anyone great at math?

yes, exactly
And the only variables are the car length and the conveyor speed.
It actually works quite well as is.
Even washed a stretch limo and the bridge reverse speed crawled and spent the maximum amount of time on the sides of the limo and reached the end right in time.
I need to understand the formula to convince the customer how the first one works and the second one doesn't, but I don't have a clue.
 
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Okay, your original formula was roughly correct. Conveyor speed should not be in the denominator, unless the idea is to wash an extra second's worth of conveyor movement beyond the back end of the car.


Let's say

  • Mstart is the position of the washer arms when they start the reverse move;
    • it is probably easiest to call this 0;
    • it would be that leftmost position of the washer arms of the video, where they were at 2m52s
  • Cstart is the position of the front of the car when the washer arms start the reverse move at Mstart
    • Cstart > Mstart
  • Cend is the position of the back of the car when the washer arms finish the reverse move at Mend
    • Cend > Cstart
  • Mend is the position of the washer arms when they finish the reverse move
    • Mend > Cend
  • Cnv_InMin is the speed of the car to the left, in inches per minute
  • L is the length of the car
BRS_InSec = (Mend - Mstart) * (Cnv_InMin / 60) / (L + Cstart - Cend)




Derivation; TL;DR


Δt = (L - (Cend - Cstart)) / (Cnv_InMin / 60) ;;; Time, in seconds, for conveyor to move car, from nose at Cstart, to tail at Cend


BRS_InSec = (Mend - Mstart) / Δt ;;; Speed to move washer arms from Mstart to Mend in time Δt




And here's a thousand words ...

xxx.jpg
 
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So, based on the video, your descriptions, and the comments/suggestions, you have a formula to prove and (if you are interested), a new implementation to try.

If the video depicts your system and you are looking only to prove the formula, then let's look at a few things.

If we are under the assumption that the second half of the video represents the system you are working on/installed, then you have a three part move:
1. The bridge starts in the middle and begins a forward move at a speed equal to the conveyor speed until it reaches its front/tunnel discharge EOT (basically, a length of BTD/2). EOT means end of travel, or a stop position, if you will.
2. It then switches to the calculated speed you mentioned in the first post and moves reverse until it reaches its rear/tunnel entrance EOT (a length of BTD).
3. Finally, it switches back to a forward move at a speed equal to the conveyor speed until it returns to the middle position (a length of BTD/2).

A simple analysis of the formulas shows that your physics buddy isn't considering the units, so there would need to be more to their equation to be correct. The original formula results in a value of X inches/sec. The suggested formula results in a value of Y (inches/sec)/(W inches - Z inches/sec), so without some time component for W (the car length - 28inches), you are stuck. The original formula did have an overshoot, but it is evidently a design feature.

Consider this example:
Assume: Conveyor speed = 600 in/min, Car Length = 1072 in, Bridge Travel Distance (BTD) = 100 in
1. Numerator: Conveyor Speed (in/sec) x BTD (in) -> ((600in/min)/(60sec/in)) x 100in = 1000in2/sec [1000 inches squared per second]

2. Denominator: Car Length + 28in - BTD -> 1072in + 28in - 100in = 1000in
3. Result: (1000in2/sec)/(1000in) = 1 in/sec
4. Conclusion: the bridge would need to move at a speed of 1 inch/second to reach its end of travel roughly at the rear end of the vehicle. I say roughly because in my example, the bridge will take 100 seconds to complete its reverse move of 100 inches and the car will have moved 1000 inches during that time. Relative to each other, the bridge moved 100 inches against the motion of the car, resulting in a relative difference of 1100 inches, so the 72 inches of remaining car length were actually surpassed by 28 inches (your PE length).

That 28 inches is actually shown in system in the second half of the video. It is a design feature for ensuring that when the third stage of the move is executed the spray nozzles get the back of the vehicle. As the bridge moves along with the vehicle during that third stage, it's really just spraying the same area of the car. When it finally stops in the middle position, the nozzles continue to spray, getting the rest of trunk/rear.

So, my final verdict is that your original formula works. The other one is incomplete. The 28 inches and 5 inches are factors you can add/remove/tweak to maximize the performance of your unit.

The only question I can think of is why it didn't work "in the shop," as you said. What was wrong with it there?
 
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A simple analysis of the formulas shows that your physics buddy isn't considering the units, ...The suggested formula results in a value of Y (inches/sec)/(W inches - Z inches/sec)


Dollars to doughnuts the physics buddy multiplied Z by 1sec, so there is 1s more modeled car sprayed to allow for a short measurement of the car, and simply dropped the (* 1sec) from the (Z inches/sec * 1sec) in the final formula.
 

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