Problem 5.3 -- FIRST, save yourself time by LOOKING AT LECTURE 10 NOTES
Look at Prof. Crandall's hand-out "Whirling Instability due to DAMPING IN THE ROTOR" from Lecture 10!
(If you do not have this, I'll bring a few extra copies to my office
hours Oct 23.)
In particular, I strongly suggest that before you start
this problem, you examine the last equation:
where:
You are told that IF Bs is ZERO, the system becomes unstable when
Now, PLUG IN:
to find poles at the critical frequency (on the brink of instability, with Bs=0). (Solve for lambda!)
Since the system is JUST on the margin of stability here, at least ONE of the poles has a real part equal to ZERO. Check that (in fact) one (or more) of your poles DOES have ONLY an imaginary part when you solve for Bs=0 at this critical frequency.
Now, to actually answer problem 5.2, you will have to solve for a NEW critical frequency, given Bs is NOT necessarily zero.
Note that in the limit, when you plug in
for YOUR MORE COMPLETE SOLUTION, it should STILL result in the same
poles [whatever you got] and the same critical frequency:
