ANALYSIS OF THE OPERATION OF SYNCHRONOUS MOTORS UNDER VIBRATION LOAD

G. A. SIPAILOV, V. N. DMITRIEV, V. F. KULAKOV

Presented by the scientific seminar of the departments of electrical machines and general electrical engineering

Vibration technology, in particular the modern technique of geophysical seismic exploration, poses the problem of exciting oscillations coordinated in time, with regulation of frequency and amplitude. At frequencies from units to hundreds of hertz, the most effective and simplest device for creating powerful synchronized oscillations is a centrifugal vibrator driven by a synchronous motor. The choice of such a vibration-drive system is determined by the possibility of obtaining circular and polarized oscillations, by the purity of the harmonic spectrum, by the low weight of centrifugal vibration exciters in combination with motors that ensure in-phase operation and synchronism of oscillations, by high efficiency, and by other advantages.

In comparison with linear-polarized vibrators of electrodynamic and induction systems, in centrifugal vibrators coordination with the characteristics of the vibrating medium is usually accomplished without additional transmission links. The direct coupling of linear electric vibrators, however, is ineffective because of the small modulation of inductance and low efficiency.

A distinctive feature of the operation of synchronous motors is a considerable pulsation of torque, which has a substantial influence on dynamic stability. The known [1] method for taking elastic links into account in an alternating-current drive must be substantially supplemented in solving this problem. The article analyzes the operation of a vibration system intended for exciting frequency-controlled seismic oscillations in geophysical investigations.

A schematic representation of the vibration platform and the action of elastic and dissipative links is given in Fig. 1a. The paired centrifugal vibration exciter creates longitudinally polarized harmonic oscillations.

According to an electromechanical analogy of the first kind, the electrical analogue of the vibration system may be the circuit shown in Fig. 1b. The emf of the source is equivalent to the harmonic force of the vibrator \(F(\omega)\); the analogue of the current in the circuit is the vibration velocity \(\dot{x}\); the inductances are equivalent to the physical masses \(m\); the capacitances are proportional to the stiffnesses \(1/k\) of the elastic links, etc.

With linear elastic and mass resistances, the vibration system (Fig. 1) has two natural resonant frequencies:

\[ \omega_{01} = \sqrt{\frac{k_1}{m_1}};\qquad \omega_{02} = \sqrt{\frac{k_2}{m_2}}, \tag{1} \]

Fig. 1. Schematic representation and electrical analog of the vibration system

and the frequency response for the displacements of the vibration platform is expressed by the following relation:

\[ x(\omega)= \frac{1}{ \sqrt{ \left[ 1-a^{2}\left(1+\frac{\beta^{2}\mu}{\beta^{2}-a^{2}}\right) \right]^{2} +4\gamma^{2}a^{2} } }, \tag{2} \]

where

\[ a=\frac{\omega}{\omega_{01}};\qquad \frac{m_{2}}{m_{1}}=\mu;\qquad \beta=\frac{\omega_{02}}{\omega_{01}}; \]

\[ \gamma=\frac{h}{\omega_{01}};\qquad h=\frac{b}{2m_{1}}. \]

The phase shift of the vibration displacement relative to the disturbing force is

\[ \varphi(\omega)=\operatorname{arc\,tg} \frac{2\gamma a}{ 1-a^{2}\left(1+\frac{\beta^{2}\mu}{\beta^{2}-a^{2}}\right) }. \tag{3} \]

An example of the frequency characteristics of a seismic vibration exciter for \(\mu=7.3\), \(\delta=0.08\), \(\gamma=0.41\) is shown in Fig. 2. The parameters of the soil are calcu—

Fig. 2. Frequency characteristics

calculated according to [2] and confirmed experimentally. The high-frequency resonance, caused by the elastic properties of the soil, is expressed only implicitly because of the considerable energy losses due to inelastic deformation of the soil and radiation. The resonance of the system of masses \(m_1\) and \(m_2\), located in a narrow band of low rotational speeds, is not considered in the present article.

With an in-phase orientation of the rotating unbalances of the vibration exciters and of the magnetic axes of the rotors, the mechanical load on the motors is uniform, and the frequency dependence of the shaft torque is

\[ \overline{M}(\omega)=\frac{x(\omega)}{m_1\omega_{01}^{2}} e^{j\left[\frac{\pi}{2}+\varphi(\omega)\right]}\cdot F^{2}(\omega). \tag{4} \]

The power is

\[ \overline{S}(\omega)=\frac{d}{dt}\,\overline{M}(\omega)=P(\omega)+jQ(\omega), \tag{5} \]

where \(Q(\omega)\) is the reactive component of the power, and \(P(\omega)\) is the value of the active power of the motor due to the constant component of the torque:

\[ M_0=\frac{x(\omega)}{2m_1\omega_{01}^{2}}\cdot F^{2}(\omega)\sin\varphi . \tag{6} \]

The variable component of the torque has double frequency:

\[ M_{\mathrm{p}}=\frac{x(\omega)}{2m_1\omega_{01}^{2}}\cdot F^{2}(\omega)\cos(2\omega t-\varphi). \tag{7} \]

With a horizontal or inclined arrangement of the axes of the unbalances, a reactive static torque of the unbalances, varying with the rotational frequency, also acts on the drive motors,

\[ M_{\mathrm{p}}=(m_0eg)_n\cos\omega t. \tag{8} \]

The force characteristic of the vibration exciter \(F(\omega)\) predominantly determines the loading of synchronous motors. The use of unregulated centrifugal vibration exciters, whose frequency dependence is

\[ F(\omega)=m_0e\,\omega^{2} \tag{9} \]

causes a considerable nonuniformity of motor loading over the frequency range. At the same time, most vibrational technological processes do not require such a substantial increase in the disturbing force when the frequency changes.

