METHOD AND DEVICE FOR DYNAMIC BALANCING OF ELECTRIC MACHINE ROTORS WITH ALLOWANCE FOR THEIR FLEXIBILITY¹

P. T. MAL'TSEV

(Presented by the scientific seminar of the Electromechanical Faculty)

Dynamic balancing of the rotors of modern electric machines is an indispensable operation in the technological production process. Balancing is usually carried out at comparatively low rotational speeds, by installing balancing weights in two transverse planes selected near the supports. The main objective in this is to eliminate dynamic pressures on the supports. Existing methods and balancing equipment ensure the solution of the problem of balancing rigid rotors. However, the application of methods and equipment developed for balancing rigid rotors does not give positive results when balancing flexible rotors.

In recent years an ever greater number of high-speed machines has been created, the rotors of which operate near the first, and sometimes beyond the first and even beyond the second critical speed. Meanwhile, in many cases the working drawings and process sheets for flexible rotors do not specify the method and regime of their balancing. In the technical literature, some attention has begun to be paid to questions of balancing flexible rotors \[1, 2, 3, 4\], but up to the present time there is no special equipment for their balancing.

A real rotor, as a rule, has unbalance distributed along its length. Therefore, for a flexible rotor the most effective balancing will be that in which simultaneous elimination of dynamic pressures on the supports and reduction of the dynamic deflections of the shaft in a specified range of rotational speeds are achieved. Such balancing can be achieved only by eliminating the initial unbalance through the installation, at diametrically opposite points, of a system of weights approximating the initial unbalance as accurately as possible.

F. M. Dimentberg \[3\] gave a fundamental solution to the problem of balancing a flexible rotor. As a first approximation he

showed the possibility of balancing a flexible rotor by placing concentrated balancing weights at a limited number of points, which ensure the elimination of dynamic pressures on the supports and a substantial reduction in the flexural deformations of the shaft. Proceeding from theoretical considerations [3] and from the basic solution of the problem, V. A. Zenkovich [4] proposed a method for determining and eliminating the unbalance distributed along the length of a flexible rotor. The method is based on the use of trial and balancing systems which, in form, correspond with some approximation to the forms of the elastic line of the shaft for the critical speeds lying within the operating range of rotor rotational speeds. The initial unbalance is determined by solving a system of equations into which are substituted the results of measuring, in the required number of planes of rotation, the dynamic deflections in two mutually perpendicular planes during a run with the initial unbalance, and the coefficients determined during runs with trial systems of weights of the first rotor from a given series.

To implement this method in industry, it is necessary to develop and manufacture special run-up balancing devices. In addition, measuring the deflections of actual rotors presents known difficulties.

The equation of the steady-state motion of a shaft of constant cross section with uniformly distributed mass and hinged supports may be written in the following form [3]:

\[ Z=e^{i\omega t}\left[ \frac{A_1+iB_1}{1-\dfrac{\omega^2}{\omega_1^2}}\cdot \frac{\omega^2}{\omega_1^2}\cdot \sin \frac{\pi s}{l} + \frac{A_2+iB_2}{1-\dfrac{\omega^2}{\omega_2^2}}\cdot \sin \frac{2\pi s}{l} + \right. \]

\[ \left. + \frac{A_3+iB_3}{1-\dfrac{\omega^2}{\omega_3^2}}\cdot \frac{\omega^2}{\omega_3^2}\cdot \sin \frac{3\pi s}{l} +\ldots \right] = e^{i\omega t}\sum_{n=1}^{\infty} \frac{A_n+iB_n}{1-\dfrac{\omega^2}{\omega_n^2}}\cdot \frac{\omega^2}{\omega_n^2}\cdot \sin \frac{n\pi s}{l}, \tag{1} \]

where \(A_1, B_1, A_2, B_2,\ldots\) are coefficients depending on the initial conditions of motion;

\(\omega\) is the angular speed of rotation of the rotor;

\(\omega_1, \omega_2\) are the first, second, etc., critical angular speeds;

\(s\) is the variable coordinate along the rotor axis;

\(l\) is the distance between the rotor supports.

