Unwind/Rewind Math Questions

[Blue92]

Hi all. First time programming a machine of this type. A center driven unwind feeding to a center driven rewind. No dancers or load cells, just used to pull coiled metal off of a core. I have laser diameter sensors on each roll, so calculating speed based on circumference is easy enough. Where Im getting hung up is the math behind getting the machine to decelerate so that we end up with the same diameter roll on the rewind as we started with on the unwind, minus the core thickness. I would like for the machine to reach a certain point then start to decelerate until it reaches the finished roll size. Any suggestions?

I read up on distance to stop formulas and jerk rates and what not but got lost quickly. Linear distance of material doesn't really matter here, since Im just pulling material off of a core and winding it back up sans core. I think we can do it based on roll diameter minus core thickness just unsure how.

The unwind/rewind shafts only spin at a maximum 400 rpm, accel/decel rates will be operator inputs, but assume pretty slow.

 

[drbitboy]

getting the machine to decelerate so that we end up with the same diameter roll on the rewind as we started with on the unwind, minus the core thickness.

Do the two cores, unwind and rewind, have the same diameter?

 

[Blue92]

Do the two cores, unwind and rewind, have the same diameter?

No core on the rewind.

 

[Blue92]

No core on the rewind.

Wind/unwind shafts are the same though.

 

[Tinine]

Have a look through these app notes I believe the link puts you on page #5. I haven't been through this stuff in a long time and tied-up right now. Check all pages, might be something of use.






Craig

 

[drbitboy]

Part I

A model for the cross-sectional area of a roll:

A = π × d² ÷ 4

A = (L + L_p) × t​

Where

π = PI
A = cross-sectional area
L = Length of material around the shaft or [shaft + core]; L is also equivalent to circumference
L_p = Phantom length, constant, representing equivalent length of material theoretical wound about nobbut itself (no core or shaft) that would make a roll of the same diameter as the shaft (+ core).
t = average thickness of a layer on the roll, including any space between layers, a measured constant, which is an implementation detail.
d = diameter of the roll, measured
​

Differentiating:

dA = (π × d ÷ 2) × dd

dA = t × dL​

so

(π × d ÷ 2) × dd = t × dL
​

rearranging

(π ÷ (2 × t)) × dd = dL ÷ d
​

(L ÷ d) is the the ratio of circumference to diameter, which is angle, θ, so (dL ÷ d) = dθ,

(π ÷ (2 × t)) × dd = dθ

k × dd = dθ, where k = π ÷ (2 × t), a constant
​

Converting to time derivatives

k × dd/dt = dθ/dt

k' × dd/dt = RPM
​

because dθ/dt is rotation rate, which scales to RPM; k' is k with the scaling factor from (radian/[time unit]) to RPM included.

N.B. k and k' are negative, assuming dθ/dt and RPM are positive.

End of Part I

 

[drbitboy]

Part I (updated)

Updated a few things, also disambiguated thickness and time variables, and used a better font:

An ideal model for the cross-sectional area of a roll:

A = π × d² ÷ 4

A = (L + L_p) × h​

Where

π = PI i.e. ratio of a circle's circumference to its diameter
d = diameter of the roll, measured in real-time
A = cross-sectional area
L = Length of material around the shaft or [shaft + core]; L is also equivalent to circumference
L_p = Phantom length, constant, representing equivalent length of material theoretical wound about nobbut itself (no core or shaft) that would make a roll of the same diameter as the shaft (+ core).
h = average thickness (height) of a layer on the roll, including any space between layers; this is a measured and/or empirical constant; determining its value is an implementation detail.
​

Differentiating:

dA = (π × d ÷ 2) × dd

dA = h × dL​

so

(π × d ÷ 2) × dd = h × dL
​

rearranging

(π ÷ (2 × h)) × dd = dL ÷ d
​

(L ÷ d) is the the ratio of circumference to diameter, which is angle, θ, so (dL ÷ d) = dθ,

(π ÷ (2 × h)) × dd = dθ

k × dd = dθ, where k = π ÷ (2 × h), a constant, probably determined empirically
​

Converting to time (τ) derivatives:

k × dd/dτ = dθ/dτ

k' × dd/dτ = RPM
​

because dθ/dτ is rotation rate, which scales to RPM; k' is k with the scaling factor from (radian/[time unit]) to RPM included.

N.B. k and k' are negative, assuming dθ/dτ and RPM are positive.

End of Part I

 

[drbitboy]

Part II, deceleration model

Using that ideal model, here is a jerk-limited profile to bring the rate of change of diameter (d); starting at diameter D₀ and time τ = 0; ending at diameter D_core and τ = Δτ. This is taken from this wikipedia link.

