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# Re: Structural vibration question

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• To: "SEAOC Newsletter" <seaint(--nospam--at)seaint.org>
• Subject: Re: Structural vibration question
• From: Christopher Wright <chrisw(--nospam--at)skypoint.com>
• Date: Sat, 10 Jul 04 15:10:05 -0500

```>Could someone please explain to me the difference
>between the ìdamping ratioî and the ìdamping factorî?
There's some ambiguity here. I checked out a half-dozen textbooks, and one, Vierck, uses the term 'damping factor,' at all. Vierck uses the term for what I'm calling 'critical damping ratio.' The following explanation foolws all the other books I checked.

The damping coefficient is the proportionality factor that multiplies into the velocity to give the viscous damping force. It's defined in terms of the response of a simple (single DOF) system. You write the equation of motion as
Total force = F(t) -Kx -Cv = Ma
F(t) is the applied force, x, v and a are the displacement, velicity and acceleration of the mass, M and K is the spring rate. C is the damping coefficient,sometimes called the damping constant or simply 'damping.' It has units of force/velocity (lb-sec/in).

It turns out that there is a certain value for the damping coefficient where the system won't oscillate, like a car with good shock absorbers. That value is the so-called critical damping coefficient and it equals 2*sqrt(KM).  The damping ratio is the ratio of the damping coefficient to the critical damping coefficient and is usually expressed as a percentage. For example for welded structures at working stress levels the damping ratio is 2-3%.

>Iíve seen both of them defined as the ìpercentage of
>vibrational energy that is lost per cycle of vibrationî.
That's incorrect. You can show that the ratio of the peaks of successive maximum displacements is e^(-2*pi*damping ratio), which doesn't work out to a percentage of the energy loss. The fractional energy loss is proportional to the square of the fractional decrease in displacement per cycle.

>This means that they are different termsfor the same thing.  I also
>read in a book that the damping ratio is equal to one half of the damping
>factor.
That's wrong, too. I have a feeling that you've mixed up terminology between different texbooks, which have themselves been a little sloppy. My favorite is 'Structural Dynamics for the Practicing Engineer by H. M. Irvine.

Christopher Wright P.E.    |"They couldn't hit an elephant at
chrisw(--nospam--at)skypoint.com        | this distance"   (last words of Gen.
___________________________| John Sedgwick, Spotsylvania 1864)
http://www.skypoint.com/~chrisw

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