ON THE QUESTION OF MEASURING THE DEPTH OF A MINE SHAFT WITH A WIRE

(Abridged presentation)

A. I. VOLKOV

Introduction

Measuring the depth of a mine shaft is part of the complex of major mine-surveying operations. By means of this task, the elevation coordinate is transferred through a vertical shaft into the mine. The elevation coordinate is of great importance both in the matter of the spatial representation of a deposit and in solving certain responsible mine-surveying tasks, above all in carrying out breakthroughs. In order to ensure the appropriate accuracy in determining the elevation coordinate, the work of measuring the depth of a mine shaft is carried out with allowance for a number of factors that affect the accuracy of solving this task.

At present, the depth of a mine shaft is measured by the following methods: 1) with a long steel tape, 2) with a short steel tape, 3) with a steel wire.

Of the indicated methods, the first provides the highest accuracy, is simple to perform, and is the most productive. Its relatively limited use in practice is explained by the necessity of having special mine tapes up to 1000 m long. In second place in terms of accuracy, simplicity of execution, and productivity stands the third method. This method, as one that provides the required accuracy, sufficiently high productivity, and does not require special apparatus, is almost the only method applied at present. In last place in terms of accuracy, productivity, and complexity of execution stands the second method. For the reasons indicated above, this method is almost never applied in practice.

The technique for carrying out field work for each of the indicated methods has been developed sufficiently well; however, with respect to processing the results of measuring depth by wire, the situation is far from favorable. This unfavorable state of affairs attracts attention even upon a cursory review of the literature on this question. Indeed, the Technical Instructions for the Production of Mine-Surveying Work and Prof. D. N. Ogloblin’s course Mine-Surveying Survey of Mine Workings require that the following corrections be introduced into the measured length of the wire: a) for the stretching of the wire due to its own weight¹, b) for the temperature of the wire, and c) for comparison.

According to the Mine-Surveying Forms for the Production of Connecting Surveys, published in No. 11 of Transactions of TsNIMB, the measured length of the wire must be corrected for: a) stretching of the wire due to its own weight, b) temperature of the wire, c) temperature of the tape, and d) comparison.

¹ The formula for calculating this correction is not given in the instructions.

A more detailed study of this question in the present work made it possible to establish a number of shortcomings in the calculation of temperature corrections and to reject entirely the formula recommended by the above-mentioned literature for calculating the correction for the elongation of the wire under the action of its own weight.

The aim of the present work is to study the sources of errors affecting the result of measuring the depth of a mine shaft with a wire, and to indicate ways of eliminating or mitigating this influence.

Sources of Errors in Measuring the Depth of a Mine Shaft with a Wire

The principal disadvantage of the method of measuring the depth of a mine shaft with a wire is that the separate measuring operations are carried out non-simultaneously and under different states of the wire. In confirmation of this it is sufficient to point out that the marks are made on the wire when it is in a plumb position, at a temperature corresponding to the air temperature of the mine shaft, and under a tension in which, in addition to the load, the wire’s own weight acts; whereas the measurement is made in parts, at the surface, with the wire in a horizontal position and, consequently, at another temperature and tension.

A more detailed study of this question makes it possible to establish the following sources of errors¹ separately for the first and second methods of measuring the depth of a mine shaft with a wire.²

For the first method of measurement these sources will be due to:

1) an incorrect estimate of the temperature of the wire in the mine shaft and at the surface;

2) the temperature difference between the wire and the tape when measuring individual sections of the wire;

3) the action of the wire’s own weight;

4) additional elongation (with time) of the wire after the marks have been made (from the action of the load and its own weight);

5) friction in the pulleys;

6) longitudinal oscillations of the wire;

7) the change of the force of gravity with depth;

8) inaccurate knowledge of the length of the tape (comparator).

For the second method of measurement, all the above-listed sources of error of the first method remain. In addition, here we have the following additional sources:

1) due to the difference in the sags of the wire in the section \(D — T\) at the beginning and at the end of the measurement (Fig. 2),

2) due to displacement of the comparator tape in the process of measuring individual sections of the wire.

In considering the question of the sources of errors, it was assumed that the measurement is carried out with observance of the basic conditions, which are as follows: a) during measurement the wire is in a horizontal position; b) the tension of the tape is equal to its tension during comparison; c) during measurement the tape occupies a position analogous to that which it occupied during comparison (horizontal or with free sag).

When the indicated conditions are not met, additional sources of errors arise, which are taken into account by known formulas.

Error Due to an Incorrect Estimate of the Temperature of the Wire in the Mine Shaft and at the Surface

Determination of the Time Required for the Wire to Acquire the Temperature of the Surrounding Air

To perform various measuring operations, the wire is transferred from one temperature condition to another. Thus, for making marks it is lowered into the mine, having the surface temperature, and for measuring the length between marks it is raised to the surface, having the temperature of the air in the mine shaft. Entering different temperature conditions, the wire gradually changes its temperature and, over time, acquires the temperature of the surrounding air. It is at this moment that its measurement or the making of marks should be performed, as well as the fixing of the end of measurements in the second method of measurement. Premature performance of one or another measuring operation leads to distortion of the result.

Considering the temperature regime of the wire in the process of measuring depth, Prof. I. M. Bakhurin wrote in his work Questions of Mine Surveying (a supplement to the course Mine Surveying): “… moreover, it is not easy to take this temperature change into account, since the wire does not immediately, but with some delay, assume the temperature of the surrounding medium.” In later works by other authors considering the measurement of mine depth, we likewise find no answer to the question of how to determine the time during which equality of the temperatures of the wire and the surrounding air is reached.

