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In Feynman Lectures Vol.1 on transients, when calculating the power of a forced oscillation it is written that the stored energy of the oscillation (the spring's kinetic energy and potential energy) doesn't change if it is averaged over a long time. And so its derivative wrt time is zero.

But there is no explanation given for this. So why doesn't the stored energy change over time?

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Work is done on the oscillator by the driving force. This puts energy into the oscillator. At the same time energy is taken away as the oscillator does work against the damping force.

Initially the energy stored in the driven oscillator increases with time because the work done on it by the driving force exceeds the work done against the damping force. However the average work done by driving force increases more slowly, in proportion to the amplitude, whereas the work done against the damping force increases more quickly, in proportion to the square of velocity. In addition the driving force gradually gets out of phase with the motion of the oscillator, reaching a maximum when the velocity is low, whereas the damping force is always in phase with the velocity, reaching a maximum when the velocity is maximum.

Eventually the work done against the damping force increases enough to equal the work done by the driving force, so that the average work done on the oscillator is zero.

This is similar to an object falling through the air. Its velocity increases until it reaches terminal velocity. Then the work being done on it by gravity equals the work being taken away by air resistance. (However in this case the driving force like the damping force is always in phase with the velocity of the falling object.)

In the case of an undamped oscillator there is no resistance. However the phase difference between the driving force and the velocity of the oscillator gradually increases until it is exactly $\pi$ radians out of phase with the velocity of the oscillator. Then the average work done by it per cycle is zero. Like the restoring force, on one quarter cycle it is doing positive work, increasing the velocity and kinetic energy of the oscillator; on the next quarter cycle it is doing negative work, slowing the oscillator and taking energy away from it.

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    $\begingroup$ Why does the driving force get out of phase with the motion of the oscillator gradually? And what will happen if the driving force is out of phase with motion initially? $\endgroup$
    – Theoretical
    Commented Apr 13, 2020 at 14:02
  • $\begingroup$ How I have worded the description is misleading : the frequency and phase of the driving force is fixed, it is the frequency and phase of the oscillator which changes in response to the driving force. I shall try to add some more mathematical details to make the situation clearer. Meanwhile I refer you to Rob's answer in the linked question. $\endgroup$
    – sammy gerbil
    Commented Apr 13, 2020 at 17:28
  • $\begingroup$ Some mathematical details will be very helpful for me. $\endgroup$
    – Theoretical
    Commented Apr 14, 2020 at 4:49
  • $\begingroup$ In the next page of the book, it is written that the energy stored per oscillation is $2\pi \frac{1}{2}m({\omega}^2+{{\omega_0}}^2).<x^2>$. How does this make sense? If energy is being stored per cycle, then shouldn’t the stored energy of the oscillator increase and so give non zero derivative wrt time? $\endgroup$
    – Theoretical
    Commented Apr 14, 2020 at 11:36
  • $\begingroup$ The average power going into the oscillator after a long time can be zero while the energy stored is a non-zero constant. The average power going in was non-zero during the transient phase but in the steady state it is zero. It is not extra energy being stored in each cycle. $\endgroup$
    – sammy gerbil
    Commented Apr 14, 2020 at 14:03
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Once the oscillator is out of the transient regime, its behaviour is periodic with a certain amplitude (and a phase shift) depending on the parameters of the system and the driving. As the motion is periodic, any quantity averaged over a period - such as the kinetic energy or potential energy - will be constant.

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