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# Calculating Reflected Inertia for Linear Systems (cont.)

In the last several postings we saw how a ball screw changed the rotary motion of the stepper motor to linear motion. We calculated the rotational equivalent of a 200 pound linear load that was effectively reduced by the inverse square of the pitch or more accurately:

J load = (W/P2)* 0.0253 where the J load inertia units are pound-in2

A gearbox does the same thing. It will reduce the output shaft’s rotary load by the inverse square of the gear ratio. Let’s go all the way back to our conveyor belt system where we calculated the reflected inertia as 0.529 lb-in2 and use that value for an example. What we’re doing is adding a gearbox in between the motor shaft and the conveyor system.

We’re also going to assume that we have a 5:1 gear ratio (gr). Just so you know, I’m not growling, I’m just using gr as an abbreviation for gear ratio. If I was growling I’d use something like grrrr.

The reflected inertia to the input side of the gearbox will be:

J total reflected = J load/gr2 + the inertia of the internal components of the gearbox.

Plug and chug:

0.529/52 = 0.0212 lb-in2 plus whatever the internal components of the gearbox adds.

The gearbox manufacturer should define that value, but you can estimate what that value is by getting the dimensions of the gear that is going to be attached to the motor shaft.

Let’s say the sun gear is steel and is 1” in diameter and ½” wide. What’s a sun gear you ask? Google “sun and planet gears” and you’ll find all sorts of great information.

Our sun gear will be attached to the motor shaft and using our disk inertia equation we get:

J sun gear = W r2/2. But first we need to calculate the sun gear’s weight.

The “but first” in the above sentence always reminds me of a “M*A*S*H” ©™®* episode where two of our heroes are trying to defuse a bomb. One of them is reading the instructions while the other one is doing the work. They get to the point where they’re inside the bomb and he reads the instructions: “Cut the red wire” and you hear the sound of the wire being snipped. He immediately continues to read: “but first, make sure that it’s connected to the positive side of the battery, otherwise cut the black wire.”

So where were we? Oh yea, calculating the weight.

π*r2*length*density = weight in pounds.

3.14159*0.52*0.5*0.283 = 0.111 pounds.

Then the inertia.

J sun gear = (0.111 * 0.52)/2 = 0.0139 lb-in^2

Our total inertia is the sum of the reflected load inertia and the inertia of the sun gear.

We also have the planetary gears to be concerned about, but they’ll be reduced by the gr squared and won’t add all that much to the total. Again the manufacture of the gearbox should publish those numbers. You could always round the sun gear’s inertia up a little to compensate for the planetary gears.

We have:

J reflected total = 0.0212 + 0.015 = 0.0362 lb-in2

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