Using data from heat engineering, the calculation of the time during which equality of the temperatures of the wire and the surrounding air is reached can be performed by the following formulas:

When the wire is heated,

\[ t^h=\frac{0.576\,dc\gamma}{\alpha}\cdot \lg \frac{t_b^0-t_0^0}{t_b^0-t_h^0}. \tag{1} \]

When the wire is cooled,

\[ t^h=\frac{0.576\,dc\gamma}{\alpha}\cdot \lg \frac{t_0^0-t_b}{t_h^0-t_b^0}, \tag{2} \]

where
\(\alpha\) — coefficient of heat transfer from air to metal;
\(c\) — heat capacity of the metal;
\(\gamma\) — specific weight of the material;
\(t_b^0\) — temperature of the air surrounding the wire during measurement;
\(t_h^0\) — temperature of the wire after a certain interval of time \(t_h\) after it has been raised to the surface;
\(t_0^0\) — initial temperature of the wire;
\(d\) — diameter of the wire.

Figure 1 gives curves characterizing the time during which heating (or cooling) of the wire occurs at different differences between the temperatures of the wire and the surrounding air. From these curves it follows that heating (cooling) at a temperature difference of \(20^\circ\)

takes place over approximately 15 min, decreasing to 10 min when the temperature difference is 5°.

The theoretical considerations presented concerning the time of heating (cooling) of the wire show that neglecting this circumstance may lead to serious distortion of the result.

Graph of the heating time of a steel wire \(d = 1.5\) mm for a given temperature difference \(t_b^\circ - t_0^\circ\)

Fig. 1

In constructing the curves, the following values of the quantities entering into the formulas were adopted: \(d = 0.0015\) m, \(a = 9 \dfrac{\mathrm{cal}}{\mathrm{m}^2\,\mathrm{h}\,^\circ}\) at an air-current velocity of 1 m/sec, \(c = 0.13 \dfrac{\mathrm{cal}}{\mathrm{kg}\,^\circ}\), \(\gamma = 7800 \dfrac{\mathrm{kg}}{\mathrm{m}^3}\).

Determination of the Mean Temperature of the Wire in the Shaft and at the Surface

From the graphs given below of the temperature regime of the air in the mine shaft (Figs. 3–7), it is seen that the air temperature in the upper part of the shaft does not remain constant, but changes with the change in horizon. Thus, the question of determining the temperature of the wire in the mine shaft is a complex one, and an incorrect solution of it may lead to a significant distortion of the result.

At present, it is recommended that the temperature of the wire in the mine shaft be taken as the mean value from its measurements at the mouth of the mine shaft

and at the level of the mine yard, or at the level of the mine yard and in the region of the upper mark (point “a,” Fig. 2). Thus, this method of determining the mean temperature assumes that the change in the temperature

Fig. 2

of the wire from the upper mark to the level of the mine yard occurs according to a straight-line law.

Graph
of the temperature regime of the air in the shaft of borehole 116is
mine 5/7 of the “Anzherougol” trust

Temperature of the air at the surface horizon above the mouth of the shaft: \(+26^\circ\).

The shaft lining is wooden.

Direction of the air stream: descending, velocity \(1.4\ \text{m/sec}\).

Conventional designations:

Symbol	Meaning
horizontal hatching	dry part of the shaft
cross-hatching	zone of weak water seepage
diagonal hatching	zone of strong water seepage

Fig. 3

In order to ascertain the actual pattern of the change in the temperature of the wire on the segment “upper mark—mine yard,” in August 1945 we carried out special observations at the mines of the Anzherougol trust to study the temperature regime of the air in a mine shaft.

From the graphs in Figs. 3–7, constructed from the results of the observations indicated above, it follows that this change is, in the general case, subject to a law represented by a curve. From the same graphs it is evident that the principal factors affecting the temperature regime of the air in a mine shaft are the following:

1) the difference between the air temperatures in the mine and at the surface,
2) the direction of the air current,
3) the velocity of the air current,
4) the presence of dripping water in the shaft.

The factors listed affect the temperature regime of the air in the mine shaft in different ways. Some of them flatten the temperature-regime curve, while others, on the contrary, increase its contrast. A high

[Graph]
of the temperature regime of the air in the shaft of No. 116 bis prospecting pit, Anzherougol trust

Fig. 4

velocity of the air current, its descending direction, and the absence of dripping water flatten the temperature curve into a straight line (Fig. 7). An insignificant velocity of the air current, its ascending direction, and a high air temperature above the shaft mouth intensify the curve, as is seen from a comparison of the graphs (Figs. 3 and 4). The presence of dripping water sharply lowers the temperature and, consequently, increases the contrast of the curve for the distribution of air temperatures in the mine shaft (Figs. 5 and 6).

The investigations carried out show that the mean temperature of the air in the mine shaft should be taken not as the arithmetic mean from observations at two points, but as the weighted mean, by lengths, from observations of the air temperature at a number of points, according to the formula

\[ t_c=\frac{t_1L_1+t_2L_2+\ldots+t_nL_n}{L_1+L_2+\ldots L_n}; \tag{3} \]

where the values \(t_1, t_2,\ldots,t_n\) are the mean temperature of the wire on the sections \(L_1, L_2,\ldots,L_n\), so that

\[ t_1=\frac{t^\circ+t'}{2},\qquad t_2=\frac{t'+t''}{2}, \]

and so on (Fig. 8). \(L_1, L_2,\ldots L_n\) are chosen on the basis of a calculation such that the temperature variation within each of these sections is close to a straight-line law.

Graph
of the temperature regime of the air in the timbered shaft
of mine 9/15 of the “Anzherugol” trust

Fig. 3

For comparison of the results of calculating the mean temperature of the wire in the mine shaft by the method currently accepted and by formula (3), let us use the data of the graph in Fig. 3. On the basis of these data we obtain:

a) by the existing method, \(t_c=17.8^\circ\),

b) by formula (3), for a depth of 43 m, \(t_c=12.4^\circ\),

which constitutes a difference in the mean wire temperatures in the initial ...

position is \(5.4^\circ\). At a depth of 200 m, under the condition that lower down the temperature remains equal to \(10.5^\circ\), this difference reaches \(7^\circ\), which corresponds—

Fig. 6.

—to a difference in the corrections to the measured length of the wire of 1.7 cm. From the example given it follows that the accuracy of the result of measuring the depth of a mine shaft depends to a considerable extent on the accuracy with which the mean temperature of the wire in its initial position is determined.

