# Inertia change through a gearbox question

• mech_rocks
In summary, the change in mass moment of inertia is due to the rotational energy of the input and output shafts. The gearbox ratio affects how much of that rotational energy is "seen" by the engine.

#### mech_rocks

Hello,

This is my first post on the webpage I hope you guys can help me out. I'm trying to understand the concept of the change of mass moment of inerita through a gearbox. I know the formula, but why does the gear ratio affect the mass moment of inertia [kg m^2]. Let's say you have a input shaft rotating at 540 rpm, a gearbox with a 2:1 ratio, and a disk on the output shaft so why is the inertia "seen" at the beginning of the input shaft equal to the inertia produced by the disk multiplied by 4 considering (effective inertia = inertia x gear ratio^2 ) ?

Maybe you could describe the physics behind this... I sure would appreciate it. Thanks!

I'm not sure this is the textbook answer, but try consider the rotational energy with and without the gearbox. First, let's apply an 10 W engine to the shaft without the gearbox for 10 seconds. The rotational energy of the disc must now be 100 J, disregarding all friction. Now insert the gear box and again apply the 10 W engine for 10 sec. The rotational energy of the disc must still be 100 J since the gearbox neither absorb nor introduce energy, and since it is the same disc with the same moment of inertia, its rotational speed must also be the same. But this means the shaft on the engine side only rotates with half angular speed (or double, depending on the "orientation" of the gearbox) compared to before and so from the engines perspective the shaft seems to have four times the moment of inertia.

So in general, if you equate the rotation energy in the two situations now using the general gearbox ratio r, you get

$$1/2 I_e (\omega/r)^2 = 1/2 I_d \omega^2$$

which after cancelling $\omega$ implies that
$$I_e = r^2 I_d$$

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Ok I kind of understand what your saying. I appreciate your time. Thanks!

## 1. How does a gearbox affect inertia?

When a gearbox is added to a system, it introduces additional mass and rotational resistance. This can increase the overall inertia of the system, making it harder to accelerate or decelerate.

## 2. Can a gearbox reduce inertia?

Yes, in some cases a gearbox can reduce inertia by using gear ratios to decrease the rotational speed and increase the torque. This can make it easier to accelerate or decelerate the system.

## 3. Is inertia affected differently depending on the type of gearbox?

Yes, different types of gearboxes (e.g. spur, helical, worm) have different gear ratios and efficiencies, which can affect the overall inertia of the system in different ways. Additionally, the size and weight of the gearbox itself can also impact the inertia.

## 4. How can I calculate the inertia change through a gearbox?

To calculate the inertia change through a gearbox, you need to know the inertia of the input and output shafts, as well as the gear ratios and efficiencies of each gear in the gearbox. You can then use the formula In = I1 + (I2 / r^2), where In is the total inertia, I1 is the inertia of the input shaft, I2 is the inertia of the output shaft, and r is the gear ratio.

## 5. Can inertia change through a gearbox be a problem in a system?

Yes, inertia change through a gearbox can be a problem in certain systems, particularly those that require precise and quick movements. The added inertia can make it more difficult to control and can impact the overall performance of the system. In these cases, it may be necessary to consider alternative gearbox designs or strategies to minimize the effect of inertia.