Page 179 - The Mechatronics Handbook

0066-frame-C09 Page 53 Friday, January 18, 2002 11:01 AM

~
p
e x1 q1 GY
1 q q
E q1 1 e
x 1 q1
1 ~ C
TF p qn GY
E qn
1 e qn
e xm q
1 n
x
m
FIGURE 9.42 Lagrange subsystem model.
momentum state equations for this Lagrange subsystem are given by

˜ ˙
p = – e i + E i (9.42)

and the state equations for the q i must be found by inverting the generalized momentum equations,
(9.38). In some cases, these n equations are coupled and must be solved simultaneously. In the end, there
are 2n first-order state equations. In addition, the final bond graph element shown in Fig. 9.42 can be
coupled to other systems to build a complex system model.
Note that in order to have the 2n equations in integral causality, efforts (forces and torques) should
be specified as causal inputs to the transforming relations. Also, this subsystem model assumes that only
holonomic constraints are applied. While this might seem restrictive, it turns out that, for many practical
cases, the physical effects that lead to nonholonomic constraints can be dealt with “outside” of the
Lagrange model, along with dissipative effects, actuators, and so on.

References
1. Arczewski, K. and Pietrucha, J., Mathematical Modelling of Complex Mechanical Systems, Ellis Horwood,
New York, 1993.
2. Beaman, J.J. and Rosenberg, R.C., “Constitutive and modulation structure,” Journal of Dynamic
Systems, Measurement, and Control (ASME), Vol. 110, No. 4, pp. 395–402, 1988.
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Press, in progress.
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Institute, Vol. 319, No. 1–2, pp. 1–36, 1985.
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Menlo Park, CA, 1999.
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Electromechanical Systems, McGraw-Hill, New York, 1968 (Reprinted by Krieger Publishing Co.,
Malabar, FL, 1982).
9. Den Hartog, J.P., Advanced Strength of Materials, McGraw-Hill, New York, 1952.
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12. Greenwood, D.T., Principles of Dynamics, Prentice-Hall, Englewood Cliffs, NJ, 1965.
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