On the Calculation of a Starting Rheostat for Mine Hoisting Machines
Abstract Generated abstract
This paper presents an analytical method for calculating stepped starting rheostats for asynchronous mine hoisting machines under contactor control, addressing errors and practical difficulties associated with earlier graphical methods based on rectified mechanical characteristics. The method specifies special resistance stages for smooth skip departure from and final approach to unloading curves, then divides subsequent stages to maintain constant minimum and maximum starting torques and equal torque jolts during switching. Formulae are derived for determining the critical slip, individual stage resistances, and the geometric progression used to distribute rotor circuit resistances. Application to a Kuzbass mine hoist reportedly reduced starting time, lowered current surges, and enabled motion through curves without auxiliary braking.
Full Text
On the Calculation of a Starting Rheostat for Mine Hoisting Machines
V. D. Petunov
(Presented by the scientific seminar of the Department of Mining Mechanics)
In order to eliminate jerks under contactor control, the stepped metallic rheostat and the control system must ensure a smooth departure and entry of the skip into the unloading curves, and, in cage hoisting, smooth acceleration and raising of the cage above the cams. For this purpose, Prof. I. A. Balashev proposed a grapho-analytical method for calculating starting resistances with allowance for the operating mode of the hoisting installation \[1\]. However, Prof. I. A. Balashev’s method assumes the use of rectified mechanical characteristics, which, in graphical construction, sometimes gives significant errors in determining the number of rheostat stages. If the number of stages is specified, then graphical subdivision is difficult.
In the present work an analytical method for calculating resistances is proposed. The essence of the method is as follows: 1) the specified speed of the skip’s departure from the curves and the speed of final approach are ensured by special rheostat characteristics; 2) after the skip leaves the curves, switching of the stages is carried out at constant values of the minimum and maximum starting torques.
The mechanical characteristics of the stages ensuring the departure of the skip from the curves and its final approach may be taken as rectified; in this case the calculation equations have a simpler form and give results quite suitable for practical calculations.
The equation of motion during the starting period of a hoisting machine with a constant winding radius is expressed, in relative units, as
\[ \gamma = \alpha \gamma_{\text{st.nach}} + Cj, \tag{1} \]
where \(\gamma\) is the torque developed by the motor;
\(\gamma_{\text{st.nach}}\) is the torque of the static resistance at starting, reduced to the motor shaft;
\(\alpha\) is a coefficient taking into account the additional resistances during motion of the skip in the unloading curves, which may be taken equal to \(1.15\text{–}1.20\);
\(j\) is the acceleration of the hoisting conveyance, \(\text{m}/\text{s}^2\);
\(C\) is a constant coefficient for the hoisting installation, equal to
\[ C = \frac{m D_{\sigma}}{2 M_{н} i^{2} \eta_{p}}, \tag{2} \]
where \(m\) is the mass of the moving parts of the hoisting machine reduced to the winding radius, \(kg\,sec^2/m\);
\[ D_b \]
is the diameter of the machine drum, \(m\);
\[ M_n \]
is the rated torque of the motor, \(kg\cdot m\);
\[ i \]
is the gear ratio of the reducer;
\[ \eta_r \]
is the efficiency of the reducer.
The moment of static resistance during the starting period may be taken as a constant quantity. Indeed, at maximum hoisting speeds in the Kuzbass of from 4 to \(8\,m/sec\), the hoisting conveyance, during starting, travels only a short distance, and the difference in the weights of the rope branches has almost no effect on the resisting torque.
The equation of the rectilinear mechanical characteristic may be expressed in relative units as
\[ \gamma=\frac{\gamma_m+\sqrt{\gamma_m^2-1}}{s_k}\cdot s, \tag{3} \]
where \(\gamma_m\) is the overload capacity of the motor;
\(s_k\) is the critical slip;
\(s\) is the slip corresponding to the torque developed by the motor.
Solving equations (1) and (3) simultaneously, we obtain
\[ s_k=s\,\frac{\gamma_m+\sqrt{\gamma_m^2-1}}{\alpha\gamma_{\mathrm{st.nach}}+Cj}. \tag{4} \]
Figure 1 shows the rectilinear mechanical characteristic at which the skip runs down from the curves. The initial
Fig. 1. Mechanical characteristic for the skip running down from the curves.
and final torques of the motor may be replaced by a constant mean value. In this case the run-down characteristic must pass through a point having a speed equal to one half of the run-down speed, and a torque equal to
\[ \gamma=\alpha\gamma_{\mathrm{st.nach}}+Cj_{\mathrm{skh}}, \tag{5} \]
where \(j_{\mathrm{skh}}\) is the mean acceleration during the motion of the skip in the curves during run-down, \(m/sec^2\).