It follows from equation (1) that, under the action of the unbalance forces and inertia forces, the rotor shaft is deflected; its spatial elastic line is a superposition of harmonics of successive orders (1, 2, 3). Each of the harmonics lies in a definite plane, and the planes of the harmonics are different, since the coefficients are complex. With a change in the angular speed \(\omega\), the ratios of the moduli of the harmonics, as well as the angles between the planes in which they are located, are disturbed. The dynamic reactions corresponding to one or another harmonic lie in the plane of that harmonic. The system of reactions from each harmonic will rotate with the same angular speed as the rotor rotational speed. For odd harmonics, the reactions of both rotor supports are parallel and directed to one side (symmetry), while for even harmonics the reactions at the right and left supports will be directed in opposite directions (skew symmetry).

dynamic reactions of each support will constitute the geometric sum of the reactions corresponding to the individual harmonics. By installing certain balancing weights in planes located near the supports, it is possible to eliminate the radial dynamic pressures on the supports at a fixed number of revolutions.

However, when the rotational speed changes, the balance achieved will be disturbed. In addition, installing balancing weights near the supports of a flexible rotor will not eliminate shaft deflection and, consequently, the vibration of the electric-machine frame will not be eliminated.

Under production conditions we encountered the need to balance, with a high degree of accuracy, rotors whose balancing quality on the existing balancing equipment did not ensure smooth operation of electric machines at the nominal rotational speed, and, as a result, the products did not satisfy the specifications. The rotors of these machines were subjected to dynamic balancing on a balancing machine with an accuracy of up to 3–4 microns. The rotational speed of the rotors during balancing was 1000 rpm. The balancing weights were installed in planes of rotation located near the supports. As observations and experiments showed, after assembly of electric machines with such rotors, the latter had a symmetrical (static) unbalance of the order of 30–50 microns, manifested at the operating rotational speed.

By analytical and experimental methods, the frequencies of the natural flexural oscillations of the rotors were determined. It was established that the operating speed of the rotors of these electric machines is 0.6–0.67 of the first critical speed. Therefore, for shaft deflection the first harmonic plays the principal role, since the influence of higher harmonics, as is seen from expression (1), owing to the decrease of the ratio

\[ \frac{\omega^2}{\omega_n^2} \]

becomes insignificant. Consequently, when operating near the first critical speed, the part of the unbalance whose form coincides with the first form of the elastic line of the shaft is manifested most strongly. Thus, the deflection of the rotor shaft of the electric machine under consideration occurs mainly along a half-wave of a sinusoid. At the same time, at every instant, dynamic pressures directed to one side act on the supports. This explains the fact that, in rotors carefully balanced in the end planes at low speeds, at operating speeds near the first critical speed a significant symmetrical unbalance appears.

Balancing of a flexible rotor operating near the first critical speed can be carried out by installing a balancing weight at the crest of the half-wave of the sinusoid, or a pair of weights on the half-wave. Near the supports, in this case, on the opposite generatrix it is necessary to install weights whose centrifugal forces at low rotational speeds will be equal and oppositely directed with respect to the force from the balancing weights installed in the middle part of the rotor length [5]. To perform such balancing, the rotor design must provide for the possibility of installing balancing weights in the indicated planes of rotation.

The design of the rotors under consideration permits balancing weights to be installed on retaining rings in three planes

rotation. The distances from the left support to the planes in which the weights are installed (Fig. 1) are \(l_1 = 0.19l\), \(l_2 = 0.5l\), \(l_3 = 0.81l\). The radii at which the weights are installed in all three planes are identical.

Dynamic balancing of rotors operating near the first critical speed should be carried out in two stages. In the first stage, the rotors are subjected to dynamic balancing at a low rotational speed. At this stage, compensation of the total effect of all components of unbalance is achieved by installing balancing weights in the end planes (Plane I and Plane III, Fig. 1). The balancing process in this case is analogous to the balancing of rigid rotors.

Fig. 1. Diagram of the arrangement of balancing planes on the rotor.

