A Damped Oscillator and Negative Damping Force

In summary, the damping force opposes motion and dissipates energy over time. The total energy of the oscillator remains constant, no matter what the damping force is. This is due to the kinetic and potential energy being described by separate terms.
  • #1
Nanofan01
5
0
A damped oscillator is described by the equation m(x'') + b(x') + kx = 0, where the damping force is given by F = -b(x'). Show that the rate of change of the total energy of the oscillator is equal to the (negative) rate at which the damping force dissipates energy.
 
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  • #2
Hi there,

to me, damping intrinsically means that the force opposes motion. So if you have a negative damping force, it's either confusing, or redundant.

This also seems like a homework question, in which case it's not posted in the right forum.

It seems to me like you could use common sense for this question then match the math up to it. If there is no damping, that means that the oscillation will be periodically identical (like a sine wave). No energy will be lost. Once you add that damping, that is the only cause of dissipating energy.

If you calculate the total change in energy of that equation of motion, then do the same without the damping force, you should find that potential + kinetic will be constant. Therefore you can conclude that the damping term is solely responsible for the loss of energy.
 
  • #3
dacruick said:
to me, damping intrinsically means that the force opposes motion. So if you have a negative damping force, it's either confusing, or redundant.

The force does oppose the motion (assuming b > 0).

m(x'') + b(x') + kx = 0
m(x'') + kx = - b(x') = F
 
  • #4
It makes sense that the rate of change of the oscillator is equal to the negative damping force but how do you mathematically derive [-b(x')] from [m(x'') + b(x') + kx]?
 
  • #5
Also, which would be the right forum?
 
  • #6
Nanofan01 said:
It makes sense that the rate of change of the oscillator is equal to the negative damping force but how do you mathematically derive [-b(x')] from [m(x'') + b(x') + kx]?

the kinetic energy is described by the m(x'') term, and the potential energy is described by the kx term. That leaves one term out right? Which just so happens to be your answer.

And there is a specific forum for homework. I don't think you'll have too much trouble finding it if you look.
 

1. What is a damped oscillator?

A damped oscillator is a physical system that exhibits oscillatory motion, such as a mass on a spring, that is subject to a damping force. This force causes the amplitude of the motion to decrease over time, eventually reaching a steady-state or equilibrium position.

2. What is negative damping force?

Negative damping force is a type of damping force that acts in the opposite direction of the motion of the oscillator. This can cause the amplitude of the oscillation to increase over time, leading to unstable behavior.

3. How does negative damping force affect a damped oscillator?

Negative damping force can cause a damped oscillator to exhibit unstable behavior and can increase the amplitude of the oscillation over time. This can result in the oscillator moving farther and farther from its equilibrium position.

4. What causes negative damping force?

Negative damping force can be caused by several factors, such as friction, air resistance, or an external force applied in the opposite direction of the motion. In some cases, it can also be intentionally introduced to a system to achieve certain effects, such as in electronic circuits.

5. How is negative damping force different from positive damping force?

Positive damping force acts in the same direction as the motion of the oscillator and causes the amplitude of the oscillation to decrease over time, leading to a stable equilibrium position. Negative damping force, on the other hand, acts in the opposite direction and can cause unstable behavior and an increase in amplitude over time.

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