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.
A mass on a wire (like you saw last week, but here it is not overdamped, so it obeys a second-order equation)
(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)
(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`2θ¨= ζθ˙ + 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)
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.
1
For each oscillator, choose a fixed point that is stable in some parameter regime andwrite linearized equations.
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.
