Research Papers

Power Conserving Bond Graph Based Modal Representations and Model Reduction of Lumped Parameter Systems

[+] Author and Article Information
Loucas S. Louca

Department of Mechanical and
Manufacturing Engineering,
University of Cyprus,
75 Kallipoleos Street,
Nicosia 1678, Cyprus
e-mail: lslouca@ucy.ac.cy

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received February 6, 2013; final manuscript received June 12, 2014; published online August 8, 2014. Assoc. Editor: Yang Shi.

J. Dyn. Sys., Meas., Control 136(6), 061007 (Aug 08, 2014) (13 pages) Paper No: DS-13-1061; doi: 10.1115/1.4027900 History: Received February 06, 2013; Revised June 12, 2014

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.

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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. [CrossRef]
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.
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.
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. [CrossRef]
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.
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. [CrossRef]
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. [CrossRef]
Louca, L. S., 1998, “An Energy-Based Model Reduction Methodology for Automated Modeling,” Ph.D. thesis, The University of Michigan, Ann Arbor, MI.
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. [CrossRef]
Golub, G. H., and Van Loan, C. F., 1983, Matrix Computations, 1st ed., John Hopkins University Press, Baltimore, MD.
Moore, B. C., 1981, “Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction,” IEEE Trans. Autom. Control, 26(1), pp. 17–32. [CrossRef]
Skelton, R. E., and Yousuff, A., 1983, “Component Cost Analysis of Large Scale Systems,” Int. J. Control, 37(2), pp. 285–304. [CrossRef]
Meirovitch, L., 1967, Analytical Methods in Vibrations, Macmillan Publishing Inc., New York.
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. [CrossRef]
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. [CrossRef]
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.
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. [CrossRef]
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.
Brown, F. T., 2006, Engineering System Dynamics: A Unified Graph-Centered Approach, 2nd ed., CRC Press, Boca Raton, FL.
Karnopp, D. C., Margolis, D. L., and Rosenberg, R. C., 2006, System Dynamics: Modeling and Simulation of Mechatronic Systems, 4th ed., Wiley, New York.
Rosenberg, R. C., and Karnopp, D. C., 1983, Introduction to Physical System Dynamics, McGraw-Hill, New York.
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. [CrossRef]


Grahic Jump Location
Fig. 1

Activity index sorting and element elimination [6]

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Fig. 5

Complex modal bond graph—zero causal path

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Fig. 4

Real modal bond graph

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Fig. 6

Real modal bond graph—zero causal path

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Fig. 2

Complex modal bond graph

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Fig. 3

Complex and real bond graph for a single mode

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Fig. 7

Physical quarter car mode

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Fig. 8

Modal decomposition bond graph

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Fig. 9

Activity and activity indices of modes

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Fig. 10

Reduced model comparison

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Fig. 14

Activity indices of modes—no causal path

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Fig. 15

Reduced model comparison—no causal path

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Fig. 16

Modal elements activity analysis—no causal path

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Fig. 11

Modal elements activity analysis

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Fig. 12

Included elements, threshold = 99%

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Fig. 13

Real modal decomposition bond graph—no causal path

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Fig. 17

Included elements, threshold = 99.9%—no causal path



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