Page 479 - Dynamics of Mechanical Systems

0593_C13_fm Page 460 Monday, May 6, 2002 3:21 PM

460 Dynamics of Mechanical Systems

where C is a constant that can be evaluated by initial conditions. Suppose, for example,
that the pendulum is displaced through an angle θ and released from rest. Then, the
0
initial conditions are:

=
when t = 0 , θθ and θ ˙ = 0 (13.9.5)
0
Thus, the first integral of Eq. (13.9.4) becomes:

˙ 2
θ 2 −( ) l cosθ =−( ) l cosθ 0 (13.9.6)
g
g
θ
˙
By solving Eq. (13.9.4) for we have:
θ =± ( [ 2g l )( cosθ − cosθ )] / 12 = d θ dt (13.9.7)
˙
0

Thus, we have a nonlinear first-order differential equation to solve. By separating the
variables, we have:

g (
/
dt =±−cosθ ) −12 dθ (13.9.8)
θ
l
2
cos
0
Suppose we are interested in determining the time of descent of the pendulum from the
θ
ˆ
angle θ to some smaller angle . Hence, by integration of Eq. (13.9.8) we have:
0
0 θ
∫
/
t = l 2 g ( cosθ −cosθ 0 ) −12 dθ (13.9.9)
ˆ θ
where we have selected the negative sign and eliminated it by interchanging the limits
on the integral.
Unfortunately, the integral of Eq. (13.9.9) cannot be evaluated in terms of elementary
functions. We could, however, expand the integrand in a series (say, the binomial series)
and integrate term by term. Alternatively, we can change the variables and convert the
integral into the form of an elliptic integral (see, for example, Reference 13.14). To this end,
let us introduce the constant k and parameter z as:

D
k = sin θ 2 ) , k < 1 (13.9.10)
( 0
and

θ
D
ksin = sin( ) 2 (13.9.11)
z
(The motivation for such substitutions is to convert the integral of Eq. (13.9.9) into a
standard form of elliptic integrals.)*

* The authors initially learned of this substitution in a course given by Professor T. R. Kane at the University of
Pennsylvania.