On the Choice of the Weight Magnitude and Plumb Wire Diameter in Mine Orientation
Abstract Generated abstract
This paper addresses the selection of plumb line weight and wire diameter for mine orientation, a problem affecting orientation accuracy and lacking definite practical guidance. It develops a calculation method that combines errors from air current pressure in the shaft and elastic forces in the wire, deriving formulas for class P and class B wires and identifying the wire diameter at which the expected projection error is minimized. The paper shows that increasing wire diameter and weight improves accuracy only up to a limit, after which elastic effects increase the error, and it proposes nomograms to simplify determination of optimal diameters and corresponding loads for given shaft conditions.
Full Text
On the Selection of the Weight Magnitude and Wire Diameter of Plumb Lines in Mine Orientation
G. F. Lysov
(Presented by the scientific seminar of the Department of Mine Surveying and Geodesy)
The selection of the magnitude of the weight and the diameter of the wire of plumb lines in mine orientation is of very great importance, since the correct solution of the problem of orientation depends on it.
Meanwhile, this question has not been sufficiently investigated. Prof. D. N. Ogloblin [1] notes that “there are no definite instructions on this question either here or abroad.” In another work [2] a table is proposed by means of which it is possible to find the minimum weights for mine orientation. However, because this table was compiled without taking into account certain factors that cause errors in orientation by mine plumb lines, it can be used only for solving the question of the degree of influence of the velocity of air movement in the mine shaft on the accuracy of orientation.
The present article contains the development of a method for selecting the magnitude of the weight and the diameter of the wire of plumb lines that ensure the solution of the orientation problem with minimum error.
According to modern concepts, the expected error of orientation by mine plumb lines can be calculated by the formula:
\[ \Theta = \pm \sqrt{\Theta_1^2 + \Theta_2^2}, \tag{1} \]
where \(\Theta_1\) is the expected orientation error depending on the velocity of motion of the air current in the mine shaft; \(\Theta_2\) is the expected additional orientation error caused by the influence of the elastic forces of the plumb-line wires.
The orientation error \(\Theta_1\) (in sec), as is known [2], can be determined by the formula:
\[ \Theta_1 = \frac{2\rho'' phH}{cQ}, \tag{2} \]
where \(p\) is the pressure of the air current, kg/m; \(h\) is the length of the lower part of the plumb line subjected to air pressure, m; \(H\) is the length of the entire plumb line (shaft depth), m; \(c\) is the distance between plumb lines, m; \(Q\) is the magnitude of the acting weight, kg; \(\rho'' = 206265''\).
Our investigations have established that the value of the pressure of the air flow \(p\) (in \(g/m\)) can be represented by an empirical formula of the following form:
\[ p = 0.0947\, d v^{1.714}, \tag{3} \]
where \(v\) is the velocity of air motion, \(m/sec\), and \(d\) is the diameter of the plumb-line wire, \(mm\).
The value of the maximum permissible weight of the load \(\Theta\) (in \(kg\)), equal to 60% of the breaking strength of the wire, can be computed from the formulas:
a) for wire of class “P”:
\[ Q = 91.45\, d^2 \tag{4} \]
b) for wire of class “B”:
\[ Q = 111.7\, d^2. \tag{5} \]
Substituting the values of \(p\) and \(Q\) from formulas (3), (4), and (5) into expression (2), we obtain:
a) when projecting with plumb lines made of class “P” wire:
\[ \Theta_1 = \pm \frac{E_{\mathrm{p}}}{d} \tag{6} \]
b) when projecting with plumb lines made of class “B” wire:
\[ \Theta_1 = \pm \frac{E_{\mathrm{v}}}{d}, \tag{7} \]
where
\[ E_{\mathrm{p}} = \frac{0.428\, h H v^{1.714}}{c}\; sec \cdot mm, \tag{8} \]
\[ E_{\mathrm{v}} = \frac{0.350\, h H v^{1.714}}{c}\; sec \cdot mm. \tag{9} \]
The expected additional projection error due to the influence of the elastic forces of the wire, \(\Theta_2\), in seconds, can be computed [3] by the formula
\[ \Theta_2 = \pm \frac{3 \cdot 6.28 \cdot \rho'' d^4}{64 \cdot R c Q}; \tag{10} \]
where \(R\) is the radius of curvature of the unloaded wire (the radius of the winch drum), in \(cm\).