Fig. 7.

Formula (3) makes it possible to calculate the mean temperature of the wire in its initial position (in the shaft) with any desired degree of accuracy. This is achieved by increasing the number of points at which the air temperature in the mine shaft is measured.

Fig. 8

The measurement of the length of the wire between the marks on the comparator is, as is known, carried out in parts. It follows from this that the temperature of the individual sections during measurement may be different owing to fluctuations in the air temperature.

The mean temperature of the wire when it is measured on the comparator can be calculated by the formula

\[ t_u=\frac{l_1t_1+l_2t_2+\cdots l_nt_n}{l_1+l_2+\cdots l_n}, \tag{4} \]

where \(t_1, t_2,\ldots t_n\) are the temperatures of the individual sections \(l_1, l_2, l_3,\ldots l_n\) at the moment of their measurement on the comparator.

Since usually \(l_1 \simeq l_2 \simeq \cdots l_n=l\), the mean temperature of the wire at the moment of measurement can be calculated by the formula

\[ t_u=\frac{t_1+t_2+\cdots t_n}{n}. \tag{5} \]

Calculation of temperature corrections

Knowing the mean temperature of the wire in the shaft and during its measurement on the comparator, one can calculate the correction to the measured length of the wire for the change in temperature by the formula

\[ \Delta L_{t_n}=Lk_1\,(t_c-t_u), \tag{6} \]

where \(L\) is the total measured length of the wire,
\(k_1\) is the linear coefficient of expansion of the wire material.

The length of the wire on the comparator is measured at the mean temperature \(t_u\), whereas the length of the comparator tape may be known at another temperature, for example \(t_0\). The temperature difference thus formed constitutes a new source of error in measuring the depth of the mine shaft with a wire. This error can be taken into account by the formula

\[ \Delta L_{t_\ell}=Lk_2\,(t_u-t_0), \tag{7} \]

where \(k_2\) is the linear coefficient of expansion of the tape material.

The total temperature correction to the measured length of the wire is equal to the sum of the corrections for the temperature of the wire and the tape, i.e.

\[ \Delta L_t=\Delta L_{t_n}+\Delta L_{t_\ell}. \]

Since, in measuring the depth of a mine shaft, a steel wire is used, the linear coefficient of expansion of the wire and of the tape is taken to be the same, i.e.,

\[ k_1 = k_2 = k. \]

Under these conditions, the total temperature correction to the measured length of the wire can be calculated by the formula

\[ \Delta L_t = Lk(t_c - t_0). \tag{8} \]

It follows from formula (8) that if the linear coefficient of expansion of the material of the wire and of the tape is the same, then the temperature \(t_u\), at which the wire is measured on the comparator, has no effect on the determination of the depth of the shaft. In formulas (6) and (7), in this case, i.e. when \(k_1 = k_2\), any arbitrary value of \(t_u\) may be substituted.

The accuracy of determining the total temperature correction to the measured length of the wire by formula (8) depends mainly on the accuracy with which \(t_0\) is determined.

Error due to the action of the wire’s own weight

Analysis of the formula currently used for computing \(\Delta L_q\)

Regarding the influence of the wire’s own weight \(q\), Prof. I. M. Bakhurin wrote in the work Problems of Mine-Surveying Art (supplement to the course Mine-Surveying Art): “As the load \(P\) is raised (the counterweight), the tension of the sections of wire that have passed over the pulley \(D\) (Fig. 2) will decrease, and the wire will decrease its length in comparison with that which it had in the vertical position \(DP\). The magnitude of the error arising from this cause is subject to study and is not yet sufficiently known.”

In the Mine-Surveying Formularies for the Production of Connecting Surveys published in 1940 (Izvestiya TsNIMB, No. 11), and in the above-mentioned course by Prof. D. N. Ogloblin, it is proposed that this change in the length of the wire be taken into account by the following formula:

\[ \Delta L_{q.m} = \frac{\gamma}{20E} L^2, \tag{9} \]

where \(L\) is the measured length of the wire,
\(\gamma\) is the specific weight of the wire material,
\(E\) is the modulus of elasticity of the wire material.

The analysis presented below shows that formula (9) does not express the true law of change of the length of the wire under the influence of its own weight \(q\) during its measurement and gives a significantly exaggerated value of the correction. To verify the conclusion reached, let us analyze formula (9). The elongation of an elementary segment of wire \(dx\) (Fig. 9) under the action of the part of length \(x\) suspended below it is expressed, according to Hooke’s law, as follows:

\[ \Delta(dx) = \frac{\gamma x dx}{E}, \]

whence the total elongation \(\Delta L_q\) of the wire under the action of its own weight \(q\) is equal to

\[ \Delta L_q = \frac{\gamma}{E} \int_0^L x dx = \frac{\gamma}{E}\frac{L^2}{2}. \]

In the equation obtained, the quantities entering it are expressed in the CGS system. Taking \(E\) in \(\mathrm{kg}/\mathrm{cm}^2\), and linear quantities in m, we shall have

\[ \Delta L_{q,n}=\frac{\gamma}{20E}L^2. \]

From the derivation of the formula it follows that its use for calculating the correction for the stretching of the wire under the action of its own weight is possible only in the case when the measured segment \(L\), which is in a vertical position when the marks are made, is first (before measurement) transferred to a horizontal position, and then measured. In this case the own weight \(d\) of the measured segment of wire \(L\) will cease to act, and it will shorten by the amount \(\Delta L_q\), computed by formula (9). However, as is known, such a method of measuring the length of a wire is not used in practice.