​

Jerk, d³d/dτ³

Jerk is positive J_MAX for the first half of the period Δτ, and negative J_MAX for the second half:

τ < Δτ / 2: = +J_MAX
τ ≥ Δτ / 2: = -J_MAX
​

Acceleration, d²d/dτ²

Acceleration is 0 at the start and end of the period Δτ, and driven by Jerk above.

τ < Δτ / 2: = J_MAX × τ
τ ≥ Δτ / 2: = J_MAX × (Δτ - τ)
​

Velocity, dd/dτ

Velocity starts at (400RPM ÷ k')* at the beginning of the period Δτ, and is zero at the end, and is driven by Acceleration above.

τ < Δτ / 2: = (400RPM ÷ k') + J_MAX × τ² ÷ 2
τ ≥ Δτ / 2: = -J_MAX × (Δτ - τ)² ÷ 2

* N.B. assumes the process, which is near the end of a rewind, is limited by the smaller diameter of the unwind roll running at maximum speed i.e. 400RPM

N.B . the rate of change of the diameter is negative, as is the constant k'
​

Diameter, d

Diameter is D₀* at the start of the period Δτ, and D_core at the end.

τ < Δτ / 2: = D₀ + (400RPM ÷ k') × τ + J_MAX × τ³ ÷ 6
τ ≥ Δτ / 2: = D_core + J_MAX × (Δτ - τ)³ ÷ 6
​

N.B. as request by OP, this is a model based on the deceleration of the measured diameter, not the length of material remaining on the unwind shaft + core. The diameter and length are related (by square root and the phantom length), so the characteristic will be different using diameter instead of length, but the end point will be the same, because remaining length will be zero when the measured diameter is equal to D_core.

 

[drbitboy]

Part III

Here is a graphical version of the jerk-limited acceleration model. Because both Jerk and Acceleration (of diameter) start and end at zero, and the Velocity (rate of change of diameter) starts at a known value (400RPM÷ k') and ends at 0, the math is fairly straightforward, with the area of each curve over the [0:Δτ] deceleration period being (i) trivially derived graphically from symmetry, and (ii) equal to the change in the quantity of the next curve.

​

 

[Peter Nachtwey]

The unwind/rewind shafts only spin at a maximum 400 rpm, accel/decel rates will be operator inputs, but assume pretty slow.

@drbitboy, what happens if the acceleration/deceleration limit is reached? Then there is an intermediate constant acceleration or deceleration phase/state.

 

[drbitboy]

@drbitboy, what happens if the acceleration/deceleration limit is reached? Then there is an intermediate constant acceleration or deceleration phase/state.

Patience, Peter, patience: I only made it to Part III so far.

We still have to set the parameters, which is where we deal with that problem. Then there is coding the algorithm, which involves solving d = D0 + (400RPM ÷ k') × τ + J_MAX × τ3 ÷ 6 for τ with a known (measured) value for d.

There is also a torque limit, which will involve estimation of inertia, but that should overlap with any acceleration/deceleration limits.

Bottom line answers to your queries: (i) the answer to "what happens if the accel/decel limit is reached" is that we use that constraint when setting the Δτ over which the decel happens and the J_MAX value; (ii) I chose to do this without a "step 6" (cf. the wiki) intermediate constant acceleration phase/state, or stated another way its duration is zero. Obviously, if we increase J_MAX (jerk a.k.a. jolt) so there is a constant acceleration phase of non-zero duration, then that changes the details, but not the basic approach, and the graphically-derived integration is still trivial because the velocity would have a linear characteristic during the constant acceleration phase, which would maintain the symmetry.

 

[drbitboy]

Actually, what if we don't care about jerk/jolt and constant acceleration (infinite jerk) is good enough? That solution is simpler.

 

[Blue92]

Actually, what if we don't care about jerk/jolt and constant acceleration (infinite jerk) is good enough? That solution is simpler.

Yes, lets do that. Chances are the customer may want to stop before the roll runs out and jog it in the rest of the way anyways.

Also, Im way lost in all of that differential math haha.

 

[Peter Nachtwey]

Since you are only winding metal you probably will get by with that. I wonder if it will make much difference is some layers are wound a little tighter than others. I wouldn't do that if winding something that stretches. Keith might have something to say about this.

@debitboy, you dodge a bullet. It is amazing how much effort you have put into so far. Taking into account all the different cases for 3rd order ramps gets messy. I wouldn't want to write that code in a PLC.

Anyway, I was preparing to pop a whole lot of popcorn but not now.

If the linear ramps don't work, then it is time to get a real motion controller. Trying to do 3rd order ramps in a PLCs is asking for abuse.

https://deltamotion.com/peter/wxMaxima/Seg1234567.html
And that is only ONE case.

 

[drbitboy]

Also, Im way lost in all of that differential math haha.

Which is why I provided the graphical approach to the same solution. Was that lost on you too?