Substituting into formula (10) the values of \(Q\) from expressions (4) and (5), we shall have:
a) when projecting with plumb lines made of class “P” wire:
\[ \Theta_2 = \pm \alpha_{\mathrm{p}} d^2; \tag{11} \]
b) when projecting with plumb lines made of class “B” wire:
\[ \Theta_2 = \pm \alpha_{\mathrm{v}} d^2; \tag{12} \]
where
\[ a_{\mathrm{p}}=\frac{665}{Rc}\ \text{sec}/\text{mm}^2, \tag{13} \]
\[ a_{\mathrm{v}}=\frac{545}{Rc}\ \text{sec}/\text{mm}^2. \tag{14} \]
Substituting the found values of \(\Theta_1\) and \(\Theta_2\) into formula (1), we shall have:
\[ \Theta=\sqrt{\frac{E_{\mathrm{p(v)}}^{2}}{d^{2}}+a_{\mathrm{p(v)}}^{2}\cdot d^{4}}. \tag{15} \]
Considering formula (15), it may be said that the dependence between the magnitude \(\Theta\) and the diameter of the plumb-line wire \(d\) will be expressed by a certain parabolic curve. If one plots graphically the variation of \(\Theta\), \(\Theta_1\), and \(\Theta_2\) as functions of the change in wire diameter (in Fig. 1 this dependence is shown for \(H=200\ \text{m}\), \(h=5\ \text{m}\), \(v=0.7\ \text{m/sec}\), \(c=4\ \text{m}\), \(R=17.5\ \text{cm}\)), it is easy to see that increasing the wire diameter \(d\) (and consequently the weight \(Q\)) in order to improve the accuracy of projection is advantageous only up to a certain limit, since increasing the wire diameter, while reducing the projection error \(\Theta_1\), at the same time leads to an increase in the additional projection error \(\Theta_2\).
Let us use formula (15) and find that value of the diameter of the plumb-line wire for which the magnitude \(\Theta\) will be minimal. Applying the known rule, we obtain:
\[ 2a_{\mathrm{p(v)}}^{2}\cdot d^{6}-E_{\mathrm{p(v)}}^{2}=0. \]
Whence
\[ d_0=\sqrt[3]{\frac{E_{\mathrm{p(v)}}}{a_{\mathrm{p(v)}}\sqrt{2}}}. \tag{16} \]
Formula (16) gives the value of the diameter of the plumb-line wire \(d_0\) (and of the corresponding maximum permissible weight \(Q_0\)) at which the projection error will be minimal. The use, in orientation, of plumb lines made of wire of larger or smaller diameters (in comparison with the wire diameter \(d_0\) found from formula (16)) will lead to an increase in the projection error and will reduce the overall accuracy of orientation. It is interesting to note that the minimum value \(\Theta_0\) does not correspond to the point of intersection of the curves \(\Theta_1\) and \(\Theta_2\) (Fig. 1), but is always shifted somewhat to the left of this point, approximately by the amount \(0.2d_{\mathrm{p}}\), where \(d_{\mathrm{p}}\) is the diameter of the plumb-line wire corresponding to the point of intersection of the curves \(\Theta_1\) and \(\Theta_2\).
To facilitate the calculation of the quantities \(E_{\mathrm{p}}\), \(E_{\mathrm{v}}\), \(a_{\mathrm{p}}\), \(a_{\mathrm{v}}\) from the above formulas, two nomograms have been compiled, by means of which it is possible, quite simply and with sufficient accuracy, to find the values of the indicated quantities.
Thus, the quantities \(E_{\mathrm{p}}\) and \(E_{\mathrm{v}}\) are found from the nomogram (Fig. 2) as functions of the given \(H\), \(h\), \(v\), and \(c\). The quantities \(a_{\mathrm{p}}\) and \(a_{\mathrm{v}}\) can be determined from the nomogram (Fig. 3) as functions of the adopted \(c\) and \(R\).
