METHOD FOR CHECKING THE QUALITY OF ROTOR BALANCING IN ASSEMBLED ELECTRICAL MACHINES

P. T. MALTSEV

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

The reliability of operation of modern electrical and other machines depends, to a certain extent, on the quality of balancing of their rotating parts. Under actual production conditions, complete elimination of unbalance is impossible; therefore it becomes necessary to assign rational tolerances for the residual unbalance of rotors. At present, however, there is no unified method for assigning permissible unbalances.

A number of authors [1, 2, 3, 4, 5], when assigning permissible residual unbalance, proceed from the premise that the dynamic pressures on the bearings, created by the unbalance of the rotor, must be less than the static pressures.

Different values are adopted for the unbalance coefficient

\[ \kappa = \frac{P_i}{Q_i} < 1, \]

where \(P_i\) is the dynamic pressure on the given bearing due to the action of rotor unbalance;
\(Q_i\) is the static pressure on the given bearing.

The value of the recommended unbalance coefficient ranges from \(\kappa = 0.01\) [1] to \(\kappa = 0.5\) [5].

It is expedient to evaluate unbalance by the displacement \(e\) of the rotor’s conditional center of gravity from the axis of rotation. Evaluation of unbalance by the displacement of the center of gravity from the axis of rotation makes it possible to compare the unbalances of rotors of different weights.

On the basis of its own experience and an analysis of published domestic and foreign materials, VNIIEM recommends [6] values of permissible residual unbalances for rigid rotors of electrical machines with rotor weights up to 1000 kg. Depending on the requirements for the quality of rotor balancing, electrical machines are divided into three groups: micromachines (with rotor weight up to 100 g), small machines (with rotor weight 100 g–3 kg), and medium machines (with rotor weight over 3 kg). In addition, two classes are introduced for small and medium electrical machines. The first class includes those...

include electrical machines with increased requirements for smooth running. The second class includes all other electrical machines with ordinary vibration requirements.

The permissible unbalance of rotors of micromachines and small machines of the 1st class is determined by the relation \(e = \dfrac{12000}{n}\), of rotors of small electrical machines of the 2nd class and medium machines of the 1st class, \(e = \dfrac{30000}{n}\), and for rotors of medium electrical machines of the 2nd class,

\[ e = \frac{60000}{n}, \]

where \(n\) is the number of rotor revolutions per minute.

For most types of electrical machines, the assignment of permissible residual unbalances according to the relations recommended by VNIIEM will be acceptable. However, for a number of machine types they will require correction and refinement, taking into account the specific operating conditions.

In connection with the increased requirements for the quality of manufactured electrical machines, in a number of cases it becomes necessary to check the balancing of rotors in assembled housings. In the technical specifications for some electrical machines there is a requirement to monitor the quality of rotor balancing by the criterion of whether the electrical machine “moves” or “does not move” on a smooth steel plate at the rated rotational speed. In this case, for some types of machines “movement” on the plate is not allowed; for other types a certain speed of “movement” is permitted. For some machines, when checking the quality of rotor balancing, a stand is used under the machine, the weight and design of which are not clearly specified; for other types, a stand is not permitted.

Practice and special experiments show that sometimes one and the same electrical machine, with a greater rotor unbalance, “does not move” or “moves” more slowly than with a smaller rotor unbalance. Some specimens of electrical machines of a given type, at a certain rotor unbalance (the centrifugal forces produced by the unbalance do not exceed a value approximately equal to 0.3–0.4 of the rotor weight), “move” on the plate, whereas others, with the same rotor unbalance and the same rotational speed, “do not move.”

When monitoring the quality of rotor balancing by the criterion of whether the electrical machine “moves” or “does not move” on the plate at the rated rotational speed, and still more when using a stand or permitting a certain speed of movement, one cannot be certain that the rotor has been balanced with the required degree of accuracy.

