Page 115 - Dynamics of Mechanical Systems

0593_C04*_fm Page 96 Monday, May 6, 2002 2:06 PM

96 Dynamics of Mechanical Systems

FIGURE 4.8.1
The gyro of Figure 4.7.5.

Because the N (i = 1, 2, 3) are fixed in the yoke Y, their derivatives may be evaluated
Yi
from the expression:

R
R dt= ωω Y × Yi ( )
dN N i = , ,3 (4.8.16)
12
Yi
Y
R
where from Eqs. (4.7.12) and (4.7.13) ωω ωω is:
R Y ˙ = + ˙
ωω= ˙ ψN + φN s ˙ ψ N c ˙ ψ N + φN (4.8 17)
R2 S3 φ Y1 φ Y2 Y3
R
Hence, the dN /dt are:
Yi
R dt = φ ˙ − ˙ ψ
dN N c N
Y1 Y2 φ Y3
R dt =−φ ˙ + ˙ ψ
dN N s N (4.18)
Y2 Y1 φ Y3
R dt = ˙ ψ − ˙ ψ
dN c N s N
Y3 φ Y1 φ Y2
D
and, by substitution into Eq. (4.8.15), α becomes:
R
φψ
˙
− ˙ ˙
˙
˙
˙˙
+ ˙˙
D
R αα= ( Ω ψ s + ˙ c φ Y ) N 1 +( ψ c + Ωφ ψφ s φ Y ) N 2
φ
φ
(4.19)
+( φ − Ωψ c ˙ φ Y ) N 3
˙˙
D
Comment: Observe that by comparing Eqs. (4.8.14) and (4.8.19) we see that αα αα is
R
R
considerably longer and more detailed than ωω ωω . This raises the question as to whether a
D
D
R
different unit vector set might produce a simpler expression for αα αα . Unfortunately, this
is not the case, as seen in Problem P4.8.2. It happens that the N are the preferred unit
Yi
vectors for simplicity.