DETERMINATION OF THE ACCELERATION TIME OF AN ELECTRIC DRIVE WITH A SAFETY TURBO COUPLING

A. V. MURIN

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

In a number of cases it is necessary to know the acceleration time and the nature of the change in speeds during this period for a drive with a safety turbo coupling. This is required, for example, when determining the productivity of a machine, selecting an electric motor, or assessing the thermal regime of the turbo coupling. Such calculations are especially necessary for drives of machines that operate with frequent starts and stops.

B. A. Gavrilenko and V. A. Minin [1] give an analytical calculation of the starting conditions of a drive with a turbo coupling having folding blades at the turbine wheel. For this purpose they use its static mechanical characteristic, linear over the entire range of slips. The driven machine is started, when such a coupling is present, by turning the blades; therefore the authors perform the calculation of the starting processes for the case in which the pump wheel has a definite rotational speed, and they do not consider the acceleration period of the driving motor. In the same work an analytical study is made of the transient regimes of a drive with a turbo coupling, taking into account its dynamic properties. The dynamic properties of the coupling are taken into account by introducing into the equation of motion the parameters of the hydraulic flow in the working cavity for the unsteady motion of its wheels. The equation of motion, taking into account the flow parameters under unsteady motion, is obtained for a single-cavity turbo coupling with a torus whose working wheels have symmetrical profiles. The authors also analyze the solution of the resulting equation for certain types of load produced by the driven machine for the case in which the pump wheel has a definite speed and the mechanical characteristic of the turbo coupling is linear.

In our opinion, calculation of transient regimes with allowance for the flow parameters for a safety turbo coupling with asymmetrical wheels and additional volume on the side of the pump wheel cannot be carried out because of the complexity of the processes in its working cavity during this period. With sufficient accuracy such a calculation can be performed by graphical methods and, in some cases, also by analytical methods, using the static mechanical characteristic of the turbo coupling. In [2] an indication is given for solving this problem by a graphical method. In our view, preference should be given to the analytical method, since the quantities obtained in solving this …

task dependencies ensure a lower labor requirement for practical calculations in such cases.

The use of safety turbo couplings is most expedient in electric drives with a squirrel-cage induction motor, especially when type A electric motors are used, for which the starting torque in most cases is equal to, or differs only slightly from, the rated torque. Therefore, below we present a calculation of the acceleration time and the determination of the laws of variation of angular velocities during this period for a drive with a squirrel-cage induction motor and a safety turbo coupling, carried out by an analytical method. The calculation is performed taking into account the acceleration time of the motor.

The study of the acceleration process of the drive motor can be performed most correctly when electromagnetic and mechanical processes are considered jointly. At present it has been established that the mechanical characteristics of induction motors under large accelerations differ significantly from characteristics taken in steady-state operation. However, in most cases the electromagnetic processes in an induction electric drive proceed much faster than the mechanical ones. The calculation of dynamic regimes in this case can be made using the static characteristic of the electric motor. The question of taking electromagnetic transients into account when calculating the starting regime of an electric drive can be resolved, for example, using the criterion given in [3]. In that work it is emphasized that, when calculating the acceleration process, the electromagnetic process may be neglected if the mechanical time constant of the drive is 2–3 times greater than the electromagnetic one. Calculations carried out by us show, for example, that all induction motors of type A satisfy this condition (even without taking into account the moment of inertia of the rotating masses of the drive connected to the motor). In view of the above, in calculating the motor acceleration process we used its static mechanical characteristic.

Figure 1 gives a diagram of an electric drive with a turbo coupling. In this diagram the following are denoted:

\(M_1\) and \(\omega_1\)—the torque and angular velocity of the electric motor;

Fig. 1. Diagram of a drive with a turbo coupling.

\(I_1\)—the total moment of inertia of the motor rotor, the connecting elements between it and the coupling, and the pump wheel with part of the working fluid;

\(M_2\)—the torque created by the resistance forces of the system;

\(\omega_2\)—the angular velocity of the turbine-wheel shaft;

\(I_2\)—the total moment of inertia, reduced to the turbine-wheel shaft, of the machine links set in motion and of the turbine wheel with part of the working fluid;

\(M\) is the torque transmitted by the turbo coupling.

