Different Dynamic Formulations for a Mechanism using Bond Graph
For modeling dynamics of mechanisms, various classical formulations are available in the literature. The equations of dynamics given by various classical formulations can also be derived from the bond graph. The bond graph is a convenient graphical representation for modeling dynamics of physical systems in multi-energy domains.
In this paper, various alternative causality assignment procedures in the bond graph are used to derive different classical formulations such as the Lagrange’s equations of the first kind (with multipliers), Lagrange’s formulation of the second kind, and Hamiltonian formulations. An example of the quick return mechanism has been modeled using the bond graph technique, and various alternative causality assignment procedures are applied to derive the various formulations. Simulation coding has been done using MATLAB and results have been analyzed and discussed. The purpose of this paper is to show how the various formulations can be obtained from bond graph using various alternative causality assignment procedures.
Keywordsclassical formulations, modeling, system dynamics, bond graph,
References1. F.T. Brown, Lagrangian bond graphs, Journal of Dynamic Systems, Measurement, and Control, 94(3): 213–221, 1972.
2. R.M. Murray, Z. Li, S.S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC Press, 1994.
3. H. Goldstein, Classical Mechanics, Second Edition, Narosa Publishing House, 1998.
4. F.T. Brown, Bond graphs for nonholonomic dynamic systems, Journal of Dynamic, Measurement, and Control, 98(4): 361–366, 1976, doi: 10.1115/1.3427052.
5. D. Karnopp, Understanding multibody dynamics using bond graph representations, Journal of the Franklin Institute, 334(4): 631–642, 1997, doi: 10.1016/S0016-0032(96)00083-X.
6. A. Mukherjee, R. Karmarkar, Modelling and Simulation of Engineering Systems through Bondgraphs, Narosa Publishing House, 2000.
7. D.C. Karnopp, D.L. Margolis, R.C. Rosenberg, System Dynamics: Modeling and Simulation of Mechatronic Systems, John Wiley & Sons Inc., 2012.
8. A. Mukherjee, R. Karmarkar, A.K. Samantaray, Bond Graph in Modeling, Simulation and Fault Identification, I.K. International Publishing House Pvt. Ltd, 2006.
9. D.C. Karnopp, D. Margolis, R.C. Rosenberg, System Dynamics: Modeling and Simulation of Mechatronic Systems, New York: Wiley-Interscience, 2006.
10. W. Borutzky, Bondgraph Methodology: Development and Analysis of Multidisciplinary Dynamic System Models, London: Springer-Verlag, 2010.
11. R. Merzouki, A.K. Samantaray, P.M. Pathak, B.O. Bouamama, Intelligent Mechatronic Systems: Modeling, Control and Diagnosis, London: Springer-Verlag, 2013, doi: 10.1007/978-1-4471-4628-5.
12. W. Marquis-Favre, S. Scavarda, Alternative causality assignment procedures in bond graph for mechanical systems, Journal of Dynamic Systems, Measurement, and Control, 124(3): 457–463, 2002.
13. D. Karnopp, Lagrange’s equations for complex bond graph systems, Journal of Dynamic Systems, Measurement, and Control, 99(4): 300–306, 1977, doi: 10.1115/1.3427123.
14. F.T. Brown, Hamiltonian and Lagrangian bond graphs, Journal of the Franklin Institute, 328(5–6): 809–831, 1991.
15. L.F. Shampine, M.W. Reichelt, The MATLAB ODE Suite, SIAM Journal on Scientific Computing, 18(1): 1–22, 1997.