Transient Modes of an Induction Motor in Drives with an Electromagnetic Friction Clutch

M. F. Filippov

Introduction

All new and reconstructed enterprises of our socialist industry are characterized by a high degree of electrification, and the further development of the power supply of industrial enterprises will proceed along the path of an even deeper introduction of electric drive into all branches of the national economy.

The electric motor, as a converter of electrical energy into mechanical energy, is used for the most diverse actuating mechanisms.

The theory of electric drive, which deals with questions of the interaction between the motor and the actuating mechanism, is still a young branch of electrical engineering; it is developing continuously and rapidly, but the practical development of electric drive is proceeding at such a rapid pace that very many questions have not yet found corresponding reflection in theory.

Meanwhile, practice urgently demands the solution of these questions, since a rational choice of the elements of an electric drive is possible only when there is a clear understanding of the behavior of these elements in their interrelation, when it is known which factors have a positive effect on the operating conditions of the electric drive and which have a negative one.

The most important thing is not only to know these factors, but also to be able to evaluate them quantitatively.

As is known, for the proper selection of a motor according to power it is necessary to have, especially under intermittent-duty operating conditions, the graph of its load. Under intermittent-duty conditions with short cycles, the transient modes of the motor have a very great influence on the character of the load diagram, and recently, in the theory of electric drive, ever greater attention has been devoted to the question of transient modes.

The study of the transient modes of a motor in various installations is, at the present time, one of the most pressing problems in the theory of electric drive.

To take transient modes accurately into account and to create optimal conditions for them means to achieve maximum utilization of the equipment and to increase the productivity of the actuating mechanism.

The present work is devoted to the question of the transient modes of an induction motor in drives with electromagnetic clutches. This question has not yet received a proper solution.

KINEMATIC DIAGRAM

Figure 1 shows the kinematic diagram of a drive with a reversible electromagnetic friction clutch. For clarity in deriving the basic equations, the simplest possible diagram has been taken, but all the relationships derived below also apply to more complex diagrams.

In the clutch shown in Fig. 1, the left core of the clutch \(C_B\) serves for “forward” motion and is connected to the motor shaft through a gear transmission with gear ratio \(i_B\). The right core \(C_H\) is engaged during “reverse” motion and is connected to the motor shaft through an idler gear \(\delta\); the gear ratio during “reverse” motion is equal to \(i_H\).

Fig. 1.

When using the equations derived below to solve specific problems, it is necessary to take into account, when determining the “reduced” quantities, that the value of the gear ratio during “forward” motion and during “reverse” motion may be different.

STARTING PROCESS OF A DRIVE WITH AN ELECTROMAGNETIC FRICTION CLUTCH

Before the clutch is engaged, the electric motor operates at no load, rotating with angular velocity \(\omega_1\), and overcomes the moment \(M_1\)—the static moment of the resistances to rotation of the motor shaft and of the elements connected to it, i.e., of the entire nonreversing part of the drive. The cores—the left \(C_B\) and the right \(C_H\), connected to the motor shaft, rotate continuously in opposite directions. The disk \(D\), rigidly connected to shaft II, is stationary.

After the magnetizing coil of one of the clutch cores is switched on, for example \(C_B\), disk \(D\) is attracted to the rotating core, is accelerated by friction, and sets the working shaft (II) of the drive into rotation.

At first the angular velocity \(\omega_D\) of the disk is less than the velocity \(\omega_{CB}\) of the core \(C_B\), and, consequently, slip of the disk relative to the core takes place; in this case a friction torque \(M_{TD}\) arises between them. This torque, loading the motor, causes a decrease in the latter’s speed as the kinetic energy of the rotating masses of the nonreversing part of the drive is expended, and, consequently, also causes a decrease in

speed of the core \(C_B\). When the decreasing speed \(\omega_{CB}\) of the core and the increasing speed \(\omega_D\) of the disk reach the same value, the slip between the disk and the core will cease, i.e., rigid coupling of the clutch will occur.