Fig. 9            Fig. 10

The measurement of the wire, as indicated above, is carried out in parts, by successively raising or lowering it. The considerations given above concerning the inapplicability of formula (9) for calculating the correction \(\Delta L_q\) when measuring the depth of a mine shaft with a wire can be confirmed in the following way. Suppose that the measurement of the length of the wire between marks \(a\) and \(b\) is made from point \(D\) (Fig. 10). The correction for stretching from the own weight \(q\) of the wire for each separately measured segment \(l\), according to formula (9), is equal to

\[ \Delta l_{q,n}=\frac{\gamma}{20E}l^2. \]

The total correction \(\Delta L_{q.l}\) to the measured length \(L\) is equal to the sum of the separate corrections to the segments \(l\)

\[ \Delta L_{q.l}=\frac{\gamma}{20E}\sum l^2. \]

If, in the measurement, the separate segments \(l\) were equal to one another, then we shall have

\[ \Delta L_{q.l}=\frac{\gamma}{20E}\,Ll, \tag{10} \]

where \(L=nl\).

The formula (10) obtained is rigorous, however, only for the case shown in Fig. 10, i.e., when the measurement of the separate segments is made from point \(D\) (the point of suspension of the wire).

Comparison of formula (10) with formula (9) shows their sharp difference. This is explained by the circumstance that formula (10) takes into account the method of measuring the length of the wire in parts, whereas formula (9) does not take this into account. This is the principal shortcoming of formula (9).

Derivation of the correction formula \(\Delta L_q\)

In Figs. 11, 12, and 13 schematic diagrams are given of the tensile forces acting on the wire under the influence of various factors. From consideration of the diagrams it follows:

Conventional designations:

• tensile force from the load \(P\);
• tensile force due to the weight of the wire, segment \(AD\);
• tensile force due to friction of the pulley;
• \(f_1, f_2\) — frictional force.

Fig. 11

1) the tensile effect of the load \(P\) is constant along the entire length of the wire and does not depend on the position of the wire: vertical or horizontal (neglecting friction in the pulleys);

* Proceedings of TPI, vol. 67, no. 2.

2) the stretching effect of the wire’s own weight \(q\) is a variable quantity—it changes (within the limits of the given section) from bottom to top (increasing) and, moreover, depends on the position of the wire: in the vertical position it is maximal, while in the horizontal position (without sag) it is equal to zero;

3) any vertically situated section of the wire is stretched by a force equal to: a) the weight of the lower-hanging part of the wire, b) one half of the weight of the wire of the given section;

4) the force with which an inclined section of the wire acts on neighboring sections is determined by the weight of a wire whose length is equal to the difference of the elevations \(h\) of the ends of the inclined section;

5) the stretching of the wire under the action of its own weight \(q\) and of the load \(P\) to the right of point \(D\) (see the graphs) is distorted by friction in the pulleys.

This distortion, random in magnitude and sign, is not amenable to precise accounting.

From the analysis presented it follows that the correction \(\Delta L_q\) depends to a considerable degree on the measurement scheme—the position of the wire to the right of point \(D\).

Figure 14 shows the initial moment in measuring the depth of the shaft by the first method, with the blocks \(D\) and \(T\) located on the same horizon;

Fig. 14

Fig. 15

in Figs. 15 and 16—the moment of measuring the first and the last segment \(l\). In connection with the raising of the wire, it is shortened on the sections \(D\)—I, I—II (the comparator), and II—III; however, the result of measuring \(L\) is affected by the shortening of the wire only on the sections \(D\)—I and I—II. A change in the length of the wire to the right of point II causes equal displacements of the extreme marks without changing the length between them.

The shortening of any segment \(l\) brought into the horizontal position is caused by a change in tension due to the cessation of the action of the wire’s own weight over a length \(s+\dfrac{l}{2}\). This change in tension for the segment \(l\) is equal to

\[ Q=\gamma sF+\gamma \frac{l}{2}F, \]

where \(F\) is the cross-sectional area of the wire.

The shortening \(\Delta l_q\) of the segment \(l\) corresponding to the force \(Q\), according to Hooke’s law, is equal to

\[ \Delta l_q=\frac{\gamma}{E}ls+\frac{\gamma}{2E}l^2. \]

Fig. 16

The total correction \(\Delta L_q\) to the length \(L\) is equal to the sum of the corrections \(\Delta l_q\) to the individual segments \(l\).

\[ \Delta L_q=\frac{\gamma}{2E}(2sL+\Sigma l^2). \tag{11} \]

For \(l_1=l_2=\ldots=l_{n-1}=l,\ l_n=P\), where \(P\) is the remainder smaller than \(l\), we shall have

\[ \Delta L_q=\frac{\gamma}{2E}(2sL+lL'+P^2), \tag{12} \]

where \(L'=L-P\).

For \(P=0\) or \(P=l\), formula (11) will have the form

\[ \Delta L_q=\frac{\gamma}{2E}L(2s+l). \tag{13} \]

When the pulleys \(D\) and \(T\) are located at different levels, the influence of the wire’s own weight on the measurement result will be somewhat different from what occurred when they were located at the same level. In Fig. 12 the pulley \(T\) is located below the pulley \(D\). The change in the tensile force in this case, due to the lifting of the segment \(l\), will be equal to

\[ Q=\gamma F\left(s+h+\frac{l}{2}\right). \]

When the pulley \(T\) is located above the pulley \(D\), \(Q\) is expressed by the formula

\[ Q=\gamma F\left(s-h+\frac{l}{2}\right). \]

Accordingly, formulas (11), (12), (13) will have the form

\[ \Delta L_q=\frac{\gamma}{2E}\left[2L(s\pm h)+\Sigma l^2\right] \tag{14} \]

\[ \Delta L_q=\frac{\gamma}{2E}\left[2L(s\pm h)+L'l+P^2\right] \tag{15} \]

\[ \Delta L_q=\frac{\gamma}{2E}L\left[2(s\pm h)+l\right]. \tag{16} \]

In formulas (14)—(16), the plus sign before \(h\) corresponds to the position of pulley \(D\) above pulley \(T\); the minus sign before \(h\) is taken when pulley \(D\) is located below pulley \(T\).

For \(h=0\), formulas (14)—(16) reduce to formulas (11)—(13). Thus formulas (14)—(16) are general.

The obtained formulas are valid when the quantities entering into them are expressed in the CGS system.

