Replace a conveyour belt of belt scale by means of an ideal string

This page is about the conveyor belt replacementof belt scale.

Those who has read an articles had got imagination of conception "to replace a conveyor belt by means of series isolated joint-supported beams (fig.1). For this conception it does not important a tension state of belt as well as there is absolutely indifferent to matter whether preciously or not were aligned idlers. Everyone who has a bit of intuition can guess that it does not go quite so. We shall be introduced here with some advanced model of conveyor belt, where more realistic picture of key factors' affects is represented. These factors are:

Belt tension;
Installation accuracy;
Stiffness/deflection of weighing idler.

$\[width=7in]{fig1.eps}$
Figure 1: Initial model of the belt conveyor as multi-joint-supported beam.

So, there is a model from immemorial times where the conveyor belt is represented as absolutely inextensible and absolutely flexible weightless string. Basically as well as the absolutely rigid body for theoretical mechanics or the absolutely elastic one for application mechanics, a flexible string -- is a mathematical fiction not existing in the nature but it helps us to understand essence of a matter.

Let's consider a mechanical system as a flexible weightless string stretched along legs ideally located at one level (fig.2). Legs are representing the carrying idlers of a conveyor belt so further it shall be called as idlers. The idlers' span is equal . The level of the upper points of idlers and the line of a string coincide. Contact spot of a string to idlers is sliding without abrasion. It is obvious, that by means of such system we substitute a real conveyor belt.

Figure 2: Model of the conveyor where conveyor belt is represented as absolutely flexible and weightlessly string, having strained by tension forces $\sigma$ .

The string is stretched by horizontal forces of a tension $\sigma$ . As the string is weightlessly, no forces are affected to idlers.

To allocate a section as a place of prospective installation the one-idler belt scale (fig.3). Actually, it means that idler is implied as weighing one. While it is admissible, the idler in essence is differed nothing from remaining, i.e. idler is absolutely rigid.

Well, we shall applying force over a weigh idler . It is obvious, that the response reaction of an idler will be equal to an applied force and directed oppositely, that is . The reactions leave former for other idlers.

Figure 3: Model of one-idler belt scale accepted for consideartion. Belt is as flrxible string. Weigh idler is given by applied force

. Now we shall change our model closer to a severe reality, having supposed, that the weigh idler

has a final rigidity/stiffness, i.e. it has ability to be deflected under an action of the applied loading as shown on fig.4. It is accepted as quite justified, that the law of this deflection is presented by known Hooke's law: $\displaystyle N_B=\vert Q\vert=ch,$

where - idler reaction; - idler's stiffness factor , ; - applied force. There is opposite direction of reaction/response force to applied force .

Figure 4: Improved model of one-idler belt scale more close to real. Belt is as flrxible string. Weigh idler has deflected down under applied force

It is easy to guess, that:

(i) if there would be no string between an applied force and a leg , the response was the same as well as in (1);
(ii) in view of presence the stretched string between idler and an applied force , the response shall be a little bit less;
(iii) whereas the string has no a transversal rigidity, and force just is transversal for it, so presence of force on a section does not influence the next sections at all .

The item (i) is obvious and, I assume, it does not demand additional explanations. The item (ii) demands clearing up as far as loading perceived by an idler will decrease. This case will be surveyed some later in more detail.

Here we shall explain item (iii). To consider a section loaded by in a deformable point and next adjacent section . Ours first interest is: whether the response after application of force to a point will be varying? Let's argue from an inverse. We shall put, that the stretched string all the same has transversal stiffness, but it means, that the strain of a string on a idler will look as shown on fig.5! Or it goes approximately so. In this case a response on an idler will be appear unconditionally.

Figure 5: If would the belt was given some trnasverse stiffness it should be cause apperaing of response force

of adjacent idler

But, sirs! We had been commited to agree right recently that we're considering a string as essence not possessing even a hint on a transversal rigidity. From here the strain at system or belt scale controller should look as shown in fig.5 . Even assuming, the magnitude of a tension $\sigma$ has a little varied, it in any the image won't call any additional gains in an idler as one is capable to perceive only vertically directed transversal loading. Isn't it remind something to you? Certainly, it reminds us assumptions when conveyor belt was considered as fulcrum-joint supported beam in (). That is, if we have accepted, that the string has no transversal rigidity -- it in the total has formed the foundation to suppose, that vertical loadings affected perpendicularly to a line of a string in span does not influence the next spans.

This page is from www.beltweigher.org,author is Vladimir Sin