INFLUENCE OF THE MOMENT OF INERTIA AND THE RESISTANCE TORQUE OF THE DRIVE ON THE TRANSIENT PROCESSES OF STARTING AND REVERSING ROLLER-TABLE MOTORS

E. V. Kononenko, B. A. Danchinov

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

Depending on the mechanism being driven, the motor may operate with various moments of inertia referred to its shaft and various load resistance torques. Therefore, an assessment of the influence of different flywheel masses and resistance torques in asynchronous motors, in particular in roller-table motors, is of practical interest.

These studies were carried out on an analog computer of the MN-14 type for a roller-table motor having the following basic parameters in relative units: \(r_s = 0.0545\), \(r_r = 0.206\), \(x_m = 2.05\), \(J_{\mathrm{rot}} = 61.5\). In the calculations, the equations and model circuit given in [1] were used.

The use of analog computing machines for these purposes eliminates the influence of a number of random factors that are unavoidable in experimental studies, and makes it possible to carry out investigations quickly and simply with variation of the drive parameters over wide ranges.

In the general case, the equation of motion of an electric drive has the form:

\[ M - M_c = J_\Sigma \frac{d\omega}{dt} + \frac{\omega}{2}\cdot\frac{dJ_\Sigma}{dt}, \tag{1} \]

where \(M\) is the driving torque of the motor,
\(M_c\) is the load resistance torque,
\(J_\Sigma\) is the total moment of inertia of the drive, referred to the motor shaft,
\(\omega\) is the angular velocity of the motor,
\(t\) is time.

The right-hand side of equation (1) consists of two terms: the first is associated with the change in the speed of motion, and the second is caused by the change in the kinetic energy of the system due to the change in the moment of inertia as a function of the angle of rotation.

In roller-table drives the moment of inertia is constant and does not depend on the angle of rotation of the roller-table rollers; consequently, the second term on the right-hand side of equation (1) drops out. In this case the equation of motion of the drive takes the form:

\[ M - M_c = J_\Sigma \frac{d\omega}{dt}. \tag{2} \]

The static load torque \(M_c\) for different working machines has different patterns of variation. The main factors on which it dep—

depends on the magnitude of the resisting torque are speed, distance, time, and the characteristics of the technological process itself.

In accordance with the existing classification of actuating mechanisms according to the nature of the static torque, the drives of the rollers of working and transport roller tables should be assigned to the class of mechanisms with a constant resisting torque [2]. Therefore, in studying the influence of the resisting torque on transient processes, it was assumed that \(M_c=\mathrm{const}\). In addition, in roller-table drives the load resisting torque is reactive, changing its sign when the direction of motion changes. In this connection, in the model circuit [1] provision is made for changing the sign of \(M_c\) depending on the sign of the speed, by means of the operational relay OPR-1.

Figures 1 and 2 present oscillograms of starting and reversing for different values of the drive moment of inertia referred to the motor shaft, where

\[ k_J=\frac{J_\Sigma}{J_{\mathrm{rot}}} \]

is the ratio of the total moment of inertia \(J_\Sigma\) to the rotor moment of inertia \(J_{\mathrm{rot}}\).

Fig. 1. Oscillograms of starting with different values of the reduced moment of inertia \((M_c=0)\):
a — \(J_\Sigma=0.5J_{\mathrm{rot}}\);
b — \(J_\Sigma=J_{\mathrm{rot}}\);
c — \(J_\Sigma=2J_{\mathrm{rot}}\);
d — \(J_\Sigma=5J_{\mathrm{rot}}\).

From the oscillograms it is evident that, as the drive moment of inertia increases, the number of torque pulsations in the initial stage of the transient condition increases; this is explained by the greater influence of electromagnetic transient processes caused by the appearance of free components of the transient torque. As is known [3], the damping coefficients of these torque components, as well as their frequencies, vary depending on the rotor speed; moreover, as the speed increases, the coefficients of the rapidly damped components decrease, while those of the slowly damped components, which exert the main influence on the transient processes, increase. Therefore, the greater the drive moment of inertia, the more slowly the rotor speed rises; the motor operates longer in the region of low speeds; the electromagnetic transient processes proceed for a longer time, which ultimately leads to a larger number of pulsations of the transient torque.

Despite the increasing torque pulsations, the oscillations of the rotor rotational speed decrease, since the considerable flywheel masses smooth out these oscillations.

Fig. 2. Oscillograms of reversal with different values of the referred moment of inertia (\(M_c=0\)).
a — \(J_\Sigma=0.5J_{\mathrm{rot}}\);
b — \(J_\Sigma=J_{\mathrm{rot}}\);
c — \(J_\Sigma=2J_{\mathrm{rot}}\);
d — \(J_\Sigma=5J_{\mathrm{rot}}\).

At the end of the transient process, as the moment of inertia of the drive increases, pulsations in the torque curve disappear, since the motor speed increases ever more slowly.

With small flywheel masses, the motor rotor receives a large acceleration (deceleration), and as a result of the occurrence of free currents caused by the change in speed, alternating pulsations arise in the torque curve at the end of the transient process, especially when \(k_J<1\). At the same time, significant oscillations are observed in the speed curve, corresponding to the change in transient torque.

The practically impossible case in which \(k_J<1\) gives an indication of the nature of the qualitative and quantitative changes in the transient process and may be used at the stage of motor design.

The quantitative changes in the transient characteristics are shown in Fig. 3, from which it is seen that the time of the transient regime (for the starting time \(t_p\) and the reversing time \(t_r\), the time taken for the rotor to reach a speed equal to 0.95 of the steady-state value is adopted) increases almost in proportion to the total moment of inertia.

