Re: mill threading, Acceleration, Axis

Hi,

My 2 cents on the acceleration debate. To paraphrase an unamed
notable who made this argument famous, "It depends on what your
definition of 'best is' is."

1) Best for simplicity: Linear ramp
2) Best for shortest time to speed: First integral of the speed-
torque curve.
3) Best for minimum "jerk factor": S-shaped ramp.


1) The best time to speed using a constant rate of acceleration is
based on using 100% of the motor's available torque at the terminal
speed. Since motor torque is much higher at the beginning of
acceleration, this available torque goes unused.


2) A step motor speed-torque curve is a two-part curve; constant
torque from zero speed to the motor's corner speed. The corner speed
is where the motor's natural load-line is intersected. The motor's
natural speed-torque curve is the 1/x function.

Suffice it to say the motor torque is high at low speeds and low at
high speeds. For the shortest time to speed, accelerate rapidly at
first, then more gradually with increasing speed as torque falls off.

The best fitting rate of acceleration is the speed-torque curve. The
resultant velocity with time (velocity profile) will be the first
integral of this curve.


3) Abrupt changes in acceleration induce ringing in the load. This is
called the "jerk factor". Stated differently, the second derivative
of the velocity is an infinate-impulse function.

The best function for the acceleration ramp is the definate integral
of sine from 0 to pi. The result is an S-shaped curve (sine from -
pi/2 to pi/2 + 1). Since all higher order derivatives of the sine
function are finite, the second derivative is also.

By analogy, we unconsciously use S shaped acceleration curves to keep
the second derivative finite in our everyday lives. Consider what you
do when you see the traffic light turn red as you approach an
intersection.

The brakes are applied gradually at first, then harder to establish a
constant rate of deceleration. As speed diminishes, the pressure on
the brake pedal is gradually released until brake action and speed
simultaneously arrive at zero.

If the brakes are applied abruptly, (linear ramp) your head bobs
forward until you adjust to force of deceleration. When the car comes
to a stop, deceleration ceases and your head bobs backward because it
was tensed against the suddenly absent deceleration.

This is the "jerk" factor in action, 2nd derivative as well as the
driver.

Mariss