On the maximum dynamic forces in the rope at the rim of the drum of a hoisting machine
Abstract Generated abstract
This paper examines maximum dynamic forces in the hoisting rope at the drum rim during unsteady starting of mine hoisting machines. It distinguishes three stages of rope loading, from initial drum rotation through slack take-up and accelerated motion, and presents calculation formulas for dynamic forces, including the effects of rope slack, terminal load, acceleration jumps, and deflecting sheave inertia. Using practical data for Kuzbass skip hoists with asynchronous drives and metallic rheostats, the paper calculates recommended design values for standard two-drum hoisting machines. It argues that limiting initial accelerations and acceleration jumps, and using suitable rheostat steps, can reduce rope overloads and wear of mechanical components.
Full Text
On the Maximum Dynamic Forces in the Rope at the Rim of the Drum of a Hoisting Machine
N. I. Klykov
(Presented by the scientific seminar of the Department of Mining Mechanics, Faculty of Mining)
Dynamic forces in the elements of a hoisting installation arise during periods of unsteady motion. Proper allowance for dynamic forces is necessary for substantiating the strength dimensions of drums, shafts, parts of coupling devices, and other mechanical units of the hoisting machine.
The maximum dynamic stresses in the rope under normal operation of the installation correspond to the beginning of the hoisting operation. Three stages of rope loading should be distinguished: 1) from the beginning of rotation of the drum to the taking-up of slack in the rope; 2) from the moment the slack is eliminated to the beginning of lifting of the load; 3) the period of accelerated motion.
The dynamic forces in the first stage of loading are small and are determined only by the mass of the rope and the acceleration
\[ F_{\mathrm{d}.1} = j \frac{p\left(H_0 + L_{\mathrm{spr}}\right)}{g}, \tag{1} \]
where \(j\) is the greatest acceleration, \(m/\mathrm{sec}^2\);
\(p\) is the weight of \(1\,m\) of rope length, \(kg/m\);
\(H_0\) is the length of the suspended rope, \(m\);
\(L_{\mathrm{spr}}\) is the length of the rope strand, \(m\);
\(g\) is the acceleration of gravity, \(m/\mathrm{sec}^2\).
The dynamic forces in the second stage of rope loading are
\[ F_{\mathrm{d}.2} = \varepsilon_2 \cdot j \frac{Q_{\mathrm{p}} + Q_{\mathrm{m}}}{g} + F_{\mathrm{d}.1}, \tag{2} \]
where \(\varepsilon_2\) is the coefficient of dynamic loading of the second stage;
\(Q_{\mathrm{p}}\) is the weight of the payload, \(kg\);
\(Q_{\mathrm{m}}\) is the dead weight of the skip, \(kg\).
The coefficient \(\varepsilon_2\) does not take into account the preliminary slack and is calculated by the approximate formula proposed by Prof. F. V. Florinsky,
\[ \varepsilon_2 = \left(10 + 0.93 \frac{\Theta g}{j}\right)\left(0.3 + 0.16\alpha\right), \tag{3} \]
where \(\Theta\) is the ratio of the pressure of the terminal load on the landing devices to the full terminal load;
\(z\) is the ratio of the weight of the hanging part of the rope to the weight of the terminal load.
In those cases when the terminal load is fully supported by the landing devices, the coefficient \(\Theta\) is equal to unity, and
\[ \varepsilon'_2 = \left(10 + 0.93 \frac{g}{j}\right) \left(0.3 + 0.16 \alpha\right). \tag{4} \]
When rope slack is present, the dynamic forces in the second stage of loading are
\[ F_{\partial 2\mathrm{n}} = \varepsilon_{2.\mathrm{n}} \cdot j \frac{Q_{\mathrm{n}} + Q_{\mathrm{m}}}{g} + F_{\partial 1}, \tag{5} \]
where \(\varepsilon_{2.\mathrm{n}}\) is the coefficient of dynamic loading in the presence of rope slack.
To determine the coefficient of dynamic loading, the following formula is proposed, derived on the basis of works [1] and [2]:
\[ \varepsilon_{2.\mathrm{n}} = k_{\mathrm{n}} \varepsilon'_2 + \frac{g}{j} \left(k_{\mathrm{n}} - 1\right) \left( \frac{k' + n_c}{1 + n_c} + \alpha \right), \tag{6} \]
where \(k_{\mathrm{n}}\) is the slack coefficient;
\(n_c\) is the ratio of the weight of the cage to the useful load;
\(k'\) is the coefficient of mine resistances for one strand
(\(k' = 1.075\) for skip hoists and \(k' = 1.1\) for cage hoists).
The slack coefficient is
\[ k_{\mathrm{n}} = 1 + \frac{220 \Delta l_{\mathrm{n}}}{H_0 + L_{\mathrm{spr}}}, \tag{7} \]
where \(\Delta l_{\mathrm{n}}\) is the absolute magnitude of free slack, m.
The dynamic forces in the third stage of rope loading are
\[ F_{\partial 3} = \varepsilon_3 \cdot j \frac{Q_{\mathrm{n}} + Q_{\mathrm{m}}}{g} + F_{\partial 1}. \tag{8} \]
To determine the coefficient of dynamic loading in the third stage of loading, the formula proposed is
\[ \varepsilon_3 = 1 + \frac{\Delta j}{j} \left(1 + 1.7\alpha\right), \tag{9} \]
where \(\Delta j\) is the acceleration jump at starting, m/sec\(^2\).
From formulas (8) and (9) it is evident that the presence of an acceleration jump significantly increases the dynamic forces in the rope, thereby accelerating its wear. Large acceleration jumps may cause breakdowns of parts of the mechanical units of the hoisting machine, increased wear and breakage of reducer teeth, etc. In this respect, hoisting installations equipped according to the G—D system and with an asynchronous motor controlled by a liquid rheostat have significant advantages.
Table 1
Maximum dynamic forces in the rope at the winding rim of two-drum hoisting machines
| Type size of hoisting machine | 1st loading stage, at $j=0$, $7\,\mathrm{m/sec^2}$ | 1st loading stage, at $j=1$, $5\,\mathrm{m/sec^2}$ | 2nd loading stage, at $j=0$, $7\,\mathrm{m/sec^2}$ | 2nd loading stage, at $j=1$, $5\,\mathrm{m/sec^2}$ | 3rd loading stage, at $j=0$, $7\,\mathrm{m/sec^2}$ | 3rd loading stage, at $j=1$, $5\,\mathrm{m/sec^2}$ | Additional forces caused by the inertia of the rope, kg | Proposed design values, kg |
|---|---|---|---|---|---|---|---|---|
| $2 \times 2 \times 1$ | 40 | 85 | 3350 | 7200 | 950 | 2000 | 145 | 8500 |
| $2 \times 2.5 \times 1$ | 60 | 135 | 4250 | 9500 | 1250 | 2850 | 155 | 11500 |
| $2 \times 2.5 \times 1.2$ | 70 | 150 | 4200 | 9450 | 1300 | 2900 | 155 | 11500 |
| $2 \times 3 \times 1.2$ | 95 | 200 | 5500 | 12350 | 1700 | 3800 | 160 | 14500 |
| $2 \times 3 \times 1.5$ | 115 | 245 | 5400 | 12000 | 1750 | 3950 | 160 | 14500 |
| $2 \times 3.5 \times 1.5$ | 155 | 325 | 6450 | 14450 | 2250 | 5000 | 185 | 17500 |
| $2 \times 4 \times 1.7$ | 280 | 600 | 7400 | 16500 | 2550 | 6000 | 230 | 19500 |
| $2 \times 5 \times 1.7$ | 300 | 625 | 8800 | 19600 | 3350 | 7500 | 340 | 24000 |
| $2 \times 5 \times 2.3$ | 400 | 850 | 8700 | 19500 | 3500 | 8000 | 340 | 24000 |
| $2 \times 6 \times 2.4$ | 550 | 1130 | 10350 | 23150 | 4500 | 9850 | 490 | 28000 |
When determining the maximum dynamic forces acting on the rope and the mechanical components of the machine, we assume that the installation has an asynchronous motor with a metallic rheostat, that starting is carried out with rope slack present, and that the terminal load and the length of the rope plumb have the maximum value for the given hoisting machine.
In addition, an additional dynamic force, caused by the inertia of the deflecting sheave, acts on the section of rope between the sheave and the drum:
\[ F_{\mathrm{dyn}}=\frac{GD^2}{gD_{\mathrm{sh}}^2}\,j, \tag{10} \]
where \(GD^2\) is the flywheel moment of the sheave, \(\mathrm{kg\,m^2}\);
\[ D_{\mathrm{sh}} \]
is the sheave diameter, \(m\).
Calculations for determining the maximum dynamic forces for standard two-drum hoisting machines are given in Table 1. The maximum dynamic forces for single-drum hoisting machines are determined by an analogous method.
In the calculations presented, the values of the maximum accelerations and acceleration jumps have been adopted on the basis of practical data for skip hoists of the Kuzbass, equipped with an asynchronous drive and a metallic rheostat. The possibility of starting with full utilization of the motor overload capacity has also been taken into account, when the accelerations reach their greatest values (on average, for the indicated hoists, \(1.5\text{--}2.0\ \mathrm{m/sec^2}\)). The design value of the rope slack has been taken as \(0.2\ \mathrm{m}\).
The formulas given may be used to determine the maximum dynamic forces when checking the strength of the mechanical parts of an operating hoisting machine. The initial data for the calculation are adopted on the basis of theoretical or experimental studies.
In order to limit possible overloads of the rope and reduce wear of the mechanical part of the installation in the second stage of loading, it is advisable to limit the value of the average acceleration to \(0.3\text{--}0.35\ \mathrm{m/sec^2}\), and the maximum to \(0.6\text{--}0.7\ \mathrm{m/sec^2}\), which should be taken into account when determining the resistance of the first running step of the rheostat.
Starting the motor with full utilization of the overload capacity meets with no fundamental objections if the acceleration jumps are then insignificant (of the order of \(0.6\text{--}0.7\ \mathrm{m/sec^2}\)). In this case it is advisable to provide two or three initial running steps of the rheostat in order to achieve the maximum accelerations with jerks not exceeding the indicated values.
The values of the maximum dynamic forces proposed in Table 1 are recommended for use in calculating the strength of the elements of two-drum hoisting machines.
Literature
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Florinskii F. V. Dynamics of a Mine Hoisting Rope. Ugletekhizdat, 1955.
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Georgievskaya V. V. Influence of rope slack on dynamic forces in them (second stage). Applied Mechanics, Academy of Sciences of the Ukrainian SSR, vol. II, no. 2, 1956.