The moment of inertia of the fluid filling the coupling, in translational motion (about the axis of rotation of the coupling), may be taken into account by increasing the moment of inertia of the corresponding wheel by 3–10% [1].

The acceleration of a system with a turbo coupling (Fig. 1) may be represented as the following processes:

1. Acceleration of the motor and of the pump wheel connected to it to the angular velocity \(\omega_{1D}\) with the turbine wheel stationary. At the speed \(\omega_{1D}\), the torque \(M\) produced by the turbo coupling becomes equal to the torque of the resistance forces of the driven system \(M_2\) (Fig. 2, a).

2. Continued acceleration of the motor to the speed \(\omega_{1p}\), corresponding to the working portion of the characteristic, and the beginning of acceleration of the driven system from \(\omega_2 = 0\) to \(\omega_2 = \omega_{2-2}\).

3. Completion of acceleration of the motor to the rated speed \(\omega_{1n}\) and of the driven system to \(\omega_{2n}\). During this period the speed of the electric motor changes only slightly, by approximately 5–10% (since its acceleration takes place on the working portion of the characteristic). For the driven system this period corresponds to a significant change in speed and is the main one, especially for large values of \(I_\text{e}\) (Fig. 2, b).

The acceleration process of a system with a turbo coupling can be described by two equations of motion (in the form of Lagrange equations). For rotational motion, in the case of a constant moment of inertia, the equation of motion for the driving part of the drive will have the form

\[ M_1 - M = I_1 \frac{d\omega_1}{dt} \tag{1} \]

and for the driven part

\[ M - M_2 = I_2 \frac{d\omega_2}{dt}. \tag{2} \]

Determination of the Acceleration Time of the Electric Motor

The acceleration time of the motor can be determined by solving equation (1), provided that analytical expressions are known for its torque \(M_1 = f(\omega_1)\) and for the torque transmitted by the turbo coupling, \(M = \varphi(\omega_1)\). In calculating transient processes in electric drives with a squirrel-cage induction motor, it is recommended [4] to approximate the starting and working portions of its characteristic (curves 1 and 2, Fig. 2, a) by parabolas (curves 3 and 4). Thus, the motor torque on the starting portion of its characteristic, up to the critical speed \(\omega_{1k}\), may be represented in the form

\[ M_1 = a + h\omega_1^2, \tag{3} \]

and on the working portion from \(\omega_{1k}\) to \(\omega_{1c}\),

\[ M_1 = a_1 + h_1\omega_1^2. \tag{4} \]

Here \(a\), \(a_1\), \(h\), \(h_1\) are constant quantities.

If the nonworking portion of the characteristic has the form represented by curves 5 and 6 in Fig. 2, a, then in this case also, in our opinion, a similar approximation may be made and the motor torque expressed by relations of the form (3). For a motor characteristic in which the minimum torque \(M_{1\min}\) is less than the initial torque \(M_{1\text{init.}}\) (curve 6), the nonworking portion of the characteristic should be replaced by two parabolas of the form (3). Therefore the relations obtained below, as applied to the variant of the nonworking portion shown by curve 1, will also be valid for motors with characteristics 5 and 6.

Fig. 2. Mechanical characteristic of an asynchronous electric motor (a); the nature of the change in the speeds of the driving and driven shafts of the turbo coupling during acceleration (b), and its generalized mechanical characteristic (c).

The torque transmitted by the turbo coupling during motor acceleration can be expressed with sufficient accuracy as

\[ M=b\omega_1^2 . \tag{5} \]

In this expression \(b\) is a constant quantity.

Substituting the expressions for \(M_1\) and \(M\) into equation (1), we obtain, for the starting portion of the motor characteristic (from \(\omega_1=0\) to \(\omega=\omega_{1k}\)),

\[ a+h\omega_1^2-b\omega_1^2=I_1\frac{d\omega_1}{dt}. \]

After separation of variables,

\[ dt=-\frac{I_1}{b-h}\cdot \frac{d\omega_1}{\dfrac{a}{b-h}-\omega_1^2}. \]

Denoting \(\dfrac{a}{b-h}=\kappa^2\) and integrating over this interval, taking into account the initial conditions at \(|\omega_1|<\kappa^{1)}\), we have

\[ t_k=\frac{I_1}{2\kappa(b-h)}\cdot \ln\frac{\kappa+\omega_{1k}}{\kappa-\omega_{1k}} . \tag{6} \]

Here \(t_k\) is the time of motor acceleration to the critical speed \(\omega_{1k}\). The angular velocity of the motor on this interval varies according to

\[ \omega_1=\frac{\kappa\left(e^{\theta t}-1\right)}{e^{\theta t}+1}, \tag{7} \]

where \(\Theta=\dfrac{2\kappa(b-h)}{I_1}\) is a constant quantity.

