Total energy of a damped oscillator

In summary, the total energy of a damped linear oscillator can be expressed as a function of time by using the equation E(t) = 1/2kx^2 + 1/2m(dot)x^2. The oscillation will continue indefinitely in an ideal situation, but in practical terms it can be considered to stop when the amplitude is comparable to thermal noise or a set criterion. The time constant, sqrt(m/k), can also be used to estimate the time when the oscillator will stop oscillating by multiplying it by a factor, such as five. However, it is important to note that dE/dt is not always negative, as the energy can be dissipated at specific points in the cycle. The solution for
  • #1
Signifier
76
0
Is it possible to express the total energy of a damped linear oscillator as a function of time? I'm confused here. I'd like to find E(t). As the oscillation is damped, dE/dt should everywhere be negative (energy being dissipated as radiation or heat). By setting E(t) equal to zero, shouldn't I be able to solve for the time at which the energy of the oscillating system is zero, and thus the time at which the system stops oscillating? And shouldn't this time be finite?

Is there another way to find the time at which the damped oscillator will stop oscillating?

Thanks!
 
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  • #2
Yes. The peak amplitude of the oscillation, that is, the envelope, decays exponentially. Since average or rms energy is related to the peak amplitude, it also decays exponentially. In theory it never exactly reaches zero so you can't say when the oscillator "stops." In practice you can say it stops when the amplitude is comparable to thermal noise or some other criterion. It is more common to specify the time constant, which is the time for the envelope to decay to 1/e of its initial amplitude.
 
  • #3
Well... you can simlpy solve the differential equation for a damped oscillator, then use
[tex]E=\frac{1}{2}kx^2+\frac{1}{2}m\dot{x}^2[/tex]
 
  • #4
An ideal damped oscillator won't stop oscillating until infinite time has elapsed. However, practically the easiest way to find the time when the damped oscillator will top oscillating would be to determine the time constant, sqrt(m/k), and then multiply it by five because after five time constants the motion will be reduced to 1% (or something close to that) of its initial amplitude.
 
  • #5
Signifier said:
As the oscillation is damped, dE/dt should everywhere be negative (energy being dissipated as radiation or heat).

The above posts are correct in saying the oscillation continues for ever, but dE/dt is not always negative.

Taking a mechanical damped oscillator for example, with equation of motion M x'' + C x' + K x = 0, the energy is dissipated by the the damper. The work is (force times velocity) = C x'^2. That is zero twice every cycle, when the velocity becomes zero.

If you evaluate E from tim_lou's equation you will get an exponential decay multiplied by a something looking like (A + B sin pt), which is a curve that "wobbles" around the "average" exponential decay in the energy.
 
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  • #6
BTW, when you consider the different solutions for under-damped, over-damped or critically damped oscillation, you get different solution.

I think there won't be any wobbling in the over-damped or critically damped cases.
 

1. What is a damped oscillator?

A damped oscillator is a physical system that experiences oscillations over time, but these oscillations gradually decrease in amplitude due to the presence of energy-dissipating forces such as friction or air resistance.

2. How is the total energy of a damped oscillator defined?

The total energy of a damped oscillator is the sum of the kinetic energy (energy of motion) and potential energy (stored energy) of the system. It remains constant as long as there is no external energy input or loss.

3. What factors affect the total energy of a damped oscillator?

The total energy of a damped oscillator is influenced by the amplitude of the initial oscillation, the strength of the damping force, and the natural frequency of the system. These factors determine the rate at which energy is dissipated and the overall behavior of the oscillator.

4. How does the total energy of a damped oscillator change over time?

As the damped oscillator undergoes oscillations, its total energy decreases due to the dissipation of energy from the system. Eventually, the oscillator will come to rest at its equilibrium position, where the total energy is at its minimum value.

5. Can the total energy of a damped oscillator ever increase?

No, the total energy of a damped oscillator can never increase because energy is constantly being dissipated from the system. Even if external energy is added to the system, it will eventually be dissipated through the damping force, resulting in a decrease in total energy.

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