Regulation of the Length of Hoisting Ropes
Abstract Generated abstract
This paper addresses the periodic adjustment of rope length in mine hoisting installations, where plastic elongation produces slack that can affect safe operation. It derives formulas for the permissible adjustment step from rope deformation, unloading overtravel, hoisting height, drum geometry, and the design constraints of two-drum machines, including correction for an integral number of teeth in the drum-resetting mechanism. The paper also proposes a method for estimating the interval between adjustments by modeling relative plastic rope elongation as a power function of operating time, with parameters obtained from measurements. Experimental data from Kuzbass mines are used to give representative parameter values, and the author suggests using nomograms based on the derived relations for practical application.
Full Text
REGULATION OF THE LENGTH OF HOISTING ROPES
N. I. Klykov
(Submitted by the scientific seminar of the Department of Mine Mechanics, Mining Faculty)
Periodic regulation of the length of ropes on mine hoisting installations is caused by the formation of slack as a result of plastic elongation of the ropes. In the general case the magnitude of the slack is
\[ \Delta l_n=\Delta l_{n.l}+h_n+\Delta l_y, \tag{1} \]
where \(\Delta l_{n.l}\) is the total plastic deformation of the ropes, m;
\(h_n\) is the height by which the cage exceeds the unloading level, m;
\(\Delta l_y\) is the elastic deformation of the rope, m.
After regulation of the rope length,
\[ \Delta l'_k=h_n-\Delta l_y, \tag{2} \]
i.e., in the process of regulation one of the ropes is tightened by the amount of the total plastic deformation. We assume that the drums in this case are turned relative to one another by one regulating step (Fig. 1)
\[ s_p=\Delta l_{n.l}. \tag{3} \]
To avoid the harmful consequences of excessive slack, the magnitude of the latter must be limited to
\[ \Delta l_n \leq 0.2\ \text{m}. \tag{4} \]
Fig. 1. Regulation step of a hoisting machine with a toothed mechanism for resetting the drums.
According to our studies, the elastic deformation of the rope and the permissible height by which the unloading level may be exceeded are, respectively,
\[ \Delta l_y=0.84\cdot 10^{-6}\cdot \kappa \cdot H_0 (L_{pr}-H_0); \tag{5} \]
\[ h_n=0.3+0.0004H, \tag{6} \]
where \(\kappa\) is the coefficient of end load;
\(L_{pr}\) is the ultimate length of the rope, m;
\(H_0\) is the greatest length of the rope suspension, m;
\(H\) is the hoisting height, m.
The coefficient of end load when an empty conveyance is landed is
\[ \kappa=\frac{Q_m+aQ_p}{Q_m+Q_p}, \tag{7} \]
where \(Q_m\) is the dead weight of the conveyance, kg;
\(Q_p\) is the payload, kg;
\(a\) is the load coefficient of the conveyance in motion.
The ultimate length of the rope is
\[ L_{pr}=\frac{\sigma_{pu}\cdot 10^6}{\gamma_0\cdot m}, \tag{8} \]
where \(\sigma_{pu}\) is the breaking strength of the rope, kg/mm\(^2\);
\(\gamma_0\) is the fictitious weight of 1 m\(^3\) of rope, kg/m\(^3\);
\(m\) is the safety factor of the rope.
The greatest length of the rope suspension is
\[ H_0=H+h_0, \tag{9} \]
where \(h_0\) is the length of the rope suspension when the conveyance is in the upper position, m.
Substituting the values (3), (4), (5), and (6) into expression (1), we obtain the adjustment increment (Fig. 2)
\[ s_p=0.84\cdot 10^{-6}\cdot \kappa\cdot H_0(L_{pr}-H_0)-0.0004H-0.1. \tag{10} \]
Fig. 2. For determining the adjustment increment of the rope length. Dependences: 1—allowance; 2—elastic deformation; 3—height of excess; 4—adjustment increment as a function of the rope suspension length.
When determining the adjustment increment for a particular hoisting machine, it is necessary to establish the smallest hoisting height on the basis of rational use of the winding surface of the drums. From the characteristics of two-drum hoisting machines of the standard series it follows that the difference between the winding surfaces of adjacent machines is 20%, i.e., at a minimum, 0.8 of the drum width can be occupied by the rope. In this case
\[ 0.8B=\left(\frac{H+l_u}{\pi\cdot D_b}+n_{BT}\right)(d+\varepsilon), \tag{11} \]
where \(B\) is the drum width, m;
\(D_b\) is the drum diameter, m;
\(l_u\) is the rope reserve for testing, m;
\(n_{BT}\)—number of friction turns;
\(d\)—rope diameter, m;
\(\varepsilon\)—clearance between rope turns on the drum, m.
