**In order for a motor to accelerate or decelerate a load, it must overcome the load’s inertia, or resistance to change in motion, as explained in Newton’s First Law.**

In belt-driven linear motion systems, the motor has to overcome not only the inertia of the applied load, but also the inertia of the belt, pulleys, and motor coupling.

The inertia of each component can typically be estimated with sufficient accuracy by using the standard inertia equations for simple shapes. Since inertia depends upon the axis around which the component rotates, we can start by considering the applied load and the belt together, since they both rotate around the axis of the driven pulley.

The applied load and the belt can be modeled as a point mass that rotates around the driven pulley, and their inertia can be calculated as:

*J _{L} = inertia of belt and applied load (kgm^{2})*

*m = mass of belt and applied load (kg)*

*r = radius of driven pulley (m)*

Belt manufacturers typically provide mass (or weight) information per unit length, so the mass of the belt can be found by multiplying the mass per unit length by the *total* length of the belt. (Be sure to use the full, circular belt length — not just the length of the stroke.)

Also, remember that the applied load is typically mounted to the belt via a carriage or table, so the mass of this part should be included in the mass of the applied load.

The pulleys and the coupling can be treating as solid cylinders that rotate about their own axes, and their inertia can be calculated as:

*J _{p} = inertia of solid cylinder (pulley, coupling) (kgm^{2})*

*m = mass of cylinder (kg)*

*r = radius of cylinder (m)*

Keep in mind that although the pulleys may have the same diameters (and radii), if one pulley is toothed (driven) and the other is smooth (idler), as is the case in many belt driven actuators, they will have different masses and, therefore, different inertias.

Although the solid cylinder approximation shown above is typically sufficient, more accurate inertia values for the pulleys and coupling can be found by taking into account that these components have a center bore and using the inertia equation for a *hollow* cylinder:

*J _{ph} = inertia of hollow cylinder (pulleys, coupling) (kgm^{2})*

*m = mass of cylinder (kg)*

*r _{o} = outer radius (m)*

*r _{i} = inner radius (m)*

It’s common for belt driven systems to use a gearbox to increase torque, reduce speed, and reduce the inertia of the load reflected to the motor. In this case, the total inertia of the moved mass (applied load, belt, pulleys, and coupling) should be divided by the square of the gear reduction, and then the inertia of the gearbox should be added. This will give the total inertia reflected back to the motor, which can be used for motor sizing and selection.

*J _{total} = total inertia reflected to motor (kgm^{2})*

*J _{L} = inertia of belt and applied load (kgm^{2})*

*J _{p1} = inertia of first pulley (kgm^{2})*

*J _{p2} = inertia of second pulley (kgm^{2})*

*J _{c} = inertia of coupling (kgm^{2})*

*i = gear reduction*

*J _{g} = inertia of gearbox (kgm^{2})*

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