#### Moment of Inertia

A basic understanding of moment of inertia serves well in ensuring proper motor sizing. It is one thing to look at static points on torque curves, but it is altogether different when considering the dynamic aspects of loads being accelerated at high rates.

The Inertial mass of an object is a measure of its resistance to a change in its velocity.

The moment of inertia of an object is at a point of reference of rotation, which is at the pivot point or axis of rotation.

The moment of inertia can therefore be thought of as a measure of the resistance to any change in rotational speed.

For linear systems, the rate of change of speed, (acceleration) is proportional to the force applied. Double the mass and the force needs to be doubled for the same acceleration. Similarly for rotational systems, the angular acceleration of the load is proportional to the torque applied. Double the moment of inertia and the torque needs to be doubled for the same angular acceleration. Moment of inertia is therefore a measure of a load’s resistance to angular speed change; of how much effort (torque) is required to cause acceleration or deceleration.

#### Matching Motor To Load:

A common rule of thumb for SmartMotor™ sizing is that the load should have no more than 10 times the moment of inertia of the motor rotor that is driving it. This gives a good starting point and

typically allows for safe sizing over a wide range of applications. Since a rotating load wants to maintain the same velocity, then when a motor attempts to accelerated or decelerate the load, it must overcome the moment of inertia of that load by applying enough torque to accelerate it or decelerate it.

It takes more torque to change speed than it does to maintain a given speed.

In the same manner, for the motor to slow down a load, the load’s moment of inertia will keep the motor going the same speed and will, in effect, back-drive the motor turning it into a generator. In extreme cases, this can result in over-voltage damage to the Drive stage.

#### How to Improve Moment of Inertia Ratio Between Motor and Load :

Adding gear reduction to a motor gives it more leverage to prevent back driving and also gives it a better advantage in accelerating a load up to speed.

For any given change in gear reduction, you get a proportional change in speed and static torque but you get a squared change in acceleration and dynamic rate of change of torque. The result is that by adding gear ratio you gain a squared decrease in the ratio of moment of inertia between motor and load.

Therefore the motor has a greater advantage in both accelerating and decelerating the load. It adds protection against damage to the system as a whole.