Computer Algebra Recipes

206 CHAPTER 4. NONLINEAR ODE MODELS
y-axis at t =0,whatarethex-, y-, and z-coordinates of the cage at time t?
Using the Lagrangian approach, show that the cage's motion is described by
Äμ + ! 2 1
0 sin μ ¡
2 !2 sin(2 μ) =0;
with !0 = p g=`. Numerically solve the equation of motion for !0 =1and
varying values of ! and discuss the behavior to which the cage is subjected.
Problem 4-44: Ride into the jaws of chaos
The pivot point O for the simple pendulum is undergoing vertical oscillations
given by A sin(!t) as indicated in Figure 4.21.
Asin(ωt) O
Figure 4.21: An example of parametric excitation.
Show that the relevant equation of motion is
Äμ + ! 2 μ
0 1 ¡ A!2
sin(!t) sin μ =0;
g
with !0 = p g=`. This nonlinear ODE with a time-dependent coe±cient is
referred to in the mathematics literature as an example of parametric excitation.
Taking !0 =1and! = 1, numerically study the e®ect of changing the ratio
A=g. Then take this ratio equal to 1, and study the e®ect of changing !.
Problem 4-45: Horizontally oscillating pivot point
The pivot point O in the previous problem is undergoing horizontal oscillations
given by A sin(!t). Derive the relevant equation of motion. Taking !0 =1,
! =1,andg = 10, numerically study the e®ect of changing the amplitude A.
θ
m