Page 513 - Dynamics of Mechanical Systems

0593_C14_fm Page 494 Tuesday, May 7, 2002 6:56 AM

494 Dynamics of Mechanical Systems

kinematical quantities in terms of n , n , and n . To this end, the configuration graph may
2
1
3
be used to obtain the relations:
d = cosψ n − sinψ n
1 1 3
d = n (14.6.1)
2 2
d = sinψ n + cosψ n
3 1 3
ˆ n = n
1 1
ˆ n = cos n −θ sin n (14.6.2)
θ
2 2 3
θ
ˆ n = sin n +θ cos n
3 2 3
and

φ
N = cosφ n − sin cosθ n + sin sinθ n
φ
1 1 2 3
φ
N = sinφ n + cos cosθ n − cos sinθ n (14.6.3)
φ
2 1 2 3
N = sinθ n + cosθ n
3 2 3
Using the procedures of Chapter 4 and the configuration graph of Figure 14.6.2, we
readily find the angular velocity of D in R to be (see Eq. (4.7.6)):

R D ˙ ˙
ωω= ˙ ψn + θn + ˆ (14.6.4)
φn
2 1 3
Then, by using the third expression of Eq. (14.6.2) to express n in terms of n and n ,
3
2
3
R ωω ω ω becomes:
D
φ
˙
R D ˙ +( ˙ ψ + sinθ φ ˙
ωω= θn )n + cosθn (14.6.5)
1 2 3
Observe in Eq. (14.6.4) that in computing the angular acceleration of D in R we will
need to compute the derivatives of the unit vectors n , n , and n . Observe further from
1
3
2
the configuration graph of Figure 14.6.1 that the n (i = 1, 2, 3) are fixed in reference frame
i
ˆ
D . Then, also from the configuration graph, we see that:
R D ˆ ˙ + φ ˙ + φ ˙
ωω = θn sinθn cosθn (14.6.6)
1 2 3
Hence, from Eq. (4.5.2) the derivatives of the n in R are:
i
φ
˙
˙
φ
R
dn dt = ωω D ˆ × n = cosθ n − sinθ n
1 1 2 3
φ
˙
R
dn dt = ωω D ˆ × n = − cosθ n + θ ˙ n (14.6.7)
2 2 1 3
R
dn dt = ωω D ˆ × n = −θ ˙ n + sinφ ˙ θ n
3 3 2 1