catenary like curve - conveyor application

huangyhg

catenary like curve - conveyor application
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.
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hi,butelja,
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
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
botelja,
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
i'm not sure to understand your point.
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
one thing not mentioned in my original post.  the conveyor belt does not run on a bed of rollers.  the bed is continuous uhmw polyethylene wear strips.
questions.   are you using a fixed type take up or a gravity take up for pretension of the belt?  
have you tried verticle edge rollers to prevent the belt from 'flatting out'  in the area of the spillage ?
rod
if the conveyor belt spits material at the curve it may well be happening that the required tension in the conveyor to lift all the material for the design height is too much, then when scarce material is at the curve it gets straight and ejects the material. only a continuously charged enough curve can keep the bottom down.
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
evelrod,
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.
yes, but nor the tension nor the load stay in an equilibrium in the present situation not causing spill of material, nor the load allows for the conveyor stay down without rollers.
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.
my questions had a point.  i insalled a system similar to what you describe, belt width of 4 feet and length of 120 feet +/- with a total rise of about 9 ft. in a run of  15 ft. used to handle dry material (borax powder).  the problems you describe are almost exactly as we faced then (a lot of years ago).  the fixed take up did not work due to slippage on the drive roller when the belt was running at capacity.  the gravity take up failed to work due to it's inability to adjust to varying loads and belt speed.  the end cure was a hydraulic/pneumatic take up roller(s) that were calibrated (via a computerized processor) to vary tension based on load/belt capacity.  this in conjunction with a drive roller with a high friction coating solved the spillage and wear problems. a 'high dollar' cure, for sure.
i know this is not a technical solution, but is intended as insight to the problem.  i hope this helps.
rod