Page 401 - Dynamics of Mechanical Systems

0593_C11_fm Page 382 Monday, May 6, 2002 2:59 PM

382 Dynamics of Mechanical Systems

n 1 θ n z

n n
θ 2 θ 2
1
n 1r
n
FIGURE 11.10.4 θ 1
2 n
Unit vector geometry for the double- 2r
rod pendulum.

n = cos θ 2 ( − θ n ) + sin 2 (θ − θ n ) = cosθ n + sinθ n
2r 1 1r 1 1θ 2 1 2 2
(11.10.14)
n 2θ = sin 2 (θ − θ 1 n ) 1r + cos 2 (θ − θ 1 n ) 1θ =−sinθ 2 n +cosθ 2 n 2
1

Let the inertia forces on B and B be represented by forces F * * passing through
1 2 1 and F 2
*
*
F
G and G together with couples having torques T * and T * . Then , , T * , and T * are:
1 2 1 2 1 F 2 1 2
˙˙
F =− ma G r = m(l 2)θ 2 ˙ n − m(l 2) n
θ
*
1 1 r 1 1 1θ
or
˙˙
˙˙
*
m
m
F = (l 2)( θ 2 ˙ 1 cosθ 1 + θ 1 sinθ 1) n + (l 2)( θ 1 2 ˙ sinθ 1 − θ 1 cosθ 1) n 2 (11.10.15)
1
1
˙˙
˙˙
*
m
m
F =−m a G 2 = mlθ 2 ˙ n − mlθ n + (l 2)θ 2 ˙ n − (l 2)θ n
2 1 1r 1 1θ 2 2r 2 2θ
or
2 ˙
F = ml θ 1 [ sinθ + θ 2 cosθ 1 ( θ 2 ) 2 ( θ 2 ) 2] n
+
2 sinθ
+
˙˙
2 cosθ
˙˙
*
2 1 1 1 (11.10.16)
1 [ 2 ˙ 2 ˙ 2]
+
2 sinθ
˙˙
˙˙
2 cosθ
−
+ ml −θ cosθ 1 + θ 1 sinθ 1 ( θ 2 ) 2 ( θ 2 ) n 2
˙˙
*
T =−(ml 2 12) θ n (11.10.17)
1 1 3
and
˙˙
*
T =−(ml 2 12) θ n (11.10.18)
2 2 3
From Eq. (11.10.12) the partial velocities of G and G and the partial angular velocities
1 2
of B and B are (see also Eqs. (11.4.23)):
1 2