After the rigid coupling of the clutch, when the friction torque \(M_TD\) disappears, the motor will begin to increase its speed to the value corresponding to the total torque of the static resistances to rotation of the reversing and non-reversing parts of the drive, i.e., at the end of acceleration

\[ M_{dv}=M_1+M_2, \]

where \(M_2\) is the torque of the static resistances to rotation of the elements connected with the disk shaft (the reversing part of the drive), referred to the motor shaft.

Thus, the starting process of a drive with an electromagnetic clutch can be divided into two periods.

The first period (the coupling process) takes time \(t_1\) (Fig. 2), during which the angular speed \(\omega_D\) of the disk is less than the speed \(\omega_{CB}\) of the core.

The second period, from the moment of complete coupling of the clutch to the end of acceleration, takes time \(t_2\); in this case the speed of the disk and the core are the same. The total duration of starting is

\[ t_n=t_1+t_2. \]

Fig. 2.

In the first period of the starting mode, corresponding to time \(t_1\), the equation of motion for the motor shaft will have the form

\[ M_{dv}=(M_1+M_T)+J_1\frac{d\omega}{dt}, \tag{1} \]

where \(\omega\) is the angular speed of the motor;
\(M_T\) is the friction torque in the clutch (between the disk and the core), referred to the motor shaft;
\(J_1\) is the moment of inertia of the parts continuously rotating together with the motor (the non-reversing part of the drive), referred to the motor shaft.

In this same first period of the starting mode, for the disk shaft, when it is accelerated by friction between the disk and the core, the equation of motion will be:

\[ M_T=M_2+J_2\frac{d\omega'_D}{dt}, \tag{2} \]

where \(\omega'_D\) is the angular speed of the disk, referred to the motor shaft;
\(J_2\) is the moment of inertia of the part of the drive connected with the disk shaft (the reversing part of the drive), referred to the motor shaft.

In the second period of the starting mode, during time \(t_2\), when the disk and the core, rigidly coupled, rotate at the same speed, the equation of motion common to the motor shaft and to the disk shaft will be

\[ M_{dv}=(M_1+M_2)+(J_1+J_2)\frac{d\omega}{dt}. \tag{3} \]

The values of the static moments of resistance to rotation, i.e., \(M_1\) and \(M_2\), as well as the values of the moments of inertia \(J_1\) and \(J_2\) entering into the equations of motion, may be regarded as constant.

The law of variation of \(M_{dv}\) as a function of the rotational speed, or of the slip \(s\), for a three-phase induction motor can be expressed by the well-known Kloss equation

\[ M_{dv} = \frac{2M_{opr}} {\dfrac{s_{opr}}{s} + \dfrac{s}{s_{opr}}}; \tag{4} \]

however, application of this equation excessively complicates the solution of the differential equations of motion.

In a drive with an electromagnetic clutch, when the motor load does not go beyond the working part of the characteristic \(M_{dv}=\varphi(s)\), it is quite possible to confine oneself to the equation

\[ M_{dv}=0.9\,\frac{M_H}{s_H}\cdot s = A\cdot s, \tag{5} \]

where \(M_H\) is the rated torque of the motor;

\(s_H\) is the slip under rated conditions.

As for the torque \(M_T\), caused by friction between the disk and the core during slipping, it depends on the current in the magnetizing coil of the core and on the coefficient of friction between the surfaces of the disk and the core.

