On Determining the Maximum Forces in the Brake Rods of Mine Hoisting Machines
Abstract Generated abstract
This paper addresses the calculation of maximum forces in the brake rods of mine hoisting machines, a requirement for assessing brake component strength and maximum braking torque. It distinguishes statically determinate and indeterminate brake systems, derives formulas for service and safety braking that account for rod geometry, drive force, and dynamic loading, and relates indeterminate-system calculations to experimentally measured nonuniform force distribution. Based on indicator and strain-gauge measurements, it recommends approximate dynamic coefficients for different brake and damping arrangements, and notes that unequal rod loading can produce severe overloads in faulty or poorly adjusted systems. The authors emphasize periodic monitoring of rod tensions, particularly after brake linkage adjustment, to improve safe operation.
Full Text
On the Determination of Maximum Forces in the Brake Rods of Mine Hoisting Machines
N. I. Klykov, V. S. Udut
(Presented by the scientific seminar of the Department of Mining Machines, Mine Transport, and Mining Mechanics)
The maximum forces in the brake rods of mine hoisting machines must be taken into account both when checking the strength of the parts of braking devices and when determining the greatest braking torque.
To determine the maximum forces in the rods, let us divide braking systems into: 1) statically determinate (Figs. 1 and 2) and 2) statically indeterminate (Fig. 2).
Fig. 1. Statically determinate braking system of a hoisting machine
In statically determinate braking systems, no more than two forces are applied to each lever, not counting the reaction at the support point (shoes with angular displacement and an individual drive; shoes with angular displacement and a common drive with one central
rod, shoes with parallel displacement without separation of the force flows on the levers). In this case, the maximum forces in the rods are determined from the moment equations with respect to the support points of the corresponding levers; moreover, the calculation should be begun with the lever to which, of the two, one known force is applied (the total pressure of air or oil on the piston, the weight of the brake load). The arms of the corresponding forces are determined from the dimensions of the levers and the position of the rods at the end of braking. In the case of dynamic application of loads, it is necessary to introduce a correction factor \(K_d\), whose value can be determined experimentally. For example, for the brake system shown in Fig. 1, under service braking,
Fig. 2. Statically indeterminate brake system of a hoisting machine
\[ F_{\max\cdot 1}= \frac{K_d m p_{\max}\cdot \pi D_{\mathrm{cyl}}^{2}} {4(m+n)\cos \varphi_1}, \tag{1} \]
\[ F_{\max\cdot 2}= \frac{K_d a m p_{\max}\cdot \pi D_{\mathrm{cyl}}^{2}} {4b(m+n)\cos \varphi_2}, \tag{2} \]
where \(F_{\max\cdot 1}\), \(F_{\max\cdot 2}\) are the greatest forces, respectively, in the first and second brake rods;
\(p_{\max}\) is the maximum pressure of air or oil in the cylinder of the service brake;
\(D_{\mathrm{cyl}}\) is the diameter of the cylinder of the service brake;
\(\varphi_1\) is the angle between the vertical and the first rod;
\(\varphi_2\) is the angle between the horizontal and the second rod;
\(a, b, m, n\) are the dimensions of the brake levers;
\(K_d\) is the dynamic factor (the ratio of the maximum force to the force established at the end of braking).
In emergency braking, using a falling load as the drive, the greatest forces in the rods for the brake system shown above are found, respectively, from the formulas
\[ F_{\max\cdot 1}= \frac{K_d m Q_m} {n\cos \varphi_1}, \tag{3} \]
\[ F_{\max.2}=\frac{K_{\mathrm{d}} a m Q_{\mathrm{m}}}{b n \cos \varphi_2}, \tag{4} \]
where \(Q_{\mathrm{m}}\) is the weight of the braking load;
\(K_{\mathrm{d}}\) is the dynamic coefficient during safety braking.
The dynamic coefficient, both in service braking and in safety braking, was determined by various experimental methods: recording diagrams with the aid of the brake indicator of Prof. I. A. Balashev; oscillograph recording with the aid of an N-10 loop oscillograph and wire strain gauges glued directly to the rods.
