We found last class that all solutions to
with converge to the particular solution
We also saw that this particular solution could be rewritten
where the gain is
and the phase shift is
Finally, we found that the gain is maximized at the resonance angular frequency
which is close to the unforced angular frequency
Of course, the last two definitions make sense only if is sufficiently small. Otherwise, the gain is maximized at zero and the unforced ODE does not have an oscillatory solution. What do these results say about the motion of a forced linear oscillator? Show the MIT video.
Consider . (To get the Greek symbol in an input cell, type \omega
and the tab key.)
a. Find the particular solution for arbitrary . Compare with our earlier work.
b. Plot the gain and phase shift on the interval . Compare with our earlier work.
c. Find the resonant radial frequency. Compare with the previous graph.
d. Plot the forcing term and the particular solution for values of less than, equal to, and more than the resonant radial frequency Compare with the earlier parts.
Find the solution to with initial conditions and .. Interpret.
Open two copies of the tone generator. Play one at 440 hertz and the other at 439 hertz to hear concert A with 1 second frequency beats. Decrease the second to hear the beat frequency increasing until two separate tones are perceived.
Memorex comercial. Play as a preview to an exercise in A06.