## PSN-L Email List Message

Subject: Re: Zero Length Spring
From: Brett Nordgren Brett3mr@.............
Date: Fri, 29 Dec 2006 17:50:03 -0500

```Geoff,

The relationships between mechanical damping, Damping Factor and Q aren't
too involved.

If you assume an undamped spring-mass system with a mass of M, kg, and a
spring constant K, Newtons/m , it will vibrate with a natural frequency of
omega-zero = 2 sqrt(K/M) radians per second. (You can divide radians per
second by 2 Pi to get cps or Hz ).

If you then add velocity damping, which means you add a resistance force
which is exactly proportional to the instantaneous velocity of the mass, of
an amount = R, Newtons / meter/second, you can then calculate the Damping
Factor, zeta and Quality factor, Q .

Quality Factor, Q, = M / R or Q = 1 / (2 * omega-zero * zeta)

and

Damping Factor, zeta, = R / ( 2 * sqrt (M * K)) or zeta = R / (2 * M *
omega-zero) = 1 / (2 * omega-zero * Q)

Regarding dimensions, since one kilogram meter/sec^2 = 1 Newton, the units
all cancel for both Q and zeta, so they are both dimensionless.

The zero in omega-zero signifies the *undamped* natural frequency.  As you
add damping, looking at the frequency response of the spring-mass system,
its amplitude peak gets lower and at the same time moves somewhat in
frequency before the damping gets so great that the peak disappears altogether.

You can see the same equations in a much prettier form near the end of

http://bnordgren.org/seismo/feedback_in_seismic_sensors.pdf   (underscored
spaces)   Though it's pretty big - a couple of MB, I think.

Regards,
Brett

At 08:29 AM 12/28/2006 -0700, you wrote:
>I understand Damping and Q
>are simply inverses of each other.
>Q is quite complex in that it
>requires the knowledge of the
>3db (0.7071) cutoff points as
>relating to the frq of interest.
>
>Can anyone tell me the proper way
>to express damping ??
>I sort of understand Q but not
>how you arrive at the proper numbers
>for damping alone.
>
>Regards;
>geoff

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