At the same time, in a number of cases, checking the quality of rotor balancing on a balancing machine after disassembly of an electrical machine that “moved” on the plate at the rated rotational speed shows that the rotor has a residual unbalance within the tolerances specified in the technical specifications. Rebalancing of the rotor did not always eliminate arbitrary movements of the electrical machine on the plate. These experiments indicate that the residual unbalance of the rotor is not the sole cause of the “movement” of an electrical machine on the plate. Consequently, the criterion of whether an electrical machine “moves” or “does not move” on the plate cannot be an indicator by which—

therefore one can judge unconditionally the degree of balance of the rotor. Moreover, it is impossible to say what rotor unbalance is present when the electrical machine begins to “shift,” or “moves” at some speed.

The displacement of an electrical machine on the plate during operation may occur as a result of its own vibrations, caused by various reasons. It is known that the vibration of an electrical machine comprises a broad frequency spectrum. Figure 1 shows oscillograms of the vibrational acceleration of an electrical machine of type U-600 at a rotational speed \(n = 6000\) rpm. In the upper photograph the vibration of the machine is recorded with displacement of the armature’s center of gravity \(e\) equal to \(7.5\) microns; in the lower photograph, the vibration at \(e = 20\) microns. As can be seen from the oscillogram (Fig. 1, a), high-frequency oscillations due to the operation of rolling bearings and magnetic forces occupy a considerable place in the vibrations of this machine.

Fig. 1. Oscillograms of vibrational acceleration of an electrical machine of type U-600:
1 — on the generator side; 2 — on the motor side; 3 — time marker, 500 Hz; a — total vibrational acceleration; б — component of vibrational acceleration at the rotational frequency.

In the oscillograms of Fig. 1, б, the components characterizing the vibrational accelerations at the rotational frequency (100 Hz) are recorded, separated from the total signal by suddenly switching on a PF-1 filter tuned to a transparency band from 85 to 120 Hz. Here we see that the amplitude of the vibrational acceleration at \(e = 20\) microns is considerably greater than at \(e = 7.5\) microns.

From the analysis of vibrations of electrical machines (Fig. 2) it follows that the levels of certain frequency components of vibrational acceleration with armature unbalance \(e \simeq 1\ \mu\) and \(e = 7.5\ \mu\) are higher than the component at the rotational frequency (100 Hz).

With an unbalance of \(e = 34\ \mu\), the level of the component at the rotational frequency proves to be higher than the components at other frequencies, and in this case the overall level is determined by the vibration acceleration at the rotational frequency.

The oscillograms in Fig. 1, b illustrate the sinusoidal character of the vibrational acceleration at the rotational frequency, caused by the unbalance of the rotor.

For sinusoidal oscillation, the relation between the level of vibration acceleration and the acceleration is described by the following expression:

\[ L = 20 \lg \frac{a_{\mathrm{e}}}{a_0}, \tag{1} \]

where \(L\) is the vibration level in db;
\(a_{\mathrm{e}}\) is the effective value of the acting vibration acceleration in \(\mathrm{cm/sec^2}\);
\(a_0 = 3 \cdot 10^{-2}\ \mathrm{cm/sec^2}\) is a constant value taken as the reference point (zero level).

Most instruments for measuring vibrations, including the IVPSh-type instrument with the VZCh vibration pickup, which we used in the tests, show the effective value of the vibration acceleration, equal to 0.707 of the maximum. To obtain the maximum (amplitude) value, three decibels must be added to the measured value of the vibration-acceleration level. Then the amplitude

Fig. 2. Effect of unbalance of the armature of a U-600 electric machine on the frequency composition of vibration acceleration.

the value of the effective vibration acceleration can be determined from the following expression:

\[ \lg a = \frac{L + 3}{20} + \lg a_0, \tag{2} \]

where \(a\) is the amplitude value of the effective vibration acceleration, in \(\text{cm}/\text{s}^2\).

The amplitude of the vibrational displacement at the rotational frequency, caused by rotor unbalance, is determined from the known relation

\[ A = \frac{a}{\omega^2}, \tag{3} \]

where \(\omega = \frac{\pi n}{30}\) is the angular frequency of the forced oscillations, in \(1/\text{s}\).

