LABORATORY AND THEORETICAL STUDIES OF THE ACCURACY OF ALIGNMENT WITH A PLUMB LINE DEPENDING ON THE WIRE DIAMETER

G. F. LYSOV

(Presented by the scientific seminar of the Department of Mine Surveying and Geodesy)

It is known that, in order to reduce the surveying error caused by the action of an air current moving in a mine shaft, it is advantageous, when orienting the shaft, to increase the size of the weight. However, when using a heavier weight, we must use a thicker wire. When aligning with plumb lines made of thicker wire, the accuracy of aiming the line of sight of the instrument at the middle of the plumb-line wire decreases and, consequently, the alignment error increases to some extent.

In the present article, the question is considered of the additional error in aligning with the line of plumb lines as a function of the wire diameter, and data are presented from experimental investigations carried out to solve this question.

Aiming the vertical thread or the bisector of the instrument at the middle of a stationary or oscillating plumb-line thread \(A\) of diameter \(d\) is carried out with a certain error \(m_d\) (Fig. 1). This is explained by the fact that the human eye detects the asymmetry in the position of one line (the vertical thread or the bisector of the theodolite) between two other parallel lines (the edges of the plumb-line wire) only up to a certain limit. From Fig. 1 it follows that the error of one direction \(CO\) in angular measure \(\Delta\), caused by inaccurate aiming of the theodolite line of sight at the middle of the plumb line \(A\), can be determined by the formula:

\[ \Delta = \frac{m_d \rho''}{l}, \tag{1} \]

where \(l\) is the distance from the instrument to the plumb line.

In order to determine the magnitude of the mean error of dividing “by eye” the distance between two lines, we carried out many—

multiple division of distances by the unaided eye and with the MGT-30 theodolite between two parallel lines drawn at various distances from one another. The results of this work are presented in Tables 1 and 2.

Table 1

Results of dividing the distance between two parallel lines “by eye” with the unaided eye

Table 2

Results of dividing the distance between two parallel lines “by eye” with the aid of the MGT-30 theodolite

From Tables 1 and 2 it follows that the value \(m_d\) is proportional to the distance between the parallel lines \(d\), while the ratio \(\dfrac{m_d}{d}\) is a constant quantity, equal on average to 0,024. Comparing the data of Tables 1 and 2, one may also assert that magnification of the telescope in this case has no effect on the accuracy of sighting with a vertical thread (bisector) “by eye” midway between two given parallel lines.

Thus, it appears possible to determine the magnitude of the error of a single direction caused by inaccurate aiming of the theodolite line of sight at the middle of the plumb-line wire, by the formula:

\[ \Delta = \pm \frac{k d \rho''}{l}, \tag{2} \]

where

\[ k = \frac{m_d}{d}. \]

The value of \(\Delta\) can also be determined in another way. The mean error of one pointing \(\Delta\) can be calculated by the formula:

\[ \Delta = \pm \sqrt{\mu_d^2 - \mu_0^2}, \tag{3} \]

where \(\mu_d\) is the mean error of one pointing when observing a wire of diameter \(d\),

\(\mu_0\) is the mean error of one pointing on a very thin line \((d\) equal to 0).

The error of one pointing \(\mu_d\) when sighting on a wire of diameter \(d\) can be determined by repeatedly sighting with the vertical thread (bisector) of the theodolite on the middle of the plumb-line wire, followed by calculation of the value \(\mu_d\) by the formula:

\[ \mu_d = \pm \sqrt{\frac{[\varepsilon_d \varepsilon_d]}{n - 1}}, \tag{4} \]

where \(E_d\) are the deviations of the individual limb readings from the mean reading,

\(n\) is the number of all readings taken (the number of sightings).

The error of one pointing \(\mu_0\) can also be determined by repeated sighting with the vertical thread of the instrument on a fixed, sharp, and distant signal, while simultaneously taking readings on the limb. In this case \(\mu_0\) can be found by the formula:

\[ \mu_0 = \pm \sqrt{\frac{[\varepsilon_0 \varepsilon_0]}{n - 1}}, \tag{5} \]

where \(E_0\) are the deviations of the individual limb readings from the mean reading.

By the method described, we determined in the laboratory several values of \(\Delta\) for fixed and oscillating plumb lines of various diameters, with an oscillation amplitude of not more than 2–3 mm and under different observation conditions. The observations were made with an OT-10 theodolite.

Using the values of \(\Delta\) found, we then calculated, by formula (2), a series of values of \(k\) for fixed and oscillating plumb lines of various diameters. The results of this work are presented in Tables 3, 4, and 5.