For the case where \(E\) is expressed in \(\mathrm{kg}/\mathrm{cm}^2\), and the lengths in m, we shall have

\[ \Delta L_{q,m}=\frac{\gamma}{20E}\left[2L(s\pm h)+\Sigma l^2\right] \tag{17} \]

\[ \Delta L_{q,m}=\frac{\gamma}{20E}\left[2L(s\pm h)+L'l+P^2\right] \tag{18} \]

\[ \Delta L_{q,m}=\frac{\gamma}{20E}L\left[2(s\pm h)+l\right]. \tag{19} \]

\(^1\) This formula was first obtained by me in 1938.

Formulas (17)—(19) remain valid also for the case of measuring the depth of a mine shaft by the second method, both when lowering the wire into the mine and when raising it to the surface.

An examination of formulas (17)—(19) shows that the correction \(\Delta L_q\) in all practical cases (when \(l_1 \cong l_2 \cong l_3 \cong \ldots l_{n-1}\) and for any value of the remainder \(P\)) is sufficiently calculated by formula (19).

On the degree of distortion of \(\Delta L_q\) calculated by formula (9)

Having proved the fundamental inaccuracy of formula (9), it is necessary to emphasize the inadmissibility of applying this formula from a practical standpoint as well. For this purpose, Table 1 below gives the values of \(\Delta L_q\) calculated by formulas (9) and (19), the difference of the corresponding corrections, and their ratio.

In calculating \(\Delta L_q\) by formula (19), it was assumed that

\[ l = 10\ \text{m}, \]

\[ s \pm h = 5\ \text{m}. \]

Table 1

\(L\), in m	\(\Delta L_q\), in mm — by formula (9)	\(\Delta L_q\), in mm — by formula (19)	Difference \(\Delta L_{q_1} - \Delta L_{q_2}\)	Ratio \(\dfrac{\Delta L_{q_1}}{\Delta L_{q_2}}\)
100	2	0.4	1.6	5
200	8	0.8	7.2	10
300	18	1.2	16.8	15
400	32	1.6	30.4	20
500	50	2.00	48.0	25

As can be seen from Table 1, the correction \(\Delta L_q\) determined by formula (9) is obtained with a distortion by tens of times in comparison with its actual value determined by formula (19). The degree of this distortion is proportional to the depth of the shaft and is equal to the ratio

\[ \frac{L}{2(s \pm h)+l} \tag{20} \]

Error due to the additional (time-dependent) stretching of the wire under the action of the load and its own weight

In the book Questions of Mine-Surveying Practice (supplement to the course Mine Surveying), published in 1936, Prof. I. M. Bakhurin says: “... during measurement, which lasts from an hour to several hours depending on the depth of the shaft, the wire may acquire an additional (time-dependent) elongation beyond that which existed at the time the marks were made.” This additional time-dependent elongation of the wire is one of the sources of error in measuring the depth of a mine shaft.

In the literature on elastic deformation it is stated that the velocity of deformation is equal to the velocity of sound, and that time-dependent elongation, although it does occur, is nevertheless a quantity of the second order, even over a very long interval of time, in comparison with which the time required to measure the depth of a mine shaft is negligible. From these considerations one may come to the conclusion that the error under consideration has no practical significance. However, in view of the fact that the magnitude of this error is influenced not only by the time factor, but also by the length of the wire, which during measurement

since the depth of a mine shaft is usually expressed in hundreds of meters, a final resolution of the question of the influence of elongation over time on the result of measurement can be made only on the basis of experimental data, which is a task for further experiments.

Error Due to Friction in the Pulleys

In the “Technical Instructions for the Production of Mine-Surveying Work” currently in force, and in the special mine-surveying literature, the influence of friction in the pulleys on the accuracy of measuring the depth of a mine shaft with a wire is not indicated. In practice this circumstance leads to the use of arbitrary pulleys, including pulleys of small diameter, with unpolished axles and without the use of lubricant. Rusty pulleys and pulleys with dirty axles, possessing a high coefficient of friction, are also used.

Analysis of the influence of the friction force in the pulleys on the accuracy of measuring the depth of a mine shaft with a wire shows that in some cases it may reach considerable magnitudes and must be classified as a random error. The friction force increases with an increase in the load acting on the pulley and in the coefficient of friction of the pulley. In addition, it depends on the diameter of the pulley and of its axle: it is directly proportional to the diameter of the axle and inversely proportional to the diameter of the pulley.

The error due to friction is the result of deformation of the wire on the section I—II (Fig. 12)—for the first method of depth measurement, and \(D—T\) and \(T—I\)—for the second method of measurement, under the influence of an additional force on the indicated sections, directed toward overcoming the friction force.

The tension of the wire on the sections \(D—T\) and \(T—I\) during the process of measurement does not remain constant, but changes (increasing when the wire is lowered into the mine and decreasing when it is raised to the surface) owing to the change in the weight of the freely hanging wire.

In Fig. 17 the load \(P\), equal to the weight of the plumb bob and the weight of the freely hanging wire, is held in equilibrium by the force \(R\). The load \(P\) and the force \(R\) act on the pulley \(D\) with a force \(N\). The indicated forces are related to one another by the following functional dependence (from the equation of moments of rotation):

Fig. 17

\[ R = N \frac{d_0}{d_\sigma}\mu + P, \tag{21} \]

where \(d_\sigma\) is the diameter of the pulley, \(d_0\) is the diameter of the axle, \(\mu\) is the coefficient of friction.

The force \(N\), as the resultant of \(P\) and \(R\), is equal to

\[ N^2 = R^2 + P^2 + 2 P R \cos \gamma . \tag{22} \]

From equations (21) and (22) we can write

\[ R^2 + P^2 + 2 P R \cos \gamma = (R - P)^2 \frac{d_\sigma^2}{d_0^2\mu^2}, \]

or

\[ R_2 + 2 P R \frac{d_0^2\mu^2 \cos \gamma + d_\sigma^2}{d_0^2\mu^2 - d_\sigma^2} + P^2 = 0, \]

whence

\[ R=P\left[\frac{d_0^2\mu^2\cos\gamma+d^2\sigma}{d^2\sigma-d_0^2\mu^2} \pm \sqrt{\left(\frac{d_0^2\mu^2\cos\gamma+d^2\sigma}{d^2\sigma-d_0^2\mu^2}\right)^2-1}\right]. \tag{23} \]

The plus sign before the radical in formula (23) corresponds to a measurement when the wire is being raised, and the minus sign when it is being lowered into the mine.