The amplitude of the vibrational displacement at the rotational frequency is found as a function of the displacement of the rotor’s center of gravity caused by the unbalance.

The vibration-measurement procedure must provide reliable measurement results. The method of mounting the machine when measuring vibration is of substantial importance. As theoretical and experimental studies by the Central Research Institute named after Academician A. N. Krylov [7], VNIIEM [8], foreign scientists [9, 10], and our own experiments have shown, measuring vibration with the product mounted on a shock-absorbing fixture provides data that make it possible to compare the vibrations of different machines and mechanisms. The shock-absorbing fixture must ensure that the natural frequencies of the product under study on the shock absorbers, in the directions of the mutually perpendicular axes \(X, Y, Z\), as well as the torsional oscillations about these axes, do not exceed \(0.25 \div 0.3\) of the frequency of the lowest rotational speed. With this method of mounting, considerable vibration isolation of the product under study from disturbances is achieved, freedom of oscillation of the product under the action of rotor unbalance is ensured, and the possibility of the test stand operating in the resonance region of the “product—shock absorbers” system is excluded, since the frequencies of the principal disturbing forces in an electrical machine lie in the range from the rotational frequency and higher.

The layout for mounting products on shock absorbers for vibration measurement is shown in Fig. 3. Vibration is measured in the direction perpendicular to the supporting surface of the product (the vertical direction). Sensors 4, rigidly fastened to platform 2, detect vertical oscillations along the \(Z\) axis caused by the action of the unbalanced force reduced to the center of gravity of the installation, and torsional oscillations about the \(Y\) axis caused by the unbalanced moment. For this system, differential equations are formulated and solved for the oscillations of the product under the action of rotor unbalance. It is assumed here that the machine oscillates on the shock absorbers as a rigid body and that there is no clearance in the bearings. As experience has shown, this assumption is valid for electrical machines of normal design having a rigid frame, with rotational speeds up to 8–10 thousand revolutions per minute.

As a result of solving the differential equations, formulas [7] were obtained for the amplitudes of oscillation of the machine under the action of rotor unbalance.

The amplitude of vertical oscillations along the \(Z\) axis is expressed by the formula

\[ A_z=\frac{m}{M}\cdot \beta_z \cdot e_c . \tag{4} \]

The amplitude of rotational oscillations about the \(Y\) axis of a point lying in the lower plane of the platform at a distance \(B\) (Fig. 3) is

\[ A'_\psi=\frac{m}{M}\cdot \frac{l}{B}\cdot \beta_\psi \cdot e_d , \tag{5} \]

The amplitude of rotational oscillations about the \(Y\) axis of the point at which the oscillations are measured will be expressed by the formula

\[ A_\psi=\frac{m}{M}\cdot \frac{l\cdot B_1}{B^2}\cdot \beta_\psi \cdot e_d , \tag{6} \]

where \(\dfrac{m}{M}=\mu\) is the ratio of the rotor mass to the mass of the product and the platform;

\[ e_c=\frac{m_1 e_1+m_2 e_2}{m} \]

is the displacement of the rotor center of gravity under static unbalance;

\[ e_d=\frac{m_1 e_1\cdot a+m_2 e_2\cdot b}{ml} \]

is the conventional displacement of the rotor center of gravity under dynamic unbalance;

\(m_1\) and \(m_2\) are the rotor masses reduced to the bearing planes;

Fig. 3. Schematic of the installation of electrical machines when measuring vibration:
\(1\) — machine, \(2\) — platform, \(3\) — shock absorbers,
\(4\) — sensors.

