Lab 8: skyscraper dynamics

Qualitative analysis of a nonlinear 2D autonomous system

No computers and no internet today. This should all be done by hand and brain.

A. The model

Here is a model of a swaying skyscraper in which x(t) represents the horizontal deflection of the top of the building. (This DE is called Duffing's equation.)

(d2x)/(dt2) =  − x + x3 − (dx)/(dt)
duffing_s24.png

The model is actually just Newton's 2nd law of motion (F = ma, or a = F ⁄ m, where a is the acceleration d2x ⁄ dt2, m is the mass, and F is the total force), with 3 contributing forces:

(i) a frictional force proportional to the velocity.

(ii) a restoring force proportional to the deflection (due to the rigidity of the building), and

(iii) a force due to gravity which tends to pull the building over further when it's leaning,

1. Which force corresponds to which term in the DE?

B. Qualitative analysis

Your task: Describe the effect of blasts of wind of various strength which give the builidng a kick (i.e. give it a non-zero velocity at position x=0). Here are some steps that will assist you ...

2. Convert the 2nd order DE into a pair of 1st order DEs by defining a second dependent variable v as

v = (dx)/(dt)

3. Sketch the nullclines in the phase plane of your 2D autonomous system (x horizontal, v vertical).

4. Decorate the nullclines with appropriate horizontal/vertical mini-tangents.

5. Determine the general direction of the vector field in each region of the phase plane.

6. Sketch solution curves starting on the v axis for a several different v(0)'s. Describe the motion of the building corresponding to each of the solution curves.

Turn in your work as part of this week's homework.