[Solved] 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, youll 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 x = x/X and t= 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>= + mg`sin() (3)

A non-dimensional form of this equation is (note that this should be in terms of x = and t= t/T to relate to the previous equation)

x = x x + sin(x) (4)

3. Duffings oscillator (a model for a slender metal beam interacting with two magnets, which we will likely revisit), in non-dimensional form

x = x + xx³ (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, > 0 and > 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.
3. Use Matlab to check your work. Pick value of and 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).