Partial Solutions to Brainfood #12
These are partial answers so that you can check to see if you are on target.
We expect your actual answers to be "fleshed out" a bit more.
- lambda = -10 +/- sqrt(80). The eigenvectors are (1, lambda).
- k1 exp(lambda1 t) (eigenvector1) + k2 exp(lambda2 t) (eigenvector2)
- When t=0, the trajectory is at k1 (1, lambda1) + k2 (1, lambda2).
Set theis equal to (1,0) to find k1=1/2 + sqrt(5)/4,
- set x(t)=0.2 and numerically solve for t. I find t=1.5788.
- In this case k1=1/2+sqrt(5)/2 and k2=1/2-sqrt(5)/2
and the time to get back to x=0.2 is about t=0.75
- The eigenvalues in this case are complex. You can
use numerical simulation, read ahead in the book, use your
physical intuition, etc. The shock absorber in this case
experiences damped oscillations.
Last modified: Fri Nov 22 12:06:06 1996