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  1. #1
    Join Date
    Feb 2012
    Posts
    39

    Rack and Pinion Calculations

    Hi All,

    I have been working on a rack and pinion router design and am currently starting the calculations to size my motors. I'm having a little bit of trouble finding some good information on torque transfer in rack and pinion systems, so I thought I would post my theoretical assumptions for comments from those intelligent that myself!

    Here's my system:

    Pinion pitch diameter (Module 1): 22 mm

    Pulley 1 pitch diameter (attached to motor): 25.47 mm

    Pulley 2 pitch diameter (attached to pinion): 66.85 mm

    Therefore, my reduction ratio is 66.85 / 25.47 = 2.62:1

    If my motor torque is 2 N.m at the desired RPM, could I then assume that the above system would be delivering 5.24 N.m (2 x 2.62) of torque?

    Then, theoretically, if the required linear force to accelerate my load is 200 N, could I assume that with the above pinion (22 mm pitch diameter) I would require 2.2 N.m (200 x 0.011) to accelerate my load to the desired speed at the desired rate?

    Am I on the right track here? Any comments would be much appreciated!

    Thanks,

    Toby

  2. #2
    Join Date
    Aug 2011
    Posts
    388
    Yes, those calcs are correct. 2 N-m at the motor will produce 476 N at the rack if zero losses. 85% overall efficiency is probably typical, so about 400 N.

    Also keep in mind (assuming it's a stepper motor), to reliably not lose steps, most industry sources recommend an extra 30-50% of torque available, all the way to the specified top speed. So, the numbers you describe would have a ~100% torque margin--plenty safe, or you could accelerate faster if wanted.
    David Malicky

  3. #3
    Join Date
    Feb 2012
    Posts
    39
    Thanks for the reply and good information David! It's been hard to find solid information on rack and pinion system motor sizing. I have found a wealth of information in the engineering texts on calculating tooth loads/forces, however these mostly seem redundant given the relatively small forces I'm dealing with (most calcs give large kN load limits).

    I will write up my full machine calcs and post them here soon. If you or anyone else would like to peruse them and point out any obvious errors it would be much appreciated! Once finalised I'm sure my calcs may come in handy for anyone else designing a R&P machine.

  4. #4
    Join Date
    Aug 2011
    Posts
    388
    Glad it's helpful. I'd suggest posting specs for your plan on steppers (A, mH, oz-in, # wires, and ideally the torque-rpm graph), drivers (V, A), and power supply (V, A) -- it's easy for 1 component to limit torque to much less than anticipated.
    David Malicky

  5. #5
    Join Date
    Feb 2012
    Posts
    39
    Hi All,

    I have done some further work on my machine design/calculations and I have decided to go with 5:1, 90% efficiency planetary gear boxes for my design. I have also changed my pinion diameter to 30mm instead of 22mm.

    Using these variables with a 100% factor of safety (perhaps overkill) I have calculated my peak axis motor torque requirements as:

    X Axis - 0.911 N.m (per motor using two motors, R & P)

    Y Axis - 1.537 N.m (one motor, R & P)

    Z Axis - 0.460 N.m (one motor, 10mm pitch ball screw)

    If anyone is interested, I have attached a PDF of my full machine specs and calculations. It would be much appreciated if someone more knowledgeable than myself could run their eye over my calcs and let me know what they think! I'd much rather hear that they're way out (and I'm an idiot) before I invest in ill fitting motors

    David,

    Given that I have changed my specs somewhat, I am now re-researching which motors I'm going to use. I am also going to develop some motion profiles for the axes of my machine so I can really drill-down on a well chosen set of motors. Once I have settled on something I will post the motion profiles and motor specs.
    Attached Files Attached Files

  6. #6
    Join Date
    Apr 2007
    Posts
    1955
    Thank you for posting the information. It is always great to see people post real calculations.

    I did not double check all of your calculations, but I have a couple of comments:

    a) I would consider to use a holding force for cutting of 2 x what you have in there. Not every one agrees with this, but I measured approx. 320 N when I did some bench testing at home. I admit that the measurements were crude. You might still be ok, since you have a 100% safely factor.

    b) Your accel rate is essentially 0.1 G, which you you might find a bit annoying with the high velocities you are targeting. When I read about the cutting times for complex items, it seems like the most common limiting factor is Z axis accel. I would be tempted to raise this to 0.3 - 0.4 G and reduce the Z velocity goal by 50 - 70%.

    c) You probably know this, but just be careful with how rapidly the stepper motors loose torque with rpm.