It should be noted here that the quantities \(E_{\mathrm{p}}\) and \(E_{\mathrm{v}}\) can also be found from the given nomogram in those cases when the plumb-line wire is enclosed in pipes [3]. In this case, \(h\) should be taken as the segment of wire enclosed between the centering plate and the point of suspension of the weight. For example, when \(h\) is equal to \(70\ \text{cm}\), it is necessary to find from the nomogram (Fig. 2) \(E_{\mathrm{p}}\) or \(E_{\mathrm{v}}\) for \(h\) equal to \(7\ \text{m}\), and to reduce the found value of \(E_{\mathrm{p}}\) or \(E_{\mathrm{v}}\) by a factor of 10.
The use of the proposed nomograms considerably simplifies the process of determining the optimal value of the diameter of the plumb-line wire—
…under which orientation can be carried out with the minimum error.
Example: It is required to determine the weight magnitude and the wire diameter of plumb lines that ensure solution of the orientation problem with minimum error for the following specific conditions: \(H = 200\ \text{m}\), \(h = 5\ \text{m}\), \(v = 0.7\ \text{m/sec}\), \(c = 4\ \text{m}\), \(R = 17.5\ \text{cm}\); projection will be carried out with plumb lines made of class “P” wire, without shielding.
Fig. 1.
Solution: 1. From the nomogram (Fig. 2), we find \(E_{\text{p}}\) as a function of the given \(H\), \(h\), \(v\), and \(c\): \(E_{\text{p}} = 58\ \text{sec}\cdot\text{mm}\).
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From the nomogram (Fig. 3), we find \(\sigma_{\text{p}}\) as a function of the given \(c\) and \(R\): \(\sigma_{\text{p}} = 9.5\ \text{sec}/\text{mm}^2\).
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Using formula (16), we calculate the desired value of the plumb-line wire diameter \(d_0\):
\[ d_0 = \sqrt[3]{\frac{58}{9.5\sqrt{2}}} = 1.6\ \text{mm} \]
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From the table given in [4], we find the value of the weights for these plumb lines, that is, \(Q_0 = 230\ \text{kg}\).
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Using formula (15), we calculate the expected projection error \(\Theta_{\min}\):
\[ \Theta_{\min} = \pm \sqrt{1250 + 610} = \pm 43'' \]
Conclusions
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It has been established that increasing the weight magnitude and the wire diameter of plumb lines in mine orientation is advantageous only up to a certain limit.
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Formulas have been determined for finding the wire diameter of plumb lines at which the projection problem is solved with minimum error.
Fig. 2.
Left scale: coefficient \(E_p\) for guide ropes, sec·m; coefficient \(E_b\) for wire of class \(B\), sec·m.
Scale \(C\): distance between plumbs, m.
Scale II: air-flow velocity, m/sec.
Scale \(V\): frequency of the lower part of the plumb, min.
Scale \(h\): length of the lower part of the plumb, m.
Scale \(I\): diameter of the plumb, m.
Scale \(H\): length of the plumb, m.
Usage scheme: \(m, E, C, II, V, h, I, H\).
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A method has been developed for calculating the wire diameters of plumb lines for which the design problem is solved with minimum error for any specific orientation conditions.
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Nomograms are proposed that simplify the process of determining the optimal wire diameter of plumb lines.
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The proposed method for finding the optimal values of the wire diameter and load of plumb lines requires experimental verification under production conditions.
Fig. 3.
LITERATURE
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Ogloblin D. N. Orientation of an underground survey through one vertical shaft. ONTI, 1938.
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Ogloblin D. N. Mine-surveying work in the underground development of mineral deposits. Metallurgizdat, Part III, 1953.
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Filatov S. A. Experience in carrying out mine orientation. Ugletekhizdat, 1958.
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Technical instruction for carrying out mine-surveying work. Ugletekhizdat, 1959.