1. 1

2. Damped SHM k “Natural Frequency” “Damping Parameter” (rad/s) (s-1) m
“Damping Constant” b
(kg/s)
EOM: damped oscillator

3. Guess a complex solution:
“trivial solution”
Actually 2 equations:
Real = 0
Imaginary = 0
… also trivial !

4. Try a complex frequency:
Real
Imaginary
A, f are free constants.

5. * amplitude decays due to damping
* frequency reduced due to damping

6. Sometimes write solution in terms of wo and Q
How damped?
Quality factor: unitless ratio of natural frequency to damping parameter
Sometimes write solution in terms of wo and Q
Sometimes write EOM in terms of wo and Q:

7. 1. “Under Damped” or “Lightly Damped”:
Oscillates at ~wo (slightly less)
Looks like SHM (constant A) over a few cycles:
wo = 1, g = .01, Q = 100, xo = 1
Amplitude drops by 1/e in Q/p cycles.

8. 2. “Over Damped”:
imaginary!
part of A
Still need two constants for the 2nd order EOM:
No oscillations!

9. Over Damped
wo = 1, g = 10, Q = .1, xo = 1

10. 3 “Critically Damped”:
…really just one constant, and we need two. Real solution:

11. Critically Damped
wo = 1, g = 2, Q = .5, xo = 1
Fastest approach to zero with no overshoot.