Actually, you need to know the master position, velocity and acceleration. One can simplify things if you can assume the master line speed is constant. Then you can assume the acceleration is constant. I find this to be a bad assumption.
What we need to find is the slave(knife) acceleration. Lets start with the formulas from above and add the master acceleration term which we have assumed to be 0 before.
vs(m,m',m''):=g(m)*m'
Now we differentiate using the product rule.
va(m,,m',m''):=g(m)*m''+g'(m)*m'
The problem here is that m'' ( master or line acceleration ) must be determined by the line encoder feedback and it is difficult enought to calculate m' ( master or line velocity ). I haven't seen that covered on this forum yet.
The other partg is to calculate g'(m). This is the rate of change of the gear ratio with respect to the master position. I showed the formula for g(m) so we must take the derivate of that or the second derivative of r*arctan(m/h). This isn't very hard because one can find the answer on the internet by search for derivative of arctan. I found.
http://www.opensky.ca/~jdhildeb/arctan/arctan_diff.html
I would usually use my Mathcad but I am in the field so I used the net instead.
The second derivative is:
-2*x/((1+x^2)^2)
Note that in our case x=m/h. So we had to apply the chain rule once for every derivative which means we must mulitply by the derivative m with respect to x which is 1/h. So the derivative of r*arctan(m/h) is r/(h*(1+x^2)) and the derivative of that is -2*r/(h^2*(1+x^2)^2). So if you want to calculate the slave acceleration it is.
sa(m,m',m''):=r/(h*(1+(m/h)^2))*m''+2*r/(h^2*(1+(m/h)^2)^2)*m'
With out a good estimate of the slave acceleration you don't have a ghost of a chance because the gear ratio is always changing. You should also run the drive in torque mode and not velocity mode to take advantage of the calculated acceleration, othewise you must depend on the quality of the drive and its ability to differentiate an analog signal. I don't think any PLC is fast enough to run a VFD or servo motor directly in torque mode. Those of you that say 'use sercos' must also jump through extra hoops to make sure the position, velocity and acceleration is downloaded at every 2 millisecond interval. How can the sercos motion controller calculate accurate accelerations from the positions downloaded every two milliseconds? Some of the better sercos controllers can download position and velocity each update and that is probably good enough because the difference in velocity can be used to calculate the acceleration.
Oasis, are you really going to try to do the control in a SLC5/04? There is a lot to these 'simple' rotary knives. Hopefully this thread will save people a lot of time by pointing them in the right direction as opposed to wasting time trying to do the impossible in a PLC.
