Page 451 - Dynamics of Mechanical Systems

0593_C12_fm Page 432 Monday, May 6, 2002 3:11 PM

432 Dynamics of Mechanical Systems

˙˙
θ
˙˙
θ
12
∂K ∂θ = ( )ml 2 [ − θ θ sin (θ − ) + θ θ sin (θ − )]
3
2 12 2 1 2 3 3 2 (12.4.15)
[ ˙ ˙ − )]
θ
2
2
2
+( ) Ml − θ θ sin (θ 2 − ) + θ θ sin (θ 3 θ 2
12
2
1
2
1
3
˙˙
˙˙
θ
θ
∂K ∂θ 3 = ( )ml 2 [ −θ θ sin (θ 3 − ) + θ θ sin (θ 2 − )]
12
3
3
23
2
1
[ ˙ ˙˙ − )] (12.4.16)
θ
2
2
+( ) Ml − θθ sin (θ 3 − ) + θθ sin (θ 1 θ 3
2
12
2
3
1
3
2
θ
θ
3
∂K ∂θ ˙ = ( )ml 2 ( [ 14 3 )θ ˙ + θ ˙ cos (θ − ) + θ ˙ cos (θ − )]
12
1 1 2 2 1 3 1 3 (12.4.17)
[ ˙ ˙ ˙ − )]
θ
2
2
+( ) Ml 2 θ 1 + θ 2 cos (θ 2 − ) + θ 3 cos (θ 1 θ 3
12
2
1
θ
θ
3
∂K ∂θ ˙ = ( )ml 2 [ ( )θ8 3 ˙ + θ ˙ cos (θ − ) + θ ˙ cos (θ − )]
12
2 2 1 2 1 3 3 2 (12.4.18)
[ ˙ ˙ ˙ − )]
θ
2
+( ) Ml 2 θ 2 + θ 1 cos (θ 2 − ) + θ 3 cos (θ 3 θ 2
2
12
2
1
θ
θ
∂K ∂θ ˙ = ( )ml 2 [ ( )θ2 3 ˙ +θ ˙ cos (θ − ) + θ ˙ cos (θ − )]
12
3 3 2 3 2 1 1 3 (12.4.19)
[ ˙ ˙ ˙ − )]
θ
2
2
+( ) Ml 2 θ 3 + θ 2 cos (θ 3 − ) + θ 1 cos (θ 1 θ 3
12
2
2
By substituting from Eqs. (12.4.8), (12.4.9), and (12.4.10) and Eqs. (12.4.14) through
(12.4.19) into Lagrange’s equations, Eq. (12.3.3), we obtain the governing equations:
( 7 3) θ ˙˙ 1 +( 32) θ 2 cos 1 (θ − θ 2) +( 1 2) θ 3 cos 1 (θ − θ 3) +( 32)θ 2 ˙ 2 sin 1 (θ − θ 2)
˙˙
˙˙
+( 12)θ 2 ˙ 1 (θ − θ )[ ˙˙ +θ ˙˙ 1 (θ − θ 2) +θ ˙˙ 1 (θ − θ 3)
θ
3 sin 3) +(Mm 1 2 cos 3 cos (12.4.20)
1 (
1 (
θ
+θ sin θ − ) + θ 2 ˙ sin θ − θ 3)] ( 52)( ) l sinθ − M + M = 0
+
2
˙ 2
g
2 2 3 2 2 3
( 43) θ 2 +( 3 2) θ 1 cos 2 (θ − θ 1) +( 1 2) θ 3 cos 2 (θ − θ 3) +( 3 2)θ 1 ˙ 2 sin 2 (θ − θ 1)
˙˙
˙˙
˙˙
+( 12)θ 2 ˙ 2 (θ − θ )[ ˙˙ +θ 2 (θ − θ 1) +θ 2 (θ − θ 3)
˙˙
θ
˙˙
3 sin 3) +(Mm 2 1 cos 3 cos (12.4.21)
2 (
2 (
+θ sin θ − ) + θ 2 ˙ sin θ − θ 3)] ( 32)( ) l sinθ − M + M = 0
+
θ
˙ 2
2
g
1 1 3 2 2 3
( 13) θ 3 +( 12) θ 1 cos 3 (θ − θ 1) +( 12) θ 2 cos 3 (θ − θ 2) +( 12)θ 2 ˙ 1 sin 3 (θ − θ 1)
˙˙
˙˙
˙˙
+( 12)θ 2 ˙ 3 (θ − θ )[ θ +θ 3 (θ − θ 1) +θ 3 (θ − θ 2)
˙˙
˙˙
˙˙
2 sin 2) +(Mm 3 1 cos 2 cos (12.4.22)
3 (
3 (
θ
+θ sin θ − ) + θ 2 ˙ sin θ − θ 2)] ( 12)( ) l sinθ − M = 0
+
2
˙ 2
g
1 1 2 3 3
If we let the joint moments (M , M , and M ) be zero and if we let the point mass M
1
3
2
also be zero in Eqs. (12.4.20), (12.4.21), and (12.4.22), we see that the equations are identical