Catenary Like Curve - Conveyor Application
Catenary Like Curve - Conveyor Application
(OP)
Here is the problem. I've posted it to this forum because of the similarity to cable structures.
I have a gravity belt thickener (belt conveyor used to dewater sludge) that has the first portion of belt horizontal. The second section is inclined at roughly 30 degrees above horzontal. It is one continuous belt driven from the top of the inclined section. The transition between the two sections has a small radius of curvature. The belt tension under load tends to lift the belt off its bed and cause operational problems (sludge spillage).
A set of rollers above the conveyor belt has been added to keep the belt from lifting. They do prevent lifting, but cause additional problems as they plow through the sludge.
It is desired to re-design the transition area between the flat and inclined conveyor sections to prevent belt lifting under load. If belt tension and weight were constant along the length, then the ideal shape of this transition region would be a catenary curve like that of a cable hanging under its own weight. However, belt tension is not constant, but increases along the belt in the direction of travel in proportion to the supporting force normal to the belt.
What is the easiest way to determine the proper shape of the transition curve? Can this problem be solved in any way other than a computer finite difference model of the belt? I'm hoping that the proper shape can be closely approximated by a circular arc or parabola. I'd also like to be able to verify any computer results via simplified manual calculations. Any help or additional insights are appreciated.
I have a gravity belt thickener (belt conveyor used to dewater sludge) that has the first portion of belt horizontal. The second section is inclined at roughly 30 degrees above horzontal. It is one continuous belt driven from the top of the inclined section. The transition between the two sections has a small radius of curvature. The belt tension under load tends to lift the belt off its bed and cause operational problems (sludge spillage).
A set of rollers above the conveyor belt has been added to keep the belt from lifting. They do prevent lifting, but cause additional problems as they plow through the sludge.
It is desired to re-design the transition area between the flat and inclined conveyor sections to prevent belt lifting under load. If belt tension and weight were constant along the length, then the ideal shape of this transition region would be a catenary curve like that of a cable hanging under its own weight. However, belt tension is not constant, but increases along the belt in the direction of travel in proportion to the supporting force normal to the belt.
What is the easiest way to determine the proper shape of the transition curve? Can this problem be solved in any way other than a computer finite difference model of the belt? I'm hoping that the proper shape can be closely approximated by a circular arc or parabola. I'd also like to be able to verify any computer results via simplified manual calculations. Any help or additional insights are appreciated.






RE: Catenary Like Curve - Conveyor Application
My intuition (with no mathematical research behind it) tells me:
1. A circular arc does not fit well with varying belt tension, since constant radius implies constant radial load intensity.
2. A parabola similarly does not fit too well. The parabola is the 'correct' curve if you suspend a cable between two points and apply a uniform VERTICAL (ie not radial) load intensity.
3. You 'should' be able to solve your problem by formulating and solving the relevant differential equation, particularly if you can approximate the varying tension to a linear variation horizontally (rather than along the curve). (Note, I am not claiming that I could solve it
Good luck
RE: Catenary Like Curve - Conveyor Application
butelja,
A great little book by Max Irvine called "Cable Structures" has several solutions to problems similar to yours.
eg it has the solution for the shape of a towed boom of logs - use that one all the time - :), inflatable dams, flying foxes etc...all hand/manual solutions.
HTH
RE: Catenary Like Curve - Conveyor Application
I think that to calculate the catenary of the system you describe, with variable tensions, weights and supports, would be rather difficult.
What I would do in your case is to try to determine the actual shape of the belt while in movement and to adjust upwards those rollers that do not make contact with the belt.
If the lifting of the belt is just due to the tension, I would run the conveyor without or with a nominal load (material) on top, and measure the gap at each roller where a separation occur.
If the small catenaries formed between rollers when the sludge is on, is a large contributor to the lifting of the belt, then I would run the conveyor with a load similar to the sludge weight, and then measure the gaps.
Of course, the set of rollers above the belt should be removed before taking any measurement.
Good Luck!
AEF
RE: Catenary Like Curve - Conveyor Application
Weight per unit length should indeed be constant to have a catenary curve, but tension in the catenary is not constant, as the supported weight at each section is of course different, and tension also depends on local slope.
So I think that your belt behaves like a catenary, though only theoretically of course, as the supported weight is not necessarily constant.
