Nonlinear Oscillations, ME 863
Due Thurs. Jan. 31, 2008
HW Problem 1:
For the linear oscillator mx" + cx' + kx = 0
(or
0
2
2
=
+
+
x
x
x
ϖ
ςϖ
, with
ϖ
2
=
k/m
and
mk
c
2
=
ς
)
classify the stability/eigenvalue cases for different ranges of parameter values and give
physical interpretations (i.e. relate the eigenvalue cases to overdamped, critically damped,
and underdamped situations).
HW Problem 2:
Consider a friction on a belt drive, for the case in which the friction has
the form sketched below.
Perform an analysis of the local stability of fixed points.
Under what parameter values is the system unstable?
Locally oscillatory?
What happens
if you add viscous damping with
)
(
v
f
c
′

?
HW Problem 3:
Compute periods of motion for the pendulum for
g
= 9.81 m/s,
l
= 1m,
and amplitudes of various sizes (
A
= 10, 30, 60, 120, etc.).
Use the table of complete
elliptic functions of the first kind.
Compare to the linearized period, and to the results
from Lindstedt’s method.
HW Problem 4.
Qualitatively sketch the phase portrait by hand for a particle with a
potential energy
3
4
2
3
4
2
bx
ax
kx
V
+
+
=
,
with
k
< 0,
a
> 0, and
b
> 0.
What effect does the value of
b
have on the behavior?
What happens if you add small linear damping to the oscillator?
HW Problem 5 (from S. L. Hendricks).
The gravitational potential between two bodies is
inversely proportional to the distance,
r
, between the bodies.
Under this potential, planets
orbit stars, and moons orbit planets, with elliptic, parabolic, and hyperbolic orbits.
What
if the universe were designed with a gravitational potential of the form
2
/
r
k
V

=
?
(Here
r
is the distance of an orbiting planet to its sun).
Sketch the phase portraits (
r
vs.
r
) and draw conclusions about life, the universe, and everything.