Converting from lb_f/in to N/m issues

[Dustinsfl]

Homework Statement

I am trying to find a damping ratio. I know $k=50$ lb$_f$/in, $m=50$ lb, $c = 0.75$ lb$_f$-sec/in.

Homework Equations

$1$ lb$_f$/in = $175.1268$ N/m
$1$ lb = $0.4536$ kg

The Attempt at a Solution

Then $k = 8756.34$ N/m, $c = 131.345$ N/m-sec, and $m = 22.675$ kg.
$$
\zeta = \frac{c}{2\omega_nm} = \frac{131.345}{2(0.00259)(22.675)} = 1000.26
$$
but this seems awfully large. Are my conversion factors incorrect?

 

[vela]

How is $\omega_n$ defined?

 

[Dustinsfl]

How is $\omega_n$ defined?

Thanks. I divided m by k instead of the other way around.