Damping constant issue using English units

[tmorgan4]

I'm likely making a dumb mistake but I'm having issues coming out with the correct units for damping constant in US units. The confusion likely lies in the conversion between lbm and lbf but I can't seem to figure it out.

In a damped vibration system the critical damping constant C[sub]c[/sub] has units lbf-sec/in. If damping ratio ζ = 1.0 then C[sub]c[/sub] = 2*√(J[sub]o[/sub] * K[sub]t[/sub]) where J[sub]o[/sub] is the mass moment of inertia (lbf-in/sec[sup]2[/sup]) and K[sub]t[/sub] is the torsional spring constant (lbf-in/rad).

Using the above formula I get:

C[sub]c[/sub] = √[(lbf[sup]2[/sup] * in[sup]2[/sup])/(sec[sup]2[/sup] * rad)]

Radians disappear and the square root is taken, leaving me with:

C[sub]c[/sub] = lbf-in/sec (should be lbf-sec/in!!!)

To further add to the confusion, I tried the same method with a translational system and got the correct units. The critical damping constant for a translational system is C[sub]c[/sub] = 2*√(K*m) where K is the spring rate (lbf/in) and m is the mass (lbf-sec[sup]2[/sup]/in). This gives me the correct units of Cc = lbf-sec/in.

Can anyone spot the error???

 

[drawoh]

tmorgan4,

The English unit for mass is slugs, best ignored. Substitute m=w/g. The pound is a unit of force, equivalent to Newtons. Units of lb[sub]f[/sub] and lb[sub]m[/sub] are evil, as are kg[sub]m[/sub] and kg[sub]f[/sub]. This should straighten out your units.

--
JHG

 

[Denial]

TMorgan4,

I think there are two corrections required for your post, both related to the fact that you are talking about a rotational system rather that a translational one.

Firstly, you state that the mass moment of inertia has units FL/T[sup]2[/sup].
It should have units ML[sup]2[/sup] which (if you must insert force where mass makes things easier) is equivalent to FLT[sup]2[/sup].

Secondly, you state that the units for damping in a rotational system are FT/L.
The units should be torque per unit of rotational speed, or (FL)/(R/T), which simplifies to FLT.

In the above I have used M, F, L, T & R as generic (and consistent) units for mass, force, length, time and rotation respectively.[ ] Provided R is measured in natural units, ie radians, it can be ignored (as you have pointed out).

 

[tmorgan4]

Thank you both for the replies. I'm looking into the above suggestions at the moment. The reason for attempting to use lbm instead of slugs was due to a suggestion made in this document:

http://www.rotorlab.me.vt.edu/Unitsformass_2004_SS.pdf

I'm working on a problem where units of length are inches instead of feet so using slugs is just one extra conversion.

 

[Denial]

As gets illustrated regularly in Eng-Tips posts, there are (at least) two approaches to the units that are best used in dynamics problems.[ ] I'm in the school that says that all you have to do is use "consistent" units, and everything comes out in the wash.[ ] That document you cite comes from the other school, and serves only to confuse people like me.

Newton's Second Law of Motion states that "the acceleration of a body is directly proportional to, and in the same direction as, the net force acting on the body, and inversely proportional to its mass."[ ] (Definition straight out of Wikipedia.)[ ] For the purposes of this discussion, "consistent" units are those which result in Newton's constant of proportionality being exactly unity.

Denial's First Law of Dynamics Problems states that if you do not use consistent units in your dynamics problems, you WILL come unstuck sooner or later.

 

[drawoh]

Denial,

When I was college, we learned two sets of dynamics equations, one for English units, and one for metric. If you treat pounds as a unit of force, you can work with one set of equations.

F = ma = w/g[×]a

The nasty thing about Newtons and slugs is that they are not basic units. They are calculated from something else, and you have to be very careful with the units you combine them with.

1N = 1 kg.m/s[sup]2[/sup]

F = ma
m = F/a
1 slug = 1 lb.sec[sup]2[/sup]/ft, I think. A slug is based on feet, right?

In metric calculations, it is very important to convert everything to meters, kilograms and seconds. Millimetres and centimetres are dangerous. In the English system, you can ignore slugs and substitute w/g for mass.



--
JHG