Dynamic analysis is extensively used to study the behavior of continuous and lumped parameter linear systems. In addition to the physical space, analyses can also be performed in the modal space where very useful frequency information of the system can be extracted. More specifically, modal analysis can be used for the analysis and controller design of dynamic systems, where reduction of model complexity without degrading its accuracy is often required. The reduction of modal models has been extensively studied and many reduction techniques are available. The majority of these techniques use frequency as the metric to determine the reduced model. This paper describes a new method for calculating modal decompositions of lumped parameter systems with the use of the bond graph formulation. The modal decomposition is developed through a power conserving coordinate transformation. The generated modal decomposition model is then used as the basis for reducing its size and complexity. The model reduction approach is based on the previously developed model order reduction algorithm (MORA), which uses the energy-based activity metric in order to generate a series of reduced models. The activity metric was originally developed for the generic case of nonlinear systems; however, in this work, the activity metric is adapted for the case of linear systems with single harmonic excitation. In this case closed form expressions are derived for the calculation of activity. An example is provided to demonstrate the power conserving transformation, calculation of the modal power and the elimination of unimportant modes or modal elements.

References

References
1.
Stein
,
J. L.
, and
Wilson
,
B. H.
,
1995
, “
An Algorithm for Obtaining Proper Models of Distributed and Discrete Systems
,”
ASME J. Dyn. Syst., Meas., Control
,
117
(
4
), pp.
534
540
.10.1115/1.2801111
2.
Ferris
,
J. B.
, and
Stein
,
J. L.
,
1995
, “
Development of Proper Models of Hybrid Systems: A Bond Graph Formulation
,”
Proceedings of the International Conference on Bond Graph Modeling
, Las Vegas, NV, January, SCS, San Diego, CA, pp.
43
48
.
3.
Ferris
,
J. B.
,
Stein
,
J. L.
, and
Bernitsas
,
M. M.
,
1994
, “
Development of Proper Models of Hybrid Systems
,”
Proceedings of the ASME International Mechanical Engineering Congress and Exposition—Dynamic Systems and Control Division, Symposium on Automated Modeling: Model Synthesis Algorithms
, Chicago, IL, November,
New York
, pp.
629
636
.
4.
Walker
,
D. G.
,
Stein
,
J. L.
, and
Ulsoy
,
A. G.
,
2000
, “
An Input-Output Criterion for Linear Model Deduction
,”
ASME J. Dyn. Syst., Meas., Control
,
122
(
3
), pp.
507
513
.10.1115/1.1286819
5.
Stein
,
J. L.
, and
Louca
,
L. S.
,
1996
, “
A Template-Based Modeling Approach for System Design: Theory and Implementation
,”
Trans. Soc. Comput. Simul. Int.
,
13
(
2
), pp.
87
101
.
6.
Louca
,
L. S.
,
Stein
,
J. L.
, and
Hulbert
,
G. M.
,
2010
, “
Energy-Based Model Reduction Methodology for Automated Modeling
,”
ASME J. Dyn. Syst., Meas., Control
,
132
(
6
), p.
061202
.10.1115/1.4002473
7.
Louca
,
L. S.
,
Rideout
,
D. G.
,
Stein
,
J. L.
, and
Hulbert
,
G. M.
,
2004
, “
Generating Proper Dynamic Models for Truck Mobility and Handling
,”
Int. J. Heavy Veh. Syst.
,
11
(
3/4
), pp.
209
236
.10.1504/IJHVS.2004.005449
8.
Louca
,
L. S.
,
1998
, “
An Energy-Based Model Reduction Methodology for Automated Modeling
,” Ph.D. thesis, The University of Michigan, Ann Arbor, MI.
9.
Rideout
,
D. G.
,
Stein
,
J. L.
, and
Louca
,
L. S.
,
2007
, “
Systematic Identification of Decoupling in Dynamic System Models
,”
ASME J. Dyn. Syst., Meas., Control
,
129
(
4
), pp.
503
513
.10.1115/1.2745859
10.
Golub
,
G. H.
, and
Van Loan
,
C. F.
,
1983
,
Matrix Computations
, 1st ed., John Hopkins University Press, Baltimore, MD.
11.
Moore
,
B. C.
,
1981
, “
Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction
,”
IEEE Trans. Autom. Control
,
26
(
1
), pp.
17
32
.10.1109/TAC.1981.1102568
12.
Skelton
,
R. E.
, and
Yousuff
,
A.
,
1983
, “
Component Cost Analysis of Large Scale Systems
,”
Int. J. Control
,
37
(
2
), pp.
285
304
.10.1080/00207178308932973
13.
Meirovitch
,
L.
,
1967
,
Analytical Methods in Vibrations
,
Macmillan Publishing Inc.
,
New York
.
14.
Margolis
,
D. L.
, and
Young
,
G. E.
,
1977
, “
Reduction of Models of Large Scale Lumped Structures Using Normal Modes and Bond Graphs
,”
J. Franklin Inst.
,
304
(
1
), pp.
65
79
.10.1016/0016-0032(77)90107-7
15.
Li
,
D. F.
, and
Gunter
,
E. J.
,
1981
, “
Study of the Modal Truncation Error in the Component Mode Analysis of a Dual-Rotor
,”
ASME J. Eng. Gas Turbines Power
,
104
(
3
), pp.
525
532
.10.1115/1.3227311
16.
Liu
,
D.-C.
,
Chung
,
H.-L.
, and
Chang
,
W.-M.
,
2000
, “
Errors Caused by Modal Truncation in Structure Dynamic Analysis
,”
Proceedings of the International Modal Analysis Conference—IMAC
, Society for Experimental Mechanics Inc., Bethel, CT, Vol.
2
, pp.
1455
1460
.
17.
Louca
,
L. S.
, and
Stein
,
J. L.
,
2002
, “
Ideal Physical Element Representation From Reduced Bond Graphs
,”
J. Syst. Control Eng.
,
216
(
1
), pp.
73
83
.10.1243/0959651021541444
18.
Borutzky
,
W.
,
2004
, “
Bond Graphs—A Methodology for Modeling Multidisciplinary Dynamic Systems (Frontiers in Simulation)
,” Vol.
FS-14
,
SCS Publishing House
,
Erlangen, Germany/San Diego, CA
.
19.
Brown
,
F. T.
,
2006
,
Engineering System Dynamics: A Unified Graph-Centered Approach
,
2nd ed.
,
CRC Press
, Boca Raton, FL.
20.
Karnopp
,
D. C.
,
Margolis
,
D. L.
, and
Rosenberg
,
R. C.
,
2006
,
System Dynamics: Modeling and Simulation of Mechatronic Systems
,
4th ed.
,
Wiley
,
New York
.
21.
Rosenberg
,
R. C.
, and
Karnopp
,
D. C.
,
1983
,
Introduction to Physical System Dynamics
,
McGraw-Hill
,
New York
.
22.
Rosenberg
,
R. C.
,
1971
, “
State-Space Formulation for Bond Graph Models of Multiport Systems
,”
ASME J. Dyn. Syst., Meas., Control
,
93
(
1
), pp.
35
40
.10.1115/1.3426458
You do not currently have access to this content.