Literally: Weight x Radius squared. Weight in pounds and the radius in feet from the axis of gyration. This term is more commonly known as WK2, they are synonymous. It is a measurement of the inertia of the rotor, which combined with the inertia of the load, is used to determine acceleration time and torque requirements.
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The radius of gyration is the distance from the axis of rotation at which the mass can can be considered to be concentrated. If r is the physical radius of the rotating object; K^2 = r^2 for a thin walled hollow cylinder K^2 = 0.5*r^2 for a uniform solid cylinder; and K^2 = 0.4*r^2 for a uniform solid sphere.
I didn't know that last detail. So WR2 would be inaccurate for a motor rotor then. That would explain why we see WK2 used. Someone picked out the wrong reference, probably learned it in the same textbook I did (way back when) and like me, repeated it. Dealing with motors, not cylinders, I don't see it used often and have never needed to really know the difference since then because I always knew what they meant when using the term (incorrectly as it turns out) for rotors. Today, I learned there is a difference. Thanks for that CJCPE.
Now I can be even nerdier when correcting people in the future!
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By uniform, I mean homogeneous. To use the formula for a cylinder, the density of the cylinder must be the same at every point. To calculate the WK^2 of an induction motor, you would need to consider that some of the cylinder is steel and some is aluminum. For an accurate calculation, you would need to separatly calculate the WK^2 of various aluminum and steel shapes and then add them together.
But remember everyone that moments of inertia are in units of MASS*length^2 (e.g. kg-m^2), not weight*length^2. If you are using units of weight (e.g. lbs), the units of moment of inertia are weight*length*time^2 (e.g. lb-in-sec^2).
The reason you want to know moment of inertia is to be able to calculate the relationship between torque and angular acceleration: T = J * alpha.
In typical SI units, you get J(kg-m2)*alpha(sec^-2) =T(kg-m^2-sec^-2) = T (N-m), so no conversion factors are necessary.
In typical English units, you get J(lb-in-sec^2)*alpha(sec^-2) = T (lb-in), and again, no conversion factors are necessary.
But you must be really careful if you start using expressions for MOI like those above. You will need a conversion factor of 1/g (g=32.2 ft/sec^2) in the T=J*alpha equation if you use W*K^2 for moment of inertia. I've seen people get very wacky results if they leave that out.
People also often screw up if they use non-official units of "pounds-mass" or "kilograms-force" as they try to convert between SI and English. Again, they are usually off by a factor of "g" if they are not careful when they do this.
The English unit of mass is a slug. It is the equivalent unit to the metric kilogram. Mass moment of inertia is expressed in slug-ft^2.
A pound is a unit of force, paralleled by the "newton" in metric.
1 lb = 1 slug-ft/s^2
As long as you use slugs for mass, feet for length and pounds for force, you can use all of the same simple physical formulas (like F = ma, etc.) as in metric without having to include a conversion factor. You're better off avoiding lbm and kgf entirely.
Gentlemen, I thank you for the help. If WR^2 is wrong for rotors, how did it motor manufacturers ever adopt this term. (They also use the WK^2 on motor data as well)
WR^2 is not wrong, but it is not part of the more widely adopted system of units. There are formulae in which WR^2 or WK^2 is used. Those formulae contain constants that make WR^2 the proper figure to use. If you don't have those formulae at hand, you can get tripped up if you don't remember the "g" when converting units.
Certainly care should be taken while using these parameters.
Wk2 is a particular “simplified” constant for moment of inertia.
It has “weight” instead of “mass” and it is used on the expression:
t = Wk2 (rpmf - rpmi)/(308*Tavg)]
t = accelerating time in seconds
rpmf = final rotating speed in revolutions per minute
rpmi = initial rotating speed in revolutions per minute (normally zero at stand still)
Tavg = Average accelerating torque in Lb*Ft; the Motor torque minus The Load Torque.
By the other hand, in the general expression of mechanics; T = j*alpha
"j" is given as a “mass = weight/g “ times radius of gyration squared.
“g” is the gravity acceleration in length per seconds squared
"T" is the torque applied in force times distance
"Alpha" is the angular acceleration in rad/sec2
The units must be consistent for the selected system of measurement; MKS, cgs, English Imperial, etc.
Motor manufacturers presumably publish WK^2 data because they believe that is preferred by their customers. Customers who use WK^2 presumably think it is easier to work with formulae that use weight data and avoid mass. The difficulty arises when those who are accustomed to using WK^2 need to communicate with those who are accustomed to the more scientific approach.
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