In the second stage, that part of the unbalance which, at operating speed, causes dynamic deflection of the rotor shaft is compensated. The unbalance is determined at the rated rotational speed from the oscillations of a damped platform on which a temporary stator with the rotor being balanced is mounted. After the first run, the determined balancing weight \(M\) is distributed over three planes. The magnitudes of the correcting weights in the corresponding planes are in the ratio

\[ m_1 : m_2 : m_3 = 0.21 : 0.6 : 0.19, \]

with

\[ m_1 + m_2 + m_3 = M. \]

After the second run, the balancing is refined by installing balancing weights in the end planes.

For balancing rotors with allowance for their flexibility, a special balancing device has been developed and introduced at one of the enterprises of the electrical industry; its general appearance is shown in Fig. 2. The device has the following main assemblies: a damped platform 3 with a temporary housing of the electric machine 9, into which the rotor to be balanced is installed; two sensors 4, an illuminating head with photocell 5; a preamplifier 7, mounted in the free space of an EO-53M oscilloscope 6. The mounting frame 1 is rigidly fastened to a massive vibration-isolated foundation slab 8. In the middle part of the temporary housing there is a special window that provides access to the middle balancing plane for soldering on the balancing weight.

Fig. 2. General view of the balancing device.

Fig. 3. Block diagram of the balancing device.

Under the action of the rotor unbalance, manifested at the nominal rotational speed, oscillations arise in the frame of the electric machine together with the platform, which is mounted on soft shock absorbers that ensure a natural-oscillation frequency of the system on the order of 0.08–0.1 of the rotor rotational frequency. The oscillations of the platform are perceived by piezoelectric sensors. The signals from the sensors \(D\), in turn, are fed by means of switch \(P\text{-}L\) to the lamp filter \(LF\) (Fig. 3), the preamplifier \(U\), and then to the input of the oscilloscope \(OSC\). On the oscilloscope screen a sinusoid is displayed, characterizing the dynamic pressure from the unbalance either on the right support of the armature or on the left, with the corresponding position of switch \(P\text{-}L\). Thus, from the amplitude of the sinusoid on the oscilloscope screen, using the calibration curve, the magnitude of the unbalance is determined.

The angular position of the unbalance is determined with the aid of an illuminating head with a photocell. The signal from the photocell \(FE\), after amplification by \(UM\), is fed to the oscilloscope modulator. A light mark is applied to the dark disk of the fan. The light beam from the lamp of the illuminating head, after being reflected from the surface on which the light mark has been applied, falls on the photocell. At the moment when, during rotation of the rotor, the light mark falls under the light beam, a bright point appears on the sinusoid. By manually turning the illuminating head about an axis coinciding with the axis of rotation of the armature, the bright point (mark) can be set at any position on the unbalance sinusoid.

Piezoelectric sensors respond to acceleration, as a result of which the high-frequency components in the signal, caused by bearing operation and by pulsations of the magnetic field, have a considerable level. Therefore the unbalance-signal amplifier includes a double \(T\)-type filter, which provides transmission of a comparatively narrow frequency band with steep roll-offs at the edges. The use in the \(T\)-type filter of potentiometers \(R_4\), \(R_5\), \(R_6\) (Fig. 4), with sliders coupled on one common axis, makes it possible, by means of the “filter” knob, to vary the mean frequency of the pass band from 50 to 130 c/s and thereby to isolate the signal at the rotational frequency. But since the filter does not ensure complete attenuation of the high frequencies, an integrating circuit \(R_{16}—C_{11}\) is placed at the output of the preamplifier. Thus, a velocity signal caused by the rotor unbalance is fed to the oscilloscope input. The “phase” switch (Fig. 4) makes it possible, when adjusting the balancing device, to obtain the image of the half-period of the sinusoid corresponding to the “heavy spot” in the lower part of the screen of the oscilloscope cathode-ray tube. When determining the angular position of the unbalance, the illuminating head is turned so that the bright spot on the sinusoid is at the minimum of the sinusoid (Fig. 5). In this position the illuminating head is fixed. After the machine is stopped, by turning the armature by hand, the “beam” point on the disk (Fig. 3) is brought into coincidence with the light beam \(h\), and in this way the “heavy spot” on the armature, which proves to be at the bottom, is determined.

