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0593_C08_fm Page 263 Monday, May 6, 2002 2:45 PM

Principles of Dynamics: Newton’s Laws and d’Alembert’s Principle 263

and

β = θ , β = θ − θ , β = θ − θ (8.11.11)
1 1 2 2 1 3 3 2

The governing equations are then:
+ [
1 (
˙˙
+
43cosβ 2 + cosβ 3 + cos 2 (β + β 3)] [ 5 3) +( 3 2)cosβ 2 + cosβ 3
β
+( 12)cos 2 (β + β 3)] β 2 ( [ 13) +( 12)cosβ 3 ( 1 2)cos 2 (β + β 3)] β 3
+
˙˙
+
˙˙
+
−( 32)( β ˙ 1 + β ˙ 2) 2 sinβ 2 ( 32)β ˙ 2 sinβ 2 +( )β ˙ 2 1 sin (β 2 + ) (8.11.12)
+
β
12
3
) 1
β
12 β ˙ + β ˙ ) 2 sinβ 12 β ˙ + β ˙ + β ˙ ) 2 sinβ 12 β ˙ + β ˙ + β ˙ ) 2 sin (β + )
2 3 2 3 3 2 3 2 3
+( )( 1 −( )( 1 −( )( 1
1 1 (
+( )( )sinβ 1 +( )( )sin β + ) +( 12)( ) l sin 1 (β + β 2 + β 3) = 0
β
52 g l
32 g l
g
2
( [ 53) +( 32)cosβ 2 + cosβ 3 +( 1 2) (β 2 + β 3)] [ 53) + cosβ β 2
3]
+ (
˙˙
˙˙
β
cos
1
+ ( [ 13) +( 12)cosββ ˙˙ 3 +( 3 2)β 1 2 ˙ sinβ 2 +( 12)β 2 ˙ 1 (β 2 + β 3)
3]
sin
(8.11.13)
+( 12)( β ˙ + β ˙ 2) 2 − 12 β ˙ + β ˙ + β ˙ ) 2 sinβ 3 2 g l sin β + )
β
3
1 sinβ −( )( 1 2 3 3 +( )( ) ( 1 2
β
12 g l sin β + β + ) = 0
+( )( ) ( 1
2 3
and
( [ 13) +( 12)cosβ 3 +( 12) (β 2 + β 3)] [ 13) +( 12)cosβ β 2
3]
+ (
˙˙
˙˙
β
cos
1
+( 13)β +( 12)β 2 ˙ 1 (β + β 3) +( 12)( β ˙ + β ˙ 2) 2 sinβ
˙˙
3 sin 2 1 3 (8.11.14)
+( 12)( ) (βg l sin + β + β 3) = 0
1 2
Although these equations (developed using Kane’s equations; see Chapter 11), are not
necessarily in their simplest form, they are nevertheless far more detailed than Eqs. (8.11.1),
(8.11.2), and (8.11.3).
8.12 A Rotating Pinned Rod
For an example illustrating nonplanar motion, consider the rotating pinned rod B depicted
in Figure 8.12.1, where S is a shaft rotating with an angular speed Ω about a vertical axis.
Let k be a unit vector parallel to the axis of S as shown. Let the radius of S be small. Let
B be attached to S at one of its ends O by a frictionless pin whose axis is fixed on a radial