The belt tension influence of belt weigher

This page is about the belt tension influence of belt weigher . Let's proceed out an equilibrium equation. An algebraic sum of projections of all system forces on ordinates axis should be equal to null

$\displaystyle \sum F_{ky}=0,$

$\displaystyle \dfrac{N_B}{2}+N_A-\dfrac{Q}{2}=0,$
$\displaystyle \dfrac{Q}{2}=\dfrac{N_B}{2}+N_A$

The equation (5) means, that the response of a weigh idler is less than applied force on magnitude = $\Delta Q$ , where $\Delta Q$ --the magnitude of the force has been "eaten" by a string tension or an error of a load transfer to the deflected idler .

It is visible on fig.10, that a belt scale system of forces $N_A,\sigma,\sigma+\Delta\sigma$ forms a right triangle similar to a right triangle $AB\lq B$ . Proceeding from here it is easy to make similitude relationship:

$\displaystyle \dfrac{N_A}{\sigma}=dfrac{\Delta y}{l}$

From here

$\displaystyle \dfrac{\Delta Q}{\sigma}=\dfrac{N_A}{\sigma}=dfrac{\Delta y}{l}; \quad\qquad \Delta Q=\sigma \dfrac{\Delta y}{l}.$

Having substituted (7) in (5) after simple transformations it has been finally obtained

$\displaystyle N_B=Q-2\sigma\dfrac{\Delta y}{l}$

The formula 8 is key and displays as far as the response of a weigh idler can vary being caused by its deflection.

The force can be presented as where - a material span loading. Force $N_B=q_{BW} l$ , where $q_{BW}$ is that that display us . Magnitude $\Delta =-2\sigma\dfrac{\Delta y}{l}$ represents absolute belt scale error caused by a belt tension.

In view of above said, we shall copy (8)

$\displaystyle q_{BW}l=ql-2\sigma\dfrac{\Delta y}{l}$

$\displaystyle q_{BW}=q-2\sigma\dfrac{\Delta y}{l^2}.$

Expression (10) looks already enough for making expression of belt weigher error estimation, caused by both a belt tension and a deflection of weigh idler.

Relative belt weigher's error is presented an expression

$\displaystyle \delta=\dfrac{q_{BW}-q}{q}100,\%.$

Having substituted in (11) (10) we shall receive in the total expression :

$\displaystyle \delta=\dfrac{2\sigma\Delta y}{ql^2}100,$

The formula (12) has been obtained for the first time by B.A.Kuznetsov. His work has been published in 1954 and it is classical notion now. Having used (12) it is possible to estimate both errors caused either a deflection of a weigh idler(s) or an inaccuracy of its installation.

Basically under the given formula it is possible to understand a trend of belt weigher development in the second half of 20 centuries when manufacturers have gone on paths of weighing idlers` stiffness increasing (at the moment of the publication of Kuznetsov article majority of belt scales had rather "soft" idlers with deflection under loading up to several millimetres). Now all belt weighers have very rigid weigh idlers (something about 0.1-0.5 mm deflection and even less under maximum loading). However and unfortunately, such approach has not yet resolved completely all problems of belt weigher accuracy matched to weighing instrument measuring standards . Moreover, such rigidity of a weigh idler demands belt weigher installing tolerance requirements bared for be done in hazardous site conditions. Recalling the trigonometry.

by Vladimir Sin www.beltweigher.org