Rotary Motion Calculator

These calculators are used to calculate rotary motion. If you need to calculate linear motion, please use the linear motion calculators.

Unit Converter
Conversion Type		Units		Units

$$ T = J \times \alpha $$

Torque = Inertia x Angular Acceleration

To size a stage properly the torque must be known. If torque is not known it must be calculated from this equation. The inertia is the total inertia of the customer payload plus the inertia of the moving components of the stage. If the acceleration component is not known it must be calculated. Calculators are provided under the Acceleration tab for estimating the acceleration of a system. Once the torque is determined, the duty cycle for all of the specific torques must be determined to calculate the RMS force, which is the average required torque. The Torque RMS tab provides the tools needed to determine the RMS Torque of a motion profile.

Variable	Description	Units	Calculation
J	Inertia
α	Acceleration	(radians/s²)
T	Torque		0
	*accelerations up to 10,000 radians/s² are possible

Triangular Acceleration
Sinusoidal Acceleration
Torque RMS
Duty Cycle
Torque At Duty Cycle

$$ \Large \alpha = \frac{4 \times \theta }{t^2} $$

Variable	Description	Units	Calculation
θ	Angular Displacement
t	Time to complete move	(sec)
α	Acceleration	(radians/s²)	0
	*accelerations up to 10,000 radians/s² are possible

$$ \Large \alpha = 2\pi^2 f^2 \theta $$

Known Variables
	Frequency and Displacement are known
	Velocity and Frequency are known
	Displacement and Velocity are known

Variable	Description	Units	Calculation
θ	Angular Displacement
f	Frequency	(Hz)
v	Velocity
α	Acceleration	(radian/s²)	0
	*accelerations up to 10,000 radians/s² are possible

$$ \Large T_{rms} = \sqrt{\left ( T_{a} \right )^2 \times t_{a} + \left ( T_{c} \right )^2 \times t_{c} + \left ( T_{d} \right )^2 \times t_{d} \over \left ( t_{on} + t_{off} \right )} $$

Variable	Description	Units	Calculation
T_α	Acceleration Torque
T_c	Constant Velocity Torque
T_d	Deceleration Torque
t_a	Time to accelerate	(sec)
t_c	Time at constant velocity	(sec)
t_d	Time to decelerate	(sec)
t_on	t_a + t_c + t_d	(sec)
t_off	Dwell time	(sec)
T_rms	Average required torque		0

$$ \Large Duty Cycle\left ( \% \right ) ={ t_{on} \over \left ( t_{on} + t_{off}\right )} \times 100 $$

Variable	Description	Calculation
t_on	Time with power applied (sec)
t_off	Dwell time (sec)
Duty Cycle (%)	Percentage of total time spent active	0

Example
Duty Cycle = 1 sec on, 3 sec off Duty Cycle = 1/(1+3) = 1/4 Duty Cycle = 25%

Note: Duty Cycle is only for DC motors.

$$ \Large T@100\% = Torque At DutyCycle \div \sqrt{ 1 \over DutyCycle} $$

Variable	Calculation
Duty Cycle %
Torque at Duty Cycle
Torque at 100 %	0

Example
Torque at 10% Duty Cycle = 1 N-m X (1/10%)^1/2 Torque at 10% Duty Cycle = 1 N-m X 3.16 Torque at 10% Duty Cycle = 3.16 N-m

Note: This calculation is only for DC motors.
Use the following formula for AC motors:

AC Duty Cycle Calculations
Torque @ 50% = Torque @ 100% * 1.75
v@ 15% = Torque @ 100% * 5
Torque @ 3% = Torque @ 100% * 8