The first step in determining the requirements of a motion-control system is to analyze the mechanics — including friction and inertia — of the load to be positioned. Load friction can easily be determined either by estimating or by simply measuring with a torque wrench.

Inertia — the resistance of an object to accelerate or decelerate — defines the torque required to accelerate a load from one speed to another, but it excludes frictional forces. Inertia is calculated by analyzing the mechanical linkage system that is to be moved. Such systems are categorized as one of four basic drive designs: direct, gear, tangential, or leadscrew.

In the following analyses of mechanical linkage systems, the equations reflect the load parameters back to the motor shaft. A determination of what the motor “sees” is necessary for selecting both motor and control.

Cylinder inertia

The inertia of a cylinder can be calculated based on its weight and radius, or its density, radius, and length.

Solid cylinder, Figure 1.

Based on weight and radius:

Based on density, radius, and length:

Hollow cylinder, Figure 2. Based on weight and radius:

Based on density, radius, and length:

With these equations, the inertia of mechanical components (such as shafts, gears, drive rollers) can be calculated. Then, the load inertia and friction are reflected through the mechanical linkage system to determine motor requirements.

Example: If a cylinder is a leadscrew with a radius of 0.312 in. and a length of 22 in., the inertia can be calculated by using Table 1 and substituting in equation 2:

Direct drive

The simplest drive system is a direct drive, Figure 3. Because there are no mechanical linkages involved the load parameters are directly transmitted to the motor. The speed of the motor is the same as that of the load, so the load friction is the friction the motor must overcome, and load inertia is what the motor “sees.” Therefore, the total inertia is the load inertia plus the motor inertia.

Gear drive

The mechanical linkage between the load and motor in a gear drive, Figure 4, requires reflecting the load parameters back to the motor shaft. As with any speed changing system, the load inertia reflected back to the motor is a squared function of the speed ratio.

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