Why does the amplitude of an undamped driven oscillator not vary with time?

In summary, the conversation discusses the behavior of a driven oscillator and how its amplitude changes over time. The equations obtained from solving the differential equations show that the amplitude does not vary with time, but the behavior becomes more complicated when the driving force is introduced. The conversation then explores two possible explanations for this behavior and the mathematical reasoning behind each one.
  • #1
adamjts
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The equations I'm getting when I solve the differential equations seem to imply that the amplitude of oscillation does not vary in time.

For example, if I have

x'' + ω02x = cos(ωt)

If we suppose that ω≠ω0,

then the general solution should look something like:

x(t) = c1cos(ω0t) + c2sin(ω0t) + (1/(ω22))cos(ωt)

This is okay with me mostly. But then thinking about what happens when ω→ω0 AND ω≠ω0, then obviously the amplitude of the oscillator should be huge. However, It would seem that the amplitude does not depend on time. Which is to say, that the exact moment that we introduce this driving force, the amplitude of the oscillator instantaneously becomes enormous. Which is hard to believe, because I would expect the object to start deviating from its simple oscillations more slowly and grow in time.

I know that when ω=ω0 that there is a factor of t in the amplitude, but that is not the case here.

Is it because the superposition of the two sinusoids makes it seem like the initial amplitudes are small. So when the driving force is introduced, the waves align such that the oscillating body does not seem to have a huge amplitude. But over some time, the waves will align such that the body does have evidently huge oscillations. This would imply, though, that the oscillations would become small again. In other words, we would expect long beats. Is this correct?

Or, maybe it's more likely that after the driver begins, the motion converges onto that of the driven oscillation? Why and how would one describe so mathematically?
 
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  • #2
adamjts said:
Which is to say, that the exact moment that we introduce this driving force, the amplitude of the oscillator instantaneously becomes enormous
You do have initial conditions to determine c1 and c2 that determine the starting situation. Try to plot a simple case to see how it looks.
Without damping the ##\omega_0## oscillation goes on forever, indeed.
 
  • #3
In order to understand what happen when ##\omega## tends to ##\omega_0## , let ##\omega=\omega_0+\epsilon##
##\omega^2-\omega_0^2=(\omega+\omega_0)\epsilon \simeq 2\omega_0\epsilon##
##\sin(\omega t)=\sin(\omega_0 t+\epsilon t)=\sin(\omega_0 t)\cos(\epsilon t)+\cos(\omega_0 t)\sin(\epsilon t) \simeq \sin(\omega_0 t)+\cos(\omega_0 t)\epsilon t##
##\cos(\omega t)=\cos(\omega_0 t+\epsilon t)=\cos(\omega_0 t)\cos(\epsilon t)-\sin(\omega_0 t)\sin(\epsilon t) \simeq \cos(\omega_0 t)+\sin(\omega_0 t)\epsilon t ##
The solution : ##x(t)=c_1 \cos(\omega t)+c_2\sin(\omega t)+\frac{1}{\omega^2-\omega_0^2}\cos(\omega t)## becomes :
##x(t) \simeq c_1 \cos(\omega_0 t)+c_2\sin(\omega_0 t)+c_2\cos(\omega_0 t) \epsilon t+\frac{1}{2\omega_0\epsilon}\left(\cos(\omega_0 t)+\sin(\omega_0 t)\epsilon t\right)##
##x(t) \simeq \left(c_1+\frac{1}{2\omega_0\epsilon} \right)\cos(\omega_0 t)+c_2\sin(\omega_0 t)+\frac{t}{2\omega_0}\sin(\omega_0 t)+c_2\cos(\omega_0 t)\epsilon t##
At ##t=0## the starting value is ##x(0)=x_0=\left(c_1+\frac{1}{2\omega_0\epsilon} \right)##
In fact, ##c_1## depends on ##\epsilon## so that the initial condition be fulfilled. : ##c_1=x_0-\frac{1}{2\omega_0\epsilon}##
##x(t) \simeq x_0\cos(\omega_0 t)+c_2\sin(\omega_0 t)+\frac{t}{2\omega_0}\sin(\omega_0 t)+c_2\epsilon t \cos(\omega_0 t)##
If ##\epsilon=0## the solution is : ##x(t) = x_0\cos(\omega_0 t)+c_2\sin(\omega_0 t)+\frac{t}{2\omega_0}\sin(\omega_0 t)##
which is exacly the solution of the equation ##x''+\omega_0^2 x=\cos(\omega_0 t)##
 
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  • #4
There are several typo in my first answer; I suppose that you can correct them.
 
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1. What is an undamped driven oscillation?

An undamped driven oscillation is a type of periodic motion where a system experiences a repeated back-and-forth movement due to an external force. The motion continues indefinitely without any loss of energy due to damping.

2. What is the difference between undamped and damped oscillations?

The main difference between undamped and damped oscillations is the presence or absence of damping. In undamped oscillations, there is no loss of energy and the amplitude remains constant. In damped oscillations, energy is gradually lost due to friction or other dissipative forces, causing the amplitude to decrease over time.

3. How is an undamped driven oscillation affected by the driving force?

The amplitude of an undamped driven oscillation is directly proportional to the amplitude of the driving force. This means that as the driving force increases, the amplitude of the oscillation also increases. However, the frequency of the oscillation remains unchanged.

4. How does resonance occur in undamped driven oscillations?

Resonance occurs in undamped driven oscillations when the frequency of the driving force matches the natural frequency of the system. This causes the amplitude of the oscillation to increase significantly, and can even lead to the system becoming unstable if the driving force is too strong.

5. What are some real-life examples of undamped driven oscillations?

Some common examples of undamped driven oscillations include a child on a swing, a pendulum, and a tuning fork. In each of these cases, the system experiences a periodic motion due to an external force, without any loss of energy due to damping.

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