Calculating the minimum vacuum force for a climbing robot

amrbekhit

1. The problem statement, all variables and given/known data
Hello. I am trying to calculate the minimum vacuum force required to keep a climbing robot attached to a ceiling. The robot body consists of a cuboid with two vacuum cups attached at either end. The robot will move using inchworm-style motion, attaching one of the cups, extending its body, attaching the second cup, retracting etc. In order to calculate the minimum vacuum force that the robot needs to generate, I have constructed the following free body diagram:



To calculate the maximum force, I am consider the situation where the robot has only one of its suction cups active, which in this case is the one on the right.

Description of forces:

• F: The suction force produced by the vacuum cup.
• R: The reaction force produced due to the robot pressing against the ceiling
• uR: Force due to friction between the ceiling and the robot's suction cup
• D: Drag force caused by fluid moving past the robot. This is 1N
• mg: Weight of the robot. This is 0.2*9.81=1.962

List of dimensions:

• r: Radius of the suction cup. 15mm
• H: Distance between the C.o.M of the robot and the ceiling: 17.5mm
• L: Distance between the suction cup centres. 56mm

In order to perform the calculation, I have made the following assumptions:
I have assumed that as the suction force decreases, the robot will not just simply fall vertically, but will behave almost as if it was hinged at point A in the diagram. Based on this assumption, I have assumed that the only reaction force is the one shown on the diagram. This is because if the suction force is at its minimum, I would imagine that the inactive suction cup would be barely touching the ceiling and so no reaction forces would be generated. Likewise, I am guessing that the side of the left hand side of the active suction cup would also be barely touching, leaving the very right hand side in firm contact with the ceiling.

2. Relevant equations
[tex]Pressure = \frac{Force}{Area}[/tex]
[tex]Circle area = \pi r^{2}[/tex]

3. The attempt at a solution
Resolving vertically:

[tex]F = R + mg[/tex]

Resolving horizontally:

[tex]D = uR[/tex]

Taking moments about A:

[tex]Fr = DH + mg(\frac{L}{2} + r)[/tex]

Solving the above equation gives a value of [tex]F = 6.79N[/tex]

Substituting this value back into the first equation gives a value of [tex]R = 4.83N[/tex]

Finally, inserting this value into the second equation gives a value of [tex]u = 0.2[/tex]

In order to calculate the pressure required in the suction cup:

[tex]P = \frac{F}{A}[/tex]
[tex] A=\pi 0.015^2 = 7.069*10^{-4}m^2[/tex]

giving a pressure of

[tex]P=9606Pa[/tex]

What I am unsure about is whether this is the correct way of calculating the minimum force. I am especially unsure about the assumption I have made regarding the reaction force. Should there be reaction forces on both suction cups? Should I consider those forces as one force for each cup, or would I need to consider the reaction forces on both sides of each cup? Am I correct in assuming that the robot would rotate around point A as the suction force decreases? Would I then need to consider any friction produced at the inactive suction cup?

Thanks

--Amr

diazona

Well.... I think you made a reasonable decision to consider the force when only one suction cup is touching the ceiling. That's when the full reaction force from the suction needs to be used, so it determines what the minimum suction pressure needs to be. Obviously, if one cup is not touching the ceiling, there will be no reaction force or friction on it.

The assumption that the robot would rotate around point A also seems reasonable, I think - definitely the rotation would be taking place around some point within the suction cup. You could try doing the calculation assuming that the axis of rotation is at the forward edge of the suction cup, just to see how much your answer for the pressure differs; if the difference is small, that tells you that the assumption about the rotation axis probably doesn't matter that much.