Quite an Unusual one

[Jonathan]

Just spotted I forgot to attache the calculations to the previous post - I've attached them to this post instead.

RotatingNut_inertia.txt

I'll make some general points which I think should answer the questions - do let me know if I've missed anything.

• Acceleration, why have it?

A CNC machine rarely moves at a constant velocity, but does generally aim to move at a constant speed. The critical point here is the difference between velocity and speed - you can have the machine moving at a constant speed, whilst the velocity of each axis is varying greatly. Simple examples are cutting arcs or simply moving round a corner. This changing velocity means you have an acceleration.

The acceleration value you enter in the motor tuning is the limiting value - the controller should never command the drives to exceed it. You can therefore work out the minimum radius of a corner the machine can achieve for a given feedrate and feedrate. From the formulas for circular motion we can express the acceleration caused by the change in direction in terms of the feedrate and radius of the path:

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For example, suppose you're cutting at 10m/min (e.g. cutting MDF) and you want to cut squares with 10mm radius corners, without the feedrate reducing at the corners, we can work out the required acceleration:

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You can play with the numbers to see when this matters - but as a general point we can say that the required acceleration is proportional to the velocity squared, so if you're mainly cutting at low speeds (e.g. cutting aluminium) then the acceleration is not so important. This should also explain the differences in toolpath you see between using constant velocity (G64) and exact mode (G61).

• Inertia ratio

It seems their limit for the inertia ratio is set by the capacitance present in the motor driver dc-bus. When the motors are decelerated, some of the energy stored in the total inertia of the system is transferred into the capacitors, since the drivers are supplied from a simple rectifier which only lets power flow one way. The problem with this is it causes the voltage on the capacitors to rise and if the energy transferred is to high (due to high inertia ratio), the capacitors will be damaged. It's actually quite easy to work out roughly how much the voltage rises, just equate the energy stored in the inertia, . to the capacitor energy, . and re-arrange for voltage. The things we can do to alleviate this problem are add more capacitance, add a breaking resistor to dissipate the energy as heat or reduce the input energy by lowering the system inertia.
All of these seem reasonable options. Recall from my previous post that the equivalent inertia of the system depends on the square of the drive ratio, so by changing the drive ratio a small amount you can lower the inertia ratio. e.g. going to 18:30 instead of 20:30 would be sufficient. You should still be able to get 20m/min as the motor has sufficient torque at 3333rpm. A more interesting way round it could be to connect the DC-buses of all the servo drivers in parallel, as in general when one axis decelerates another is accelerating so will absorb the energy. You might still need a braking resistor to handle an e-stop event though.

• Equivalent inertia of gantry

In my previous post I stated the following formula, without deriving or referencing it, which I agree was not good practice so I'll explain where it comes from:
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When I use the word inertia, for a rotating system I mean the moment of inertia. For a mass moving linearly, such as the gantry, the inertial mass is equal to it's mass, .. The issue here is we need to incorporate this inertial mass into a rotating system, so we can sum the inertias seen by the motor shaft. The general formula to calculate the moment of inertia, for a mass (.) rotating about an axis at a distance . from that axis is:
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For every revolution of the ballscrew (or nut) the gantry moves a distance L (the screw pitch), so we can relate the angular speed of the ballscrew (or nut) to the linear speed of the axis as follows:
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From the definition of angular speed we know that .. We can combine these equations to find the equivalent inertia:
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Substitute:
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We're not quite there though, as the ballscrew (or nut) isn't directly coupled to the shaft. This causes the angular velocity to be scaled by the drive ratio, lets call it .. If you include that in the above derivation you'll find the inertia is scaled by the square of this ratio, so:
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