Rack and Pinion System Design

Hi everyone

I'm new here and I was wondering if anybody could help me. I need to design a rack and pinion gear train at work and I am having trouble finding specific information on these gear trains. I have all the gear stress formulas and factors, but my basic problem remains:

How do I calculate the torque required on the gear, if the required linear load is known?

I thought about using the torque formula (T=r x F) and using the pitch diameter as my moment arm (r). Somehow this feels too simple. Am I on the right track here?

Once I know what diameter gears I need to look at, the rest of my design will be (relatively) straightforward.


  • Kaustubh Katdare
    Kaustubh Katdare
    Tagging @#-Link-Snipped-# , @#-Link-Snipped-# , @#-Link-Snipped-# .
  • Sekocan
    Hi and welcome,

    I believe the known linear load is on the rack. So the torque on the gear would be
    T = r * F where T is the torque on the gear, r is the radius (not diameter) of the gear and F is the load on the rack.
    You should be careful about the units. A common mistake is using kilograms instead of Newtons.

    Besides, there would be another force between the gear and the rack. This force will try to seperate gear from rack. Will try to push them away. This force is the result of the tooth profile of the gear and the rack. Please consider this while designing the bearings.
  • SteveM
    Thanks, I'll definitely consider that. I'm assuming that this force will be a function of the pressure angle I choose to use?
    (if you guys can give me other general tips, I welcome them😀, I only got my degree in December, so I'm very new on the job)
  • Sekocan
    Yes it's definietly a function of the pressure angle.

    Another tip: have a mechanical design book on your table all the time (I have Shigley's Mechanical Engineering Design which is not the simplest book but it has almost everything about mechanical engineering).

    I had my degree some years ago and I forgot %90 of everything I've learned in my undergrad years and almost %100 in grad years. As engineers, we are dealing with very technical details and without practice, it is impossible to remember everything. Besides, any mistake we do in our design may even cost several lives. So what you are doing here, being suspicious about your own calculations and asking for help from colleagues is a great thing.
  • SteveM
    Thanks Sekocan, my Shigley's book is actually on my table right now! The sections on gear design have most of the formulas needed for gear design, though I have to agree, it isn't the easiest book to use.

    I will definitely have some colleagues look at work, I always appreciate a second opinion and some constructive criticism.
  • zaveri
    Another thing:

    when power is transmitted between two gears, then either speed or torque is varied, depending on the size and the number of teeth on both the gears.

    so hence the torque which you are applying though the pinion gear, may vary as soon as it is transmitted to the rack gear.
  • SteveM
    I'm basing my calculations on the maximum external force that the rack during operation. I understand how gear trains swap out speed for torque, but I don't understand how the torque can vary as the force is transmitted to the rack.

    As I understand it, the contact point between the teeth should stay at more or less the same radius, while the motor delivers a certain torque output. This should mean that the torque demand on the motor is constant. The only variation I see in the torque demand of the motor is the variation that comes in during the initial acceleration, which is a transient effect that dies off as an equilibrium speed is reached.

    @#-Link-Snipped-#, am I missing something in the way I am seeing this?
  • Sekocan
    Yes SteveM you are right. Since Rack gear is not a circular gear but it is a linear gear, the torque on the pinion is transformed to linear force on the rack.

    If you draw the free body diagrams (I strongly recommend to draw free body diagrams for every force calculation problem) and assume the system is balanced (not moving or moving at constant speed), you will see that the shaft is giving a torque (T) to the pinion and the balancing force on the pinion is the force coming from the rack via one of the teeth. So the balancing equation is:
    T - F * r = 0
    If the F is constant (and since the "r" is also constant), T must be constant for all times.

    Of course I am also assuming that the change in contact radius that you mentioned is negligible.
  • SteveM
    Thanks, it's good to know I'm on the right track! Can anyone perhaps provide me with a link to a website that gives Lewis form factor values that I can trust?

    I'm writing a program that calculates the gear stress to help with my work, so values in tabulated form would be great. I found this website:

    Spur Gears - Roymech

    It has tables listing Lewis form factors, but I'm not sure if it is completely reliable.

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