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If you consider a motor attached to a spur gearbox with a reduction ratio of say $r=50:1$ which is then back-driven so that the motor acts as a generator, what would be the effective moment of inertia of the system? If the motor has a rotor inertia of say $1*10^{-5} kgm^2$ and the gearbox has a mass inertia of $1*10^{-6} kgm^2$ would I be correct in calculating the overall inertia through

$I_{total}=r^2I_{rotor}+I_{gearbox}=0.025 kgm^2$

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That should be correct. The rotational velocities and accelerations in the motor, caused by turning the output shaft, increase by r. Along with the acceleration, the reaction torque in the motor increases by the same amount. The reduction ratio will also apply on this torque and and hence increase it again by r at the output shaft.

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  • $\begingroup$ But take care -- at low speeds motor cogging torque and friction will dominate the torque at the gearbox shaft. $\endgroup$ – TimWescott Dec 4 '19 at 22:59
  • $\begingroup$ Is there at least an approximate method on how to simulate the motor's cogging torque and friction of the gearbox? The simulation I am trying to run would mostly be for low speeds but I can't seem to find much on calculating these parameters. $\endgroup$ – Inf_E Dec 5 '19 at 11:00
  • $\begingroup$ You didn't mention simulation in your question. What kind of simulaiton are you doing? MBD? Modelica? FEM? It's hard to give any help, without knowing where you are right now. If you have an MDB model with the gears modeled as contacts, you should be able to use a friction model, preferably LuGre or something, that includes stiction. $\endgroup$ – Ingo Dec 6 '19 at 11:27
  • $\begingroup$ Talking about cogging: If you know the effect in your motor, you can apply it as a torque that depends on the angle. If not, if you want to compute the effect itself, you might need some electormagnetics tool. You can hand forge the laws as differential equations in the MBD tool, use a subroutine, cosimulation. And there are specialized tool as well (Flux: youtube.com/watch?v=lFsRbX8noLk) $\endgroup$ – Ingo Dec 6 '19 at 11:28

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