If in formula (23) we substitute the values \(d_0\), \(d_b\), \(\mu\), and \(\gamma\) for pulley \(D\), then we obtain the tension force \(R_1\) for the wire on the section \(D—T\). The tension force \(R_2\) for the wire on the section I—II is computed from the formula

\[ R_2=R_1\left[\frac{d_0^2\mu^2\cos\gamma+d^2\sigma}{d^2\sigma-d_0^2\mu^2} \pm \sqrt{\left(\frac{d_0^2\mu^2\cos\gamma+d^2\sigma}{d^2\sigma-d_0^2\mu^2}\right)^2-1}\right], \tag{24} \]

where the parameters \(d_b\), \(d_0\), \(\mu\), and \(\gamma\) refer to pulley \(T\).

When defective pulleys are used, cases are possible of both partial and complete cessation of their free rotation about the axle (seizing). In the presence of such a condition, friction occurs between the wire and the pulley. The tension force \(R\) in this case may be determined by Euler’s formula

\[ \begin{aligned} R_1&=P e^{\mu\alpha}\\ R_2&=R_1 e^{\mu\alpha} \end{aligned} \tag{25} \]

where \(e\) is the base of natural logarithms,

\(\alpha\) is the angle of wrap (Fig. 17).

The force arising as a consequence of the friction force in the pulleys is determined from the difference between \(R\) and \(P\), i.e.,

\[ f=R-P. \tag{26} \]

The maximum value of this difference will occur when measuring the first segment, the minimum when measuring the last (if the measurement is performed during raising; during lowering the picture will be the reverse).

The linear magnitude of the error in measuring the length of the wire due to the friction force in the pulleys, when measuring by the first method, can be determined as follows. Any measured section \(l\) of the wire receives, owing to the friction force in the pulleys, an additional tension equal, as indicated above, to \(f\). The additional tension of the wire is accompanied by its deformation \(\Delta l_f\), determined by the formula

\[ \Delta l_f=-\frac{f}{EF}\,l. \]

The total (maximum) deformation \(\Delta L_f\) of a wire of length \(L\) is equal to the sum of the deformations of the separate sections \(l\). If the separate sections \(l\) are equal to one another, then we shall have

\[ \Delta L_f=-\frac{f_1+f_2+\cdots+f_n}{EF}\,l. \]

The expression \(f_1+f_2+\cdots+f_n\) is the sum of the terms of an arithmetic progression, where \(f_1\) corresponds to \(f_{\max}\), and \(f_n\) to \(f_{\min}\). Thus, the final expression for \(\Delta L_f\) may be written in the following form:

\[ \Delta L_f=-\frac{f_c}{EF}\,L, \tag{27} \]

where \(E\) is the modulus of elasticity of the wire in \(\dfrac{kg}{cm^2}\);

\(F\) is the cross-sectional area of the wire in \(cm^2\);

\(L\) is the total measured length of the wire in \(m\);

\(f_c\) is computed by the formula

\[ f_c=\frac{f_{max}+f_{min}}{2}\,n, \tag{28} \]

\(f_{max}\) is determined from the difference \(R_{max}-P_{max}\).

\(R_{max}\) is computed by formula (24), under the condition that \(R_1\) is determined for the value \(P_{max}\)

\[ P_{max}=P+q\,(L'-l-s), \tag{29} \]

where \(P\) is the weight of the plumb bob in \(kg\);

\(q\) is the weight of one running meter of wire in \(kg\);

\(L'\) is the freely hanging part of the wire (from the upper mark to the plumb bob, in \(m\)).

When computing \(f_{min}\), the value \(R_{min}\) is likewise determined by formula (24). In this case \(R_1\) is determined on the basis of \(P_{min}\)

\[ P_{min}=P+q\,(L'-L-s). \tag{30} \]

If the measurement of the depth of the shaft is carried out while the wire is being raised, but by the second method, then \(\Delta L_f\) has a form different from that given for the first method (27). The new form of the expression \(\Delta L_f\) may be obtained from the following considerations. When measuring the first segment \(l\), a wire of length \(s\) and \(l-s\) arrives at the comparator, whose tensions at the initial moment were respectively \(R_1\) and \(P_1\). On the comparator the measured segment \(l\) has tension \(R'_2\). On the section \(D-T\) (Fig. 18), at the moment of measuring the first segment, the tension is equal to \(R'_1\). The difference thus created in the tension of the wire, owing to its displacement, is accompanied by deformation, in consequence of which a segment different from \(l\), equal to \(l+\Delta l_f\), will be raised to the surface,

Fig. 18

where

\[ \Delta l_f = -\frac{R'_2-R_1^{n}}{E F}\,s - \frac{R'_2-P_1}{E F}\,(l-s) - \frac{R'_1-P_1}{E F}\,s \]

or

\[ \Delta l_f = \frac{R_1^{n}-R'_1}{E F}\,s - \frac{R'_2-P_1}{E F}\,l \]

The total maximum deformation \(\Delta L_f\) of a wire of length \(L\) is equal to the sum of the deformations of the individual segments \(l\), which, provided these segments are equal to one another, taking (26), (28) into account, is expressed by the formula

\[ \Delta L_f = \frac{\Delta R}{E F}\,s - \frac{t_c}{E F}\,L, \tag{31} \]

where

\[ \Delta R = R_1^{n} - R_1^{k}. \tag{32} \]

