Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov
Vol. 173　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1970

ON THE ISSUE OF ESTIMATING LOAD DISTRIBUTION IN THE MULTI-MOTOR DRIVE OF AN OVERHEAD CONVEYOR

A. V. MURIN

(Presented by the scientific seminar of the Department of Applied Mechanics)

As is known, the actual values of the nominal slips of asynchronous electric motors may differ noticeably from the data given in catalogs. Thus, according to GOST 183-55, for asynchronous squirrel-cage motors a deviation of the nominal slip of up to 25% is permitted. Moreover, the standard limits only the greatest limiting value of the slip. Its smallest limiting value is not established by the standard. It is practically difficult to select electric motors having the same nominal slip. Therefore the electric motors used in multi-motor systems usually have nominal slips that differ somewhat in magnitude, which in turn leads to a noticeable nonuniformity in the distribution of load among them.

An estimate of the influence of motor parameters (including deviations in the values of nominal slip) on load distribution in a two-motor asynchronous drive is given in [1]. In that work, this estimate was carried out with allowance for a certain nonlinearity of the working portion of the mechanical characteristic of an asynchronous electric motor. A general solution of this problem for a number of motors greater than two, so far as we know, is not given in the literature. Below we consider the determination of the nonuniformity of load distribution in the multi-motor drive of an overhead conveyor, caused by differences in the nominal slips of the electric motors for any number of them.

The installed power of the drive motors of overhead conveyors, for various reasons, is usually higher than required, which ensures operation of the electric motors of such a drive at slips not exceeding the nominal ones. In this connection, when solving the stated problem with accuracy sufficient for engineering practice, the indicated portion of the working characteristic of the electric motor may be considered linear and described by the known relation

\[ M = \frac{M_{\mathrm{n}}}{s_{\mathrm{n}}}\cdot s \tag{1} \]

or, in relative units,

\[ \varphi = \frac{1}{s_{\mathrm{n}}}\cdot s, \tag{1'} \]

where \(M_{\mathrm{n}}\) is the nominal torque,

\(s_{\mathrm{n}}\) — nominal slip,
\(s\) — current slip.

The magnitude of the relative error in determining the slip from this dependence, as compared with the results of calculating it from an expression that takes into account a certain nonlinearity of the working portion of the characteristic of an induction motor,

\[ \psi=\frac{2\psi_{\mathrm{k}}}{\dfrac{s}{s_{\mathrm{k}}}+\dfrac{s_{\mathrm{k}}}{s}}, \]

is, for values of the moments \(\psi<\psi_{\mathrm{n}}\), about 3%. If a protective hydrodynamic coupling is present in the drive, expression (1) can also be used to describe the working portion of the drive characteristic, since at moments up to \((1.3—1.5)M_{\mathrm{n}}\) this part of its characteristic can, with sufficient accuracy, be considered linear. Therefore, the results obtained by us for a multi-motor system with induction electric motors will also be valid for the case when the indicated couplings are used in the drive.

For the steady-state operating mode of a multi-motor system, the equation of motion in dimensionless quantities may be represented as

\[ \sum_{i=1}^{i=n}\psi_i-\sum_{i=1}^{i=n}\psi_{ci}=0, \tag{2} \]

where \(\psi_i=\dfrac{M_i}{M_{\mathrm{n}1}}\) is the moment developed by the corresponding drive motor;

\[ \psi_{ci}=\frac{M_{ci}}{M_{\mathrm{n}1}} \]

is the moment of the resistance forces of each branch of the conveyor, reduced to the shaft of the drive motor;

\(n\) is the number of drive motors (conveyor branches);
\(M_{\mathrm{n}1}\) is the nominal moment of the first motor.

Using expression (1), the moment of the \(i\)-th motor can be represented as

\[ \psi_i=\beta_i\frac{s_i}{s_{\mathrm{n}i}}, \]

where \(\beta_i=\dfrac{M_{\mathrm{n}i}}{M_{\mathrm{n}1}}\) is the relative nominal moment of this motor.