Let us find the motor acceleration time on the interval from \(\omega_1=\omega_{1k}\) to \(\omega_1=\omega_{1p}\) (Fig. 2, \(a\)). For this purpose, in equation (1) it is necessary to substitute the value of \(M_1\) from expression (4), which is analogous in form to expression (3). The form of the equation obtained as a result of this substitution and its solution will be the same as in the preceding case. Thus, the motor acceleration time on this interval is, for \(|\omega_1|>\kappa_1\),

\[ t_{1p}-t_k= \frac{I_1}{2\kappa_1(b-h_1)} \left[ \ln\frac{\omega_{1p}+\kappa_1}{\omega_{1p}-\kappa_1} - \ln\frac{\omega_{1k}+\kappa_1}{\omega_{1k}-\kappa_1} \right]. \tag{8} \]

In this expression

\[ t_{1p} \]

is the total time of motor acceleration to the speed \(\omega_{1p}\),

\[ \kappa_1=\sqrt{\frac{a_1}{b-h_1}} \]

is a constant quantity.

If it turns out that \(|\omega_1|<\kappa_1\), then in expression (8) it is necessary to interchange \(\omega_1\) and \(\kappa_1\). The motor speed on this interval varies according to the law

\[ \omega_1= \frac{\kappa_1\left[ce^{\theta_1(t-t_k)}+1\right]} {ce^{\theta_1(t-t_k)}-1}, \tag{9} \]

where \(\theta_1\) and \(c\) are constant quantities equal to

\[ \theta_1=\frac{2\kappa_1(b-h_1)}{I_1} \quad \text{and} \quad c=\frac{\omega_{1k}+\kappa_1}{\omega_{1k}-\kappa_1}. \]

\(^{1)}\) Calculations show that on this interval \(|\omega_1|<\kappa\). For \(|\omega_1|>\kappa\), in expression (6) it is necessary to interchange \(\omega_{1k}\) and \(\kappa\).

Thus, the acceleration time of the motor from the starting instant until it reaches the working portion of the characteristic is equal to the sum of the right-hand sides of expressions (6) and (8).

In view of the small changes in motor speed during acceleration on the working portion of its characteristic (from $\omega_1=\omega_{1о}$ to $\omega_1=\omega_{1н}$), this period may be disregarded. If it is necessary to study the motor acceleration process from $\omega_{1р}$ to $\omega_{1н}$, it should be borne in mind that during this period the motor torque is approximately equal to the torque transmitted by the coupling. This is explained by the fact that the right-hand side of equation (1) is small on this section (or even equal to zero) owing to the insignificant change in the motor speed $\omega_1$ and the duration of this period. Thus, the duration of motor acceleration on the working portion of its characteristic is determined mainly by the parameters of the driven system $(I_2, M_2)$ and by the mechanical characteristic of the coupling.

The properties of safety turbo couplings, including their starting properties, are usually evaluated from their mechanical characteristics obtained by braking the turbine wheel [1, 2, 5, 6, 7, 8, etc.]. However, as our investigations have shown [9], the characteristics of a safety turbo coupling with asymmetrical wheels and an additional volume on the pump-wheel side, obtained during braking and release (beginning with $\omega_2=0$), differ substantially. In this connection, for calculations of starting processes of drives with such a coupling, one should use the characteristics obtained when its turbine wheel is released. Figure 2, c shows a typical characteristic of this coupling when its driven shaft is released. Obviously, by linearizing the actual curve, dividing it into sections, the coupling characteristic can always be reduced to the form shown in Fig. 2, c (dashed line). Since the motor acceleration time on this section is determined by the acceleration duration of the driven system, then, having determined the time of its acceleration on each section of the coupling characteristic and, from the coupling torque, also the motor speed, we thereby learn the nature of motor acceleration on the working branch of its characteristic.