\(e_1\) and \(e_2\) are the displacements of the rotor centers of mass in the bearing planes, caused by unbalance;

\(a_1\), \(b_1\), \(l_1\), \(B\), and \(B_1\) are shown in the diagram (Fig. 3);

\(\beta_z\) and \(\beta_\psi\) are the dynamic coefficients for oscillations along the \(Z\) axis and rotations about the \(Y\) axis.

\[ \beta_z= \frac{\dfrac{\omega^2}{\lambda_z^2}} {\sqrt{\left(1-\dfrac{\omega^2}{\lambda_z^2}\right)^2+\gamma_z^2}}\, . \tag{7} \]

\[ \beta_z=\frac{\omega^2}{\lambda_z^2} \sqrt{ \frac{ \left(1-\frac{\omega^2}{\omega_z^2}\right)^2+\gamma^2 }{ \left(1-\frac{\omega^2}{\lambda_1^2}\right) \left(1-\frac{\omega^2}{\lambda_2^2}\right) -\gamma^2 \left[ 2-\left(1-\frac{h_0^2}{\rho_y^2}\right)\frac{\omega^2}{\lambda_\varphi^2} -\frac{\omega^2}{\lambda_x^2} \right] } }. \tag{8} \]

where \(\omega\) is the rotor rotational frequency,
\(\lambda_z, \lambda_x, \lambda_\varphi, \lambda_1\), and \(\lambda_2\) are the natural frequencies of the system.

As can be seen from Fig. 3, in the case of dynamic unbalance the measured vibration amplitude, all other conditions being equal, will be the larger the greater the distance to the point at which the sensor is mounted. From equations (4) and (6) one can determine the value of \(B_1\) at which the measured vibration amplitudes, both for static and for dynamic unbalance (in the case where \(e_c=e_d\)), will be equal to one another. Thus, by choosing an appropriate value of \(B_1\), identical instrument readings can be obtained for the same dynamic pressures on the bearings, independently of the nature of the rotor unbalance.

Analyzing formulas (4) and (6), we see that for \(e_c=e_d\) and \(A_z=A_\varphi\) we shall also have

\[ \beta_z=\frac{l\cdot B_1}{B^2}\cdot \beta_\varphi=\beta . \]

Consequently, formulas (4) and (6) can be represented in the generalized form

\[ A=\mu\cdot\beta\cdot e, \tag{9} \]

where \(e\) is the conditional displacement of the rotor center of gravity, independent of the nature of the unbalance.

The value of \(\beta\) can be calculated from formula (7).

For cushioning electrical machines during vibration measurements, steel screw springs were used, which provided free-vibration frequencies of the machine of not more than \(8\ \mathrm{Hz}\), i.e., the circular frequency was

\[ \lambda_z=2\pi f_0=50\ \frac{1}{\mathrm{sec}} . \]

The circular frequency of forced vibrations at \(n=6000\ \mathrm{rpm}\) is equal to

\[ \omega=\frac{\pi n}{30}=\frac{\pi\cdot 6000}{30}=628\ \frac{1}{\mathrm{sec}} . \]

The coefficient of inelastic resistance for spring shock absorbers is

\[ \gamma=0.01 . \]

Then \(\beta=1.006\), as calculated from formula (7); consequently, for practical calculations one may take \(\beta=1\).

Table 1 gives the results of measuring the vibration level at the rotational frequency of the electrical machine \((\mu=0.37,\ n=6000\ \mathrm{rpm})\) at various points (Fig. 4) on the housing and on the platform, for static and dynamic unbalance \((e_c=e_d=10\ \mu\mathrm{m})\).

The expected value of the vibration level at the rotational frequency for \(e=10\ \mu\mathrm{m}\), calculated from formulas (9), (3), and (1), should be \(L=71\ \mathrm{dB}\), or \(A=3.8\ \mu\mathrm{m}\).

From Table 1 we see that, in the case of static unbalance, when the sensors are placed at any of the points (0–13), the measured values of the vibration level practically coincide with the calculated ones. Devia-

Table 1

Fig. 4. Arrangement of the vibration-measurement points.

tions of the experimental data from the calculated values of the vibration level lie within the error range of the instrument. At the same time, in the case of dynami-

dynamic unbalance, when the sensor is installed at points 0 and 9, the measured vibration levels are significantly lower than the calculated ones. As the distance \(B_1\) increases, the measured values increase. At points 6, 7, 8, 11, 12, and 13 the measured values prove to be higher than the calculated ones and higher than the experimental data obtained with the same static unbalance.

