J.
, 1969, “
On the Formulation of Equations of Rotational Motion for an N-Body Spacecraft
,” Report No. TR-0200 (4133), The Aerospace Corporation, El Segundo, CA.
12.
Jerkovsky
,
W.
, 1978, “
The Structure of Multibody Dynamics Equations
,”
J. Guid. Control
0162-3192,
1
, pp.
173
182
.
13.
Lanczos
,
C.
, 1970,
The Variational Principles of Mechanics
,
Dover
,
New York
.
14.
Hughes
,
P. C.
, 1986,
Spacecraft Attitude Dynamics
,
Wiley
,
New York
.
15.
Naudet
,
J.
,
Lefeber
,
D.
, and
Terze
,
Z.
, 2003, “
Forward Dynamics of Open-Loop Multibody Mechanisms Using An Efficient Recursive Algorithm Based On Canonical Momenta
,”
Multibody Syst. Dyn.
1384-5640,
10
, pp.
45
59
.
16.
Meirovitch
,
L.
, 1970,
Methods of Analytic Dynamics
,
McGraw-Hill
,
New York
.
17.
Meirovitch
,
L.
, and
Stemple
,
T.
, 1995, “
Hybrid Equations of Motion for Flexible Multibody Systems Using Quasicoordinates
,”
J. Guid. Control Dyn.
0731-5090,
18
(
4
), pp.
678
688
.
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