According to the experimental data, the dynamic coefficient during service braking is \(K_{\mathrm{d}} = 1.0\text{–}1.1\). The presence of large clearances in the joints, increased wear of the brake shoes, and other technical faults considerably increase this coefficient. Fig. 3 shows a diagram of forces in a brake rod during service braking with the use of a pneumatic drive.
Fig. 3. Diagram of forces in the brake rod during service braking (eastern skip hoist of mine 9-15, Anzherougol trust)
Fig. 4. Diagrams of forces in the brake rods during safety braking: 1 — skip hoist of the Salair mine; 2 — skip hoist of the Tsentralnaya mine; 3 — skip hoist of the “Dalniye Gory” mine of the Kiselevskugol trust; 4 — laboratory hoisting installation.
The dynamic coefficient during safety braking depends not only on the clearances in the joints and the degree of wear of the brake shoes, but also on the nature of damping of the forces when the braking load falls. In Fig. 4 are given diagrams of forces in the brake rods during safety braking with different degrees of damping.
On the basis of experimental data, the average values of the dynamic coefficient during safety braking may be taken as:
\[ \begin{aligned} K_{\mathrm{d}} &= 1.0\text{–}1.1 &&\text{— hydraulic brake;}\\ K_{\mathrm{d}} &= 1.1\text{–}1.2 &&\text{— pneumatic brake;}\\ K_{\mathrm{d}} &= 1.3\text{–}1.5 &&\text{— oil and spring dampers;}\\ K_{\mathrm{d}} &= 2 &&\text{— falling load without pneumohydraulic drive and damper.} \end{aligned} \]
In statically indeterminate systems (shoes with angular displacement and a common drive with two central rods, shoes with pa-
with parallel displacement and with division of the force flows on the levers), the forces in the rods may be distributed in different ways, depending on the adjustment of the rod lengths, the presence of clearances, misalignments, etc.
Thus, for the brake system shown in Fig. 2, the force in the common vertical rod during service braking can be determined from formula (1), and during emergency braking from formula (3). But to find the forces in the other rods it is necessary to determine experimentally the relative deformations, stresses, or forces in two identical rods (for comparison). For example, Fig. 5 gives diagrams of the forces in the upper (above the drum) and lower (below the drum) brake rods. Having determined from these diagrams the ratio of the smaller stress to the larger, or of the smaller force to the larger,
Fig. 5. Diagrams of forces in the brake rods of the skip hoisting machine of the “Ziminka—Kapital’naya” mine of the “Prokop’evskugol’” trust: 1 — upper horizontal rod; 2 — lower horizontal rod
\[ \lambda=\frac{\sigma'_{\max\cdot 3}}{\sigma''_{\max\cdot 3}} \tag{5} \]
\[ \lambda=\frac{F'_{\max\cdot 3}}{F''_{\max\cdot 3}}, \tag{6} \]
one can find the forces in all the rods. These forces during service braking are determined from the formulas
\[ F'_{\max\cdot 2}= \frac{K_{\mathrm{d}}\,l''\,p_{\max}\cdot \pi D_{\mathrm{cyl}}^{2}\lambda\cos\varphi_{4}} {4c\,l\left(\lambda\cos\varphi_{4}+\cos\varphi_{5}\right)\cos\varphi_{2}}, \tag{7} \]
\[ F''_{\max\cdot 2}= \frac{K_{\mathrm{d}}\,m l p_{\max}\cdot \pi D_{\mathrm{cyl}}^{2}\cos\varphi_{5}} {4c n\left(\lambda\cos\varphi_{4}+\cos\varphi_{5}\right)\cos\varphi_{3}}, \tag{8} \]
\[ F'_{\max\cdot 3}= \frac{K_{\mathrm{d}}\,a m l p_{\max}\cdot \pi D_{\mathrm{cyl}}^{2}\lambda} {4bc n\left(\lambda\cos\varphi_{4}+\cos\varphi_{5}\right)}, \tag{9} \]
\[ F''_{\max\cdot 3}= \frac{K_{\mathrm{d}}\,a m l p_{\max}\cdot \pi D_{\mathrm{cyl}}^{2}\lambda} {4bc n\left(\lambda\cos\varphi_{4}+\cos\varphi_{5}\right)}, \tag{10} \]
where \(a, b, c, m, n, l\) are the dimensions of the levers (Fig. 2);
\(\varphi_1, \varphi_2, \varphi_3\) are the angles between the rods and the vertical;
\(\varphi_4, \varphi_5\) are the angles between the rods and the horizontal;
\(K_{\mathrm{d}}\) is the dynamic coefficient during service braking (see above);
\(p_{\max}\) is the maximum pressure of air (or oil) in the service-brake cylinder;
\(D_{\mathrm{cyl}}\) is the diameter of the service-brake cylinder;
\(F'_{\max\cdot 2},\ F''_{\max\cdot 2}\) are the greatest forces, respectively, in the upper and lower vertical rods;
\(F'_{\max\cdot 3},\ F''_{\max\cdot 3}\) are the same in the horizontal rods.
