Page 146 - The Mechatronics Handbook

When modeling simple translational systems or fixed-axis rotational systems, the basic set of laws
summarized below are sufficient to build the necessary mathematical models.

Basic Dynamic and Kinematic Laws
System Dynamics Kinematics
N
N
Translational ∑ i F i = 0 ∑ i V i = 0
N
Rotational ∑ i T i = 0 ∑ i w i = 0
N
Junction type 1-junction 0-junction
There is a large class of mechanical systems that can be represented using these basic equations, and
in this form it is possible to see how: (a) bond graph junction elements can be used to structure these
models and (b) how these equations support circuit analog equations, since they are very similar to the
Kirchhoff circuit laws for voltage and current. We present here the bond graph approach, which graph-
ically communicates these physical laws through the 0- and 1-junction elements.

Identifying and Representing Motion in a Bond Graph
It is helpful when studying a mechanical system to focus on identifying points in the system that have
distinct velocities (V or ω). One simply can associate a 1-junction with these points. Once this is done,
it becomes easier to identify connection points for other mechanical components (masses, springs, damp-
ers, etc.) as well as points for attaching actuators or sensors. Further, it is critical to identify and to define
additional velocities associated with relative motion. These may not have clear, physically identifiable points
in a system, but it is necessary to localize these in order to attach components that rely on relative motion
to describe their operation (e.g., suspensions).
Figure 9.20 shows how identifying velocities of interest can help identify 1-junctions at which mechan-
ical components can be attached. For the basic mass element in part (a), the underlying premise is that
a component of a system under study is idealized as a pure translational mass for which momentum and
velocity are related through a constitutive relation. What this implies is that the velocity of the mass is
the same throughout this element, so a 1-junction is used to identify this distinct motion. A bond attached
to this 1-junction represents how any power flowing into this junction can flow into a kinetic energy
storing element, I, which represents the mass, m. Note that the force on the bond is equal to the rate of
p ˙
change of momentum, , where p = mV.

J
V V 1 K V 2 1 J 2 ω
ω
1 2
m m 1 m 2
I: m I: m I: J I: J
1 2 1 2
µ
1 I: m
V
Simple translating mass defines distinct 1 V 0 V 1 1 0 ω 1
velocity. Attach the I-element to the V 1 2 V ω ω 1 2 ω
corresponding 1-junction. 1 V 2 1 ω 2
3 3
1 C: 1/K 1 R
relative x relative ω
velocity velocity
(a) (b) (c)

FIGURE 9.20 Identifying velocities in a mechanical system can help identify correct interconnection of components
and devices: (a) basic translating mass, (b) basic two-degree of freedom system, (c) rotational frictional coupling
between two rotational inertias.

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