A graph-theoretic formulation to perform sensitivity analysis on multibody systems is presented in this article. In this formulation, linear graphs are used to capture the system topologies from which a graph-theoretic formulation simultaneously generates the system equations and the sensitivity equations. This ensures the automated, accurate, and efficient generation of sensitivity equations. The basic formulation steps are outlined to illustrate the process of the generation of sensitivity equations. The salient aspects of multibody systems are presented along with a brief description of the software platform that has been used to implement the algorithm. A 3D pendulum and a double-wishbone suspension system are analyzed to demonstrate the application of the algorithm. The results are validated by using a finite difference formulation. Finally, the efficiency of the software implementation is assessed by comparing the computational costs associated with the proposed method and that of existing methods.

References

References
1.
Serban
,
R.
and
Haug
,
E. J.
,
1998
, “
Kinematic and Kinetic Derivatives in Multibody System Analysis
,”
Mech. Struct. Mach.
,
26
(
2
), pp.
145
173
.10.1080/08905459808945425
2.
Serban
,
R.
and
Freeman
,
J. S.
,
1996
, “
Direct Differentiation Methods for the Design Sensitivity of Multi-Body Dynamic Systems
,”
Proceedings of the 1996 ASME Design Engineering Technical Conferences and Computers in Engineering Conference
, pp.
18
22
.
3.
Bestle
,
D.
and
Eberhard
,
P.
,
1992
, “
Analyzing and Optimizing Multi-Body Systems
,”
Mech. Struct. Mach.
,
62
, pp.
181
190
.
4.
Cao
,
Y.
,
Li
,
S.
,
Petzold
,
L.
, and
Serban
,
R.
,
2003
, “
Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution
,”
SIAM J. Sci. Compu.
,
24
(
3
), pp.
1076
1089
.10.1137/S1064827501380630
5.
Cao
,
Y.
,
Li
,
S.
, and
Petzold
,
L.
,
2002
, “
Adjoint Sensitivity Analysis for Differential-Algebraic Equations: Algorithms and Software
,”
J. Comput. Appl. Math.
,
149
(
1
), pp.
171
191
.10.1016/S0377-0427(02)00528-9
6.
Anderson
,
K. S.
and
Hsu
,
Y.
,
2002
, “
Analytical Fully-Recursive Sensitivity Analysis for Multibody Dynamic Chain Systems
,”
Multibody Syst. Dyn.
,
8
, pp.
1
27
.10.1023/A:1015867515213
7.
Bhalerao
,
K.
,
Poursina
,
M.
, and
Anderson
,
K.
,
2010
, “
An Efficient Direct Differentiation Approach for Sensitivity Analysis of Flexible Multibody Systems
,”
Multibody Syst. Dyn.
,
23
, pp.
121
140
.10.1007/s11044-009-9176-0
8.
Barrio
,
R.
,
2005
, “
Sensitivity Analysis of ODES/DAES Using the Taylor Series Method
,”
SIAM J. Sci. Comput.
,
27
, pp.
1929
1947
.10.1137/030601892
9.
Griewank
,
A.
,
2003
, “
A Mathematical View of Automatic Differentiation
,”
Acta Numer.
,
12
,pp.
321
398
.10.1017/S0962492902000132
10.
Eberhard
,
P.
and
Bischof
,
C.
,
1999
, “
Automatic Differentiation of Numerical Integration Algorithms
,”
Math. Comput.
,
68
(
226
), pp.
717
731
.10.1090/S0025-5718-99-01027-3
11.
Carr
,
S.
and
Savage
,
G. J.
,
1995
, “
Symbolic Sensitivity Analysis of Nonlinear Physical Systems Using Graph Theoretic Modeling
,”
Proceedings of the Maple Summer Workshop and Symposium, Mathematical Computation With Maple V. Ideas and Applications
, pp.
118
127
.
12.
Savage
,
G. J.
,
1993
, “
Automatic Formulation of Higher Order Sensitivity Models
,”
Civil Eng. Syst.
,
10
(
4
), pp.
335
350
.10.1080/02630259308970132
13.
Banerjee
,
J. M.
and
McPhee
,
J. J.
,
2013
, “
Symbolic Sensitivity Analysis of Multibody Systems
,”
Multibody Dynamics
(Computational Methods in Applied Sciences) Vol.
28
,
J. C.
Samin
and
P.
Fisette
, eds.,
Springer, Dordrecht, Netherlands
, pp.
123
146
.
14.
Fisette
,
P.
and
Samin
,
J. C.
,
1996
, “
Symbolic Generation of Large Multibody System Dynamic Equations Using a New Semi-Explicit Newton/Euler Recursive Scheme
,”
Arch. Appl. Mech.
,
66
, pp.
187
199
.10.1007/BF00795220
15.
Docquier
,
N.
,
Poncelet
,
A.
, and
Fisette
,
P.
,
2013
, “
Robotran: A Powerful Symbolic Generator of Multibody Models
,”
Mech. Sci.
,
4
, pp.
199
219
.10.5194/ms-4-199-2013
16.
Schmitke
,
C.
,
Morency
,
K.
, and
McPhee
,
J.
,
2008
, “
Using Graph Theory and Symbolic Computing to Generate Efficient Models for Multi-Body Vehicle Dynamics
,”
Proc. Inst. Mech. Eng., Part K
,
222
(
4
), pp.
339
352
.
17.
Banerjee
,
J. M.
and
McPhee
,
J. J.
,
2011
, “
Graph-Theoretic Sensitivity Analysis of Multibody Systems
,”
Proceedings of the Multibody Dynamics 2011 ECCOMAS Thematic Conference
,
J.-C.
Samin
and
P.
Fisette
, eds.
18.
McPhee
,
J. J.
,
1996
, “
On the Use of Linear Graph Theory in Multibody System Dynamics
,”
Nonlinear Dyn.
,
9
, pp.
73
90
.10.1007/BF01833294
19.
Schmitke
,
C.
and
McPhee
,
J.
,
2005
, “
Forming Equivalent Subsystem Components to Facilitate the Modelling of Mechatronic Multibody Systems
,”
Multibody Syst. Dyn.
,
14
, pp.
81
110
.10.1007/s11044-005-4577-1
20.
Schmitke
,
C.
and
McPhee
,
J.
,
2008
, “
Using Linear Graph Theory and the Principle of Orthogonality to Model Multibody, Multi-Domain Systems
,”
Adv. Eng. Inform.
,
22
, pp.
147
160
.10.1016/j.aei.2007.08.002
21.
McPhee
,
J. J.
,
2000
,
Dynamics of Multi-Body Systems: Conventional and Graph-Theoretic Approaches
, Department of Systems Design Engineering,
University of Waterloo
,
Waterloo
.
22.
Banerjee
,
J. M.
,
2013
, “
Graph-Theoretic Sensitivity Analysis of Dynamic Systems
,” Ph.D. thesis, University of Waterloo, Waterloo.
You do not currently have access to this content.