Owing to the presence of the integrating circuit, a rotation of the phase of the unbalance signal takes place (practically by \(90^\circ\)). However, the plane of the first harmonic makes a certain phase angle with the plane of the unbalance distributed along the rotor length, which causes shaft deflection in the first mode. Therefore the central angle between the light mark and the “beam” point on the disk, found experimentally,

Fig. 4. Schematic circuit diagram of the electronic measuring unit.

Visible annotations in the diagram: “to the \(y\)-plates”; “Phase”; “to the modulator”; “power-supply terminal block of the EO-53 oscilloscope.”

Fig. 5. Unbalance sinusoid with a light marker.

at a rotational speed of 5400–5500 rpm, amounts to 73° (Fig. 3).

The characteristic of the filter is shown in the graph of Fig. 6, from which it is seen that when the filter is tuned, for example, to the mean frequency of 90 Hz, the side frequencies differing from the tuning frequency by a factor of 1.5–2 will pass with an attenuation of 4–7 times, i.e., by 12–17 dB.

From the graphs of Fig. 7 b and 7 c it is seen that the amplitude characteristic of the measuring channel remains linear up to an input voltage of about 40 mV (curve 1). The dependence of the beam sweep on the oscilloscope screen on the input voltage (curve 2), after 60 divisions, does not remain linear because of the optical distortion that occurs in visual observation of the sinusoid.

Fig. 6. Filter characteristic: 1—according to millivoltmeter readings; 2—according to the scale of the oscilloscope screen.

The graph of Fig. 8 shows the effect of reducing the vibration-acceleration level¹ at the rotational frequency after balancing rotors with allowance for their flexibility on the balancing device described. The envelopes of the frequency components were plotted from the average values in each band, obtained as a result of measuring the vibrations of 12 specimens of the electric machine.

Seven months’ experience in operating the proposed balancing device makes it possible to draw the following conclusions.

1. Balancing rotors whose operating speed is close to the first critical speed, carried out in two stages (at low speed and then at nominal rotational speed), makes it possible to eliminate dynamic pressures on the supports and to reduce substantially the dynamic deflection of the shaft.

2. In 2–3 starts of the rotor at the second stage of balancing, it is possible to achieve [[unclear: continuation cut off at bottom of page]]

Fig. 7. Amplitude characteristic of the electronic-measuring channel:
a—at attenuation 1:1; b—at attenuation 1:10;
1—according to the millivoltmeter readings; 2—according to the oscilloscope screen scale.

a reduction in the vibration amplitude of the electric machine at the rotational frequency from 15–20 microns to 1–1.5 microns is attained.

1. The parameters of the electronic measuring channel of the balancing device make it possible to carry out balancing at rotor rotational speeds from 3500 to 9000 rpm.

Fig. 8. Vibration spectra of an EMU-type electric machine: 1—after balancing on a balancing stand at \(n = 1000\) rpm; 2—after balancing in two stages with the proposed balancing device.

References

1. Meldahl A. Balancing elastic rotors. ZAMM, Vol. 34, No. 6, 1954.
2. Shimek G. Balancing rotors with elastic shafts. Strojirenstvi, No. 9, 1954.
3. Dimentberg F. M. Flexural vibrations of rotating shafts. Publishing House of the Academy of Sciences of the USSR, 1959.
4. Zenkevich V. A. Balancing flexible rotors. Bulletin of the Electrical Industry, No. 8, 1959.
5. Zenkevich V. A. Balancing rotors of electric machines with allowance for their flexibility. Bulletin of the Electrical Industry, No. 1, 1962.

Received by the editors
in May 1962.

1. The acceleration \(a_y = 3 \cdot 10^{-2}\ \text{cm/sec}^2\) is taken as the zero vibration-acceleration level. ↩↩