Paper roller question - FBD etc

[VFR]

Dear friends...

During the last months I am puzzled by the following problem and I would like to kindly ask for your assistance,

There is a device that I am trying to analyse and although the results do make some sense it seems that the experiments that I have done fail to replicate theoretical results.

I will try and describe the device by means of drawings and I will try to show what I have done so far.

In this device, there are for shafts. Two lower shafts and two upper shafts. One of the lower shafts is loaded with paper so that its diameter now changes to something larger than the original one.
The paper is then guided through the upper two shafts and then it is loaded onto the empty, lower shaft.

The lower left shaft is attached to a pulley and is rotated through a drive belt by a motor.
Every shaft is placed rigidly on ball bearings.
Assumptions: Frictionless contact everywhere and for the shake of argument I did not take into account any losses from the drive belts or anything.

My FBD looks like so:

I now take each shaft separately and I have the following equations.


Y positive upwards:
X positive toe the right
Moments anticlockwise

For Lower left shaft:
Moments about A > T1.R1-Mm=0 (1) knowing the motor torque I can solve for T1.
Χ> T1.cosθ1+Y1=0 (2) I know T1, hence Y1
Υ> T1.sinΘ1-X1=0 (3) I know T1, hence X1


For the upper left shaft:
Moments about B > T1.R2-T2.R2=0 (4) from this we have that T1=T2
X> T2+T1sinθ2+X2=0 (5) we can find X2
Y> T1.cosθ2-Y2=0 (6) we can find Y2


For the upper right shaft:
Moments about C > T2.R3-T4.R3=0 (7) from this one T2=T4
Χ> -T1-T4.sinθ3+Χ3=0 (8) we can find X3
Υ> -Τ4.cosθ3+Υ3=0 (9) we can find T4


For the lower right shaft
X> T4.cosθ4+Χ4=0 (10) we can find X4
Y> T4.sinθ4+Υ4=0 (11) we can find Y4
Moments about D > T4.R4+Md=0 (12) we can find Md

From all of the above I can understand that T1.R1 equals T4.R4 which means that the smaller the R1 (as the paper moves to the right) the higher the torque I have to put in order to roll the paper.

What I fail to understand from the above is the following. From my understanding, decreasing θ3 should make it easier for the paper to move the the left spool. increasing θ3 should make it more difficult to roll the paper to the left as now the paper has to do move through a higher angle.

My question is the following. I have the system, I have the paper roll numbers, and I want to find the optimum angles for that system in order to roll as smoothly as possible...

Is there something I am fundamentally missing here ?


Assistance is much appreciated,
D.

 

[eljose79]

Dear D,

Thank you for reaching out for assistance with your experiment. Based on the information and equations you have provided, it seems like you have a good understanding of the mechanics of the device. However, there are a few things that may be causing discrepancies between your theoretical and experimental results.

First, it's important to consider the assumptions you have made in your analysis. While assuming frictionless contact and neglecting losses from drive belts can simplify the equations, in reality these factors can significantly affect the performance of the device. It may be helpful to try incorporating these factors into your calculations to see if it improves the accuracy of your results.

Additionally, it's important to consider the materials and properties of the paper being used. The diameter and weight of the paper can greatly impact the amount of torque required to roll it, and may also affect the optimum angles for smooth rolling. Experimenting with different types of paper or adjusting the loading on the lower left shaft may help to better replicate your theoretical results.

Lastly, it's possible that there may be other external factors at play that are affecting the performance of the device. It may be helpful to conduct more controlled experiments and make sure all variables are consistent in order to get more accurate results.

I hope this helps and wish you the best of luck with your research.