Page 113 - Dynamics of Mechanical Systems

0593_C04*_fm Page 94 Monday, May 6, 2002 2:06 PM

94 Dynamics of Mechanical Systems

To see this, consider three reference frames R , R , and R as in Eq. (4.7.5):
1
2
0
ωω = ωω + ωω (4.8.3)
R 0 R 2 R 0 R 1 R 1 R 2
By differentiating in R we obtain:
0
R 0
R 1
R 0
R 0 d ωω R 2 dt = R 0 d ωω R 1 dt + R 0 d ωω R 2 dt
or

R 1
R 1
R 0 αα R 2 = R 0 αα + R 0 d ωω R 2 dt (4.8.4)
Consider the last term: From Eq. (4.6.6) we have:

R 0 d ωω R 2 dt = R 1 d ωω R 2 dt+ R 0 ωω × R 1 ωω R 2
R 1
R 1
R 1
(4.8.5)
= R 1 αα + R 0 ωω × R 1 ωω R 2
R 2
R 1
Then, by substituting into Eq. (4.8.4) we have:
R 1
R 0 αα R 2 = R 0 αα + R 1 αα + R 0 ωω × R 1 ωω R 2 (4.8.6)
R 2
R 1
Hence, unless R 0 ωω × R 1 ωω R 2 is zero, the addition theorem for angular acceleration does not
R 1
hold as it does for angular velocity. However, if R 0 ωω × R 1 ωω R 2 is zero (for example, if
R 1
ωω and are parallel), then the addition theorem for angular acceleration holds as
R 0 R 1 R 1 ωω R 2
well. This occurs if all rotations of the reference frames are parallel, or, equivalently, if all
the bodies (treated as reference frames) move parallel to the same plane.
In some occasions the angular acceleration of a body B in a reference frame R ( α ) may
B
R
be computed by simply differentiating the scalar components of the angular velocity of
R
B
B
B in R ( ω ). This occurs if ω is expressed either in terms of unit vectors fixed in R or in
R
terms of unit vectors fixed in B. To see this, consider first the case where the limit vectors
B
R
of ω are fixed in R. That is, let N (i = 1, 2, 3) be mutually perpendicular unit vectors
i
R
B
fixed in R and let ω be expressed as:
R B
ωω= Ω N + Ω N + Ω N (4.8.7)
1 1 2 2 3 3
Then, because the N are fixed in R, their derivatives in R are zero. Hence, in this case,
i
R α becomes:
B
R B ˙ ˙ ˙
αα= Ω N + Ω N + Ω N (4.8.8)
1 1 2 2 3 3
B
R
Next, consider the case where the unit vectors of ω are fixed in B. That is, let n (i =
i
B
1, 2, 3) be mutually perpendicular unit vectors fixed in B and let ω be expressed as:
R
R B
ωω= ω n + ω n + ω n (4.8.9)
1 1 2 2 3 3