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Math Help - Improvement on a winding machine.

  1. #1
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    Improvement on a winding machine.

    I'm sorry if this in the wrong section, this is my first post. My problem is that I need to improve the diameter measurement on a paper winding machine.

    The paper roll is sitting on top of two drums and there is another roller sitting on top of it to keep it from bouncing away from the machine (rider roll).
    The position of the rider roll is measured using a linear position sensor.
    At the start a core is inserted into the machine and the paper is taped to it. This is illustrated to the left in the picture below.
    The cores can have different diameter, but is always known.
    As paper is winded onto the paper roll, its diameter is growing and that pushes the rider roll up. A finished roll can be seen to the right in the picture.
    Obviously the distance between the Main Drums and their diameter is known. I
    t seems to me that it would be possible to make a formula that takes these values and the measurement and give you the paper roll diameter, I just can't figure it out.





    The measured distance can be calibrated to start or end somewhere else if that makes it easier.
    If you are curious here are some measurements:

    Main Drum diameter: MDd=790 mm
    MainDrum Center distance: MDcc=810 mm
    Core diameter: Cd= 70-150 mm
    Finshed roll diameter: 1000-1500 mm
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by randomswede View Post
    I'm sorry if this in the wrong section, this is my first post. My problem is that I need to improve the diameter measurement on a paper winding machine.

    The paper roll is sitting on top of two drums and there is another roller sitting on top of it to keep it from bouncing away from the machine (rider roll).
    The position of the rider roll is measured using a linear position sensor.
    At the start a core is inserted into the machine and the paper is taped to it. This is illustrated to the left in the picture below.
    The cores can have different diameter, but is always known.
    As paper is winded onto the paper roll, its diameter is growing and that pushes the rider roll up. A finished roll can be seen to the right in the picture.
    Obviously the distance between the Main Drums and their diameter is known. It seems to me that it would be possible to make a formula that takes these values and the measurement and give you the paper roll diameter, I just can't figure it out.





    The measured distance can be calibrated to start or end somewhere else if that makes it easier.
    If you are curious here are some measurements:
    Main Drum diameter: MDd=790 mm
    MainDrum Center distance: MDcc=810 mm
    Core diameter: Cd= 70-150 mm
    Finshed roll diameter: 1000-1500 mm
    Measure the distance from the level of the centres of the main rollers.

    Then consider the triangle made by the vertical through then cebtre of the paper roll, the horizontal through the main roller centres and a line conecting one of the main roller centres to the centre of the paper roll.

    (see the attachment)

    Then Pythagoras' theorem tells us that:

    (x+r)^2=d^2+(h-x)^2

    or:

    x=\frac{d^2+h^2-r^2}{2(r+h)}

    CB
    Attached Thumbnails Attached Thumbnails Improvement on a winding machine.-gash.png  
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  3. #3
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    Looking at your solution it makes perfect sense, mostly because I was annoyingly close to that.

    The current program is written in "S7 STL", the control system assembly language, which makes it a bit tricky to read. The documentation is lacking in clarity since the formula in there lacks a closing bracket, but this is what it says:

    x=2\frac{2(r*d +d^2+h^2}{2(r+h)}

    ...but by reading the program it seems that this is how it's really done:

    x=\frac{2 (\frac{2r*2d}{2}+d^2+h^2)}{2(r+h)}

    The distances r and d are not given as a radius, but as a diameter which maybe can explain part of the difference. But my followup question is does these formulas make any sense? Or are they just wrong?

    Anyway i will start to code a new program using your formula later...
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  4. #4
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    Just to be clear, the distance d is half the distance between the main drum center right? Or rather the distance from the main drum center to the center of the paper roll...
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by randomswede View Post
    Just to be clear, the distance d is half the distance between the main drum center right? Or rather the distance from the main drum center to the center of the paper roll...
    Yes

    CB
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