The values of the constants entering equations (3) and (4) may be determined by substituting into them the known values of $M_1$ and $\omega_1$ for the particular points of the motor characteristic [4]. The constant in equation (5) may be determined from the value of the starting torque $M^1_{\text{start}}$ (Fig. 2, c) transmitted by the coupling at 100% slip, and from the speed of the drive motor at this torque. Determining the constant from the value of $M^1_{\text{start}}$ is justified by the fact that the torque transmitted by the coupling in the region of large slips practically does not change [9].

Determination of the Acceleration Speed of the System at the End of Motor Acceleration

1. Purely inertial load

In this case the beginning of system acceleration coincides with the beginning of motor acceleration. In equation (2) we substitute the value of $M_t$ from (5). The equation of motion for this case will have the form

$$ b\omega_1^2 = I_2 \frac{d\omega_2}{dt}. $$

Here we replace $\omega_1$ by expressions (7, 9) and integrate. Taking into account the initial conditions, we obtain the speed of the system at the end of the section from $\omega_1=0$ to $\omega_1=\omega_{1к}$ (Fig. 2, a)

\[ \omega_{2-1} = \frac{b\kappa}{I_2}\left[t_k - \frac{2\left(e^{\theta t_k}-1\right)}{\theta\left(e^{\theta t_k}+1\right)}\right]. \tag{10} \]

Integration is carried out similarly over the interval from \(\omega_1=\omega_{1k}\) to \(\omega_1=\omega_{1p}\). The speed of the turbine wheel of the coupling at the end of the motor acceleration, when it develops the speed \(\omega_{1p}\), is equal to

\[ \omega_{2-2}=\omega_{2-1}+\frac{b\kappa_1}{I_2}\left\{(t_p-t_k)-\frac{4c\left[e^{\theta_1(t_p-t_k)}-1\right]}{\theta_1(c+1)\left[ce^{\theta_1(t_p-t_k)}+1\right]}\right\}. \tag{11} \]

If in expression (10) \(t\) is substituted for \(t_k\), and in expression (11) \(t\) for \(t_p\), then these dependences will represent the change in the speed \(\omega_2\) on these intervals.

2. Inertial load in the presence of a constant component

The equation of motion for this case will have the form

\[ b\omega_1^2-D=I_2\frac{d\omega_2}{dt}, \]

where \(D\) is the static component of the drive load.

In the presence of a constant component in the load, acceleration of the system will obviously begin at \(M=M_2=D\), but \(M=b\omega_{1D}^2\). Then the motor speed at the beginning of acceleration of the system is

\[ \omega_{1D}=\sqrt{\frac{D}{b}}. \]

Knowing \(\omega_{1D}\) and using expression (6) for \(\omega_{1D}<\omega_{1k}\), or (6) and (8) for \(\omega_{1D}>\omega_{1k}\), we find the delay time of the start of motion of the system (relative to the start of motor acceleration), \(t_D\). Then the law of variation of the motor speed from the moment the system begins accelerating, for \(\omega_{1D}<\omega_{1k}\), over the interval from \(\omega_1=\omega_{1D}\) to \(\omega_1=\omega_{1k}\), is

\[ \omega_1=\frac{\kappa\left[e^{\theta(t_D+t)}-1\right]}{e^{\theta(t_D+t)}+1}, \]

and over the interval from \(\omega_1=\omega_{1k}\) to \(\omega_1=\omega_{1p}\),

\[ \omega_1=\frac{\kappa_1\left[ce^{\theta_1(t-t'_k)}+1\right]}{ce^{\theta_1(t-t'_k)}-1}, \]

where \(t'_k=t_k-t_D\) (Fig. 2, b).

The latter expression will also be valid for the case when \(\omega_{1D}>\omega_{1k}\).

