# Refreshing gear ratio and total inertia

• azizz
In summary, the setup shown in the figure involves a motor transferring its speed and force to an actuator through gears and a spindle. The speed and force of the actuator are determined by the motor's speed and torque, respectively, using specific equations. The total inertia seen by the motor can be calculated by combining the motor's inertia with the inertia of the spindle, taking into account the gear ratios.
azizz
Suppose we've got the setup as shown in the figure (see attachment).
The idea is that the motor transfers its speed and force (rotary) to the actuator force and speed (linear) via some gears and a spindle.
Here:
J = inertia [kg m^2]
n = rotary to linear transmission [---]

If I am not mistaken, then the speed of the motor $$\phi_{motor}$$ is related to the speed of the actuator $$\phi_{actuator}$$ as follows:

$$\phi_{actuator} = \phi_{motor} \left( \frac{R_{motor}}{R_{spindle}} n_{actuator} \right)$$

The force of the actuator $$F_{actuator}$$ is related to the torque of the motor $$T_{motor}$$ as

$$F_{actuator} = T_{motor} \left( \frac{R_{spindle}}{R_{motor}} \frac{1}{n_{actuator}} \right)$$

And my main problem is the following: what is the total inertia $$J_{tot}$$ seen by motor? Is that

$$J_{tot} = J_{motor} + \frac{J_{spindle}}{ \left( \frac{R_{spindle}}{R_{motor}} \right)^2 }$$

or

$$J_{tot} = J_{motor} + \frac{J_{spindle}}{ \left( \frac{R_{spindle}}{R_{motor}} \frac{1}{n_{actuator}} \right)^2}$$

If someone could confirm/correct my formula, that would be very helpful.

Bob

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The first equation should, as far as I can see, give you the equivalent rotational inertia as if all the spindle inertia had been moved to the motor shaft. You would still need to combine all the gear ratios when transforming the motor torque or angular speed to the linear force or linear speed.

Ok thanks.

## What is gear ratio and how does it affect the performance of a machine?

Gear ratio refers to the ratio of the number of teeth on the input gear to the number of teeth on the output gear. It determines the speed and torque of a machine, with a higher gear ratio providing more torque and a lower gear ratio providing more speed.

## What is the purpose of refreshing gear ratio?

The purpose of refreshing gear ratio is to optimize the performance of a machine by adjusting the gear ratio to better suit the desired speed and torque requirements. This can help improve efficiency and prevent damage to the machine.

## How is gear ratio refreshed?

Gear ratio can be refreshed by changing the size or number of teeth of the gears, or by adding or removing gears in the gear train. This can be done manually or using computer-aided design (CAD) software.

## What is total inertia and why is it important in gear ratio refreshing?

Total inertia refers to the combined mass and rotational inertia of all the components in a system. It is important in gear ratio refreshing as it affects the acceleration and deceleration of the machine, and can impact the overall performance and efficiency.

## How do you calculate total inertia when refreshing gear ratio?

To calculate total inertia, you need to know the mass and moment of inertia of each component in the system. The total inertia can then be calculated by summing up the individual inertias using the parallel axis theorem. This calculation can be done manually or using specialized software.

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