We call these rods horizontal and vertical only conventionally, since in the general case the angles \(\varphi_1, \varphi_2, \varphi_3, \varphi_4, \varphi_5\) are not equal to zero.
and must be taken into account for a more accurate determination of the forces in the rods.
The forces in the rods for the brake system shown in Fig. 2, under safety braking, are found from the formulas:
\[ F'_{\max \cdot 2}= \frac{K_d m l Q_m \cdot \lambda \cos \varphi_4} {c n \left(\lambda \cos \varphi_4+\cos \varphi_5\right)\cos \varphi_2}, \tag{11} \]
\[ F''_{\max \cdot 2}= \frac{K_d m l Q_m \cdot \cos \varphi_5} {c n \left(\lambda \cos \varphi_4+\cos \varphi_5\right)\cos \varphi_3}, \tag{12} \]
\[ F'_{\max \cdot 3}= \frac{K_d a m l Q_m \cdot \lambda} {b c n \left(\lambda \cos \varphi_4+\cos \varphi_5\right)}, \tag{13} \]
\[ F''_{\max \cdot 3}= \frac{K_d a m l Q_m} {b c n \left(\lambda \cos \varphi_4+\cos \varphi_5\right)}. \tag{14} \]
In practice, other variants of statically indeterminate systems are also encountered, using hydraulic, pneumatic, or weight drives both for service and for safety braking. In such cases the forces in the rods can be determined by analogous formulas.
The experiments carried out showed that in normally operating brake systems \(\lambda = 0.8\text{--}0.9\). In emergency cases the value of this coefficient may decrease to zero, which corresponds to an almost twofold overload of some rods and complete unloading of others. Such a nonuniform distribution of forces, together with other unfavorable factors (excessive clearances, sticking of the load with its subsequent breakaway, incomplete contact of the brake shoes with the rim, etc.), may cause a serious accident in the hoisting installation. Therefore special attention should be paid to the loading of the rods during service and safety braking, using various means of monitoring rod tensions—strain gauging, installation of pointer deformation indicators, etc. It is especially necessary to monitor the distribution of forces in the rods after adjustment of their length.
Conclusions
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For a more accurate determination of the maximum forces in the brake rods of mine hoisting machines, it is necessary to take into account the position of the rods at the end of braking, the dynamic nature of load application, and, in statically indeterminate systems, also the nonuniformity of the distribution of forces in the rods.
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The coefficient of dynamic load application, as well as the coefficient of nonuniformity of force distribution in the rods, are determined experimentally. In the absence of experimental data, we recommend using the above-mentioned values of these coefficients.
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If it is necessary to check the strength of brake rods under emergency loads, the value of the dynamic coefficient should be taken as maximal, and the coefficient of nonuniformity as equal to zero.
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For normal operation of braking devices, it is necessary to introduce periodic monitoring of brake-rod tensions (use of deformation indicators, strain gauging), especially after adjustment of the lever brake system.