Page 103 - Dynamics of Mechanical Systems

0593_C04*_fm Page 84 Monday, May 6, 2002 2:06 PM

84 Dynamics of Mechanical Systems

B
Y
Z
L
n
n
Z θ
Z X
R
B
FIGURE 4.4.1 FIGURE 4.4.2
A body B rotating in a reference frame R. End view of body B.

Observe that θ measures the rotation of B in R. It is also a measure of the orientation of
B in R. In this context, the angular velocity of B in R is a measure of the rate of change of
orientation of B in R.
The simple angular acceleration α of B in R is then defined as the time rate of change of
the angular velocity. That is,

˙˙
αα = d ωω dt = θn (4.4.2)
If we express α and β in the forms:

αα = αn and ω = ωωn (4.4.3)

then α, ω, and θ are related by the expressions:

˙
ω = θ and α = ˙ ω = θ ˙˙ (4.4.4)
By integrating we have the relations:

θ = ∫ ωdt + θ and ω = ∫ αdt + ω o (4.4.5)
o

where θ and ω are values of θ and ω at some initial time t .
o
o
o
From the chain rule for differentiation we have:
2
α = ˙ ω = d ω dt = (d ω θ)(d θ ) = d d ω ) dt (4.4.6)
dt
2
ω ω θ = ( d
d
Then by integrating we obtain:
ω = 2 ∫ α θ ω o 2 (4.4.7)
+
2
d
where, as before, ω is an initial value of ω when θ is θ . If α is constant, we have the
o
o
familiar relation:
ω = ω + 2 αθ (4.4.8)
2
2
o