Belt-driven linear systems are common in applications that require long travel and high speed, such as gantry robots and material handling and transport. The motors chosen for these systems are often servo motors, due to their ability to precisely control position, speed and torque.
Servomotor sizing and selection requires determining the continuous and intermittent drive torques required for the application. Continuous torque is calculated by calculating the root mean square of all torque requirements in the application: torque required for acceleration, torque for constant speed, and torque for deceleration. In most applications, maximum (intermittent) torque occurs during acceleration.
To determine the root mean square torque (continuous), we first calculate the torque values required during each phase of the motion profile.
Torque required for constant speed
For a belt drive system, the motor torque required during constant speed is simply the total axial force (F for ) on the belt multiplied by the radius (r 1 ) of the drive pulley.
T c = torque required during constant speed (Nm)
F for = total axial force (N)
R 1 = radius of the driving pulley (mm)
η = belt drive system efficiency
Note that the efficiency (η) of the belt drive system is included in the torque equation. This efficiency takes into account losses such as friction between the belt and pulleys. Also note that we assume that the drive pulleys and idler (driven) pulleys have the same radius, as is often the case with belt-driven linear motion systems.
Unlike screw drives, which often face axial forces from external operations such as pressing or drilling, belt drives are not designed to resist external axial forces. Therefore, the total axial force for a belt drive system consists only of the force required to move the load, which is the weight (m * g) of the load (both the external load and the belt) multiplied by the coefficient of friction (μ ) of the guide that supports the load.
m = mass of the load moved (external load plus belt) (kg)
g = gravity (m/s 2 )
μ = guide friction coefficient
Torque required for acceleration
The acceleration phase of the motion profile is normally the period when maximum torque is required from the motor, and this torque value, T para is often considered an intermittent couple.
The torque required during acceleration includes the torque required at constant speed plus the torque required to accelerate the load.
T to = total torque required during acceleration (Nm)
T counts = torque required due to acceleration (Nm)
The torque due to acceleration is obtained by multiplying the total inertia of the system (J t ) by the angular acceleration (α). 
J. t = total inertia of the system (kgm 2 )
a = angular acceleration (rad/s 2 )
Total system inertia includes motor inertia (as the motor must overcome its own inertia), coupling, pulleys, and load.
J. I = engine inertia (supplied by the manufacturer) (kgm 2 )
J. c = joint inertia (supplied by the manufacturer) (kgm 2 )
J. p1 = inertia of the driving pulley (supplied by the manufacturer or calculated) (kgm 2 )
J. p2 = inertia of the idler pulley (supplied by the manufacturer or calculated) (kgm 2 )
J. I = load inertia (kgm 2 )
Although we assumed above that the drive and idler pulleys have the same radius, their inertias may be slightly different since the drive pulley is toothed and therefore has a slightly larger radius and greater mass than the idler pulley.
The inertia values of the motor, coupling and pulleys are generally specified by their respective manufacturers. However, it is necessary to calculate the inertia of the load. Remember that the load includes the mass of both the external load and the belt, as the motor must generate enough torque to overcome the inertia of the belt.
l l = mass of the external load (kg)
l b = belt mass (kg)
R 1 = radius of the driving pulley (mm)
For angular acceleration, we assume that the system is accelerating from zero to some maximum velocity, where No is the maximum angular velocity and t is time to accelerate.
N = maximum angular velocity (rpm)
t = acceleration time (s)
If the system is accelerating from a non-zero velocity, the equation would simply incorporate the change in velocity (ΔN) divided by the time during which the increase in velocity occurred (Δt).
Torque required for deceleration
The motor drive torque required for deceleration is equal to the constant speed torque minus the torque due to acceleration.
T d = torque required during deceleration (Nm)
Continuous couple
Now that we know the motor drive torques required during acceleration, constant speed, and deceleration, we can calculate the root mean square of these values to determine the continuous torque required by the motor.
T RMS = root mean square torque (continuous) (Nm)
t for = time for acceleration (s)
t c = time for constant speed (s)
t d = deceleration time(s)
t total = total movement time (including any idle time between movements) (s)
Feature image credit: Rollon SpA