Then equation (2) can be written in the following form:

\[ \sum_{i=1}^{i=n}\beta_i\frac{s_i}{s_{\mathrm{n}i}}-\sum_{i=1}^{i=1}\psi_{ci}=0. \tag{3} \]

Proceeding from the assumption, introduced by V. K. D’yachkov [2] and verified experimentally by him and by other researchers, of equality of the average velocities of motion of any points of the closed chain traction element, and also assuming equal diameters of the pitch circles of the driving sprockets, it is possible from expression (3) to determine the average slip of each motor of a multi-motor overhead conveyor:

\[ s=\frac{\displaystyle\sum_{i=1}^{i=n}\psi_{ci}} {\displaystyle\sum_{i=1}^{i=n}\frac{\beta_i}{s_{\mathrm{n}i}}}. \]

Taking into account a certain possible fluctuation of the speed of the \(i\)-th motor about its mean value, the last expression can be written as follows:

\[ s = \frac{\displaystyle \sum_{i=1}^{i=n} \psi_{ci}} {\displaystyle \sum_{i=1}^{i=n} \frac{\beta_i}{s_{\mathrm{н}i}}} (1 \pm \delta_s), \]

where \(\delta_s = \dfrac{\Delta s}{s}\) is the relative change in the motor slip.

Then the expression for the torque with which any drive motor is loaded during conveyor operation will have the following form:

\[ \psi_i = \frac{ \displaystyle \beta_i (1 \pm \delta_{si}) \sum_{i=1}^{i=n} \psi_{ci} }{ \displaystyle s_{\mathrm{н}i} \sum_{i=1}^{i=n} \frac{\beta_i}{s_{\mathrm{н}i}} }. \tag{4} \]

Using, to estimate the nonuniformity of load distribution among the individual electric motors of a multi-motor system, the criterion proposed by V. K. Dyachkov [3], we have

\[ \Delta_i = \left( \frac{1}{n} - \frac{\psi_i}{\displaystyle \sum_{i=1}^{i=n} \psi_{ci}} \right)\cdot 100, \]

here \(\Delta_i\) is the deviation from uniform load distribution for the \(i\)-th motor, expressed as a percentage. Taking expression (4) into account, the deviation from uniform distribution for the \(k\)-th motor will be

\[ \Delta_k = \left[ \frac{1}{n} - \frac{\beta_k (1 \pm \delta_{sk})} {\displaystyle s_{\mathrm{н}k} \sum_{i=1}^{i=n} \frac{\beta_i}{s_{\mathrm{н}i}}} \right]\cdot 100. \tag{5} \]

From expression (5), having adopted a certain permissible value of the nonuniformity in load distribution \(\Delta_k\), one can obtain a relation for determining the value of the nominal slip of the last, \(n\)-th drive motor, given the known nominal slips of the remaining motors. The slip of the \(n\)-th motor in this case is equal to

\[ s_{\mathrm{н}n} = \frac{s_{\mathrm{н}k}\beta_n} { \displaystyle \frac{\beta_k(1 \pm \delta_{sk})} {\frac{1}{n} - \frac{\Delta_k}{100}} - \sum_{i=1}^{i=n-1} \beta_i \frac{s_{\mathrm{н}k}}{s_{\mathrm{н}i}} }. \tag{6} \]

In this expression \(\Delta_k\) is in percent. It is obvious that the value of the nominal slip of the last (\(n\)-th) motor must be determined on the basis of the permissible nonuniformity for each of the \(n-1\) motors. Comparing the obtained values of \(s_n\) will make it possible to identify the value of this slip at which the least nonuniformity of load distribution among the motors of the multi-motor system will be ensured.

Conclusions

The relationships obtained make it possible:

1. To estimate the magnitude of the nonuniformity of load distribution among individual motors (and, in the presence of fluid couplings, among the drives) of a multi-motor overhead conveyor, caused by differences in the actual values of their nominal slips.

2. Given \(n-1\) motors (fluid couplings) having somewhat different values of nominal slip, to select the final, \(n\)-th motor (fluid coupling) of the multi-motor system such that the least nonuniformity of load distribution will be ensured.

References

1. V. V. Poltev. Influence of motor parameters on load distribution in a two-motor synchronous drive. Izv. vuzov. Gornyi zhurnal, No. 6, 1965.

2. V. K. D’yachkov. Overhead Conveyors. Mashgiz, 1961.

3. V. K. D’yachkov. Factory tests of overhead conveyors with multi-motor drive. VNIIPTMASH. Collection Overhead Cableways, No. 22, 1958.