Substituting into the equation of motion the angular speed \(\omega_1\) obtained from these dependences and solving the differential equations, we find the speed of the system during motor acceleration at the end of the interval from \(\omega_1=\omega_{1D}\) to \(\omega_1=\omega_{1k}\):

\[ \omega_{2-1}-\omega_{2-0}= \frac{b\kappa}{I_2} \left[ (t'_k-t_0)+\frac{4}{\theta} \left( \frac{1}{e^{\theta(t_D+t'_k)}+1} - \frac{1}{e^{\theta(t_D+t_0)}+1} \right) \right] - \frac{D}{I_2}(t'_k-t_0), \]

but

\[ t'_k=t_k-t_D,\qquad \omega_{2-0}=0,\qquad t_0=0. \]

Then, at the motor speed \(\omega_1=\omega_{1k}\), the system speed (for \(\omega_{1D}<\omega_{1k}\)) is

\[ \omega_{2-1}= \frac{b\kappa}{I_2} \left[ (t_k-t_D)+\frac{4}{\theta} \left( \frac{1}{e^{\theta t_k}+1} - \frac{1}{e^{\theta t_D}+1} \right) \right] - \frac{D}{I_2}(t_k-t_D), \tag{12} \]

and at the motor speed \(\omega_1=\omega_{1p}\) (for \(\omega_{1D}<\omega_{1k}\))

\[ \omega_{2-2}=\omega_{2-1}+\frac{bk_1}{I_2}\left\{(t_{\mathrm p}-t_k)+\frac{4}{\theta_1}\left[\frac{1}{ce^{\theta_1(t_{\mathrm p}-t_k)}-1}-\frac{1}{c-1}\right]\right\} -\frac{D}{I_2}(t_{\mathrm p}-t_k). \tag{13} \]

If it turns out that \(\omega_{1D}>\omega_{1k}\), then the speed of the system when the motor reaches the speed \(\omega_{1\mathrm p}\) will be

\[ \omega_{2-1}=\frac{bk_1}{I_2}\left\{(t_{\mathrm p}-t_D)+\frac{4}{\theta_1}\left[\frac{1}{ce^{\theta_1(t_{\mathrm p}-t_D)}-1}-\frac{1}{c-1}\right]\right\} -\frac{D}{I_2}(t_{\mathrm p}-t_D), \tag{14} \]

but in this case, obviously, \(\omega_{2-1}\) is small and may be neglected. The laws of variation of the system speed during acceleration in the sections under consideration can be obtained from expressions (12–14), if \(t\) is substituted for \(t_k\) in (12) and for \(t_{\mathrm p}\) in (13, 14).

In the case of a load that depends on speed in various ways, the problem can be reduced to the case considered above by replacing the actual load curve with a stepped one having averaged values of the torque at each step.

Determination of the drive acceleration time after acceleration of the motor

The time spent on accelerating the system after the motor has reached the speed corresponding to the operating section of its characteristic is determined by means of equation (2). Taking into account the adopted method of simplifying the mechanical characteristic of the coupling, we find the acceleration time of the system in each section for certain types of load. The beginning of the first section on the coupling characteristic is determined by the speed \(\omega_{2-2}\), which the system acquires by the end of acceleration of the motor.

1. Inertial load in the presence of a constant component

The equation of motion for this case will have the form

\[ M-D=I_2\frac{d\omega_2}{dt}, \]

where \(D\) is the constant component of the load. Integrating this equation over the section of the coupling characteristic from \(\omega_2=\omega_{2-2}\) to \(\omega_2=\omega_{2-3}\) (Fig. 2, \(в\)), we have

\[ t_1-t_{\mathrm p}=\frac{I_2}{M-D}(\omega_{2-3}-\omega_{2-2}), \]

where

\[ M=M'_{\text{start}} \]

is the torque transmitted by the turbo coupling in the section under consideration;

\(t_1\) is the acceleration time of the system up to the speed \(\omega_{2-3}\) from the beginning of motor acceleration.

The change in speed in this section is described by the expression

\[ \omega_2=\omega_{2-2}+\frac{M-D}{I_2}(t-t_{\mathrm p}). \]

Analogous expressions for determining the time and the character of the change in the system speed during acceleration can also be written for the remaining

sections of the coupling characteristic. The total acceleration time of the system, obviously, will be equal to

\[ t_c=t_p+\sum (t_i-t_{i-1}), \tag{15} \]

where \(t_i-t_{i-1}\) is the acceleration time of the system on each of the considered sections of the coupling characteristic.

