I would like to cut some gears. I have a Hardinge UM and an Ellis Dividing Head. To synchronize the hob and gear blank I want to set up a rotary encoder on the mill and a stepper motor on the dividing head.

I have read all I can on the google and ordered the Digital Machinist 2012 summer issues detailing the building of an Electronic Dividing Head. I cannot find any documentation on how anyone came up with the encoder or stepper motor specifications. It almost seems like they grabbed really big numbers hoping they would be good enough.

I am a numbers kind of guy and would like a better idea of why the numbers recommended are what they are...

Lacking any direction I would like to embark on a thought process.

Let's say I want to build this system. I want to synchronize the hob and gear blank so that the tool path does not deviate more than 0.001"+/- of where it is supposed to be. To do this let's assume I need to be able to measure the gear blank tooth position, and hob tooth lateral position to within 0.0002".

The questions are:

How many pulses per revolution does my rotary encoder need to produce to track the hob tooth lateral movement to within 0.0002"?

How many steps per revolution does my stepper motor need to have to move the gear tooth 0.0002" per step?

Starting with the first question:

One revolution of the hob will result in the hob cutter moving one tooth laterally. I will call this lateral movement the ToothArc.

ToothArc = PitchCircumference/NumTeeth.

PitchCircumference = pi*NumTeeth/DiametralPitch.

Or ToothArc = (pi*NumTeeth/DiametralPitch)/NumTeeth = pi/DiametralPitch.

Step_Rev = Pulses / Revolution.

ToothArc = Lateral Distance / Revolution

So Lateral Distance / Pulse = ToothArc/Step_Rev = pi/(DiametralPitch*Step_Rev)

I know I want Lateral Distance / Pulse = 0.0002" and I want to solve for Step_Rev:

Step_Rev = pi/(DiametralPitch*0.0002) = 5000*pi/DiametralPitch = 15,700/DiametralPitch

So for 16DP I need 981 pulses/revolution from my rotary encoder. For 24DP I need 654 pulses/revolution.

This feels good as my cheap eBay encoder can do 2,400 pulses per revolution and is rated for 5,000rpm.

Now for the second question:

My dividing head has a 90:1 ratio. My stepper motor has Steps_Rev steps per revolution. So for one revolution of the gear blank I have 90*Steps_Rev stepper motor steps.

One revolution also move the gear teeth the DiametralCircumference distance. Rather than express this in terms of DP and NumTeeth, lets just work with gear blank diameter (Gdia).

So the circumference = pi*Gdia. So we could express the distance moved along the circumference of the gear blank per stepper motor pulse = pi*Gdia/(90*Steps_Rev).

I know I want 0.0002" = piGdia/(90*Steps_Rev) or Steps_Rev = 5000*pi*Gdia/90 = 175*Gdia.

So for a 6" gear I need 1050 steps/revolution, a 3" gear would be 525 steps/revolution, a 1" gear 175 steps/rev.

This would require microstepping for a 200 step motor, or a reduction gear between stepper motor and dividing head.

So my questions are:

Is my math correct?

Is my goal of 0.001" accuracy in tooth path high or low? I would like gears that look good and move smoothly, I am not building swiss watches.

Is my assumption that I need to measure and move 0.0002" to get 0.001" accuracy high or low?

Is there a better way to approach this (from a mathematical modeling perspective)?

I have read all I can on the google and ordered the Digital Machinist 2012 summer issues detailing the building of an Electronic Dividing Head. I cannot find any documentation on how anyone came up with the encoder or stepper motor specifications. It almost seems like they grabbed really big numbers hoping they would be good enough.

I am a numbers kind of guy and would like a better idea of why the numbers recommended are what they are...

Lacking any direction I would like to embark on a thought process.

Let's say I want to build this system. I want to synchronize the hob and gear blank so that the tool path does not deviate more than 0.001"+/- of where it is supposed to be. To do this let's assume I need to be able to measure the gear blank tooth position, and hob tooth lateral position to within 0.0002".

The questions are:

How many pulses per revolution does my rotary encoder need to produce to track the hob tooth lateral movement to within 0.0002"?

How many steps per revolution does my stepper motor need to have to move the gear tooth 0.0002" per step?

Starting with the first question:

One revolution of the hob will result in the hob cutter moving one tooth laterally. I will call this lateral movement the ToothArc.

ToothArc = PitchCircumference/NumTeeth.

PitchCircumference = pi*NumTeeth/DiametralPitch.

Or ToothArc = (pi*NumTeeth/DiametralPitch)/NumTeeth = pi/DiametralPitch.

Step_Rev = Pulses / Revolution.

ToothArc = Lateral Distance / Revolution

So Lateral Distance / Pulse = ToothArc/Step_Rev = pi/(DiametralPitch*Step_Rev)

I know I want Lateral Distance / Pulse = 0.0002" and I want to solve for Step_Rev:

Step_Rev = pi/(DiametralPitch*0.0002) = 5000*pi/DiametralPitch = 15,700/DiametralPitch

So for 16DP I need 981 pulses/revolution from my rotary encoder. For 24DP I need 654 pulses/revolution.

This feels good as my cheap eBay encoder can do 2,400 pulses per revolution and is rated for 5,000rpm.

Now for the second question:

My dividing head has a 90:1 ratio. My stepper motor has Steps_Rev steps per revolution. So for one revolution of the gear blank I have 90*Steps_Rev stepper motor steps.

One revolution also move the gear teeth the DiametralCircumference distance. Rather than express this in terms of DP and NumTeeth, lets just work with gear blank diameter (Gdia).

So the circumference = pi*Gdia. So we could express the distance moved along the circumference of the gear blank per stepper motor pulse = pi*Gdia/(90*Steps_Rev).

I know I want 0.0002" = piGdia/(90*Steps_Rev) or Steps_Rev = 5000*pi*Gdia/90 = 175*Gdia.

So for a 6" gear I need 1050 steps/revolution, a 3" gear would be 525 steps/revolution, a 1" gear 175 steps/rev.

This would require microstepping for a 200 step motor, or a reduction gear between stepper motor and dividing head.

So my questions are:

Is my math correct?

Is my goal of 0.001" accuracy in tooth path high or low? I would like gears that look good and move smoothly, I am not building swiss watches.

Is my assumption that I need to measure and move 0.0002" to get 0.001" accuracy high or low?

Is there a better way to approach this (from a mathematical modeling perspective)?

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