In expression (32), \(R_1^{n}\) is the initial tension of the wire (at the moment the marks are applied) on the section \(D - T\), computed by formula (23) for the initial value of \(P\), equal to

\[ P = p + q L'', \tag{33} \]

where \(L''\) is the freely hanging part of the wire (from the pulley to the plumb bob), in m; \(R_1^{k}\) is the final tension of the wire on the section \(D - T\) (when measuring the last segment), determined by formula (23) for the minimum value of \(P\),

\[ P_{\min} = P + q(L'' - L). \tag{34} \]

The quantity \(f_c\) entering formula (31) is determined from expression (28) by the differences of the maximum and minimum values of \(R\) and \(P\). \(R_{\max}\) and \(P_{\min}\) are determined by formula (24) for the value \(R_1\), computed on the basis of \(P_{\max}\) and \(P_{\min}\). The maximum value of \(P\) is equal to

\[ P_{\max} = P + q(L'' - l). \tag{35} \]

If the measurement of the depth of the mine shaft is carried out while lowering the wire into the mine, then, reasoning analogously to the preceding case, we may write

\[ \Delta L_f = - \frac{\Delta R_1}{E F}\, s - \frac{f_c}{E F}\, L, \tag{36} \]

where \(\Delta R_1\) is computed by formula (32) under the condition that \(R_1^{n}\) is computed for the minimum value of \(P\), determined by formula (34), and \(R_1^{k}\) for the final value of \(P\), determined by formula (33).

Since \(R_1^{n} < R_1^{k}\), and \(R_2 < P\), \(\Delta L_f\) in the given case has a positive value and is equal to the sum of the terms in expression (36). On this basis we conclude that expressions (31) and (36) differ from one another only by the sign of the second term and, consequently, may be replaced by the following general formula:

\[ \Delta L_f = \frac{|\Delta R_1|}{E F}\, s \pm \frac{|f_c|}{E F}\, L. \tag{37} \]

The plus sign in formula (37) corresponds to measurement while lowering the wire into the mine; the minus sign, to raising it to the surface.

In connection with the fact that, at the slightest reverse motion of the pulleys, the friction force changes sign to the opposite one, \(\Delta L_f\) has a random character and, thus, formulas (27), (37) make it possible to determine only the maximum value of this error.

The magnitude of the error \(\Delta L_f\), depending on the condition of the pulleys, the measurement scheme, and the depth of the shaft, varies over wide limits: from 0 to 0.5 m and more.

It follows from this that the error due to friction in the pulleys depends entirely on the condition of the pulleys themselves, their design, and their dimensions. The use of defective pulleys is completely inadmissible, since it may lead to a sharp distortion of the result.

The influence of friction in the pulleys may be sharply reduced by decreasing the ratio \(\dfrac{d_0}{d_s}\) (mainly by increasing \(d_s\)), by removing dirt and rust from the pulley axle, and by applying lubrication. It may be reduced practically to zero by using pulleys with ball bearings, whose coefficient of friction is expressed in thousandths of a unit. Determining the depth as the mean of measurements during lowering and raising also leads to a reduction in the influence of friction in the pulleys.

Error Due to Longitudinal (Vertical) Oscillations of the Wire

When the plumb bob is lowered and raised, the bob and, consequently, the wire perform longitudinal oscillations. These oscillations arise under the action of overcoming the force of inertia at rest at the beginning of the measurement and the inertia of motion at the moment when the lowering or raising of the wire is stopped. In addition, they arise in connection with a sudden change in the speed or direction of motion of the bob, and also under the action of various external causes. During oscillations of the bob, the wire is under the action of a dynamic load. The deformation of the wire in the presence of oscillatory motions is a variable quantity and changes from zero to twice its value under the static application of this load. Thus, carrying out any measuring operations at a time when the oscillations have not yet died out may lead to a distortion of the result, the absolute value and sign of which are variable quantities. The probability of an error due to oscillations of the bob is the greater, the larger the period of oscillation \(T\), determined (without taking specific conditions into account) by the formula

\[ T = 2\pi \sqrt{\frac{P L'}{E F g}}, \tag{38} \]

where

\[ P = P + \frac{q L'}{3}, \]

and \(g\) is the acceleration of gravity. For \(P = 10\ \text{kg}\), \(q = 0.014\ \text{kg}\), \(L' = 500\ \text{m}\), the period of oscillation is \(\simeq 1\ \text{sec}\).

Since the error due to oscillations of the wire cannot be taken into account during processing because of its random character, it must be eliminated in the course of measurement. For this purpose all measuring operations must be carried out after the sharpest oscillations have died out. The time required for the damping of the indicated oscillations must be determined experimentally before beginning the depth measurement. If, in calculating the weight of the bob, one does not take into account the dynamic action of the latter and the friction in the pulleys, then, as a result of oscillations, stresses may arise in the wire that lie beyond the limits of elastic deformation. In this case the wire will undergo plastic deformation, and the measurement result will be distorted.

Error Due to the Change of the Force of Gravity with Depth

Measurement of the depth of a mine shaft with a wire, as indicated above, is carried out by successively moving the wire into the mine or raising it to the surface.

Since at different horizons the force of gravity has different values, the tension of the wire, and consequently also its deformation during the measurement process, is a variable quantity. The average change \(\Delta G_{cp}\) in the force of gravity that is of interest to us, for a difference of horizons \(h\), is determined by the formula

\[ \Delta G_{cp} = \frac{1}{2}\, m \cdot \Delta g_{max}, \tag{39} \]

where \(m\) is the mass of the load (plumb bob) suspended from the wire, and \(\Delta g_{max}\) is the maximum change in the acceleration of gravity, determined by the formula

\[ \Delta g_{max} = g_0 (0.0000003\, h), \tag{40} \]

where \(g_0\) is the mean value of the acceleration of gravity at the surface of the earth.

At a shaft depth of \(1000\) m and a tension of \(10\) kg,

\[ \Delta G_{\mathrm{av}} = 1.5\ \text{g}. \]

It follows from the example given that the error depending on the change of gravity with depth has no practical significance.

