Nonlinear Oscillations
ME 863
Due Tues. Feb 3, 2004
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 c > f'(v)?
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.
Verify that the separatrix consists of a saddle point.
What
effect does the value of b have on the behavior?
What happens if you add linear damping
to the oscillator?
HW Problem 5.
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 V = k/r
2
?
(Here r is the distance of
an orbiting planet to its sun).
Sketch the phase portraits (
r
vs. r) and draw conclusions.
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Nonlinear Oscillations
ME 863
Due Thurs. Feb 19, 2004
HW 6.
Consider the van der Pol equation:
x" + x =
ε
(1x
2
)x'.
(a) Use the method of
averaging to determine a firstorder expansion and the presence of a limit cycle.
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 Spring '09
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 Limit sets, Fixed point, Periodic point, Mathieu

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