Damped Harmonic oscillator: Setup

1
Damped Harmonic oscillator Setup
Defining equation
define
Is this realistic for a harmonic oscillator in a
viscous medium most of the time?
2
Damped Harmonic oscillator Operator
Operator Method
This is an operator equation, and can be factored
This is handy because these commute, if a
solution is solved the equation Ax or Bx, then
the entire equation is solved.,
3
Damped Harmonic oscillator Solution
If A(t) solves the first, and B(t) solved the
second, then
These are each of the form
4
Damped Harmonic oscillator Solve
Apple this to the two equations and add the two
solutions
What happens when
5
Damped Harmonic oscillator Cases
There possible cases
Overdamping
Critical Damping
Underdamping
6
Damped Harmonic oscillator Overdamping
How do we determine the Cs? Is this oscillatory?
Red C10, C21 Green C11, C20 Blue C10.5,
C2-0.5
7
Damped Harmonic oscillator Overdamping
C10.5, C2-0.5
8
Damped Harmonic oscillator Critical damping
Critical damping
Whats wrong?
Go back to the operator method
Only solution, so we cant be done So lets use
a solution to one of these to find the rest.
9
Damped Harmonic oscillator Critical damping
Integrate!
10
Damped Harmonic oscillator Critical damping
Red C10, C21 Green C11, C20 Blue C10.5,
C2-0.5
11
Damped Harmonic oscillator Underdamping
However, the exponent is imaginary. Need to make
that clearer.
Eulers identity on the inside
The result must be real though
12
Damped Harmonic oscillator Underdamping
Almost the same as a normal harmonic oscillator,
but with two differences.
13
Damped Harmonic oscillator Energy
Is this a constant of motion?
What forces are accounted for in the potential
term?
Does this look familiar?
This should make sense what needs to happen in
the limit of c-gt0?
14
Damped Harmonic oscillator Q
A way to quantify the energy loss due to damped
harmonic oscillator is there a quality factor, Q.

Tau is the time constant why is there a 2 in
tau?.