A law \(F(\omega)\) differing from a quadratic one is usually obtained by means of vibration-exciter designs with variation of the static torque of the unbalances through certain kinematic schemes. Any dependence \(F(\omega)\) can be obtained with a vibrator having separate regulation of the frequency and the disturbing force [3].

In experimental studies on a model of a vibration emitter with a reduced mass \(m=170\ \text{kg}\) and \(\omega_{01}=680\ 1/\text{s}\), an adjustable unbalanced vibrator was used, the force characteristic of which \(F(\omega)\) is given in Fig. 3. The calculated mechanical characteristic of the synchronous vibration drive \(M(\omega)\) agrees satisfactorily with the points recorded in the experiment. The discrepancy with experiment can be explained by the nonlinearity of the soil parameters and by measurement error.

Fig. 3. Mechanical characteristics of the vibration drive

To assess the dynamic stability of the operation of synchronous motors under vibrational load, we shall find the oscillations of the angle \(\Delta \Theta(\omega t)\) due to the variable components of the torque, fixed with respect to frequency. Solving the equation of motion

\[ \frac{I}{P}\frac{d^{2}\Delta\Theta}{dt^{2}}+\delta\frac{d\Delta\Theta}{dt}+\sigma\Theta=M_{\mathrm{n}}+M_{\mathrm{p}}, \tag{10} \]

where \(\delta\) is the damping coefficient, determined by the damping properties of the motor;
\(\sigma\) is the constant of the synchronizing torque;
\(I\) is the moment of inertia of the motor and of the unbalances;
\(p\) is the number of pole pairs,

is

\[ \Delta\Theta=\Delta\Theta_{\mathrm{n}}\cos(2\omega t-\varphi-\varphi_{\mathrm{n}}) +\Delta\Theta_{\mathrm{p}}\cos(\omega t-\varphi_{\mathrm{p}}), \tag{11} \]

where

\[ \Delta\Theta_{\mathrm{n}}= \frac{M_{\mathrm{n}}} {\sqrt{\left(\sigma-4\frac{I}{P}\omega^{2}\right)^{2}+\delta^{2}\omega^{2}}}; \tag{12} \]

\[ \Delta\Theta_{\mathrm{p}}= \frac{M_{\mathrm{p}}} {\sqrt{\left(\sigma-\frac{I}{P}\omega^{2}\right)^{2}+\delta^{2}\omega^{2}}}; \tag{13} \]

\[ \varphi_{\mathrm{n}}=\operatorname{arc\,tg} \frac{2\delta\omega}{\sigma-4\frac{I}{P}\omega^{2}}; \tag{14} \]

\[ \varphi_{\mathrm{p}}=\operatorname{arc\,tg} \frac{\delta\omega}{\sigma-\frac{I}{P}\omega^{2}}. \tag{15} \]

Equating the derivative to zero,

\[ \frac{d}{d(\omega t)} \left[ \Delta\Theta_{\mathrm{n}}\cos(2\omega t-\varphi-\varphi_{\mathrm{n}}) +\Delta\Theta_{\mathrm{p}}\cos(\omega t-\varphi_{\mathrm{p}}) \right]=0, \tag{16} \]

we find the phase angle \((\omega t)_m\) at which the deviation of the load angle is greatest.

Substitution of \((\omega t)_m\) into (11) determines \(\Delta\Theta_{\max}\). The largest angle

\[ \Delta\Theta_{\max 1}=\Delta\Theta_{\mathrm{n}}+\Delta\Theta_{\mathrm{p}} \tag{17} \]

will occur at \(\varphi=\varphi_n=\varphi_p=0\pm k\pi\) at the instant \((\omega t)_1=0\).

In this case, the maximum negative deviation of the load angle

\[ \Delta \Theta_{\max 2} = - \left( \frac{\Delta \Theta_p^{\,2}}{4 \Delta \Theta_n} + \Delta \Theta_n \right) \tag{18} \]

occurs at the instant

\[ (\omega t)_2 = \arccos \left( - \frac{\Delta \Theta_p}{4 \Delta \Theta_n} \right). \tag{19} \]

Figure 4 shows, in polar coordinates, the graph of the variation of \(M(\omega t)\) and of the load angle \(\Theta(\omega t)\) for \(\alpha = 0.1\).

Fig. 4. Graph of oscillations of torque and load angle

As the frequency decreases, the influence of the inertial component of the motor torque weakens and the angle oscillations increase, with the static torque of the unbalances exerting the principal influence.

In the frequency range above \(\alpha = 0.3\), the influence of the damping and synchronizing torques may be neglected; in this case the calculation error does not exceed 4–5%.

The studies have shown that the use of synchronous motors in the frequency-controlled drive of vibrators will be more rational when the force characteristic \(F(\omega)\) is limited at high frequencies. The choice of the motor power and overload capacity is determined by the frequency characteristics \(x(\omega)\) and \(F(\omega)\). Owing to the considerable magnitude of the inertial component of the torque, which has a damping effect on the oscillations of the load angle, stability analysis of synchronous motors in the frequency range above 20 Hz can be performed without taking dynamic factors into account.

REFERENCES

1. E. F. Morozov et al. On the analysis of the amplitude-frequency characteristic of an AC electric drive with elastic links. Elektrotekhnika, 1973, No. 4.

2. N. D. Krasnikov. Dynamic Properties of Soils and Methods for Determining Them. Leningrad, Stroiizdat, 1970.

3. V. F. Kulakov, G. A. Sipailov, K. A. Khorkov. Source of elastic vibrations for vibroseismic prospecting. Positive decision on application No. 1754242/26–25 of September 8, 1973.