# Gear teeth force from the rotation angle

• FEAnalyst
In summary, the conversation discussed the use of formulas for forces acting on the teeth of paired spur gears in books about machine design fundamentals. However, in a specific case where only the angle of rotation of the driven gear is known, it was questioned whether this information could be used to determine the necessary torque for force calculations. The expert recommended a book for more detailed information on gear design and explained the limitations of using velocity instead of torque in finite element analysis. The conversation also touched on the differences in gear design for speed reduction vs speed increase and the challenge of convergence in nonlinear solutions.

#### FEAnalyst

TL;DR Summary
How to calculate the force acting on the teeth of paired gears knowing only the angle of rotation instead of torque?
Hi,

in books about machine design fundamentals, one may easily find the formulas for forces acting on the teeth of paired spur gears. They require torque as input. For example, for the tangential force: $$F=\frac{2T}{d}$$
where: ##T## - torque applied to the driven gear, ##d## - pitch diameter of the gear.

However, in my case, only the angle of the rotation of the driven gear is known. Can it be used to determine the torque necessary for the aforementioned force calculations?

What do you call the angle of the rotation?

Lnewqban said:
What do you call the angle of the rotation?
What I mean is that the smaller gear rotates e.g. ##30 ^{\circ}## around its axis, driving the other gear in pair. The thing is that I want to compare the results of finite element analysis with analytical calculations but in the analysis I’m not applying torque to the driven gear but prescribing its rotation as a boundary condition instead. So I know only the angle of rotation (the calculations shouldn’t use any of the simulation results as inputs so I can’t take the torque derived from the analysis). Maybe I could use the velocity even though the simulation is static and uses unit time which is not a physical time but rather a measure used for load incrementation.

Angle is meaningless if you do not know the force applied by one tooth on another. Without a torque, there is no contact force. Without a force, why FEA?

As a gear pair rotates, the velocity ratio is a constant, but with constant input torque, or constant output torque, the radii to the points of contact on the teeth change, so the tooth force is not fixed, but varies slightly.

Machine design books are written for engineers who want to calculate the strength of a gear. They only need to know the strength of a gear tooth at its worst position. They need equations that are easy to use, where the results can be compared to published material properties.

If you really want to know the details of stresses in gear teeth, I can recommend Dudley's Handbook of Practical Gear Design, now in its 4th Edition: https://www.amazon.com/dp/0367649020/?tag=pfamazon01-20. A quick read should convince you that there is far more to gear design than simply calculating stress in gear teeth. For example, can you explain why two apparently identical gear pairs with the same torque, power, speed, ratio, number of teeth, tooth pitch, size, type (helical vs spur), alloy, surface finish, tolerances, and heat treatment will have different tooth profiles if one is designed for speed reduction and the other designed speed increase? And if the text does not have enough detail, the references should.

Lischka
Baluncore said:
Angle is meaningless if you do not know the force applied by one tooth on another. Without a torque, there is no contact force. Without a force, why FEA?
So there’s know way to determine the torque analytically even knowing the velocity and all the geometric parameters of the gears? Simulations of gears very often use applied rotation angle or velocity because force-controlled loading (torque in this case) is much worse for convergence of a nonlinear solution (and here nonlinearity is present in its worst form - changing contact conditions). I could get the reaction moment from the reference node to which I apply the boundary conditions on rotational degree of freedom but my goal is to avoid using any data from the simulation for analytical calculations because they are supposed to be used for verification of FEA results.

jrmichler said:
If you really want to know the details of stresses in gear teeth, I can recommend Dudley's Handbook of Practical Gear Design
Thank you for the recommendation. This case is just a benchmark and simple FEA example to be verified with hand calculations but the book might be useful in the future. Currently, I mainly use Polish engineering books and I have only the Shigley’s book when it comes to fundamentals of machine design (not really my area since I deal with FEA on a daily basis).

FEAnalyst said:
What I mean is that the smaller gear rotates e.g. ##30 ^{\circ}## around its axis, driving the other gear in pair. The thing is that I want to compare the results of finite element analysis with analytical calculations but in the analysis I’m not applying torque to the driven gear but prescribing its rotation as a boundary condition instead. So I know only the angle of rotation (the calculations shouldn’t use any of the simulation results as inputs so I can’t take the torque derived from the analysis). Maybe I could use the velocity even though the simulation is static and uses unit time which is not a physical time but rather a measure used for load incrementation.
In that case, you are limited to the way in which the rotational velocity that one gear wheel is linked to its pair.
That only depends on the ratio of both wheel’s diameters or number of teeth.
For two instantaneously meshing teeth, the tangential velocity must be the same.

As stated above, the force of that contact, which could be so high as to be able to break or shear the root of one tooth, or both, does not depend on that tangential velocity, but only on the transferred torques.

https://lpsa.swarthmore.edu/Systems/MechRotating/RotMechSysGears.html

https://khkgears.net/new/gear_knowledge/gear_technical_reference/gear_forces.html

https://www.koreindustries.com/digg...ction-pressure-angle-and-backlash-discussion/

https://www.researchgate.net/publication/267491002/figure/fig1/AS:477485075832832@1490852721012/FREE-BODY-DIAGRAM-OF-TWO-GEAR.png

Last edited:
All right, if it's always necessary to know both input torque and velocity to calculate the force acting on the teeth and it's not possible to derive one from the other then it seems that I will have to use the torque measured in the simulation.

Lnewqban
FEAnalyst said:
All right, if it's always necessary to know both input torque and velocity to calculate the force acting on the teeth and it's not possible to derive one from the other then it seems that I will have to use the torque measured in the simulation.
Clarification: you don’t need to know velocity to calculate the force at the root of the meshing teeths, torque will always suffice.
Only when calculating tranferred power, velocity is important.

cherish