Understand F_0 & Omega in Driven Oscillators

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In summary, the conversation discusses the equation for driven oscillators and the meanings of the variables involved. The value ##F_0## represents the amplitude of a sinusoidal forcing function, while ##\omega_0## and ##\omega## represent the undamped and damped natural frequencies of the system, respectively. It is also noted that ##F_0## has dimensions of force.
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I found this equation for driven oscillators, and am unsure what the F_0 means, and how it's used. Can someone please briefly explain this?
Also, I don't get what the difference is between the initial and final omega. (how can there be a final omega if it's constantly increasing?)
 

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##F_0## is the amplitude of a sinusoidal forcing function, ##\omega_0## is the undamped natural frequency of the system, and ##\omega## is the frequency of the forcing function, which looks something like ##f(t)=f_0\sin(\omega t)## or ##f(t)=f_0\cos(\omega t)##.

The definition of ##\omega## shown is also the damped natural frequency of the system.
 
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  • #3
So are the units for F_0 meters? (So it's not a force?)
Thanks.
 
  • #4
##F_0## has dimensions of force. You can easily see this by looking at the dimension of each part of the expression. Consider [tex]f(t)=f_0 \cos(\omega t).[/tex] The dimension of ##t## is of course time, and the dimension of ##\omega## must be angle/time so that ##\omega t## has dimensions of an angle (e.g. radians). This must be so, as, for example, what is the cosine of a second? It's meaningless. Since ##f(t)## is a force, and ##\cos(\omega t)## is dimensionless, it must be that ##f_0## has the dimensions of force, e.g. Newtons for SI.
 
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F_0 in the equation for driven oscillators represents the amplitude of the driving force. This force is applied to the oscillator at a certain frequency, which is represented by the term omega (ω). The initial omega refers to the natural frequency of the oscillator without any external driving force, while the final omega refers to the frequency of the oscillator when it reaches a steady state under the influence of the driving force.

It is important to note that the final omega is not constantly increasing, but rather it reaches a steady state where the energy input from the driving force is balanced by the energy loss due to friction or other factors. In this state, the oscillator will vibrate at a constant amplitude and frequency.

Overall, the equation for driven oscillators helps us understand how an external force can affect the behavior of an oscillator and how it reaches a steady state under the influence of this force. I hope this explanation helps clarify the meaning of F_0 and omega in driven oscillators.
 

What is F_0 in driven oscillators?

F_0 represents the amplitude of the external driving force in a driven oscillator. It is the maximum force that is applied to the oscillator and determines the strength of the driving force.

What does Omega represent in driven oscillators?

Omega (ω) is the angular frequency of the external driving force in a driven oscillator. It is equal to the frequency of the driving force divided by 2π.

How does F_0 affect the behavior of a driven oscillator?

The higher the value of F_0, the greater the amplitude of the oscillations in the driven oscillator. This means that a larger driving force will produce larger oscillations in the system.

What happens when Omega is equal to the natural frequency of the oscillator?

When Omega is equal to the natural frequency of the oscillator, the system is said to be in resonance. This means that the amplitude of the oscillations will be at its maximum and the energy transfer between the driving force and the oscillator will be the most efficient.

How do F_0 and Omega affect the phase difference in driven oscillators?

The phase difference between the driving force and the oscillations of the driven oscillator depends on both F_0 and Omega. If Omega is close to the natural frequency of the oscillator, the phase difference will be small. However, if F_0 is significantly larger than Omega, the phase difference will be close to 90 degrees.

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