Page 354 - Dynamics of Mechanical Systems

0593_C10_fm Page 335 Monday, May 6, 2002 2:57 PM

Introduction to Energy Methods 335

O

θ i

θ
f

P
FIGURE 10.8.3
A simple pendulum released from
rest and falling through angle θ i – θ f . P

or

i)
(
mgl cosθ − cosθ = 1 m lθ ˙ f ( ) 2 (10.8.8)
f
2
Solving for we have:θ ˙ f

˙
θ = 2 g l ( cosθ − cosθ i) / 12 (10.8.9)
f f
The result of Eq. (10.8.9) could also have been obtained by integrating the governing
differential equations of motion obtained in Chapter 8. Recall from Eq. (8.4.4) that for a
simple pendulum the governing equation is:

˙˙ g sinθ = 0 (10.8.10)
θ +( ) l
θ
˙
Then, by multiplying both sides of this equation by , we have:
˙˙˙
˙
θθ +( ) l sinθθ = 0 (10.8.11)
g
2 ˙
θ
Because θθ may be recognized as being (1/2)d /dt, we can integrate the equation and
˙ ˙˙
obtain:
( 12)θ 2 ˙ −( )cosθ = constant (10.8.12)
g
l
Because is zero when θ is θ , the constant is –(g/ )cosθ . Therefore, we have:θ ˙ i i

2 )(
˙ 2
θ = ( g l cosθ − cosθ ) (10.8.13)
i
When θ is θ , Eqs. (10.8.9) and (10.8.13) are seen to be equivalent.
f
The work–energy principle may also be used to determine the pendulum rise angle
when the speed at the equilibrium position (θ = 0) is known. Specifically, suppose the
angular speed of the pendulum when θ is zero is θ ˙ o . Then, the work W done on the
pendulum as it rises to an angle θ is:
f
(
∆
W = mg h = − mgl 1 cosθ f) (10.8.14)
−