How to calculate acceleration torque

1. Basic formula

The acceleration torque (Ta) can be calculated using the following formula:

Ta = Jeq⋅α

in:

Ta: Acceleration torque (Nm)
J eq: equivalent moment of inertia (kg·m²)
α: angular acceleration (rad/s²)
2. Parameter calculation

(1) Equivalent moment of inertia (Jeq)
The equivalent moment of inertia is the sum of the moments of inertia of all relevant components converted to the same axis:
J eq = J load + J motor / (i ^ 2)
= J load + J motor / ( i ^ 2)
in:

J load: moment of inertia at the load end
J motor: moment of inertia at the motor end
i: reduction ratio
(2) Angular acceleration (α)
Angular acceleration is the rate of change of angular velocity and can be calculated using the following formula:
α=Δω/Δt
Δω: change in angular velocity (rad/s)
Δt: time (s)
If the angular velocity is expressed in rotational speed (rpm), it can be converted to angular velocity:

ω=2πn/60

3. Total torque

The total torque T total is the sum of the load torque (T load ) and the acceleration torque (Ta):

T load=F*r The load moment is related to the lever arm.
T load usually also includes friction torque, gravity torque, etc.
Practical Application

example

If the system has the following parameters:

J load=0.05 kg\cm²
J motor=0.01 kg\cm²
Reduction ratio i=5
The angular velocity increases from 0 to 100 rpm in 2 seconds.
step:

Calculate the equivalent moment of inertia:
Jeq=0.05+0.01/(5^2)=0.0504 kg\cm²

Calculate the angular acceleration:

ω=2π⋅100/60=10.47 rad/s

α=Δω/Δt=(10.47−0)/2=5.235 rad/s²

Calculate the acceleration torque:

Ta=J eq⋅α=0.0504⋅5.235=0.264 Nm