Comparing the data of Tables 1, 2, and 3, one may note the good agreement of the values of \(k\) found by different methods for fixed plumb lines under daylight-surface conditions, and observe a clearly expressed regularity in the relationship between the value of \(\Delta\) and the diameter \(d\) of the wire, both for fixed and for oscillating plumb lines under different observation conditions. As the final value of the coefficients \(k\) for the various observation conditions, we shall take the arithmetic mean of the values of \(k\) obtained by the more accurate second method.

Further, in order to establish the relationship between the error of scale reading and the diameter of the plumb-line wire when sighting by scale methods, we carried out in the laboratory observations of the oscillations of plumb lines of various diameters using millimeter scales. As a result of this work it was established that, when sighting on plumb lines by scale methods, there is no additional error caused by inaccurate sighting on the middle of the plumb-line wire.

Let us now turn to consideration of the question of the accuracy of alignment to the plumb-line line depending on the magnitude of the wire diameter.

In the case when alignment is carried out by the method of the connecting triangle (Fig. 2), the angle \(\gamma\) (because of inaccurate sighting—

Table 3

Results of observations of stationary plumb lines under conditions corresponding to observations on the daytime surface of a mine

Table 4

Results of observations of stationary and oscillating plumb lines under conditions corresponding to underground conditions

Table 5

Values of the coefficient \(k\) under different observation conditions

—at the midpoint of the wire of the stationary or oscillating plumb lines) is measured with an additional error \(m_\gamma\), which may be calculated by the formula:

\[ m_\gamma = \pm \sqrt{\frac{\Delta_1^2 + \Delta_2^2}{6}}, \tag{6} \]

where \(\Delta_1, \Delta_2\) are the mean errors of one direction to the plumb lines \(O_1\) and \(O_2\), caused by inaccurate pointing of the instrument’s line of sight at the midpoint of the wires of the plumb lines.

The angles \(\beta_1\) and \(\beta_2\), for the same reason, are measured with additional mean errors:

\[ m_{\beta_1} = \pm \frac{\Delta_1}{\sqrt{6}}, \qquad m_{\beta_2} = \pm \frac{\Delta_2}{\sqrt{6}}. \tag{7} \]

Since the angle \(\gamma_0\) is then recommended to be computed [1] by the formula:

\[ \gamma_0=\frac{2\gamma+(\beta_2-\beta_1)}{3}, \tag{8} \]

the additional error of the angle \(\gamma_0\) is determined from the expression:

\[ m_{\gamma_0}=\pm \frac{1}{3}\sqrt{\frac{5\Delta_1^2+5\Delta_2^2}{6}}. \tag{9} \]

The additional error of the angle \(B\), computed by the sine formula for the connecting triangle, can be found by the formula:

Fig. 2.

\[ m_B=\pm \frac{b}{3c}\sqrt{\frac{5\Delta_1^2+5\Delta_2^2}{6}}, \tag{10} \]

where \(b, c\) are the lengths of the sides of the connecting triangle.

Consequently, the error of one alignment with the plumb-line sightline in this case will increase by the amount:

\[ m_g=\pm \sqrt{\frac{5b^2}{9c^2}\left(\frac{\Delta_1^2+\Delta_2^2}{6}\right)+\frac{\Delta_2^2}{6}}. \tag{11} \]

Substituting into formula (11) the values \(\Delta_1\) and \(\Delta_2\) found from formula (2), we obtain:

\[ m_g=\pm \frac{k d \rho''}{\sqrt{6}}\sqrt{\frac{5b^2}{9c^2a^2}+\frac{5}{9c^2}+\frac{1}{a^2}}. \tag{12} \]

Denote the ratio \(\dfrac{b}{c}\) by \(\lambda\), and, taking into account that \(a \simeq b+c=(1+\lambda)c\), rewrite formula (12) in the following form:

\[ m_g=\pm A d, \tag{13} \]

where \(d\) is the diameter of the plumb-line wire in mm, and \(A\) is determined as follows:

\[ A=\frac{k\rho''}{c\sqrt{6}}\sqrt{\frac{10\lambda^2+10\lambda+14}{9(1+\lambda)^2}}\ \text{sec}/\text{mm}. \tag{14} \]

The expected additional error \(M_g\) of contact on the surface and at the oriented horizon can be found from the expression:

\[ M_g=\pm 3d\sqrt{A_{\text{п}}^{2}+A_{\text{ш}}^{2}}, \tag{15} \]

where \(A_{\text{п}}\) is a coefficient that can be found from formula (14), depending on the conditions of contact on the surface, as a function of the specific values of \(\lambda_{\text{п}}\), \(c\), and \(k_{\text{п}}\);

\(A_{\text{ш}}\) is a coefficient calculated as a function of the specified conditions of contact at the underground horizon, i.e., depending on \(\lambda_{\text{ш}}\), \(c\), and \(k_{\text{ш}}\).