The greatest length of the suspended rope, from expressions (9) and (11), is
\[ H_0=\frac{0{,}8\pi \cdot D_b \cdot B}{d+\varepsilon}-\pi \cdot n_{BT}\cdot D_b-l_u+h_0 . \tag{12} \]
Substituting the value (12) into expression (10), we determine the adjustment pitch in terms of the parameters of the hoisting machine:
\[ \begin{aligned} s_p={}&\left[0{,}84\cdot 10^{-6}\cdot \kappa \cdot L_{np} -0{,}84\cdot 10^{-6}\cdot \kappa \left(\frac{0{,}8\pi D_b\cdot B}{d+\varepsilon}-\right.\right.\\ &\left.\left.-\pi\cdot n_{BT}\cdot D_b-l_u+h_0\right)-0{,}0004\right] \left(\frac{0{,}8\pi D_b B}{d+\varepsilon}-\right.\\ &\left.-\pi n_{BT}D_b-l_u+h_0\right)+0{,}0004h_0-0{,}1 . \end{aligned} \tag{13} \]
Finally, the adjustment pitch is corrected downward so that the number of teeth of the wheel of the resetting mechanism is an integer,
\[ n_z \ge \frac{\pi D_b}{s_p}. \tag{14} \]
In view of the fact that at present there is no proper monitoring of rope slack, we consider it advisable, for practical purposes, to develop a method for determining the time after which the next adjustment of the rope lengths should be carried out.
Experimental measurements of the “elongation” of ropes over specified intervals of time, which were carried out at certain installations in the Kuzbass, made it possible to establish relationships between the magnitudes of relative plastic deformations and the operating time of the ropes. In general form these relationships are power functions,
\[ \varepsilon_{pl}=a t^n, \tag{15} \]
where \(\varepsilon_{pl}\)—relative plastic deformation of the rope;
\(t\)—operating time of the rope from the moment of its installation, months;
\(a\)—coefficient depending on the characteristics of the rope operating conditions;
\(n\)—exponent.
Since, in the general case, the operating time of individual ropes may be different, then over the time elapsed after adjustment, one rope will receive an increment
\[ \Delta \varepsilon_{pl.1}=a\left[(t_1+\Delta t)^n-t_1^n\right], \tag{16} \]
and the other
\[ \Delta \varepsilon_{pl.2}=\left[(t_2+\Delta t)^n-t_2^n\right], \tag{17} \]
where \(t_1\) and \(t_2\)—respectively, the operating time of one and the other ropes from the moment of their installation, months;
\(\Delta t\)—time elapsed after adjustment of the ropes, months.
Since in regulation the “stretch” of two ropes is selected at once, then
\[ \varepsilon_{\mathrm{sh.r.}}=\Delta \varepsilon_{\mathrm{pl.1}}+\Delta \varepsilon_{\mathrm{pl.2}}, \tag{18} \]
where \(\varepsilon_{\mathrm{sh.r.}}\) is the relative regulation step.
The magnitude of the relative regulation step is
\[ \varepsilon_{\mathrm{sh.r.}}=\frac{s_p}{H_0+L_{\mathrm{str}}}, \tag{19} \]
where \(L_{\mathrm{str}}\) is the length of the rope string, m.
Substituting values (16) and (17) into expression (18), we obtain
\[ \varepsilon_{\mathrm{sh.r.}}= a\left[(t_1+\Delta t)^n+(t_2+\Delta t)^n-(t_1^n+t_2^n)\right]. \tag{20} \]
The problem is reduced to finding, from expression (20), the time after which the next regulation of the ropes must be carried out.
The proposed method is as follows.
According to Fig. 3, which shows the power function (15),
\[ \frac{\Delta \varepsilon_{\mathrm{pl.1}}}{\Delta t}=\operatorname{tg}\alpha_1; \tag{21} \]
Fig. 3. For determining the term for regulating the rope length.
\[ \frac{\Delta \varepsilon_{\mathrm{pl.2}}}{\Delta t}=\operatorname{tg}\alpha_2. \tag{22} \]
From expression (18)
\[ \Delta \varepsilon_{\mathrm{pl.2}}=\varepsilon_{\mathrm{sh.r.}}-\Delta \varepsilon_{\mathrm{pl.1}}. \tag{23} \]
Substituting the value (23) into (22), we obtain
\[ \frac{\varepsilon_{\mathrm{sh.r.}}-\Delta \varepsilon_{\mathrm{pl.1}}}{\Delta t} =\operatorname{tg}\alpha_2. \tag{24} \]
Solving (21) and (24) jointly, we have
\[ \Delta t=\frac{\varepsilon_{\mathrm{sh.r.}}}{\operatorname{tg}\alpha_1+\operatorname{tg}\alpha_2}. \tag{25} \]
The tangents of the angles of inclination of the tangents \(MP\) and \(KQ\) are the first derivatives of the given power function at specific values of the operating time of the individual ropes, i.e.,
\[ \operatorname{tg}\alpha_1=an t_1^{\,n-1}, \tag{26} \]
\[ \operatorname{tg}\alpha_2=an t_2^{\,n-1}. \tag{27} \]
Substituting the values (26) and (27) into expression (25), we obtain
\[ \Delta t \to \frac{\varepsilon_{\mathrm{sh}\cdot \mathrm{p}}}{a n \left(t_1^{\,n-1}+t_2^{\,n-1}\right)} . \tag{28} \]
This solution is exact for infinitely small increments of the plastic deformation of the ropes and gives significant deviations for finite values of the indicated quantities.