Figure 3 shows the curve of the rise of current in the core coil after switching on. The delayed rise of current is caused by the self-inductance of the coil and by the presence of Foucault currents in the core. Point \(a\) corresponds

Fig. 3.

to the beginning of friction between the disk and the core, i.e., the disk is pressed against the core not immediately after the magnetizing coil is switched on, but after a time interval \(t_a\). During the time \(t_a\), the current in the coil, and consequently the magnetic flux, reaches a considerable value, close to the maximum value, and after contact of the disk with the core,

\(^{1)}\) Filippov M. F.—On the equation of the characteristic \(M=\varphi(s)\) of a three-phase induction motor. Izv. TII, vol. 59, issue II, 1941.

continuing to grow, quickly assumes a steady-state value. Thus, the clutch torque \(M_T\) has a steady-state value almost throughout the entire engagement process and, with accuracy sufficient for practice, it may be regarded as constant and equal to the maximum torque of the clutch, especially since excitation forcing is currently used in electromagnetic clutches.

It is, of course, assumed here that the voltage supplying the core coil and the condition of the rubbing surfaces do not change during the engagement process, which is in fact the case under normal clutch operation.

After substituting the value of the motor torque from equation (5) into equation (1), we obtain

\[ A s = (M_1 + M_T) + J_1 \frac{d\omega}{dt}. \tag{6} \]

It is more convenient to express the equation of motion in terms of slip, since slip expresses the relative change in rotational speed and, consequently, may refer to any shaft of the kinematic scheme of the drive. From the expression

\[ s = \frac{\omega_s - \omega}{\omega_s}, \]

we find

\[ \frac{d\omega}{dt} = -\omega_s \frac{ds}{dt}. \tag{7} \]

Consequently, equation (6) can be written in the form

\[ A s = (M_1 + M_T) - J_1 \omega_s \frac{ds}{dt}. \tag{8} \]

The solution of this differential equation gives

\[ s = C_1 e^{-\frac{t}{T_{k_1}}} + \frac{(M_1 + M_T)}{A}, \tag{9} \]

where \(T_{k_1} = \dfrac{J_1 \omega_s}{A}\) is the electromechanical time constant of the nonreversible part of the drive rotating with the clutch disengaged.

The integration constant \(C_1\) is easily found, knowing that at the initial instant, before the clutch is switched on, the motor slip is determined only by the torque of the static resistances to rotation \(M_1\), i.e., at \(t = 0\) we have

\[ s = s_1 \]

and

\[ C_1 = A s_1 - (M_1 + M_T); \]

but from equation (5)

\[ s_1 = \frac{M_1}{A}, \]

therefore

\[ C_1 = -M_T. \]

Consequently, the motor slip in the first period of the starting mode, i.e., during the engagement process, changes according to the equation

\[ s_1 = \frac{M_1 + M_T - M_T e^{-\frac{t}{T_{k_1}}}}{A} = \frac{M_T\left(1 - e^{-\frac{t}{T_{k_1}}}\right) + M_1}{A}. \tag{10} \]

The torque of the motor over this same period will be expressed by the equation

\[ M_{\mathrm{dv}}=A s_1=M_T(1-e^{-\frac{t}{T_k}})+M_1. \tag{11} \]

The obtained equations make it possible to construct the curve of variation of the motor slip, or of its torque, in the first period of the starting mode, i.e., over the time interval from 0 to \(t_1\).

As for the disk shaft, during this time it rotates at a speed different from the speed of the core connected with the motor, and the equation of motion of the disk shaft in this case, from equations (2) and (7), will take the form:

\[ M_T=M_2-J_2\omega_s\frac{d s_D}{dt}; \tag{12} \]

the solution of this equation gives

\[ s_D=\frac{M_2-M_T}{j_2\omega_s}\,t+C_2. \tag{13} \]

At the initial instant of starting the disk is stationary, i.e., at \(t=0\) we have \(s_D=1\), and therefore the constant of integration \(C_2=1\).

Thus, the equation for determining the slip of the disk in the first period of acceleration up to complete engagement of the clutch is

\[ s_{D1}=1-\frac{M_T-M_2}{j_2\omega_s}. \tag{14} \]

The change in motor slip according to equation (10) is presented in Fig. 4 (curve \(ak\)). The change in disk slip according to equation (14) is represented by the straight line \(mn\). For lack of space, the intersection of this straight line with the ordinate axis at the value \(s=1\) is not shown.

