Bringing Newton’s Second Law of Motion into the Motion Control World.

For any linear motion control
application, the solution to the problem requires at a minimum three points of
data:

·
How far do I need to move?

·
How heavy is the payload?

·
How fast do I need to get there?

These three points of data can be fed into Newton’s Second Law of Motion stating that the amount of acceleration is dependent upon the force acting on it and the object’s mass.

$$acceleration=\frac{force}{mass}$$

The equation above is the basic
building block of all motion profile acceleration calculations. You can utilize
this equation to determine the maximum theoretical accelerations possible by
any motor or actuator if you know the mass of the moving object, and the amount
of force the motor generates. There are several opposing forces that affect the
maximum acceleration such as friction, drag, and hysteresis. Factoring in the
losses, the maximum continuous accelerations that can be achieved through
direct drive linear motion are between 5-10 G’s for closed loop position
control applications and 10-20 G’s for open loop sinusoidally oscillating
applications.

While there are an infinite
number of motion profiles, there are three profiles that define acceleration
best for most linear motion applications.

·
triangular

·
sinusoidal

·
trapezoidal

The ** triangular motion profile**, also known as a saw tooth
profile, is the simplest as it assumes constant acceleration and deceleration
through the motion profile. This model is best to understand the basic
requirements in your motion profile. Below you can find the equation for
calculating the acceleration with a triangular motion profile.

__example__

To get the acceleration
necessary to move one inch in 0.050 seconds, you plug this into the equation
above to get the following:

A ** trapezoidal motion profile** is derived from the triangular
profile, as it assumes a constant acceleration until a desired velocity is
achieved, and then maintains a constant velocity for a time. In this case, you
will need to know two of three variables: Time to reach the target constant velocity, the targeted constant velocity, or distance available to reach the targeted constant velocity. The basic trapezoidal profile can be seen below.

· If the two known variables are ** time and distance** to achieve the constant velocity:

· then the calculation for the acceleration necessary to accelerate to a constant velocity can be determined with the formula below:

__example__

To get the acceleration
necessary to move to a constant velocity after one inch in 0.050 seconds, you
plug this into the equation above to get the following:

** velocity and distance** to achieve the constant velocity:

$$acceleration\:in\:G's=\frac{target\:constant\:velocity^2}{distance\:available\:to\:reach\:the\:targeted\:constant\:velocity\times acceleration\:due\:to\:gravity}$$

__example__

To get the acceleration
necessary to move to a constant velocity of 50 inches per second in one inch,
you plug this into the equation above to get the following:

** velocity and time** to achieve the constant velocity:

$$acceleration\:in\:G's=\frac{2\times target\:constant\:velocity}{time\:to\:reach\:the\:targeted\:constant\:velocity\times acceleration\:due\:to\:gravity}$$

__example__

To get the acceleration
necessary to move to a constant velocity of 50 inches per second in 0.050
seconds, you plug this into the equation above to get the following:

This type of profile is typically
found in long travel applications where a part needs to achieve a constant
velocity. Once you determine the acceleration in any of these applications you
can also calculate the missing variable whether it be distance, time, or
velocity.

The third common motion profile
is typically utilized in oscillatory systems, which is a ** sinusoidal motion profile**. These profiles have a constantly
changing acceleration to ensure a smooth oscillation between two end points. To
calculate the acceleration with this motion profile, it is necessary to know
two of three variables: Frequency of oscillation, displacement of oscillation,
or maximum velocity.

· If the two known variables are ** frequency of oscillation and displacement** in the oscillatory system:

· then the calculation for the acceleration necessary to oscillate can be determined with the formula below:

__example__

To get the acceleration necessary to oscillate
one inch at 20 Hertz, you plug this into the equation above to get the
following:

· If the two known variables are ** maximum velocity and frequency of oscillation** in the oscillatory system:

· then the calculation for the acceleration necessary to oscillate can be determined with the formula below:

$$acceleration\:in\:G's=\frac{2\times \pi\times frquency\:of\:oscillation\times maximum\:velocity}{acceleration\:due\:to\:gravity}$$

__example__

To get the acceleration necessary to
oscillate at 20 Hertz with a maximum velocity of 50 inches per second, you plug
this into the equation above to get the following:

· If the two known variables are ** maximum velocity and displacement** in the oscillatory system:

· then the calculation for the acceleration necessary to oscillate can be determined with the formula below:

$$acceleration\:in\:G's=\frac{2\times maximum\:velocity^2}{displacement\times acceleration\:due\:to\:gravity}$$

__example__

To get the acceleration necessary to
oscillate at 20 Hertz with a maximum velocity of 50 inches per second, you plug
this into the equation above to get the following:

No matter the motion profile the basic solutions for calculating acceleration will give an understanding for acceleration you are trying to get out of a system. With this understanding, any user can verify the best technology to suit their desired motion control application.