Let us refine the method set forth as applied to finite values of the relative adjustment step.
According to Fig. 3 (triangle \(MAB\)),
\[ \Delta \varepsilon_1=\Delta t \cdot \operatorname{tg}\alpha_1=\Delta t \cdot a n t_1^{\,n-1}. \tag{29} \]
For the same values of \(t_1\) and \(\varepsilon_{\mathrm{sh}}\) (triangle \(MA'B'\)),
\[ \Delta \varepsilon_1=\Delta t' \cdot \operatorname{tg}\alpha_1' = a\left[(t_1+\Delta t')^n-t_1^n\right], \tag{30} \]
where \(\Delta t'\) is a more exact value of the quantity being determined (taking into account the angle of the secant, and not the angle of the tangent).
Equating the right-hand sides of expressions (29) and (30), we find
\[ \Delta t'=\left(\Delta t \cdot n \cdot t_1^{\,n-1}+t_1^n\right)^{\frac{1}{n}}-t_1, \tag{31} \]
Similarly,
\[ \Delta t''=\left(\Delta t \cdot n \cdot t_2^{\,n-1}+t_2^n\right)^{\frac{1}{n}}-t_2. \tag{32} \]
The exact value of the quantity being determined lies between \(\Delta t'\) and \(\Delta t''\). Very close to the exact value is the arithmetic mean
\[ \Delta t_{\mathrm{av}}=\frac{\Delta t'+\Delta t''}{2}. \tag{33} \]
Substituting the values (31) and (32) into expression (33) and taking formula (28) into account, we finally obtain
\[ \Delta t_{\mathrm{av}}= \frac{ \left[ \frac{\dfrac{\varepsilon_{\mathrm{sh}\cdot \mathrm{p}}}{a}} {1+\left(\dfrac{t_2}{t_1}\right)^{n-1}} +t_1^n \right]^{\frac{1}{n}} + \left[ \frac{\dfrac{\varepsilon_{\mathrm{sh}\cdot \mathrm{p}}}{a}} {1+\left(\dfrac{t_1}{t_2}\right)^{n-1}} +t_2^n \right]^{\frac{1}{n}} -t_1-t_2 }{2}. \tag{34} \]
For determining the coefficient \(a\) and the exponent \(n\), theoretical dependences can be found on the basis of mathematical processing of experimental data on the operation of ropes of various constructions. Most simply, and with sufficient practical accuracy, these quantities are determined in the following way. After suspension and initial adjustment of the rope, the position of the hoisting vessel and the machine is noted for a definite position (for example, when the vessel is unloaded). After the rope has worked for a certain
time \(t_1\), from the position of the hoisting machine and the conveyance, the “elongation” of the rope \(\Delta l_{\mathrm{pl}.1}\) is determined. The relative magnitude of the “elongation” is
\[ \varepsilon_{\mathrm{pl}.1}=\frac{\Delta l_{\mathrm{pl}.1}}{H_0+L_{\mathrm{str}}}. \tag{35} \]
Similarly, the relative “elongation” of the rope is determined for another value of the rope operating time \(t_2\). Then
\[ n=\frac{\lg \varepsilon_{\mathrm{pl}.1}-\lg \varepsilon_{\mathrm{pl}.2}} {\lg t_1-\lg t_2}; \tag{36} \]
\[ a=\frac{\varepsilon_{\mathrm{pl}.1}}{t_1^{\,n}} =\frac{\varepsilon_{\mathrm{pl}.2}}{t_2^{\,n}}. \tag{37} \]
According to experimental measurements of the “elongation” of ropes under production conditions at several mines of the Kuzbass, the average value of the exponent \(n\) was 0.163, and that of the coefficient \(a\), 0.0056.
In view of the unwieldiness of the resulting formula (34), in order to speed up the solution the intermediate relations may be presented in the form of nomograms, by means of which one can construct a single nomogram for a given hoisting installation for the entire service life of the ropes (Fig. 4).
Fig. 4. Nomogram for determining the intervals for regulating rope length.
In addition to the analytical method set forth above, the problem of determining the regulation intervals can be solved graphically.
Conclusions
- The regulation step, which determines the type and structural form of the mechanism for resetting the drums of a mine hoisting machine, must correspond to the parameters of the machine being produced and be deter-
should be divided according to formula (13), with an adjustment downward (for example, in order to obtain an integral number of teeth of the wheel of the repositioning mechanism).
- To avoid excessive slack due to plastic elongation of the ropes, it is necessary to regulate the rope length in a timely manner. The time between individual adjustments is proposed to be determined by formula (34), or from nomograms compiled on the basis of this formula.