FBD of two translating rollers connected by a belt

[ramadhankd]

Hey, I'm trying to create a mechanism consisting of multiple rollers connected with belt, moving in the same linear direction with acceleration a, but I got stuck in finding the relationship between the forces and acceleration. I even got confused in building the FBD of only two rollers. The picture below is the FBD of two rollers connected with a belt that I made. As can be seen the equation is quite inconsistent. Can anyone tell me what's wrong?
Thanks

 

[256bits]

@ramadhankd
Nice drawing.
I am unclear as to how you are accelerating the rollers.
From the diagram, are you pulling on each roller with equal forces of F ( pulling force 2F )?
Or is it that the tension in the belt is drawing along the roller at the left?
In the first case, the belt is superfluous.
In the second case, there should be no F pulling the roller on the left.

Do you have a mechanical link between the roller centres, or just freewheeling under the belt.

 

[berkeman]

Reminds me of a similar mechanism...

https://image.made-in-china.com/202...4-for-ATV-Snowmobile-Tread-Conversion-Kit.jpg

 

[ramadhankd]

@ramadhankd
Nice drawing.
I am unclear as to how you are accelerating the rollers.
From the diagram, are you pulling on each roller with equal forces of F ( pulling force 2F )?
Or is it that the tension in the belt is drawing along the roller at the left?
In the first case, the belt is superfluous.
In the second case, there should be no F pulling the roller on the left.

Do you have a mechanical link between the roller centres, or just freewheeling under the belt.

The rollers are both pulled by the same force F. There is indeed a link connecting them, yet this link is static so no kinematic analysis is needed to be done. You said that the belt is superfluous, meaning that the tension is zero? I try to solve It and what I found is that T1=T2, with no actual relation to determine T. The model I attached above is a simplification of the actual model. The actual model is like this.

I simplify the model as two rollers connected with the belt to know how the angular acceleration is affected by the belt. Shall It have lower angular acceleration, same, or more than when they aren't connected by the belt? I'm curious because when I try to connect the rollers using gear mechanism (yes, the rollers are going to move in an alternate rolling direction), the angular acceleration is reduced. I attached my analysis for this case just in case you need to review It.

I try to solve it using the same approach, yet the equations involved are just too much It got me confused. Thus, I try to simplify the mechanism using the two rollers model. My aim is only to know the value of angular acceleration (thus affecting the angular velocity and displacement, of the Blue (big) wheel should It able to be accelerated by the red (small) wheel, while both are contacting the ground. I design this for a roller mop, where more angular acceleration means more angular velocity and displacement, thus yielding more mopping action and better efficiency.

 

[256bits]

2

The rollers are both pulled by the same force F

Your analysis seems to indicate that the total moment of inertia for a two cylinder system is reliant upon whether the cylinders rotate in the same direction, or in opposite direction.
Is there a sign error someplace.

 

[ramadhankd]

The sign error is the equation that states ma=F+(T1+T2)-fr and ma=F-(T1+T2)-fr. There is inconsistency at the (T1+T2) variable. It's like having y=x and y=-x and try to solve for x using the two function.
My assumptions are that both move in equal angular acceleration since the diameter are the same, and of course, the same linear acceleration.
Both rollers are identical so there's no need to calculate their relation to each other.