    In order to achieve the torque level you are targeting, it will almost force it to a nema 34 size, and those tend to loose torque and acceleration fast if your design calls for higer than 200 rpm.

    It gets really tricky to stay in the stepper motor range at this level, as you are really close to servo territory.

    Please keep posting information on your design, and I will also try to run the numbers as a double check.

    Harry

  7. #7
    Join Date
    Feb 2012
    Posts
    39
    Thanks for your suggestions Harry!

    I picked 200 N as a cutting force purely through guess work. I've searched long and hard for some data on the cutting forces associated with machining woods and MDF and have come up with nothing. If you've done some testing, I'm happy to go with your observations. It would be great if someone with access to a dynamometer could run some tests for us

    Regarding the Z axis acceleration rate, I can see what you are saying. Given my acceleration displacement is 83mm, that is a long way for the Z axis to travel to reach maximum velocity. I will go back to the drawing board here.

    I am definitely moving into servo territory! I want to build a quality machine that hits the goals I have set, so if stepper motors are the limiting factor then I am happy to go for servo motors.

    I'll do a bit more on my design/calcs and post my work soon.

  8. #8
    Join Date
    Apr 2007
    Posts
    1955
    One of the top of the line rack and pinion plus servo suppliers are these guys at atlanta drive systems.

    ATLANTA Drive Systems - The World Leader in Rack & Pinion Drive Technologies

    Even if you don't buy from them, it is worth sending your requirements in to them and see what comes back. It is easy to hit US $ 1 000 to 1 500 per motor plus gearing.

    It will actually be cheaper if you can find a used machine, put it in a container, and bring it over.

  9. #9
    Join Date
    Oct 2005
    Posts
    255

  10. #10
    Join Date
    Sep 2013
    Posts
    3
    Hi,
    I am trying to understand the rack and pinion calculations, and have no idea how you got 476N at the rack.
    Can you explain it a little more?
    Very appreciated.

    Quote Originally Posted by dmalicky View Post
    Yes, those calcs are correct. 2 N-m at the motor will produce 476 N at the rack if zero losses. 85% overall efficiency is probably typical, so about 400 N.

    Also keep in mind (assuming it's a stepper motor), to reliably not lose steps, most industry sources recommend an extra 30-50% of torque available, all the way to the specified top speed. So, the numbers you describe would have a ~100% torque margin--plenty safe, or you could accelerate faster if wanted.

  11. #11
    Join Date
    Mar 2003
    Posts
    35538
    Torque at the pinion divided by the pinion radius in meters. 5.24/.011 = 476
    Gerry

    UCCNC 2017 Screenset
    http://www.thecncwoodworker.com/2017.html

    Mach3 2010 Screenset
    http://www.thecncwoodworker.com/2010.html

    JointCAM - CNC Dovetails & Box Joints
    http://www.g-forcecnc.com/jointcam.html

    (Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)

  12. #12
    Join Date
    Sep 2013
    Posts
    3
    OK, I got it, thanks for your help.
    For my case, I am trying to figure out a design that was done by some other people:
    a 20lbs load is fixed on the top of a rack that is driven by two spur gears, the pinion is a 28-tooth with a pitch diameter 0.875", driven by a stepping motor M-2218-2.4S (holding torque 0.64N-m), the gear to move the rack assembly is a 64-tooth with a pitch diameter 2.0".
    So, the force to move the rack assembly is: (2.0/0.875)*0.64/(1*25.4/1000)=57.59N
    Well, I don't think this is right.
    Please help me out.

  13. #13
    Join Date
    Mar 2003
    Posts
    35538
    It looks right. With that setup, you'd have a little over 12lbs of holding force. Force will ro pthe faster the motor spins, but you might be able to accelerate the 20lbs at .25G, which is pretty quick.
    Gerry

    UCCNC 2017 Screenset
    http://www.thecncwoodworker.com/2017.html

    Mach3 2010 Screenset
    http://www.thecncwoodworker.com/2010.html

    JointCAM - CNC Dovetails & Box Joints
    http://www.g-forcecnc.com/jointcam.html

    (Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)

  14. #14
    Join Date
    Sep 2013
    Posts
    3
    Thanks again.
    However, I don't have any experience on this kind of mechanism, and I really don't feel comfortable about the amount of force on the rack.
    Isn't it too low?

