Linear Motion – Calculating Acceleration for Linear Motion Control Applications

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Unit Converter
Conversion Type Units Units

Newton’s Second law:

 F = m xa
Force = Mass x Acceleration

To size a stage properly the force must be known. If force is not known it must be calculated from this equation. The mass is the total mass of the customer payload plus the mass 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 force is determined, the duty cycle for all of the specific forces must be determined to calculate the RMS force, which is the average required force. The Force RMS tab provides the tools needed to determine the RMS Force of a motion profile.

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Variable Description Units Calculation
m Moving mass
a Acceleration
F Force 0
*accelerations up to 10 g’s are possible under closed loop control
*accelerations up to 20 g’s are possible under open loop control

 

        

triangular formula graph
$$ \Large \alpha = \frac{4 \times d }{t^2 \times G} $$
 

 

 

Variable Description Units Calculation
d Distance moved
t Time to complete move (sec)
G Gravitational constant
a Acceleration (g’s) 0
*accelerations up to 10 g’s are possible under closed loop control
*accelerations up to 20 g’s are possible under open loop control

 

Trapezoidal formula graph
$$\Large a = {2 \times d \over t_{a}^2 \times G} $$
Known Variables
Time and Distance are known
Velocity and Distance are known
Time and Velocity are known

 

Variable Description Units Calculation
d Distance moved
t Time to complete move (sec)
v Velocity
G Gravitational constant
a Acceleration (g’s) 0
*accelerations up to 10 g’s are possible under closed loop control
*accelerations up to 20 g’s are possible under open loop control

sinusoidal formula graph

$$ \Large a = {2 \pi^2f^2 D \over G} $$
Known Variables
Frequency and Distance are known
Velocity and Frequency are known
Distance and Velocity are known

 

Variable Description Units Calculation
d Distance moved
f Frequency (Hz)
v Velocity
G Gravitational constant
a Acceleration (g’s) 0
*accelerations up to 10 g’s are possible under closed loop control
*accelerations up to 20 g’s are possible under open loop control

force formula graph

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

 

Variable Description Units Calculation
Fa Acceleration Force
Fc Constant Velocity Force
Fd Deceleration Force
ta Time to accelerate (sec)
tc Time at constant velocity (sec)
td Time to decelerate (sec)
ton ta + tc + td (sec)
toff Dwell time (sec)
Frms Average required force 0

$$ \Large Duty Cycle\left ( \% \right ) ={ t_{on} \over \left ( t_{on} + t_{off}\right )} \times 100 $$
Variable Description Calculation
ton Time with power applied (sec)
toff 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.

$$ \Large F@100\% = Force At DutyCycle \div \sqrt{ 1 \over DutyCycle} $$
Variable Calculation
Duty Cycle %
Force at Duty Cycle
Force at 100 % 0
Example
Force at 10% Duty Cycle = 1 lb X (1/10%)1/2

Force at 10% Duty Cycle = 1 lb X 3.16

Force at 10% Duty Cycle = 3.16 lbs

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

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

For the easy to use Calculator or the App CLICK HERE

H2W Technologies, Inc. is dedicated to the design and manufacture of linear and rotary motion products that are used in the motion control industry. The complete line of linear electric motors includes: Single and dual axis linear steppers, DC brush and brushless linear motors, voice coil actuators, and AC induction motors. Also offered is a complete line of ball screw, lead screw and belt driven positioning stages.

Other motion control products include: Limited angle torque motors for compact, limited angular excursion rotary servo applications, 3 phase brushless rotary servo motors with matching digital servo amplifiers and permanent magnet linear brakes for fail-safe, zero power braking for baggage handling and people moving applications as well as amusement park rides.

With over 75 years combined experience in the linear and rotary motion field, the H2W Technologies team of engineers offers the optimal solution to the most demanding motion control, requirements.

For additional information contact Mark Wilson at H2W Technologies, 26380 Ferry Ct, Santa Clarita,  CA 91350; Tel: 888-702-0540 FREE, Fax: 661-251-2067, E-Mail: info@h2wtech.comor visit the website at http://www.h2wtech.com

 

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