Fig. 4.

The first period of the starting mode (the engagement process) ends when the slips of the disk and the motor become equal, which corresponds to point \(k\), found graphically as the point of intersection of the curve \(ak\) and the straight line \(mn\).

The abscissa of point \(k\) indicates the duration of the engagement process, and its ordinate indicates the slip of the motor and disk at the instant of rigid engagement of the clutch.

A simultaneous solution of equations (10) and (14) for the analytical determination of \(t_1\), \(s_k\) is not possible, but \(t_1\) can be calculated with very high accuracy as the abscissa of point \(n\), i.e., the point of intersection of the straight line \(mn\) with the asymptote of the curve \(ak\). In Fig. 4, in addition to the straight line \(mn\), the straight lines \(n_1m_1\) and \(n_2m_2\) are shown, representing the change in disk slip for different values of the moment of inertia \(J_2\).

As can be seen from the figure, when the curve \(ak\) is far from the asymptote, the straight line \(mn\) has a very small slope, and the projection of point \(n\) onto the abscissa axis almost coincides with the projection of point \(k\). When, however, the straight line \(mn\) becomes noticeably inclined, the curve \(ak\) practically coincides with its asymptote \(nn_2\), and the points \(n\) and \(k\) coincide.

The equation of the asymptote \(mn_2\) is easily obtained from equation (10) for \(t=\infty\):

\[ s_{ac}=\frac{M_1+M_T}{A}. \tag{15} \]

Solving equations (14) and (15) jointly, we find

\[ t_1=\frac{J_2\omega_s}{M_T-M_2}\left(1-\frac{M_1+M_T}{A}\right); \tag{16} \]

substituting the obtained value \(t_1\) into equation (10), we find the value \(s_k\).

For the second period of the starting regime, from equations (3), (5), and (7) we obtain

\[ As=(M_1+M_2)-(J_1+J_2)\omega_s\frac{ds}{dt}; \tag{17} \]

the solution of this equation gives

\[ s=\frac{C_3 e^{-\frac{t}{T_{k_2}}}+(M_1+M_2)}{A}, \tag{18} \]

where

\[ T_{k_2}=\frac{(J_1+J_2)\omega_s}{A} \tag{19} \]

and represents the overall electromechanical time constant of the drive (with the clutch engaged).

To simplify the equations, the time count for the first and second periods is introduced separately.

The integration constant \(C_3\) in equation (18) is determined if one takes into account that at the beginning of the second starting period, i.e., at \(t=0\), we have \(s=s_k\):

\[ C_3=As_k-(M_1+M_2)=M_k-(M_1+M_2). \]

Consequently, in the second period of acceleration of the drive, the equation for determining the slip is expressed as

\[ s_{II}=\frac{[M_k-(M_1+M_2)]\cdot e^{-\frac{t}{T_{k_2}}}+(M_1+M_2)}{A}; \tag{20} \]

accordingly, the motor torque in this same period is expressed by the equation

\[ M_{\partial II}[M_k-(M_1+M_2)]\cdot e^{-\frac{t}{T_{k_2}}}+(M_1+M_2); \tag{21} \]

\(s_k\), and consequently also \(M_k\), can be found graphically as the ordinate of point \(k\) (Fig. 4) or analytically from equation (10) at \(t=t_1\), i.e.,

\[ s_k=\frac{M_T\left(1-e^{-\frac{t_1}{T_{k_1}}}\right)+M_1}{A}; \tag{22} \]

\(t_1\) is found from equation (16).

In Fig. 4, equation (20) is represented in the form of the curve \(kv\).

The duration of the second acceleration period \(t_2\) can also be determined graphically and analytically; here there may be three cases.

