When sizing a motor, one of the most important factors is the required torque. In general, motor torque-speed curves outline two primary areas of permissible torque: continuous and intermittent. Intermittent motor torque is allowed only for a short time (specified by the manufacturer) and in most cases is the torque required during acceleration. A motor’s continuous torque is determined by calculating the root mean square of all the torques that occur throughout the application, which typically includes torque during acceleration, torque during constant speed, and torque during deceleration.

#### Constant speed torque

The motor torque required at constant speed is the sum of the torque needed to drive the load, the preload torque of the screw assembly, and the torque due to friction of the support bearings and seals.

T_{c} = torque at constant speed (Nm)

T_{d} = torque to drive the load (Nm)

T_{p} = torque due to preload (provided by manufacturer) (Nm)

T_{f} = torque due to friction of support bearings and seals (provided by manufacturer) (Nm)

Drive torque is primarily influenced by the axial load on the screw and the screw’s lead.

F_{a} = total axial force (N)

P = lead (mm)

η = efficiency of ball screw

The axial load is not only the process force (drilling, punching, etc.), but also includes the force required to move the load. Since most ball screw assemblies use profiled guide rails to support the load, this will simply be the force that the load exerts radially (downward) times the coefficient of friction of the guide.

F = axial process force (N)

m = mass being moved (kg)

g = acceleration due to gravity (m/s^{2})

μ = coefficient of friction of linear guide

*Note that the preload torque fluctuates due to manufacturing tolerances and lead variation, so manufacturers will either provide a range of allowable values (for example, 0.04 to 0.17 Nm), or they will indicate an allowable percentage variation from a nominal preload torque value (for example, 0.10 Nm, ±40 %).*

#### Acceleration torque

The maximum required motor torque often comes when the load is being accelerated. The total torque during acceleration takes into account the inertia of the system being moved and the motor’s acceleration.

T_{a} = total torque during acceleration (Nm)

T_{acc} = torque due to acceleration (Nm)

J = inertia of the system (kgm^{2})

ω’ = angular acceleration (rad/s^{2})

N = angular velocity (rpm)

t = acceleration time (s)

J_{m} = inertia of the motor (provided by manufacturer) (kgm^{2})

J_{s} = inertia of the screw shaft (provided by manufacturer) (kgm^{2})

J_{l} = inertia of the load (kgm^{2})

#### Deceleration torque

Deceleration torque is simply the torque at constant speed *minus* the torque due to acceleration.

*For vertical applications, the torque required to cause back driving is important for determining whether the load will “fall” on its own, or if the screw assembly provides enough resistance to hold the load in place when no brake is applied. This article explains how to calculate the back driving torque.*

Aside from being necessary to move the load or execute the process as desired, the required motor torque also determines the amount of current needed from the servo amplifier. When current is applied to a motor, it encounters resistance and, as a result, heat is generated. This heat is commonly referred to as I^{2}R losses (I = current and R = resistance). Since heat is exponentially related to current, the required motor current, and thus the required motor torque, becomes a critical parameter for motor selection.

*Feature image credit: Bishop-Wisecarver Corporation*

Ivan Ivnaov says

In the formula for Drive torque Td, what is the eta (n) coefficient? It’s not described…

Thanks.

Danielle Collins says

Hi Ivan, eta (η) is the efficiency of the ball screw.

ARVIND KADAM says

F is in N.

m is in Kg.

mg works out to be Kg.met/sec sq

Do we need to change units to get Fa in N

Danielle Collins says

Hello Arvind,

The units of a N (Newton) are kg*m/s^2. So mass, m, (units of kg) multiplied by gravity, g, (units of m/s^2), gives us force, F, with units of Newton (kg*m/s^2).

Doug says

Does the ball screw pitch diameter get canceled out in the calculations. I would have thought that as the diameter increased the torque would be reduced.

Bharadwaj says

What if the movement is horizontal, ie, the load on ball screw is not axial but radial. How would the calculations change?

Danielle Collins says

Hi Doug,

The ball screw lead does have a significant influence on the torque required.

(In ball screw terminology, we commonly refer to lead, which is the linear distance the screw moves for every complete rotation, rather than pitch, which is the linear distance between screw threads.)

The equation for drive torque: Td = (Fa * P)/(2000 * π * η) demonstrates that torque, T, is directly proportional to the lead of the screw, P.

Danielle Collins says

Hello,

Ball screws are designed to withstand axial loads, but not radial loads. This is why ball screws are typically paired with some type of linear guide. The linear guide supports any radial loading in the system.

Louis DiFrancesco says

Once the load has been accelerated to desired speed, how does one calculate the linear speed of the load with respect to pitch and rpm of ball screw? I assume Force is mainly the frictional load since the linear speed is constant.