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0593_C12_fm Page 419 Monday, May 6, 2002 3:11 PM

Generalized Dynamics: Kane’s Equations and Lagrange’s Equations 419

O
n
2
θ G 1
1 n
B 1
1
Q G
θ 2 2
FIGURE 12.2.3 B
2
Double-rod pendulum.

or

F =−( 3 mg 2)sin θ (12.2.16)
l
θ 1 1
and

F = v G 1 ⋅( mg ) + v G 2 ⋅( mg )
n
n
˙
˙
θ 2 θ 2 1 θ 2 1
or
F =−( mg 2)sin θ (12.2.17)
l
θ 2 2
In Sections 11.10 and 11.12, Eqs. (11.10.21), (11.10.22), (11.12.14), and (11.12.15), we
found the generalized inertia forces F * and F * to be:
θ 1 θ 2
m θ sin(
m θ cos(
F =−( 4 3) l 2 ˙˙ 12) l 2 ˙˙ θ − ) +( 12) l 2 ˙ 2 θ − ) (12.2.18)
m θ −(
θ
θ
*
θ 1 1 2 2 1 2 2 1
and
m θ sin(
m θ cos(
F =−( 13) l 2 ˙˙ 12) l 2 ˙˙ θ − ) −( 12) l 2 ˙ 2 θ − ) (12.2.19)
m θ −(
θ
θ
*
θ 2 2 1 2 1 1 2 1
Kane’s equations then produce the governing equations:
F + F * = 0 (12.2.20)
θ 1 θ 1
and
F + F * = 0 (12.2.21)
θ 2 θ 2
or

( 4 3) θ 1 +( 12)θ 2 cos 2 (θ − θ 1) −( 12)θ 2 ˙ 2 sin 2 (θ − θ 1) + 3 ( g 2 ) sinθ 1 = 0 (12.2.22)
˙˙
˙˙
l