First case. If the sum of the static moments of resistance to rotation is less than the rated motor torque, i.e.,

\[ M_1+M_2<M_n, \]

then acceleration should be considered complete when the slip corresponding to the rated torque of the motor is reached, which in Fig. 4 corresponds to point \(b\). Taking \(t=t_2\), \(M_{\mathrm{rot}}=M_{\mathrm{N}}\) in equation (21) and solving it with respect to \(t_2\), we obtain

\[ t_2=T_{k2}\ln\frac{M_k-(M_1+M_2)}{M_N-(M_1+M_2)} . \tag{23a} \]

In the second case, when \(M_1+M_2>M_N\), steady motion will theoretically occur, as equation (20) shows, after an infinitely long interval of time; the slip asymptotically tends to the value

\[ s_{\mathrm{II}}=\frac{M_1+M_2}{A}. \]

In practice, acceleration may be considered complete when the exponent in equation (20) reaches the value

\[ -\frac{t}{T_{k2}}=3 \quad \text{i.e.} \quad t_2=3T_{k2}. \tag{23b} \]

The third case is possible when, in order to shorten the time of the entire drive operating cycle, the working mode is begun without waiting for the end of acceleration of the drive at no load, and the sum of all static torques in the working mode is greater than the rated motor torque, i.e.,

\[ M_1+M_2+M_c>M_N, \]

where \(M_c\) is the useful working static torque.

If the working mode is begun at a slip corresponding to the motor torque

\[ M_{\mathrm{rot}}=M_1+M_2+M_c, \]

then the duration of the second period will be

\[ t_2=T_{k2}\ln\frac{M_k-(M_1+M_2)}{M_c}. \tag{23c} \]

The total starting duration of a drive with an electromagnetic friction clutch is

\[ t_n=t_1+t_2. \]

• EQUATIONS OF THE REVERSING MODE

The reversing process of a drive with an electromagnetic friction clutch can be divided into three periods, with separate equations of motion for each of them (Fig. 5).

During the braking period (time \(t_T\)) the disk shaft is braked to a complete stop by the torque due to friction between the disk and the core and then, over the time \(t_1\), is accelerated by this same torque to the speed at which rigid engagement of the clutch occurs. Consequently, during the braking period and in the first acceleration period there is slip of the disk relative to the clutch core, while the motor reduces its rotational speed. After rigid engagement of the clutch, the motor accelerates again.

In the braking period the motor equation for the disk shaft is written as:

\[ M_T+M_2=J_2\frac{d\omega'_D}{dt} \]

or

\[ M_T+M_2=-J_2\omega_s\frac{ds_D}{dt}; \tag{24} \]

the solution of this equation gives

\[ s_{DT}=C_4-\frac{(M_T+M_2)}{J_2\omega_s}t. \tag{25} \]

Before braking begins, the disk was rotating at speed \(\omega_D\); after the clutch is switched on for reverse, the disk is pressed against the other core, and if the transmission ratios from the cores to the motor shaft are not the same, then the slip \(s_{D_2}\) of the disk relative to the new synchronous speed of the core rotating in the opposite direction can be determined from the equation

\[ s_{D_2}=\frac{\omega_s+\frac{\omega_{D_2}}{i}}{\omega_s}, \tag{26} \]

where \(i\) is the transmission ratio from the motor shaft to the clutch core that comes into operation during reversing (braking).

Fig. 5.

It should be firmly kept in mind that, when changing over to operation of another core, it is necessary to change the reduced values of \(M_T\), \(M_2\), and \(J_2\) in accordance with the new transmission ratio, and, consequently, the value of the electromechanical constant \(T_k\) will also change.

The integration constant \(C_4\) in equation (25) is found knowing that at the initial instant of braking, i.e., at \(t=0\), \(s_D=s_{D_2}\); therefore \(C_4=s_{D_2}\), and the slip equation for the disk shaft during the braking period will take the form:

\[ s_{DT}=s_{D_2}-\frac{M_T+M_2}{J_2\omega_s}t. \tag{27} \]

The equation of the disk slip during acceleration from rest to the speed corresponding to rigid engagement of the clutch was found earlier:

\[ s_{DI}=1-\frac{M_T-M_2}{J_2\omega_s}t. \tag{14} \]

Time is counted separately for each period. In Fig. 6, the change in disk slip according to equation (27) is represented by the straight line \(qr\), and according to equation (14) by the straight line \(pm\).

