1. 6. Coupled Oscillators L L . . . . a k b m m D
Introduce coupled oscillators as a drawing.
2. Demo what happens when you pluck one: energy transferred back and forth.
3. Write equations of motion: also coupled!
4. Could go for a solution, but a simple sinusoid won’t do (recall demo in 2). D
coupled equations of motion x

2. Normal Modes : motions where all bodies move at the same frequency.
Sum:
. . +
Look for simpler motions first, motions where the solution would be simple sinusoid.
2. Describe first normal mode, demonstrate it.
3. Describe second normal mode, demonstrate it.
4. Show it mathematically: sum, matches NM 1
. .
Normal Coordinate 1:
Normal Frequency 1:

4. Difference: ..
Normal Coordinate 2:
Normal Frequency 2:

5. Normal Mode Solutions :
“Abnormal” motion:
Initial Conditions:
1. Solve original motion using normal modes. D Ao x

6. similarly…
Switch back to xa and xb:
xb also beats

7. m m k k k x x1 x2 Equations of motion Look for Normal Modes! Plug in….x x1 x2
Equations of motion
Demo other couplings: no spring, pendulum to mass/spring, spring torsion
Look for Normal Modes!
Plug in….

8. Simplify and group by A’s
Matrix representation
True if the Determinant is 0:

9. Normal Mode frequencies

10. Larger N: www.falstad.com/coupled
Oscillators which exchange energy are coupled, and their equations of motion are coupled. Their complex motion can be described simply in terms of normal modes in which all bodies oscillate with the same frequency.