### Re: Radial Force On Pulleys

I haven’t really thought about in detail, have only really concerned with the “overall” result.

Having said that, my understanding of Eulers equation is, its basis considers an element of belt on the pulley and the radial component of its tension which through multiplication of some artificial friction coefficient contributes to the ability of the element to transmit force. The results of the element is integrated between its limits (the pulley wrap) to find the amount of force a given installation can transmit. This relationship would seem to implicate the force would be greatest at “T1”. However as the radial force not only depends on the belt tension and the component, which contributes to the radial force, there may be other locations on the pulley, which through their geometry, contribute a larger radial component.

I guess I am not sure, that this is an answer. Plenty of people have questioned the theory of this relationship.

There have been papers written on the detailed interaction between the pulley shell and the load carrying component of the conveyor belt (lagging / bottom cover), which I cannot recall at the moment, though maybe some of them may address your query.

What does the pulley supplier use for their shell mechanical stress calculations? I thought it was average (force / belt width x wrap x diameter) though I am not sure. There are a few papers floating around about this also.

In the case of a non drive pulley, neglecting friction etc (assuming constant tension), you could derive an equation for radial force around the pulley circumference (derivative of the equation describing a circle x^2+y^2=R^2, something like dy/dx = - x/y, if I recall correctly), reasonably simply. For drive type pulleys, you would have to come to some form of agreement regarding the increase of tension from entry to exit around the pulley and apply this to your model.

This paper maybe of interest:

www.aws.org/wj/supplement/05-2003-PADILLA-s.pdf

Regards,

Lyle ■

### Re: Radial Force On Pulleys

For non-driven pulleys the answer is per your guess or so it seems:

Radial or normal force (Fn) = belt tension (T) / pulley radius (R)

This form is derived from differential equaltion as the limit of the parallelogram of forces when the angle of contact approaches zero.

The above force unit is per given width and assumes the force is constant across the width. Unfortunately, this is not so and has been measured by various researchers to show a non-uniform force.

The non-driven pulley force is also dependent on where the pulley is located. A tail or head pulley, which transitions the idler trough adds to the non-linear action.

Driven pulleys are far more complex.

The width or axial loading is the easier to address than the crossectional radial loading. One approach is to apply a design criteria which meets or exceeds the true value by some reserve. Today with modern analytic tools a Fourier Harmonic Expansion method is used to provide the axial load pattern associated with the type of loading. Typically, the harmonic loading expansion terms include the first 35 elements of the series to get the level of error to below 2%.

The radial load is assumed by many to be the e^(theta x f) drop from T1 to T2 over the wrap angle. Although, this is not true it does provide a result that is not too far from reality.

Then there is the issue of belt tracking and its concentrated force when the belt is misaligned to its true center.

What about material buildup, lagging distortion, ....... ?? ■

### Re: Radial Force On Pulleys

Dear Mr. Bowley,

For any pulley of belt conveyor, the belt is entering on to the pulley periphery at one side. Suppose here the belt tension is T1. The belt is leaving the pulley periphery at other location where say the tension is T2. In case of non-drive pulley T1 and T2 will be nearly same. The radial force acting on pulley, by the belt, is the vector sum of T1 and T2.

One has to imagine a segment of belt (belt strip) on pulley periphery. At each end of this small strip there would be certain tensions. These tensions will have a resultant force passing through pulley centre. Mathematical vector summation of forces from all such belt elements will give the resultant value T1 + T2, passing through the pulley centre. For approximation, it is like Hoop’s stress creating radial compression.

The belt design considers that there is a uniform tension / stress across the belt width. This infers that the belt force distribution across the pulley face will be uniform. This can have some alteration if there is pulley crowning or other external reason such as belt transition near pulley. But these are not of much significance.

Regards,

Ishwar G Mulani.

Author of Book : Engineering Science and Application Design for Belt Conveyors.

Author of Book : Belt Feeder Design and Hopper Bin Silo

Advisor / Consultant for Bulk Material Handling System & Issues.

Email : parimul@pn2.vsnl.net.in

Tel.: 0091 (0)20 25882916 ■

### Re: Radial Force On Pulleys

Thanks everybody for your prompt reply and insight on this matter. I am looking into horizontal curved belts and am trying to get a handle on the vectors involved for such calculations. The question i posed is related to the vector tward the center of the radius of curvature for a given section of belt. Sort of an parallel situation to the question that i posed. Other vectors being weight of the belt and normal forces from each idler. I am sure it is more complex (Stiffness, Buckling) than the above statement and im sure i will be getting into the thickof it soon.

Thanks again. ■

### Re: Radial Force On Pulleys

Pick up a publication by Prof. Grimmer 1963 and one more recent in Bulk Solids Handling where the basics are illustrated for horizontal curves. I bet Dr. Wolbier would know which articles reference horizontal curve engineering. I believe one was published in the late 1990's or early 2000 by Hans(?) Grimmer and Franz Kessler.

There is a good article on the internal forces by Prof. Oehmen from Hannover University from many years ago. ■

### Re: Radial Force On Pulleys

Isn't that the Krupp address? Surely they have a library on the horizontal curve engineering. Dr. Gizbert Schultz, I believe. ■

### Re: Radial Force On Pulleys

Nordell,

Thanks for the references and your help. I am indeed at Krupp and from what i hear i will be meeting you at some point in the future for sure. The only refence material i just found is from the most recent CEMA Belt conveyor text which did help a bit but leaves somthing to be desired. Ive been told we should have more reference material but i have not been able to locate them. I beleive our German office generally handles these calcs but ive been asked to familiarize myself with them.

The CEMA book which i imagine you are familiar with incorperates the friction of the rollers as a restoring force. I would think this is only valid in a static scenario and would have almost zero effect in a rolling dynamic scenario as the belt could walk up the idler regardless of the static friction.

Although this book does do an alright job of breaking up the vector components it does not address variations in tensions across the belt nor does it deal with material relocation in the trough. Id imagine theres alot more to it than i am considering in this short study of mine.

My purpose i believe is to estamate, and not fully design so i am unsure the level to which i am to master the curved conveyor anaylsis. ■

### Re: Radial Force On Pulleys

You will not get any joy from the German literature that I referenced. The published work I have seen places the material generated pressure and position as an artificial increase in belt density, as a membrane in the belt plance. This is from the material's initial position in the horizontally referenced, non-banked plane. This is not a true condition. It is solved as a matter of convenience, hoping the error is not too large.

You need to solve for the transverse displacement of belt and material which may act together initially and then separately as the belt and ore move over many idlers. Then you need to solve for the eventual change in ore adjusted crossectional position after is has traveled over these many rollers. Then you need to solved for these conditions with the shift in the belt and material from the horizontal radial pull due to belt tension and curvature.

Then you need to check to see the belt does not lift or curl from the outer wing or edge buckle at the inner wing assembly from excessive compression at high rate of curvature.

Then you need to solve for the compound conditions of horizontal and vertical conditions.

Once done, you are on first base. ■

## Radial Force on Pulleys

I am trying to figure out the radial force vector or normal force/unit length that a belt exerts on a pulley. I have searched the net and several texts to no avail and have been trying to derive it for over an hour. Does anyone know the general formula for this. Is it possibly

Normal force in force/unit length=tension/radius N=T/R

One concept i am debating with a coworker is whether the radial pressure of what ever you want to call it is constant over the belts contact length. Or does it have a profile similar to the force excerted on a bushing by a shaft. My argument, aided by a vector diagram is that its constant but im open to sugestions. i am also not considering this a drive pulley.

Thanks. ■