Coupled Harmonic Oscillators
This animation shows two equal masses attached to springs. The outside springs (k) are the same, and the middle spring (kp) can be different. The initial positions and velocities of each mass can be specified.
a) To excite the Symmetric Normal Mode, use X01=X02 and V01=V02.
b) To excite the Anti-Symmetric Normal Mode, use X01= -X02 and V01= -V02.
Try using a very small middle spring constant (0.4) and a very large outside spring constant (k>20) - and start only one mass initially ... can you see the motion of that moving mass transferred to the other mass? That shows the weak coupling. Now try a very strong inner spring, and a weak outer spring - and move only one mass. What happens here is an internal vibration that slowly moves from side to side.
The "X" that is plotted below the springs, and the black line in the top graph, is the Center of Mass of the system. Note that since this is a conservative system, the center of mass motion is uniform (regardless of how the two masses individually move, the center of mass of the system remains regular in its motion).
On the top right, there is a "configuration space" diagram. This plots X1 on the horizontal against X2 on the vertical. This is usually a "space-filling" type curve unless you use specific starting values :
a) Try the "normal modes" above, and see what the configuration space shows.
b) Or, to get a "lissajous" type figure, try using these values : m=1, k=25, kp=5.5, X20=3, X10=V10=V20=0 (this will create W1=5 and W2=6). Or, try k=9, kp=3.5.
c) What about m=0.1, k=10, kp=15? And then there is m=0.1, k=10, kp=40?
What conditions create the "simple" lissajous figures, and what create the more complicated graphs?
Note: This physlet seems to do a great deal of processing .. and occasionally "freezes" - just press the play button to start it again .. after a few restarts it seems to be fine - I can't figure out why at the moment.