The impact values of the transient torque in the starting regime at \(k_J>2.5\div3\) may be considered practically independent of the added masses. This can be explained by the same dependence of the damping coefficients on the rotational speed.

With significant flywheel masses, the rotor speed increases so slowly that during the time in which the transient torque reaches its first maximum (0.005–0.008 sec) it changes very insignifi-

Fig. 3. Dependences of the impact current and torque and of the time of the transient modes on the reduced moment of inertia \((M_s = 0)\). The indices denote: \(s\) — starting, \(r\) — reversing, \(b\) — plugging braking.

ingly. Therefore, the damping coefficients also change very insignificantly, and consequently, all other conditions being equal, the values of \(M_{\mathrm{imp}}\) almost do not depend on the magnitude of the flywheel masses.

A completely different picture is observed at \(k_J < 2.5\): a decrease in \(J_\Sigma\) leads to a sharp decrease in \(M_{\mathrm{imp}}\). The point is that, with small flywheel masses, the motor rotor receives a large acceleration and, during the time in which the torque reaches its maximum, the speed has time to change very substantially, and by different amounts for different values of \(J_\Sigma\). The resulting increase in the damping coefficients of the slowly decaying free components of the transient torque leads to a more rapid damping of the electromagnetic transient process and, as a result, to a decrease in the impact torques.

The difference in the values of the impact torques at different \(J_\Sigma\) in this case is determined by the difference between the initial speed and the speed at which the torque reaches its impact value.

In the reversing mode, constancy of \(M_{\mathrm{imp}}\) is observed at \(k_J < 1.5\). This can be explained by the fact that, during reversing, even with relatively small flywheel masses, at the initial instant the rotor retains an approximately constant speed for longer because of the kinetic energy acquired by the drive.

For the same reasons one can explain the practical constancy of the impact currents for any values of the total moment of inertia.

Figures 4 and 5 show oscillograms, and Fig. 6 shows the dependences of the transient characteristics during starting and reversing of the motor with various values of the load-resistance torque.

During starting (Fig. 4), an increase in the load leads to a change in the nature of the transient processes analogous to that caused by an increase in the moment of inertia: in the initial stage of starting, torque pulsations with considerable amplitudes arise, which last until the rotor reaches a speed equal to \(0.4 \div 0.5\) of synchronous speed; at the end of the transient process, slight torque oscillations are observed, which decrease with increasing load and then disappear; the number of large-amplitude pulsations increases with increasing load, and the damping of the pulsations slows down. The reasons for such changes in the nature of the transient processes when the load torque increases are the same as when the moment of inertia increases.

Fig. 4. Starting oscillograms with various values of the load torque \((J_\Sigma = 2J_{\text{rot}})\). a — \(M_c = 0\); b — \(M_c = 0.5\); c — \(M_c = 1.0\); d — \(M_c = 1.5\)

When starting the motor under load, the rotor accelerates with some oscillations, since, during the pulsating variation of the motor torque, in the intervals of dips in its curve the load-resistance torque becomes greater than the motor driving torque. The greater the resistance torque, the greater these oscillations.

During reversal (Fig. 5), damping of the torque pulsations occurs much more slowly than during starting; under the action of its braking torque and the load-resistance torque, the motor rotor decelerates sharply; reversal to the opposite direction occurs with a small (depending on the magnitude of \(M_c\)) acceleration and oscillations.

The magnitude of the load on the motor shaft, both during starting and during reversal, has practically no effect on the magnitude of the surge currents and torque (Fig. 6); the greater \(M_c\), the shorter the braking time of the opposing—

Fig. 5. Oscillograms of reversing with different values of load torque \((J_{\Sigma}=2J_{\mathrm{rot}})\):
\(a\)—\(M_c=0\); \(b\)—\(M_c=0.5\); \(c\)—\(M_c=1.0\); \(d\)—\(M_c=1.5\).

Visible plot labels: \(M\), \(n\); scale \(0.2\) sec; \(M, n\), p.u.

Fig. 6. Dependences of the peak current and torque and of the durations of the transient modes on the load torque \((J_{\Sigma}=2J_{\mathrm{rot}})\).

Visible plot labels: \(t\), sec; \(M_{\mathrm{ud}}^{p}\), \(M_{\mathrm{ud}}^{n}\), \(i_{\mathrm{ud}}^{p}\), \(i_{\mathrm{ud}}^{n}\), p.u.; \(M_c\), p.u.; \(t_p\), \(t_n\), \(t_T\).

by the current \(t_T\); the starting time \(t_{\text{п}}\) and the total reversing time \(t_p\), with an increase in \(M_c\), at first increase slowly (up to \(0.5 \div 0.6\) p.u.), and then rapidly.

Thus, it may be said that an increase in \(J_\Sigma\) and \(M_c\) leads to an increase in the duration of transient modes, while the impact values of currents and torques, for practically possible values of \(J_\Sigma\) and \(M_c\), remain constant; the transient processes of starting and reversing, at large flywheel masses and loads, are accompanied by the occurrence of considerable and prolonged pulsations of the electromagnetic torque, which adversely affects the operation of the kinematic links of the drive, reducing its reliability and service life.

REFERENCES

1. E. V. Kononenko, B. A. Danchinov. Study of the electromechanical transient processes of roller-table motors at the AVM, taking saturation of the steel into account, by scattering fields. Article in the present collection.

2. V. P. Andreev, Yu. A. Sabinin. Fundamentals of Electric Drive. GEI, 1963.

3. M. M. Sokolov, L. P. Petrov, L. B. Masandilov, V. A. Ladenzon. Electromagnetic Transient Processes in an Asynchronous Electric Drive. “Energiya,” 1967.