  15. #15
    thegolfer, I was wondering how you came to the figure for the rolling resistance in the guides of your system? I am busy designing a system with a rack and pinion and I need to find the rolling (internal) resistance of the guidance rails.

  16. #16
    Join Date
    Nov 2014
    Posts
    1

    Re: Rack and Pinion Calculations

    Hi 'Thegolfer', you probably arent checking these posts 2 years later. But Just in case you are, I was going through your wonderful calculations, its unclear of your use of the acceleration displacement as the means of measuring the torque needed:

    Is there a reason as to why the displacement (work done) is only considering the acceleration displacement?
    -My guess would be that, during that period of travel the motor feels the mac amount of stress (is this the case?) Or should i consider my actual length of travel?

  17. #17
    Join Date
    Feb 2012
    Posts
    39

    Re: Rack and Pinion Calculations

    Hi Obstructor11,

    You're right, the only reason work done is considered during acceleration is because peak power is required to accelerate a load. According to Newton's second law, force = mass x acceleration. Considering f=ma in a perfect, friction free world, if an object is moving at a constant velocity acceleration = 0 and therefore so to does the force required to move the object.

    My calculations from back then are much more complicated than they really need to be. I'll have a look at them now and simplify them. Post your machine specifications if you like and we can work on them as a group.

  18. #18
    Join Date
    Feb 2012
    Posts
    39

    Cool Re: Rack and Pinion Calculations

    Hi all,

    For anyone interested, attached is a document with a simpler approach to calculating the motor torques required to drive a CNC router that uses ball screws on the Z axis and rack and pinion on the X and Y axes. I have probably used a low acceleration rate here, but the calculation methodology is the same.

    Please note that this document does not include the friction losses in the rack and pinion gears, but does include a 100% factor of safety that would take care of these losses. There are methods to calculate these forces and one day when time permits I will have a go at them for this example.

    Hopefully these calcs can help someone select the right motors for their machine. Remember to always chose stepper motors based on a supplied torque curve graph so you can verify that the motor can deliver the required torque to move your machine at the required RPMs. Many a stepper motor has a high holding torque but low torque at high RPMs.
    Attached Thumbnails Attached Thumbnails CNC Router Calculations Version 2.pdf  

  19. #19
    Join Date
    Feb 2012
    Posts
    39

    Re: Rack and Pinion Calculations

    Hi All,

    At the request of a fellow Zoner (thanks tinmancarving) I have put my calculations into a spreadsheet, which is attached.

    To get the thing working, just enter your data into the white cells. The grey cells are where the calculations take place. Please let me know if you find any problems.

    I have included rack and pinion and ball screws for each axis, just leave the type you are not using blank (or use them both to compare). I have included critical speed and buckling loads for the ball screws. These calculations give you the fastest rotational speed for the ball screw before whipping occurs and the highest load the ball screw can carry before it buckles. These calculations were taken from the Hiwin ball screw catalogue and are converted from kilogram force to newtons.

    Please keep in mind that this spreadsheet simplifies some aspects of the calculation process and omits others. Instead of calculating a friction force for the linear guides, I have chosen to use a constant intrinsic resistance force. This method has been taken from the Thomson linear guides catalogue. Also, the moment of inertia of any rotating parts (i.e. the ball screw or pinion gear) and stepper motor detent torque have been omitted. This is because they are usually insignificantly small and can be covered off in a generous factor of safety.

    If enough people are keen I can also do a pounds, inches, ounces, hogsheads, baseballs per square knots version.

    Thanks,

    Toby
    Attached Files Attached Files

  20. #20
    Join Date
    Oct 2015
    Posts
    11

    Re: Rack and Pinion Calculations

    Quote Originally Posted by thegolfer View Post
    At the request of a fellow Zoner (thanks tinmancarving) I have put my calculations into a spreadsheet, which is attached.
    Thanks Toby for this spreadsheet. It's helped me get a ballpark idea of what size motors I'd be needing.

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