2. Load depending on speed, in the presence of a constant component

For this case the equation of motion has the form

\[ M-D-A\omega_2=I_2\frac{d\omega_2}{dt}. \]

This equation is easily reduced to the form

\[ \frac{d\omega_2}{dt}+z\omega_2=p, \]

where

\[ z=\frac{A}{I_2};\qquad p=\frac{M-D}{I_2}. \]

Integration over the section up to \(\omega_2=\omega_{2-3}\) gives, for \(\dfrac{p}{z}>|\omega_2|\),

\[ t-t_p=\frac{1}{z}\ln\frac{p/z-\omega_{2-2}}{p/z-\omega_{2-3}}. \]

For this section the value of \(p\) is determined at \(M=M'_{\text{start}}\). The change in system speed during acceleration on the section under consideration obeys the law

\[ \omega_2=\frac{p}{z}-\frac{p/z-\omega_{2-2}}{e^{z(t-t_p)}}. \]

The expressions obtained for \(t\) and \(\omega_2\) can be used to determine the time and nature of the acceleration of the system on all the remaining sections of the coupling characteristic, if the corresponding values \(M_i\), \(t_i\), \(\omega_{2-i}\) are substituted into them. The total acceleration time of the system is determined from expression (15).

3. Load depending on the square of the angular velocity, in the presence of a constant component

For such a load case the equation of motion will have the form

\[ M-D-A\omega_2^2=I_2\frac{d\omega_2}{dt}. \]

This equation can be reduced to the form

\[ \frac{d\omega_2}{dt}+z\omega_2^2=p, \]

where

\[ z=\frac{A}{I_2},\qquad p=\frac{M-D}{I_2}. \]

The solution of this equation on the section up to \(\omega_2=\omega_{2-2}\) has the following form, for \(\dfrac{p}{z}>|\omega_2|\),

\[ t_1-t_p=\frac{1}{2\sqrt{pz}}\left[ \ln\frac{p/z+\omega_{2-3}}{p/z-\omega_{2-3}} -\ln\frac{p/z+\omega_{2-2}}{p/z-\omega_{2-2}} \right]. \]

In this equation \(p\) is determined at \(M=M'_{\text{start}}\).

The speed of the system during acceleration on this segment varies according to the relation

\[ \omega_2 = \frac{p\left[m e^{\sqrt{pz}(t-t_p)}-1\right]} {z\left[m e^{\sqrt{pz}(t-t_p)}+1\right]}, \]

where

\[ m=\frac{p/z+\omega_{2-2}}{p/z-\omega_{2-2}}. \]

The acceleration time and the nature of the change in speed on the other segments of the coupling characteristic can be determined with the aid of the expressions obtained for the first segment, if the corresponding values \(M_i\), \(t_i\), \(\omega_{2-i}\) are substituted into them. The total acceleration time of the system is determined from expression (15).

Conclusions

1. The proposed method makes it possible to calculate the transient process of an electric drive with a safety turbo coupling, taking into account the time and character of motor acceleration.

2. The method can be used in calculating the starting regimes of electrohydraulic drives with all the most commonly used types of asynchronous squirrel-cage motors.

References

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2. Sturm K. — D. Anlauf- und Sicherheitskupplungen in Antrieben von Förderbändern und anderen Steigförderern. Bergbautechnik, 1956, No. 4.

3. E. Ya. Kazovskii. Some questions concerning transient processes in direct-current machines. Gosenergoizdat, 1953.

4. A. T. Golovan. Fundamentals of Electric Drive. Gosenergoizdat, 1959.

5. Selection of the type of drive for coal machines operating under various conditions. Brief scientific report. Ugletekhizdat, 1957. (All-Union Scientific Research Coal Institute).

6. Hydraulic transmissions of face machines and conveyors. Brief scientific report. Ugletekhizdat, 1958. (All-Union Scientific Research Coal Institute).

7. V. M. Bermn et al. On the question of the application of safety turbo couplings. Ugol’, No. 9, 1958.

8. D. M. Belen’kii et al. Bench tests of the “Voith” traction hydraulic coupling. Scientific Proceedings of the Karaganda Scientific Research Institute. Issue 4. Gostoptekhizdat, 1959.

9. A. V. Murin. On the influence of the position of the axis of rotation of a hydraulic coupling on its characteristics. Izvestiya TPI, vol. 96, part 2, 1961.