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Calculating reflected inertia for linear systems

Author: Bob Parente | May 22, 2019

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Review from last time

In the last several postings we talked about ball screws and acme screws along with the advantages and disadvantages of both of them. In this posting we’ll get to calculate the inertia that this type of rotary-to-linear converter reflects back to the motor.

Setting up the assumptions for the calculation

We’re going to assume that we have a steel ball screw that is 18” long and 0.5”in diameter and caries a load of 200 pounds on its movable horizontal plate. We’ll also assume that it has a lead of 0.2 inches or is 5 a pitch screw or 5 revolutions per inch, that the screw is 90% efficient and that you remember the equation for a disk as J = Wr2/2. You do remember that don’t you? Don’t fret if you don’t. You could always look it up on Google.

Calculating the weight of the screw

First, we calculate the screw’s weight in pounds using its volume, length and density.

W = (πr2)*length*density

Substituting that for W we get:

J = ((πr2)*length*density*r2)/2 = (πr4*length*density)/2

If we rewrite it using the diameter (d) divided by 2 for r and plug in the density value for steel (0.283 lbs/in3) we have:

J = (π(d/2)4*length*0.283)/2 = π*(d4/16)*length*0.283

Reducing it by using all the known values we get

J screw = d4*length*0.0278

(I did all that because this is the equation I grew up with.)

Our diameter is 0.5” and our length is 18” and of course our screw is made of steel.

Plug and chug: J screw = 0.54*18*0.0278 = 0.031275 lb-in2.

Calculating the reflected load

The equation for the load as its reflected back to the motor shaft is:

J load = Weight*(1/pitch2)*(1/2 π)2 or J load = (W/P2)* 0.0253

This effectively is the linear load inertia converted to rotary load inertia that the motor shaft sees.

We know that our load weighs 200 pounds and that we have a 5 pitch screw.

Plug and chug again: J load = (200/52)*0.0253 = 0.2 lb-in2.

The total reflected inertia to the motor shaft is the sum of the two inertias:

Or J screw + J load = 0.03128 + 0.2 = 0.2313 lb-in2

Calculating the torque required to overcome friction losses

Now if you have a 200 pound weight sliding around on a surface, one would think you might have some friction losses.
The torque to overcome that friction is:

Torque in oz-in = Friction force in pounds/ ((2 π/16)*pitch*screw efficiency)
Torque in oz-in = Friction force in pounds/ ((0.393)*pitch*screw efficiency)

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