Error Due to Inexact Knowledge of the Length of the Tape (Comparator)

The measurement of individual sections of the wire in measuring the depth of a mine shaft is usually carried out with the aid of a steel tape (measuring tape), whose length is known with a certain degree of accuracy. Inexact knowledge of the length of the tape (comparator) causes an error in the result. The magnitude of this error is expressed by the formula

\[ m = \pm \lambda L, \tag{41} \]

where \(\lambda\) is the error per unit length of the tape (comparator),
\(L\) is the measured length of the wire.

According to the technical instruction currently in force for the performance of mine-surveying work, the wire may be measured with a tape whose length is known with a relative error equal to

\[ \frac{1}{40000}; \]

consequently, the error in measuring the depth of a shaft in this case will be equal to

\[ m = \pm 0.000025\,L, \]

which amounts to \(2.5\) mm for every \(100\) m, or \(1\) cm for \(400\) m of depth. As is evident from the example given, this error has a substantial influence on the accuracy of the result of measuring the depth of a shaft, and therefore the use of more accurate measuring instruments is desirable.

Error Due to the Difference in the Sag of the Wire on the Section \(D—T\) at the Beginning and at the End of the Measurement

The section of wire \(D—T\) (Fig. 2) has a free sag. The tension of this section at the initial moment and during the measurement does not remain constant: it decreases when the wire is raised and increases when it is lowered into the mine.

The sag of this section also does not remain constant during the measurement: it increases when the wire is raised and decreases when it is lowered.

Thus, the displacement of the mark “\(a\)” (or the mark “\(b\)”) occurs not only in connection with the raising of the wire and its shortening under the influence of the change in its own weight, but also in connection with the change in the sag of the wire on the section \(D—T\).

The total displacement of the mark “\(a\)” (or “\(b\)”), caused by the change in sag, is equal to the difference in the length of the curve (wire) of this section at the end of the measurement and at the initial moment, or, what is the same, to the difference of the corrections for reducing the indicated curves to their chord, and may be determined by the formula

\[ \Delta L_f = \frac{8\left|f_1^2 - f_0^2\right|}{3l}, \tag{42} \]

where \(f_0\) is the sag of the wire on the section \(D—T\) at the initial moment,
\(f_1\) is the same at the end of the measurement,
\(l\) is the length of the measured section of wire.

The sign of the correction \(\Delta L_f\), computed by formula (42), is obtained automatically; consequently, the correction \(\Delta L_f\) is always positive and does not depend on the direction of motion of the plumb bob.

For any value of \(L\) possible in practice, the quantity \(\Delta L_f\) amounts to fractions of a millimeter and thus has no practical significance.

From the example given it follows that this source of error may practically be neglected.

Error Due to Displacement of the Tape (Comparator) During the Measurement Process

The error due to displacement of the tape during the measurement process is inherent only in the second method of measuring depth and is equivalent to a displacement of the initial graduation mark. In magnitude and sign it is random in character and cannot be expressed analytically. Firm fastening of the tape and immobility of the device on which it is fixed fully guarantee the measurements against the error indicated above. In addition, it can be detected and taken into account in processing if the measurement of individual sections of the wire is carried out by the method described by V. K. Yuzefovich, marking the position of the collimation planes of the theodolites by fixed points.

Conclusion

As indicated above, up to the present time extremely little attention has been paid to the question of measuring the depth of a mine shaft. This is especially true of the method, widely used in practice, of measuring depth with a wire. This circumstance was emphasized by Prof. I. M. Bakhurin in his work mentioned above, where he raised the question of the need to conduct research in this field.

The present work, therefore, is an attempt to solve the problem put forward by Prof. I. M. Bakhurin concerning a comprehensive study of the sources of error in measuring the depth of a mine shaft with a wire.

If one takes into account that recently erroneous ideas about certain sources of errors distorting the measurement result have entered the literature and, consequently, practice, then the timeliness and significance of the present work, in our opinion, are beyond doubt.

From the work it follows, first of all, that the temperature of the wire during the performance of the various measuring operations has the greatest influence on the accuracy of measuring the depth of a mine shaft. It is precisely to this circumstance that the most serious attention must be paid in solving this problem. The erroneousness of the formula currently used for the correction for elongation of the wire under the action of its own weight has been proved, and a new formula has been given, from which it follows that this error has almost no practical significance.

The analysis carried out made it possible to establish and practically evaluate a whole series of such sources of errors which up to the present time had not been mentioned at all in the literature (friction in the pulleys, longitudinal oscillations of the wire, etc.).

The work as a whole, in our opinion, to a considerable extent fills the gap that exists in this question and should serve as a basis for revising the corresponding section of the “Technical Instructions for the Performance of Mine-Surveying Work.”

References

1. Bakhurin I. M.—Questions of Mine-Surveying Art. Supplement to the “Course of Mine-Surveying Art,” 1936.
2. Bauman V. I.—Course of Mine-Surveying Art, part 2, 1932.
3. Ogloblin D. N.—Mine-Surveying of Mine Workings, 1944.
4. Bratgun O.—Practical Manual of Mine-Surveying Art, 1903.
5. Mine-Surveying Handbook, 1938.
6. Proceedings of TsNIMB, No. 11, 1940.
7. Transactions of the Leningrad Mining Institute, vol. 8, 1934.
8. Greber G.—Introduction to the Theory of Heat Transfer, 1933.
9. Khvolson O. D.—Short Course of Physics, part 3, 1923.
10. Timoshenko S. P.—Course of Strength of Materials, 1930.
11. Dobrovolsky V. A.—Machine Parts, 1939.
12. Hütte—Engineer’s Handbook, vol. 1, 1930.
13. Technical Instruction for the Production of Mine-Surveying Alignments, 1938.

1. The error depending on readings on the tape and rods, inherent in all measuring operations, is not considered. ↩

2. There are two methods of measuring the depth of a mine shaft with a wire, the first of which is described in the book by Prof. I. M. Bakhurin, Mine Surveying (supplement to the course Mine Surveying), and the second in the book by Prof. D. N. Ogloblin, Mine Surveying of Mine Workings. ↩