Formula (14) is inconvenient for direct calculation of the quantities \(A_{\text{п}}^{2}\) and \(A_{\text{ш}}^{2}\), but it can be used to construct a nomogram, with the help of which it is possible to find the values of \(A_{\text{п}}^{2}\) and \(A_{\text{ш}}^{2}\) from the given quantities \(\lambda\) and \(c\) quite simply and with sufficient accuracy.

Such a nomogram and the scheme for its use are presented in Fig. 3. This nomogram has two scales, “\(A^{2}\).” One scale serves to determine the values of \(A_{\text{п}}^{2}\) when contacting stationary plumb lines under surface conditions \((k_{\text{п}}=0.0233)\); the other is for determining the values of \(A_{\text{ш}}^{2}\) when contacting stationary or oscillating plumb lines under underground conditions \((k_{\text{ш}}=0.0796)\).

Using the given nomogram, let us determine the additional error of one contact under the following conditions: \(c=3.5\) m, \(\lambda=0.8\), \(d=4.5\) mm; observations will be carried out under underground conditions.

From the nomogram we find \(A_{\text{ш}}^{2}=3.5\ \text{sec}^{2}/\text{mm}^{2}\). Therefore, for the mean additional error of one contact \(m_g\) we obtain:

\[ m_g=\pm A_{\text{ш}}d=\pm 8''.4 \]

In order to establish the fact that the contact error increases with an increase in the diameter of the plumb-line wires, we carried out contact in laboratory conditions by the connecting-triangle method, without changing the centering of the instrument and the signal at point \(D\) (Fig. 2). The observed plumb lines had the following design. Plastic tubes 10 cm long and with an outside diameter of 2.8 mm were tightly fitted onto the main plumb-line wire, of diameter 0.6 mm; other tubes, with outside diameter 4.5 mm and length 5 cm, had previously been stretched over them (Fig. 4).

Contact was performed in the following order:

a) when sighting at the middle of a stationary plumb line of diameter 0.6 mm—three times;

b) when sighting at the middle of the stationary part of a plumb line of diameter 2.8 mm—three times;

c) when sighting at the middle of the stationary part of a plumb line of diameter 4.5 mm—three times.

All angles were measured in three repetitions with an MGT-30 theodolite; the observation site was the basement room of the mine-surveying laboratory, and the lighting was electric.

The results of the experiment are presented in Table 6.

The data of Table 6 definitely indicate the fact that the error of alignment increases depending on the magnitude of the diameter of the plumb-line wires. We shall use these data to calculate the magnitude of the additional

Fig. 3.
In the figure: “Alignment diagram”; “given”; “response”; “Distance between plumbs \(C, \lambda\)”.

error of alignment \(m_g\) to plumb lines with a diameter of \(4.5\) mm, applying the formula:

\[ m_g = \pm \sqrt{m^2 - m_0^2}, \tag{16} \]

where \(m\) is the total error of alignment, depending on the accuracy of aiming the theodolite line of sight at the middle of the plumb-line wire, on the accuracy of the instrument, and on the method of measuring angles;

\(m_0\) is the part of the alignment error depending only on the accuracy of the instrument and on the method of measuring angles.

Figure 4.

We shall take the value of the total mean error of contact \(m\) from Table 6, i.e. \(m=\pm 10'',0\).

The quantity \(m_0\), as is known, can in our case be computed from the formula:

\[ m_0=\pm m_i\sqrt{1+\left(\frac{b}{c}\right)^2}, \tag{17} \]

where \(m_i\) is the error of the measured angle for three repetitions, equal to \(\pm 3'',5\). Taking into account that \(b\) is equal to \(2.467\ \text{m}\), and \(c=3.512\ \text{m}\), from (17) and (16) we obtain for \(m_g\):

\[ m_g=\pm\sqrt{100-20}=\pm 9'',0. \]

Previously, for the same conditions, we had theoretically found an additional contact error \(m_g\) equal to \(\pm 8'',4\). The good agreement in the values of \(m_g\) found by different methods indicates the reliability of the established relationship between the additional contact error and the diameter of the plumb-line wire.

Table 6

Conclusions

1. The increase in the error of alignment with the connecting triangle as the diameter of the plumb-line wire increases should be regarded as an established fact.

2. Formulas have been established relating the additional alignment error on the surface and on the underground horizon to the magnitude of the diameter of the plumb-line wire.

3. A method has been developed for determining the additional error that arises when aligning to plumb lines made of wires of different diameters, and a nomogram has been proposed that simplifies the process of determining this error.

References

Technical Instructions for the Performance of Mine-Surveying Work, Ugletekhizdat, 1959.