[SOLVED] MA508- Worksheet 4

Description

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)

(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)

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`<sup>2</sup>θ<sup>¨</sup>= ζθ<sup>˙</sup> + 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.

1

1. For each oscillator, choose a fixed point that is stable in some parameter regime andwrite linearized equations.
2. 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.