# Calculating the tilt of a box on a moving roller conveyor belt through a 180 degree turn

• Engineering

## Homework Statement:

What is the formula to calculate the tilt of a box traveling along a conveyor at 65 fpm and rounding a 6 foot radius 180 degree turn.

## Relevant Equations:

DEGREES(ATAN(h/w))
I am attempting to build a calculator that will calculate the tilt of a different shapes of packages traveling along a conveyor. the variables I am most interested in considering are: length, width and height of the package; speed of conveyor; radius and degree of turns.

Can anyone provide me some guidance?

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berkeman
Mentor
Welcome to the PF.

Are you familiar with how to draw free body diagrams (FBDs)? And are you familiar with the equation that relates circular path velocity with the "centripital" acceleration to cause that motion through the circular path?

Unfortunately, I am not familiar FBDs.

berkeman
Mentor
I just now familiarized myself with the all that it entails.

gneill
Mentor
Just an observation: It seems to me that a speed of 65 fpm is fairly modest, and packages would have to be pretty tall compared to their base in order to be in danger of tilting.

Take a quick example (I won't show my calculations because the OP needs to learn this stuff to proceed himself). Suppose that the mass of the package was 10.0 lbs. Switching to metric units, the gravitational force acting through the center of mass would be about 44.5 N. The centrifugal force wanting to tip the package would be only 0.27 N, also acting through the center of mass. The center of mass would have to be pretty far from a narrow base in order for the centrifugal contribution to overcome the gravitational one.

Ok. I am with you this far. The conveyor actually has the ability to run between 45 to 250 rpm. The package weights range between 0.5 to 50 lbs.

berkeman
Mentor
The conveyor actually has the ability to run between 45 to 250 rpm. The package weights range between 0.5 to 50 lbs.
What is the range of the package dimensions?

Length = 6 to 50 inches
Width = 4 to 50 inches
Height = 4 to 50 inches

I don't believe it has any relevance, but the only rule regarding dimensions is: Width can never be larger than length.

gneill
Mentor
Are there any rules about how the package is oriented on the conveyor belt? Intuitively, it would seem that a package would be more stable if its length were oriented so that it aligned with the radius of curvature. That would give gravitation a larger "lever arm" to work with.

berkeman
Mentor
Length = 6 to 50 inches
Width = 4 to 50 inches
Height = 4 to 50 inches
I don't believe it has any relevance, but the only rule regarding dimensions is: Width can never be larger than length.
So 4x6x50 tall? That's going to fall over no matter what...

Technically, given a box can never be longer than it is wide, it would be 6(L) x 4(W) x 50(H). I acknowledge that box configuration will not ride well on a conveyor, that would be filtered out as a possible configuration because its tilt angle would be far less than 30 degrees.

berkeman
Mentor
People must be placing these packages on the belt. They should be trained to place them with the longest dimension either in x or y on the belt, certainly not in the z direction. Even robot handlers should know better than that, right?

berkeman
Mentor

They should, but in scenarios like UPS, FedEx, and Amazon, packages travel so fast that their orientation will change when combining in contact the each other, bumper rails, loss of friction, etc...

Coming in contact. Sorry!

gneill
Mentor
Nice pic. If the belt were going to go through a sharp curve, I think I'd want the parcels rotated through 90° to put the longer base edge parallel to the width of the belt.

That is the ideal world of consistent sized items. Not real given Amazon, UPS and FedEx

gneill
Mentor
That is the ideal world of consistent sized items. Not real given Amazon, UPS and FedEx
Amazon seems to have standardized box sizes. The amount of packing material versus packaged item is sometimes quite astounding.

Amen! That is exactly why I am doing what I am doing. Filtering for tilt angles is the final step in determining ideal packaging and packaging intervals for ecommerce.

gneill
Mentor
Of course, it's rather hard to determine the center of mass of the package when the item packed might be located almost anywhere withing the packing material. I've seen relatively small items sequestered near to one end of a pretty large box, held in place by those plastic bubble ribbons. In some cases the overall packaging clearly out-masses the shipped item!

I agree with all that you have said. Now you can see why it is difficult to make the perfect scenario work; however, you can mitigate the unforeseen by building a better average case scenario which is prevalent 85% of the time.