MA508 – Worksheet 4

MA508 – Worksheet 4
A damped linear oscillator is a classical mechanical system. One typically analyzes it to
death in math, physics and engineering courses. Its importance lies in the fact that, near
equilibrium, many systems behave like a damped linear oscillator. Here, you’ll see how it
works.
Here are three differential equations that govern non-linear oscillators of one sort or
another.
1. A mass on a wire (like you saw last week, but here it is not overdamped, so it obeys a
second-order equation)
mx¨ = −bx˙ − k
p
x
2 + h
2 − `0
 x
√
x
2 + h
2
(1)
A non-dimensional form of this equation is (note that this should be in terms of ˆx = x/X
and tˆ= t/T to relate to the previous equation)
x¨ = −x − βx˙ +
x
√
x
2 + α2
(2)
2. A pendulum on a torsional spring (like you saw two weeks ago, but here it is not
overdamped, so it obeys a second-order equation)
−m`2 ¨θ = ζ
˙θ + κθ − mg` sin(θ) (3)
A non-dimensional form of this equation is (note that this should be in terms of x = θ and
tˆ= t/T to relate to the previous equation)
x¨ = −βx˙ − αx + sin(x) (4)
3. Duffing’s oscillator (a model for a slender metal beam interacting with two magnets,
which we will likely revisit), in non-dimensional form
x¨ = −x˙ + βx − αx3
(5)
a) Find the fixed point(s) of each oscillator and classify them (i.e., stable node, unstable
node, saddle, stable spiral, unstable spiral, etc.). Note that, in ALL CASES, β > 0 and
α > 0.
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b) For each oscillator, choose a fixed point that is stable in some parameter regime and
write linearized equations.
c) Compare your linearization to that of a linear oscillator (¨x = −(k/m)x − (b/m) ˙x) and
determine the effective spring constant, k/m, and effective damping constant, b/m, for each
system.
d) Use Matlab to check your work. Pick value of α and β and run some simulations of the
three non-linear oscillators. Compare these with the predictions of the linear system you
found in part c, which can be solved analytically (you did this on HW 1).
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