The acceleration of descent is equal to
\[ j_{c.x}=\frac{2h_p}{v_{c.x}^{2}}, \tag{6} \]
where \(h_p\) is the travel of the skip in the unloading curves, m;
\(v_{c.x}\) is the adopted descent speed, m/sec.
The slip corresponding to half the descent speed is
\[ s'_{c.x}=\frac{v_0-0.5\,v_{c.x}}{v_0}, \tag{7} \]
where \(v_0\) is the hoisting speed corresponding to the synchronous speed of the motor.
Substituting equation (7) into equation (4), we obtain the critical slip for the descent characteristic:
\[ s_K=\frac{v_0-0.5\,v_{c.x}}{v_0}\cdot \frac{\gamma_m+\sqrt{\gamma_m^{2}-1}} {\alpha\gamma_{\mathrm{st.nach}}+Cj_{c.x}}. \tag{8} \]
The resistance of any rheostat step can be expressed by the relation
\[ R=\frac{s_K R_n}{\gamma_m+\sqrt{\gamma_m^{2}-1}}, \tag{9} \]
where \(R_n\) is the rated resistance, ohm.
Solving equations (8) and (9) simultaneously, we obtain the resistance of the step that ensures the descent of the skip from the curves:
\[ R_{c.x}=\frac{v_0-0.5\,v_{c.x}}{v_0}\cdot \frac{R_n}{\alpha\gamma_{\mathrm{st.nach}}+Cj_{c.x}}. \tag{10} \]
In an analogous manner, equations are derived for determining the resistance of the step that ensures the final approach of the skip into the curves:
\[ R_d=\frac{v_0-0.5v_d}{v_0}\cdot \frac{R_n}{\alpha\gamma_{\mathrm{st.kon}}+Cj_d}, \tag{11} \]
where \(v_d\) is the adopted final-approach speed, m/sec;
\(\gamma_{\mathrm{st.kon}}\) is the moment of static resistance at the end of motion, which is taken as constant for the entire stopping time.
If the number of contactors of the magnetic station is less than seven, or the total hoisting height is less than 100–150 m, the descent and final-approach characteristics must be combined.
The preliminary step is selected from the conditions of rope inspection and shaft revision. The torque developed by the motor on this step must be not less than 30% of the rated motor torque [2].
The resistances of all subsequent steps should be divided so that, at a constant switching torque, identical jolts of torque are obtained during shunting of the rheostat steps. In this case the step resistances will form a decreasing geometric-
geometric progression, whose first term will be the resistance of the characteristic ensuring the skip’s transition from the curves, and whose last term will be the resistance of a phase of the motor rotor. The common ratio of the geometric progression, equal to the resistance coefficient, is determined as
\[ \varphi=\sqrt[n-n_1]{\frac{R_{\text{rot}}}{R_{\text{c.x}}}}, \tag{12} \]
where \(n\) is the number of contactors of the magnetic station,
\(n_1\) is the number of preliminary stages (this also includes the stage for pulling up the skip),
\(R_{\text{rot}}\) is the internal resistance of a rotor phase, ohms.
The resistance of any stage is
\[ R_x=R_{x-1}\varphi=R_{\text{c.x}}\varphi^{x-n_1-1}, \tag{13} \]
where \(x\) is the ordinal number of the stage.
It should be borne in mind that by these formulas we obtain the total resistances included in one phase of the motor rotor. To determine the resistances of the individual sections it is necessary to find the differences between the resistances of adjacent stages.
\[ \Delta R_x=R_x-R_{x-1}. \tag{14} \]
At mine No. 8 of the “Prokop’evskugol’” trust of the “Kuzbassugol’” combine, the starting rheostat was modified according to a calculation performed by the method set out above. Whereas previously, at the 4th stage, a considerable surge of current occurred, with a total starting time of 14 sec, after the rheostat was modified the starting time became 11.5–12 sec, with smaller current surges and the same load. The current and speed diagrams obtained after modification of the rheostat are shown in Fig. 2. Owing to the softening of the mechanical characteristics, motion along curves became possible without braking, whereas previously, in order to eliminate large jerks, a mechanical brake was used to create an additional load.
Fig. 2. Current and speed diagrams during the starting of the skip hoist of mine No. 8.
LITERATURE
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Balashev I. A., Khrustalev I. K. Calculation of metal rheostats for asynchronous motors of mine hoisting machines. Tomsk, Polytechnic Institute, 1955.
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Vasilevskii M. N. Asynchronous drive of mine hoisting machines. Ugletekhizdat, 1953.