Forced Oscillations - Why Amplitude Reaches Fixed Value at Zero Frequency?

In summary, at a frequency of zero, the system is in a static state and the amplitude of the oscillator is equal to the force applied divided by the mass. At high frequencies, the acceleration is approximately constant, resulting in an amplitude that is proportional to the force and inversely proportional to the square of the frequency. It is possible to have an amplitude at zero frequency if the force applied is constant, and this only makes sense for a constrained structure.
  • #1
Jimmy87
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Hi, why is it that as the frequency of the driver decreases below the natural frequency of the oscillator it reaches a fixed amplitude when the external frequency is zero whereas whenever you go to the other extreme and have a very high external frequency the amplitude of the oscillator approaches zero? The graph that displays this information shows that the amplitude of the oscillator has a value (Fo/k) when the driving frequency is zero (i.e. the graph doesn't go through the origin). How can you have an amplitude at zero frequency? Thanks for any help given!
 
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  • #2
A frequency of zero is the same as doing statics, not dynamics. You apply a force to the spring and it stretches.

At high frequencies, the acceleration is approximately constant. From Newton's second law and ignoring the stiffness of the spring, the acceleration is close to ##F/m## so the amplitude is approximately ##F/(m\omega^2)##.
 
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  • #3
AlephZero said:
A frequency of zero is the same as doing statics, not dynamics. You apply a force to the spring and it stretches.

At high frequencies, the acceleration is approximately constant. From Newton's second law and ignoring the stiffness of the spring, the acceleration is close to ##F/m## so the amplitude is approximately ##F/(m\omega^2)##.

Thanks AlephZero. So are you saying that at the instantaneous moment you turn the signal generator on, you don't have a frequency but you still have a force therefore you are dealing with a static case at that instant so therefore the mass on the spring will get an amplitude equal to that of the vibrator? In other words are you saying that you are dealing with a case before the vibrator has completed one cycle?
 
  • #4
Well, in real life you probably can't set the vibrator to "zero" frequency. But you can imagine that if the frequency is very low compared with the natural vibration frequencies of the object (e.g. one cycle per hour or whatever), you can ignore the inertia forces (i.e. mass x acceleration) because the acceleration is very small. So this is the same as applying a "static" load that slowly increases and decreases, and measuring the static deflections it produces (and the deflections are proportional to the load, of course).

The math still makes sense if the frequency is zero. A force ##F_0 \sin \omega t## is always ##0## if ##\omega = 0##, but a force ##F_0 \cos \omega t## is a constant force ##F_0##.

Note, this only makes sense for a structure that is constrained in some way. If the object is completely free (e.g. floating weightless in space) the displacements will become to "infinitely large" as the frequency goes to zero, because you are applying a force in one direction for a long time and that will move the object a large distance.
 
  • #5


This phenomenon can be explained by the concept of resonance in forced oscillations. When an external force is applied to an oscillator, it can cause the system to oscillate at its natural frequency. This is known as resonance and results in a maximum amplitude of the oscillator.

As the frequency of the external force decreases below the natural frequency of the oscillator, the system is no longer in resonance and the amplitude decreases. However, as the frequency approaches zero, the system is still able to oscillate at a small amplitude due to the energy stored in the system. This is why the graph shows a non-zero amplitude at zero frequency.

On the other hand, when the external frequency is very high, the system is unable to keep up with the rapid changes and the amplitude decreases. This is because the system does not have enough time to build up energy and reach its maximum amplitude.

In summary, the fixed amplitude at zero frequency is a result of the energy stored in the system, while the decreasing amplitude at high frequencies is due to the inability of the system to keep up with the external force. This phenomenon is a fundamental aspect of forced oscillations and is crucial in understanding the behavior of oscillating systems.
 

1. What are forced oscillations?

Forced oscillations are periodic motions that are caused by an external force or driving force. This force acts on a system that already has its own natural frequency of oscillation.

2. What is the amplitude of forced oscillations?

The amplitude of forced oscillations is the maximum displacement of the oscillating system from its equilibrium position. It is directly affected by the frequency and magnitude of the external force.

3. Why does the amplitude reach a fixed value at zero frequency?

At zero frequency, the external force is not acting on the system and the natural frequency of the system takes over. This results in the amplitude reaching a fixed value as the system oscillates at its natural frequency without any interference from the external force.

4. How does the amplitude change with increasing frequency?

As the frequency of the external force increases, the amplitude of forced oscillations also increases. This is because the external force is providing more energy to the system, resulting in larger displacements.

5. What factors affect the amplitude of forced oscillations?

The amplitude of forced oscillations is affected by the frequency and magnitude of the external force, as well as the natural frequency and damping of the oscillating system. Additionally, the phase difference between the external force and the system's motion can also influence the amplitude.

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