On choosing the number of sensors when measuring the length of moving objects
Abstract Generated abstract
This paper addresses the problem of contactless length measurement for moving objects whose speed may vary and for which equipment reliability requires limiting the number of sensors. It analyzes a ruler-type sensor arrangement, first deriving the required sensor count when the object length is bounded between lower and upper limits, then showing how additional starting sensors can divide the measurement range more efficiently. A general expression is obtained for the number of sensors as a function of the subdivision parameter, and minimization gives an optimal count of \(2\sqrt{(L-l)/\delta}+1\). The example considered shows that this optimized arrangement can reduce the number of sensors from 302 to 36 for the stated measurement range and accuracy.
Full Text
ON CHOOSING THE NUMBER OF SENSORS IN MEASURING THE LENGTH OF MOVING OBJECTS
B. N. EPIFANTSEV
(Presented by the scientific seminar of the automation sector of the Research Institute of Electrical Measurements)
The intensive introduction of computing technology into the control of production processes has brought to the fore a number of problems of a purely applied nature. One of them is the contactless measurement of the length of moving objects under conditions where the speed of motion may vary over wide limits and where reliability requirements impose strict conditions on the amount of equipment required for this purpose.
In the general case, the measurement may be carried out by the “ruler method,” whose graduation is determined by the required accuracy \(\delta\). Then the number of sensors is found from the formula:
\[ m=\frac{L}{\delta}+1, \tag{1} \]
where \(L\) is the maximum length of the product.
In reality, the size of the object is bounded from below by the value \(l\). This makes it possible to reduce the number of sensors to
\[ m=\frac{L-l}{\delta}+2. \tag{2} \]
Fig. 1 explains (2). The sensor \(D_{\text{н}}\) records the beginning of the appearance of object \(O\) and determines the instant of time at which the decision device \(RU\) analyzes the state of the sensors \(D_1, D_2, \ldots, D_k\). The number \(j\) of the open sensor \(D_j\), nearest to the closed sensor \(D_i\), corresponds unambiguously to the measured length.
The measuring system considered is simple; however, as the difference \(L-l\) increases, the number \(m\) becomes catastrophically large, thereby determining the limited range of its application. Thus, for example, with \(L=1800\ \text{cm}\), \(l=1200\ \text{cm}\), \(\delta=2\ \text{cm}\), we have \(m=302\) pcs.
The method discussed can be improved by introducing two starting sensors, as shown in Fig. 2. Then, with the aid of the system \(D_{2\text{н}}, D_1, D_2, \ldots, D_k\), it is possible to determine the size of objects within the limits \(l \div (l+\Delta l)\), and with the aid of \(D_1, D_2, \ldots, D_k, D_{1\text{н}}\), within the limits \(L \div (L-\Delta l)\). By applying an analogous technique several times, one can ultimately arrive at the scheme in Fig. 3, in which the number of sensors is found from formula (2).
Analyzing Figs. 1, 2, 3, it is easy to give a general expression for the required number of sensors:
\[ m=\frac{L-l}{\kappa\delta}+1+\kappa. \tag{3} \]
Fig. 1
Fig. 2
Fig. 3
Function (3) has a minimum; therefore, differentiating with respect to \(\kappa\) and setting the derivative equal to zero, we obtain:
\[ \kappa = \sqrt{\frac{L-l}{\delta}}, \]
\[ m_{\min}=2\sqrt{\frac{L-l}{\delta}}+1. \tag{4} \]
Thus, for the example given above, by formula (4) we have \(m=36\), which is almost an order of magnitude smaller than the number of sensors given by formula (2).