Page 111 - The Mechatronics Handbook

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P v P v
m dm
r OP r OP

O
O
(a) (b)

FIGURE 8.1 Difinition of velocity and position vectors for single particle (a) and rigid body (b).

where we have assumed that the mass does not change over time. The angular momentum of a particle
with respect to an arbitrary reference point O is defined as
H O = r OP × ( mv) (8.3)
where r OP is the position vector between points O and P (see Fig. 8.1(a)). The balance of angular momentum
for a single infinitesimally small particle is automatically satisfied as a result of (8.1). In the case of multiple
particles (a rigid body composed of infinite number of particles), the linear and angular momenta are
defined as the sum (integral) of the momentum of individual particles (Fig. 8.1(b)):
L = ∫ v m and H O = ∫ r OP × v d m (8.4)
d
V V
The second fundamental law of classical mechanics states that the rate of change of angular momentum
is equal to the sum of all moments acting on the body:

+
H O = ∑ M i ∑ r i × F i (8.5)
˙
i i
where M i are the applied external force-couples in addition to the forces F i . If the reference point O is
chosen to be the center of mass of the body G, the linear and angular momentum balance law take a
simpler form:

mv ˙ G = ∑ F i (8.6)
i

+
I G ωω ωω ˙ = ∑ M j ∑ r i × F i (8.7)
j i
where ωω ωω is the instantaneous vector of angular velocity and I G is the moment of inertia about the center
of mass. Equations (8.6) and (8.7) are called equations of motion and play a central role in the dynamics
of rigid bodies. If there is no motion (linear and angular velocities are zero), one is faced with a statics
problem. Conversely, when the accelerations are large, we need to solve the complete system of Eqs. (8.6)
and (8.7) including the inertial terms. In mechatronic systems the mechanical response is generally slower
than the electrical one and therefore determines the overall response. If the response time is critical to
the application, one needs to consider the inertial terms in Eqs. (8.6) and (8.7).

Equations of Motion of Deformable Bodies
Rigid bodies do not change shape or size during their motion, that is, the distance between the particles
they are made of is constant. In reality, all objects deform to a certain extent when subjected to external
forces. Whether a body can be treated as rigid or deformable is dictated by the particular application.

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