That's a difficult question. Basically the mesh stiffness of a given gear pair depends on several parameters, particulartly the load and the rotational position of the gears. Provided we assume linear-elastic material properties, which is reasonable for metal gears, the contact stiffness varies with the load (as the contact zone gets larger and thus stiffer with higher contact forces) whereas the other portions, like tooth bending a.o. are constant. But the stiffness definetely depends on the point of force application on the tooth flank and on the contact ratio (i.e. how many teeth are sharing the load at a given mesh position) and thus on the angular postion of the gears. These variations are periodic.
You could determine the mesh stiffness by measurement. As it cannot be measured directly, usually the transmission error over the angular position is recorded at a constant torque (and sufficiently slow rotational speed where dynamic effects can be neglected). From this you can determine the mesh stiffness, but you would have to subtract the effects from other compliances you are not intending to be included in your mesh stiffness, like shaft torsion.
Another way is to measure the resonance of the gear pair. You can slowly accelerate the gears from 0 rpm until you pass the main resonance (parameter-excitation with meshing frequency) and measure the angular vibrations. Provided you have a dynamical model of the gear stage, you could fit the stiffness parameters to match the measurement. This usually works for the mean mesh stiffness, but won't give you the exact variations over the angular position.
You could also determine the mesh stiffness by simulating the proposed measurements with a finite-element-model. If you want to capture the effects of non-linear load dependency from the Hertzian contact, you need to account for the micro-geometry and a very fine mesh in the contact zone.
Finally there are several analytical and semi-analytical approaches to determine the mesh-stiffness which give you a more or less good approximation. For the average stiffness of a standard spur gear you can consult ISO 6336 (c_{\gamma}). You can also consult gear literature, e.g. Weber-Banaschek or Schmidt.
It is a difficult problem.
Instead of measuring the stiffness you better calculate the total compliance of the meshing gears, which sums up five compliances as calculated in the paper
Minimization of spur gear dynamic loading through the generalized theory of gearing
by Th. Costopoulos, G. Nikas. You may find this paper in my ResearchGate section.
I hope this article will help,
Improving mesh stiffness calculation of cracked gears for the purpose of vibration-based fault analysis .
- Dec 2013, Engineering Failure Analysis. By:
- 📷Omar D. Mohammed
- 📷Matti Rantatalo
- 📷Jan-Olov Aidanpää
Lots of methods can calculate meshing stiffness of gear pair, such as FEM, ISO standard, Ishikawa method and potential energy and so on. Between them, I think the potential energy method is the most convenient. It can help you understand how it is calculated to see Tian's Master's Thesis as following,
" Tian, Xinhao. "Dynamic simulation for system response of gearbox including localized gear faults /." Masters Abstracts International, Volume: 43-03, page: 0979.;Adviser: Ming J. Zuo. (2004)."
Also, in this research field, Ma wrote a paper about meshing stiffness of cracked gear, and Liang made a review about dynamic analysis of fault gear pairs. You can learn from the two reviews after you know the calculation details .
“ Ma, Hui, et al. "Review on dynamics of cracked gear systems." Engineering Failure Analysis 55(2015):224-245.”
“ Liang, Xihui, M. J. Zuo, and Z. Feng. "Dynamic modeling of gearbox faults: A review." Mechanical Systems & Signal Processing 98(2018):852-876.”
In my recent research work, according to Wan's gear meshing stiffness model, I wrote and opened the source program of the spur gear meshing stiffness that can help you to calculate it more conveniently.
" Wang, J., Yang, J. & Li, Q. Nonlinear Dyn (2018). https://doi.org/10.1007/s11071-018-4204-3 "
Hello Scholars,
Is it possible to adopt a standard for the mesh stiffness in the case of dynamic simulation of gears? Especially if there is a standard from ISO
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