What Defines the Maximum Frequency in Damped Driven Oscillations?

In summary, the conversation discusses the equation (w_max)^2 = (w_0)^2 - (1/2)y^2, which is commonly used in damped driven oscillations. The equation represents the exact resonance frequency (w_max) and the classical resonance frequency (w_0), with y representing the width or damping constant divided by the mass of the oscillator. The equation can be derived from the second order ODE for a damped, driven oscillator.
  • #1
Master J
226
0
I am looking at forced vibrations and I have come across this:

(w_max)^2 = (w_0)^2 - (1/2)y^2

Now I am not entirely sure of what the (w_max) is. ANd where does this equation come from? It was simply stated without a derivation.

THanks guys!:biggrin:
 
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  • #2
There is simply a blank page with that formula on it?

Surely there is some sort of context. As you write it I not only do not know what w_max is, I also don't know what w_0 is or what y is!
 
  • #3
W_O is the natural/resonance frequency for the system being driven by a force with frequency w.

y is the width, or the damping constant divided by the mass of the oscillator.

The equation comes up in resonance. I think it has to do with the maximum amplitude of the system ?
 
  • #4
That expression comes up a lot in damped driven oscillations- not just masses and springs, but any sort of linear oscillator.

W_max is the exact resonance frequency, w_0 is the "classical resonance frequency" (i.e. the resonance without damping present).

This expression is strightforward to derive beginning with the 2nd order ODE for a damped, driven oscillator.

http://en.wikipedia.org/wiki/Harmonic_oscillator

(about half-way down)
 

FAQ: What Defines the Maximum Frequency in Damped Driven Oscillations?

What is forced vibration and how does it differ from free vibration?

Forced vibration refers to the oscillation of a system caused by an external force or input. This force can be periodic or non-periodic and can be applied at any frequency. In contrast, free vibration occurs when a system vibrates on its own without any external influence or input.

What is resonance and how does it relate to forced vibrations?

Resonance occurs when the frequency of the external force matches the natural frequency of the system. This causes the amplitude of the forced vibration to increase significantly, potentially leading to damage or failure of the system. Resonance is a common phenomenon in forced vibrations and can be both beneficial and detrimental depending on the situation.

What factors affect the resonance frequency of a system?

The resonance frequency of a system is primarily determined by its stiffness and mass. A stiffer and lighter system will have a higher resonance frequency, while a more flexible and heavier system will have a lower resonance frequency. Other factors that can influence resonance frequency include damping, boundary conditions, and external forces.

How can resonance be controlled or prevented in a system?

There are several ways to control or prevent resonance in a system. One approach is to change the natural frequency of the system by altering its mass, stiffness, or damping. Another method is to use vibration isolators or absorbers to minimize the transfer of energy between the system and its surroundings. Additionally, careful design and analysis can help identify potential resonance frequencies and avoid them.

Can forced vibrations and resonance be beneficial?

Yes, forced vibrations and resonance can be beneficial in certain situations. For example, musical instruments rely on forced vibrations and resonance to produce sound. In engineering, resonance can be used to amplify small vibrations and signals, making them easier to detect. However, it is crucial to carefully consider and control forced vibrations and resonance to prevent potential damage or failure of systems.

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