# Is squared motor gearbox ratio proportional to inertia ratio?

I read an interesting article

It is very interesting, but I can not follow the 2nd last paragraph. I don't understand why it is true.

Gearboxes offer a significant benefit in that they affect the inertia ratio by a factor of the gearbox ratio squared.

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Probably related to the fact hat a factor $X$ increase in the gearbox ratio increases the torque by a factor $X$ and also reduces the angular acceleration of the load by a factor $X$. – Johannes Aug 8 '13 at 2:18
Any equation to support?? – Marco Aug 8 '13 at 2:40
As @Johannes says: we have a shaft that responds to a torque $\tau$ by $\tau = I \mathrm{d}_t \omega$. Now we bolt a gearbox of ratio $\lambda$ in: so now we find we have to exert torque $\tau^\prime = \lambda^{-1} \tau$ on the gearbox input to get the torque $\tau$ on the output. At the same time, the gearbox's input angular speed $\omega^\prime$ is $\lambda$ times the angular speed of the original shaft, so $\omega^\prime = \lambda \omega$. Plugging these into our original response equation: $\tau = \lambda \tau^\prime = I \mathrm{d}_t \omega = I \mathrm{d}_t \omega^\prime \lambda^{-1}$ ... – WetSavannaAnimal aka Rod Vance Aug 8 '13 at 2:40
...or $\tau^\prime = I \lambda^{-2} \mathrm{d}_t \omega^\prime$. So the effective inertia is $I \lambda^{-2}$. This kind of relationship arises whenever a system conserves a product of two variables say $V$ and $I$: if there is a proportional relationship $V = Z I$ between the two variables, and the system scales one of them $V \mapsto \lambda V$ then the system transforms the effective proportion $Z$ by $\lambda^2$. Gearboxes conserve $\tau \omega$ and electrical transformers conserve $V I$ by dint of conservation of energy. $\lambda^2 Z$ is the impedance reflected by a transformer. – WetSavannaAnimal aka Rod Vance Aug 8 '13 at 2:46
@WetSavannaAnimalakaRodVance Would you please put your comment into the answer section? I will accept that as the answer. Actually, it is a good and clear answer. – Marco Aug 8 '13 at 2:55

Gearboxes belong to a class of linear system that conserves a product of observable quantities by dint of the principle of conservation of energy. For a gearbox, the product $\tau \omega$, where $\tau$ is the torque exerted on or by a driveshaft and $\omega$ the shaft's angular speed. For a lossless gearbox at steady state, we have:

$\tau_{in} \omega_{in} = \tau_{out} \omega_{out}$

($\tau_{in}$ being the torque exerted on the gearbox's input shaft, $\tau_{out}$ being that exerted by the gearbox's output shaft). So this equation simply says power input = power output. I emphasise steady state because the gearbox can also store energy as rotational kinetic energy in its wheels and other spinning parts. The statement you make assumes that this energy stored by the gearbox is vanishingly small compared to the energy transfers of interest through the gearbox. Therefore we can assume that the above equation, for this argument, holds at all times, whether the system be at steady state or accelerating or decelerating. An "ideal" gearbox is one that is both lossless AND stores no kinetic energy.

Therefore, if the gearing changes the angular speed ratio, so that $\omega_{in} = \lambda \,\omega_{out}$ (here $\lambda > 1$ is "gearing down", so that the input shaft spins faster than the output shaft), the torques are related by the reciprocal ratio $\tau_{in} = \lambda^{-1} \tau_{out}$.

