a collection of design calculations

[russt]

Ok, I confess, I've caught the bug and I have to have a CNC router. What I want to do here is to go over some of the calculations I've been doing. The idea is basically collect a bunch of back-of-the-envelope calculations to direct the design decisions. Hopefully we can get a handle on the strengths and weaknesses of the design and build it up or pare it down accordingly.

Here are the design objectives (in no particular order):

1) Cheap, hardware store style router. All components should be easily sourced from the internet or from local stores. Shouldn't require any non-diy type tools to build, i.e. no welding, milling, or turning. Ideally just a circular saw/table saw and a drill.

2) It should be super rigid, doesn't have to be especially strong. This is a hobby machine, so machine speed isn't of the essence the way it is in a production environment. If the loads are too much for the strength of the frame, then just slow down the motion, take a lighter cut, etcetera...

3) It should be very precise, but doesn't have to be especially accurate. Basically the motion should be very repeatable, but doesn't have to be spot on relative to all the other points on the table. The idea here is that accurate components cost mucho bucks, and require lots of work to assemble and tune. On the other hand, this thing is being driven by a computer which can easily distort the toolpaths to compensate for inaccuracies in the motion. So let's build a precise, but inaccurate machine that can be calibrated such that the computer can make it accurate. The ideas for doing this will probably be posted in a separate thread.


Since it is much easier to talk concretely about a specific machine, here's a pic of what I have in mind. It's a typical moving gantry style router along the lines of JGRO's or Joe's. Cutting volume is 48" x 24" x 5", which should allow for a wide range of possibilities. The frame is constructed of Unistrut, which I arrived at after considering MDF for a long time.

I'd love to get some feedback on the calculations. The trouble with quick calculations is that you invariably assume certain things, which may not turn out to be true in reality. Hopefully some of you experienced CNCers out there will spot the problems.

Cheers, -Russ

[russt]

Ok, here's the first calculation. What's the force generated by the stepper motors turning the leadscrews?

A screw is essentially just an inclined plane that's wrapped around the surface of a cylinder. So let's unwrap a single turn. We get a triangle with a base of 2*pi*r = circumference, and a height equal to the thread pitch. For the 1/2" 10 threads per inch Acme rod I'm using, that gives a mechanical advantage of base/height 2*pi*r*tpi, or about 15.

Now the force that the stepper motor is pushing the inclined plane is equal to the torque/radius. My moderate to weak steppers generate about 118oz-in of stall torque.

Also inclined planes have friction, so let's stick in an efficiency factor of 50%.

Putting it all together, I get a force of efficiency*torque /radius *mechanical advantage = .5*118/.25*15 = 3540oz = 221 lbs. That's plenty of force to break things, I think.

As a caveat, I should note that this is maximum force generated under stall conditions. Stepper motors lose torque fairly rapidly as they start turning. Looking at the motors spec sheet (http://www.japanservo.com/digital/general/pdf/KH56K.pdf) it appears that the torque drops by a factor of two at 1000rpm. So I'll get only 100lbs of force at 1000rpm or 100inches/min. This still seems more than sufficient.

Next we should look at the the typical forces in play during router operation.

Cheers, -Russ

[russt]

Here are the three categories of forces I can think of:
1) inertial forces - equals the mass times the acceleration. For example, take a 50lb gantry (22kg), and accelerate it from 0 to 100ipm over 1 second. Switching to metric units, I get a force required of 1 N, or 5lbs.

The minimum time to go from 0 to 100ipm using the maximum force (100lbs) is 50ms, or 166 stepper motor steps. Of course, this will entail large deflections of the gantry and x rails.

2) machine weight - a 50lb gantry (22kg) has a gravitational force of 220 N, or 50lbs. Duh!

3) cutting forces - basically pretty small, but I'm going to be assuming 10lbs in the x and y direction as a nice upper limit. It can of course be adjusted by taking a larger or smaller cut. The smaller the cutting force, the less the deflection of the machine and hence the more accurate the cut. Given that a computer is going to be doing the cutting and this is a hobby machine, it's ok to leave it slow while I play with the kids. It doesn't even have to be that slow, according to the estimate above.


So to recap, the main thing is to make the machine rigid under it's own weight. Next we have to make it stiff under the typical cutting forces, and next we have to drive it slowly enough that the inertial forces don't cause deflections.

Interestingly enough, it is possible that the gantry weight type of forces can be compensated out by the calibration mechanism that I'd like to apply, since the weight is something that is repeatable.


Two major caveats that I haven't really explored yet.
1) The whole machine will be vibrating in operation, and so it isn't enough to make it rigid under the typical cutting forces, but also make it rigid enough that the resonant frequencies are much higher than the typical router speeds.
2) When the bit bites hard into something it can catch and then you have the whole rotor mass slowing abruptly which can generate substantial impulses. Haven't considered that yet.