Another assumption for the belt behaving like a catenary is that it has no bending stiffness: on a fairly long travel this shouldn't be of importance, but if the curve you are seeking for is over a short length of belt (a few times its width?) then this becomes an issue.
prex
motori@xcalcsREMOVE.com
http://www.xcalcs.com
Online tools for structural design
RE: Catenary Like Curve - Conveyor Application
RE: Catenary Like Curve - Conveyor Application
Have you tried verticle edge rollers to prevent the belt from 'flatting out' in the area of the spillage ?
Rod
RE: Catenary Like Curve - Conveyor Application
So your problem is as much to keep an uniform load being conveyed as to find what shape of the conveyor belt stays in equilibrium with the demanded tension required by fitness to use.
Tension for a catenary of own -or constant along directrix- weight is quite constant. You can see it yourself in my catenary.mcd sheet freely downloadable from the Mathsoft Collaboratory
http://collab.mathsoft.com/~mathcad2000
Search Catenary
By the way with Mathcad solving any problem of such kind should be easier, due to its having solvers and Diff Eq solvers.
RE: Catenary Like Curve - Conveyor Application
Presently a fixed tail pulley and catenary sag on the back side to keep tension. Re-design will utilize a gravity take-up to better compensate for thermal expansion. Edge rollers were tried initially, but due to the width (5 ft) of the belt, it still pulled up in the center. The differential pull damaged the belt.
Ishvaag,
The conveyor is continuously charged.
RE: Catenary Like Curve - Conveyor Application
The dynamic variation of tension and especially load causes vibration or movement in the vertical plane that causes material ejection, in the natural search of a tangent curve to the inclined part that equilibrates the actual weight.
Less material conveyed, the tangent at the inclined part gets upwards for the same tension. Inertial effects apart, and assuming an instantaneously static problem, shape won't be but under the constant weight and an equilibrating tension a catenary, and always a funicular of the extant load.
In more than studying the thing, I would give a look to see if the conveyor system is within ordinary parameters for the function: too high for the other parameters and application? too much material conveyed per second? Too irregular load per unit length?
I have the feeling of some parameter is wrong, maybe the curve is too sharp for the angle or material conveyed, or the extant tension for such curve and material is too big. From our catenary there's someone pulling from the ends, drag behind ad pulling force ahead.
Looking for advice from a conveyor systems fabricator may kill the problem in a single consultancy.
RE: Catenary Like Curve - Conveyor Application
I know this is not a technical solution, but is intended as insight to the problem. I hope this helps.
Rod
RE: Catenary Like Curve - Conveyor Application
because it has typically a uniform mass per slope length.
The constant c is related to T0 (catenary tension at the horizontal portion)
and w (mass per unit length) by
T0 = w * c
Tension at any point T = T0 + w * y.
The alternative parabolic formula only applies to suspended bridge decks etc with uniform mass per horizontal length.
You will need to modify these to account for drag force and normal reaction along the bed. The maths may get a bit tricky.
Perhaps the easiest is to set up some spreadsheet series using small increments (say 2 deg) of curve, and calculate equilibrium of forces for each increment.
You will probably find the system fairly stable for a whole range of curves and tensions provided your belt tension at the bottom of the curve is fairly low.
As a practical help to achieve stability at startup etc, use a fluid coupling at the head drive.
I agree with Evelrod on the need for a takeup that can accurately control force with minimum length change.
It may also help to use a steel-cord belt to reduce belt stretch.
RE: Catenary Like Curve - Conveyor Application
Furthermore I check Kavanagh on cable supported bridges for the catenary equation and seems give
T=w·(f+h)
where f is sagitta and h rest from bottom apex to a suitable origin of coordinates downwards, hence a constant.
Is there anyone that can deny or confirm?
RE: Catenary Like Curve - Conveyor Application
Just Wandering.
The rentapen
RE: Catenary Like Curve - Conveyor Application
Check this sketch. I think it is a simple way to resolve the problem.
http://members.aol.com/aefsr2000/documents/conveyor.jpg
AEF
RE: Catenary Like Curve - Conveyor Application
Your tension formula is the same as mine.
Combining my two formulas gives
T = w * c + W * y = w (c + y)
and yours is
T = w (f + h)
Just the symbols used are different.
RE: Catenary Like Curve - Conveyor Application
RE: Catenary Like Curve - Conveyor Application
I think your link is broken. Clicking on it sends me to the AOL member homepages search engine.
Imagineer
RE: Catenary Like Curve - Conveyor Application
Thanks for let me know about the link.
Yes, it seems to have been broken, and I don't know why.
I will try to fix it
dlew