What does this do to the "inertia" of a load linked to the gearbox's output? The mass moment of inertia $I$ of the output load defines how the load responds to torque:

$\tau_{out} = I\, \mathrm{d}_t \omega_{out}$

so that, assuming the gearbox's inertia is small compared to $I$ (this is the same as the small energy storage in gearbox assumption above), then we can plug our reciprocal relationships:

$\lambda \tau_{in} = I\, \mathrm{d}_t \left(\lambda^{-1} \omega_{in}\right)$

or, since we can bring the $\lambda^{-1}$ through our time derivative thanks to the low gearbox inertia assumption:

$\tau_{in} = \lambda^{-2} I\, \mathrm{d}_t \left(\omega_{in}\right)$

So now, if we think of our system as a "black box": as seen from the input, the load responds to torque as though there were no gearbox there, but now with moment of inertia:

$\lambda^{-2} I$

which is the statement you sought to understand. More generally, the effective inertia seen at the input for an output load with inertia $I$ is:

$I_{effective} = I_{gearbox} + \lambda^{-2} I$

where now $I_{gearbox}$ accounts for the gearbox's inertia.

Another, analogous system is the electrical transformer, which conserves the product $V I$, so that $V_{in}(t) I_{in}(t) = V_{out}(t) I_{out}(t)$. If the transformer's voltage "step down" ratio is $\lambda$ so that $V_{out} = V_{in} \lambda^{-1}$ and if it is laden at its output with an impedance $Z$ (in general, the impedance is an integro-differential operator, rather than a simple scalar: $Z$ is analogous to $I\,\mathrm{d}_t$ above), then the impedance "reflected" to the input is:

$Z_{in} = \lambda^{-2} Z$

Analogous comments apply about "ideal" transformers as opposed to real ones that both lose energy and store energy in the magnetic and electric fields inside them:

$Z_{in} = Z_{transformer} + \lambda^{-2} Z$

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Question: Gearboxes offer a significant benefit in that they affect the inertia ratio by a factor of the gearbox ratio squared. Answer: • Consider a motor running an inertia load JL via a mass-less, loss-less gearbox. The effective inertia (J) or its impedance (Z=sJ, where s is a complex variable), as seen through a gearbox of ratio G (where G > 1 for a car differential, and G < 1 for a bicycle chainwheel), can be derived from power considerations. • The kinetic energy of only the load (for example, a disk on a shaft) is KEL = (1/2) JL ωL 2 from which the power of the load is PL = d(KEL)/dt = JL ωL αL where ωL is angular velocity, and αL = d(ωL)/dt is angular acceleration. Reflecting back to the motor, using αm = αL G and ωm = ωL G, have power “reflected” to motor P = JL (ωm/G)(αm/G) = (JL/G 2 ) ωm αm = JL’ ωm αm where JL’ = JL/G 2 is the apparent load as seen by the motor. • So, if G = 10, then the effective inertia is 100 times less than the base inertia. Conversely, the effective impedance looking from the load to the motor is amplified by a factor of 100. • The total inertia of the system, ignoring the inertia of the gearbox elements, is JT = Jm + Jg + JL’ where Jm is the inertia of the motor, and Jg = 0. The motor torque is proportional to current Im through the motor, Tm = Km Im = JT αm. • This torque will be tempered by dissipative torques in the gears and load (which can change with speed), in the real case, and thus αm will be tempered [1], αm = (Tm – Tg – Tload)/JT • It can be shown [2] that the optimal transfer of power from motor to load occurs when Jm = JL’ , which is to say that the effective impedances are matched. The optimal gear ratio would then be G 2 = JL/Jm. Refs: [1] D.G. Alciatore and M.B. Histand, 2003, “Introduction to Mechatronics and Measurement Systems, McGraw-Hill, New York, NY, pg. 401. [2] D.M. Auslander and C.J. Kempf, 1996, “Mechatronics”, Prentice- Hall, NJ, pg. 122. 1/13/2015

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You should really try to format this a bit better to improve readability. Also, note that we have a system called "MathJax" which allows you to render mathematics as in LaTeX. Please use this to further improve your answer. – Danu Jan 13 at 21:31
Some general formatting would be nice as well (e.g., paragraph breaks). – Kyle Kanos Jan 13 at 21:47