As for the motor, its speed during braking and acceleration of the disk in the reverse direction up to rigid engagement of the clutch decreases—

...creases. The equation for determining the motor slip is analogous to equation (9) and has the form:

\[ s=\frac{C_5 \cdot e^{-\frac{t}{T_{k_1}}}+(M_1+M_T)}{A}. \]

We find the constant \(C_5\) from the condition that, before reversing, the motor slip was determined by the sum of the static moments resisting rotation, i.e., at \(t=0\) we have

\[ s=s_2=\frac{M_1+M_2}{A}, \]

then

\[ C_5=M_2-M_T \]

and the motor-slip equation, covering the period of braking and acceleration during the clutch engagement process, will be

\[ s_I=\frac{(M_T+M_1)-(M_T-M_2)\cdot e^{-\frac{t}{T_{k_1}}}}{A}. \tag{28} \]

The curve expressed by equation (28) is shown in Fig. 6 as curve \(ak\). The time count in equation (28) is common for the period of braking and acceleration of the disk up to rigid engagement of the clutch.

The motor torque during the clutch engagement process in reversing will vary according to the equation

\[ M_{dvI}=(M_T+M_1)- \]

\[ -(M_T-M_2)\cdot e^{-\frac{t}{T_{k_1}}} \tag{28a} \]

The braking duration \(t_T\) is found from equation (27), substituting the value of the slip at the end of braking \(s_{D_T}=1\), i.e.,

\[ t_T=(s_{D_2}-1)\frac{J_2\omega_s}{M_T+M_2}. \tag{29} \]

The total duration of the clutch engagement process is represented by the abscissa of point \(k\).

As in the case of the starting mode, it may be assumed that the abscissas of points \(k\) and \(n\) are practically equal, and then, using the equation of the asymptote of curve \(ak\) and equation (14), we find

\[ \frac{M_1+M_T}{A}=1-\frac{M_T-M_2}{J_2\omega_s}(t_k-t_T), \]

whence

\[ t_k=t_T+\left(1-\frac{M_1+M_T}{A}\right)\frac{J_2\omega_s}{M_T-M_2}. \tag{30} \]

Fig. 6.

The duration of acceleration of the disk up to the end of the engagement process is

\[ t_1=t_k-t_T=\left(1-\frac{M_1+M_T}{A}\right)\frac{J_2\omega_s}{M_T-M_2}. \tag{31} \]

For the third period of reversal of the drive, i.e., during acceleration after rigid engagement of the clutch, the equation of motion was given earlier (equation 17), whose solution is represented in the form:

\[ S_{II}=\frac{\left[M_k-(M_1+M_2)\right]e^{-\frac{t}{T_{k3}}}+(M_1+M_2)}{A}. \tag{20} \]

The torque of the motor in this same period is determined by equation (21). The values \(s_k\) and \(M_k\) during reversal are found from equations (28) and (28a) after substituting into them the value \(t=t_k\), determined from equation (30), or else are determined by graphical construction, as the ordinates corresponding to point \(k\).

When determining the duration of acceleration after rigid engagement of the clutch during reversal, the considerations stated with respect to equations (23a), (23b), and (23c) should be taken into account.

The total duration of reversal is

\[ t_p=t_k+t_2=t_T+t_1+t_2. \]

EXPERIMENTAL VERIFICATION OF THE EQUATIONS

For experimental verification of the results of this work, a direct-action electromagnetic friction clutch (non-reversing) was used.

