Damped Natural Frequency

[breaking_point]

Hey,

I've always been curious as to how the damped natural frequency is derived:

ωd = ωn sqrt(1−ξ^2)

Does anyone have any literature? Is it empirical based on experiments?

Thanks

 

[JStephen]

That discussion is several pages in my vibrations textbook. It is not empirical, it is derived from the equations of motion

 

[structSU10]

the damping VALUE used for most building structure could be considered empirical. the actual equation is a simplification, lumping all sources of damping together, and really only using the percentage of critical damping that is present.

 

[JStephen]

See the attached scan.

 

[WARose]

Like another poster said: it (the derivation of that formula) typically takes up a few pages in text of most dynamics books. (See 'Structural Dynamics: Theory and Applications', by: Tedesco, et al for one.)

Does anyone have any literature? Is it empirical based on experiments?

Do you mean the [red]value[/red] of damping? It is a pretty elusive value. It varies based on material, mode of vibration (i.e. lower modes vs. higher ones), stress level in a material, etc ,etc. We simply work with best estimates based on experiment.

 

[breaking_point]

Hi all,

Thanks for the responses.

To clarify - no, I don't mean a value of damping. I understand this is impossible to predict and depends on many, many things for a structure (though we know typical ranges for different types of structures). I am referring to the derivation of the natural damping equation. I'm going to read through the literature you've all provided, it looks like it will be helpful. I find that have a general understanding of derivations helps my knowledge from first principles.

 

[JoelTXCive]

I can't tell you exactly how the damped natural frequency is derived........BUT I did a pretty good math proof on how the regular natural frequency is derived and how the equation of motion for free vibration is solved.

In the attached math proof for free vibration, the damping force is zero (coefficient times velocity which is c times u_dot). The fact that the damping term is zero allows us to solve the equation of motion with a nice and simple (i use that term loosely lol) quadratic equation. If the damping term is not zero, then I think the math gets a lot harder. I had it on my to do list to figure out......but haven't done so yet.

Anyway, maybe the attached will help.

If someone grinds through the math on a damped equation of motion, then please post. I would love to see.

 

[breaking_point]

@JoelTXCive,

I actually have been through some nice notes of the undamped free vibration case, but yours are presented so much better that I'm going to use those instead, so thanks! Perhaps I'll make a similarly formatted doc for the damped case and share it back here.

 

[JoelTXCive]

Thanks for the compliment!

I used a couple textbooks to come up with those notes, but there was one older textbook that I thought was pretty good. I don't have it at the office though. I'll see if I can find it this weekend and post the name/author.

I add a lot commentary to my notes. If I don't, I end up looking at them a year or two after the fact and thinking "what in the heck was I doing here?"

Also, the textbooks always seem to assume that the reader is as smart as the author in the math department. They always skip steps. I like to see each step.

 

[JoelTXCive]

This is similar to what JStephen posted above.

These are excerpts from the Richart, Hall & Woods' Vibrations of Soils & Foundations, which is a great book. I found a used copy on Amazon.