Page 407 - Dynamics of Mechanical Systems

0593_C11_fm Page 388 Monday, May 6, 2002 2:59 PM

388 Dynamics of Mechanical Systems

G
where a is the acceleration of G relative to the fixed surface S (the inertia frame); where
ω and α (i = 1, 2, 3) are the n components of ωω ωω and the angular acceleration αα αα of D relative
i
i
i
to S; and, finally, where:
Ι = Ι = mr 2 4 , Ι = mr 2 2 (11.10.48)
11 33 22
From Eqs. (4.12.7) and (4.12.8), a and αα αα are:
G

r − + ˙ ˙
+ sinθ
˙˙
a = ( ψφ + φθ2 ˙ ˙ cosθ n ) + ( θ ψ φ cosθ + φ ˙ 2 sin cosθ )
˙˙
θ
G
r ˙˙
1 (11.10.49)
˙ ˙
r +− ( ψφ sinθ − φ ˙ 2 sin θ − θ ˙ 2 n )
2
3
and
˙˙
− ˙ ˙
˙ ˙
αα= ( θ ψ φ cosθ ) n 1 +( ψ + sinφ θ + θφ cosθ ) n 2
˙˙
˙˙
(11.10.50)
˙˙
˙ ˙
˙ ˙
φ
+( cosθ φθ sinθ ψ
−
+ ) θ n
3
Hence, F and T may be written as:
*
*
F =− ( [ ˙˙ + ˙˙ sin + φθ cosθ n ) +− + ψ φ cosθ
(
θ
˙ ˙
θ
˙ ˙
˙˙
mr ψφ
*
2
1
(11.10.51)
˙ ˙
+φ ˙ 2 sin cosθ n ) +− ( ψ φ sin − φ ˙ 2 sin θ θ ˙ 2 n ) ]
θ
θ
−
2
2 3
and
) ( [ ˙˙ ˙ ˙ n )
θ
*
2
T =−(mr 4 θ − 2ψ φ cos −θ φ ˙ 2 sin cosθ 1
˙ ˙
+( 2ψ ˙˙ sin + 2θ θφ cosθ n ) (11.10.52)
˙˙ + 2φ
2
+( φ cos + 2θ ψθ n ) ]
˙ ˙
˙˙
3
Finally, by using Eq. (11.9.6), the generalized inertia forces become:
*
F = v G θ ˙ ⋅F * + ωω θ ˙ ⋅T *
θ
(
θ
θψθcos +
˙˙
= mr − + ˙ ˙ θ φ sin cos θ)
2
˙ 2
−( mr )( θ − 2 ψ φcos − ˙ 2 θ θ) (11.10.53)
˙ ˙
˙˙
2
θ φ sin cos
4
[
˙˙
θ
θ
= mr −( ) +( ) 2 ψ φcos +( ) φ sin cos θ ]
θ
˙ ˙
2
˙ 2
54
3
54