The general appearance of the clutch is shown in Fig. 7. The core of the clutch is secured with a tight fit on the shaft of a three-phase induction motor. The motor has the following data:

\[ M_{\mathrm{н}}=2.25\ \text{kgm};\qquad S_{\mathrm{н}}=0.053;\qquad \omega_s=105;\qquad \lambda=2.44. \]

In the experiments, oscillograms of the starting regime were obtained (Figs. 8 and 9), \(\omega=f(t)\).

The oscillations superimposed on the principal curve are due to the fact that a UM-21 motor was used as the tachometric generator, and its toothed fields caused additional oscillations of the oscillograph loop.

The oscillograms (Figs. 8 and 9) clearly show the presence of two characteristic periods in the starting regime, and that neither of these periods may be neglected when determining the duration of the transient regime, as was done in the latest (1938) edition of Prof. S. A. Rinkevich’s book Theory of Electric Drive and in T. N. Sokolov’s work “Analytical Investigation of Transient Processes in Drives with Electromagnetic Clutches,” published in the proceedings of the Leningrad Industrial Institute for 1938, No. 2.

Figure 9 presents an oscillogram of the starting regime with the motor characteristic softened by additional resistance in the rotor. From this latter oscillogram it is evident that the motor characteristic strongly affects the nature and duration of the transient process. The softer the motor characteristic, the earlier complete engagement of the clutch occurs, but the total duration of the transient process increases.

Fig. 7.

Fig. 8.

To verify whether the equations derived above reflect the actual picture of the transient processes, let us determine the duration of the individual periods of the clutch starting mode from equations (16) and (23c). For the experimental setup the following data are available:

\[ M_T = 7.0\ \text{kgm}; \qquad M_1 = 0.21\ \text{kgm}; \qquad M_2 = 0.29\ \text{kgm}; \]

\[ J_1 = 0.021\ \text{kgm sec}^2; \qquad J_2 = 0.007\ \text{kgm sec}^2. \]

Consequently,

\[ A = 0.9 \frac{M_H}{S_H} = 0.9 \frac{2.25}{0.053} = 38.2; \]

\[ T_{k_1} = \frac{J_1 \omega_s}{A} = \frac{0.021 \cdot 105}{38.2} = 0.058; \]

\[ T_{k_2} = \frac{(J_1 + J_2)\omega_s}{A} = \frac{0.028}{38.2} = 0.0765. \]

Fig. 9.

Substituting the specified values into equations (16) and (23c), we obtain

\[ t_1 = \frac{J_2 \omega_s}{M_T - M_2} \left( 1 - \frac{M_1 + M_T}{A} \right) = \frac{0.007 \cdot 105}{7.8 - 0.29} \cdot \left( 1 - \frac{0.21 + 7.8}{38.2} \right) = \]

\[ = 0.077\ \text{sec}. \]

\[ t_2 = 3T_{k_2} = 3 \cdot 0.0765 = 0.23\ \text{sec}. \]

The results obtained differ little from the experimental ones (oscillogram in Fig. 8).

CONCLUSION

Summing up, the following conclusions may be drawn:

1. Transient modes (starting, reversing) in a drive with an electromagnetic clutch consist of two characteristic periods (the process of engagement and acceleration after engagement), and when determining the duration of transient modes both periods should be taken into account to an equal extent, rather than neglecting either of them, as has been done up to now.

2. The character and duration of transient modes are greatly influenced by the motor characteristics, which in the existing methods for determining the duration of transient modes were either not taken into account at all or were taken into account to an insufficient degree.

3. The equations proposed in this work, which correctly reflect the physical aspect of the phenomena of transient modes, give results that differ only insignificantly from the actual ones.

4. Until now the question of the character of the motor load during transient modes in a drive with an electromagnetic clutch had not been resolved. The equations obtained in this work make it possible, with accuracy sufficient for practice, to construct the motor load curve, which is of very great importance in drives with a short cycle duration.

5. The analytical expressions obtained for transient modes make it possible, in each individual case, to correctly resolve the question of the rational choice of the parameters of the motor and the clutch.