The amplitude of the vibration displacement at the points above the bearings (points 12 and 13) in the case of dynamic unbalance reaches 10 microns, whereas with static unbalance it is 3.8–4.3 microns. Therefore, for a quantitative assessment of rotor unbalance it is important to choose correctly the points at which vibration measurements are to be made.

Fig. 5. Dependence of the vibration level of a type U-600 unit on armature unbalance:
1—overall level of vibration acceleration (experimental curve);
2—calculated dependence of the vibration level at the rotational frequency (100 Hz);
○ ○ ○—experimental points for static unbalance;
△ △ △—” ” dynamic unbalance;
□ □ □—” ” general case.

The measured vibration levels at points 3, 4, and 5 in the case of dynamic unbalance practically coincide both with the calculated values and with the results of measurements for an equivalent static unbalance.

Thus, in order to assess the quality of balancing of the rotor of a given electrical machine, the sensors must be installed at point 4.

Figure 5 shows the dependence of the level of vibrational acceleration at points 4 on the unbalance of the armature of an electrical machine, type U-600 ($\mu = 0.37$, $n = 6000$ rpm). The theoretical curve 2 is constructed from calculation data using formulas (9), (3), (1). Experimental data obtained as a result of tests at various armature unbalances are plotted on the same graph. As can be seen from the graph, the experimental data agree well with the calculated data. The deviations of the experimental points from the theoretical curve lie within the instrument error. The graph shows that, with an increase in unbalance, the overall level of vibrational acceleration (curve 1) changes only slightly. With a rotor unbalance of 15 microns and above, the overall level of vibrational acceleration is determined by the level of the vibration component at the rotational frequency. Similar results were obtained by the author on a number of electrical machines of other types, having operating rotational speeds from 3000 to 10000 rpm.

Thus, by measuring the amplitude of vibrational displacement at the rotational frequency at the corresponding points on the platform or on the housing of an electrical machine mounted on a cushioning support, it is possible to give a quantitative estimate of the rotor unbalance, provided that the value of the ratio ($\mu$) of the rotor mass to the mass of the electrical machine with the platform is known.

Conclusions

1. Checking the quality of rotor balancing by the criterion of whether the electrical machine “moves” or “does not move” over a smooth metal plate at nominal rotational speed does not make it possible to give a quantitative estimate of the unbalance.

2. The vibration of an electrical machine at the rotational frequency of the rotor is in a definite dependence on the unbalance, characterized by the displacement of the rotor’s center of gravity.

3. Measurement of the vibration at the rotational frequency of an electrical machine mounted on a cushioning support, which ensures a natural-oscillation frequency of the system no greater than 0.25–0.3 of the rotational frequency, makes it possible to carry out a quantitative estimate of the rotor unbalance.

Literature

1. Dinerman A. P. Balancing of Turbine Rotors. Mashgiz, 1946.

2. Samoilov V. A. Vibrations of Power-Station Units and Balancing of Rotors. Gosenergoizdat, 1949.

3. Kolesnik N. V. Static and Dynamic Balancing. Mashgiz, 1954.

4. Borisevich V. N. Balancing of Armatures of Small Electrical Machines. Mashgiz, 1961.

5. Shitikov B. V. Dynamic Balancing of Rotors. Transzheldorizdat, 1951.

6. Aleksandrovskii V. V., Zenkevich V. A. “Permissible Unbalance of Rigid Rotors of Electrical Machines.” Vestnik elektropromyshlennosti, 1961, No. 9.

7. Krochkov Yu. S. “Vibrations of Ship Mechanisms under the Action of Rotor Imbalance.” Transactions of the Central Scientific Research Institute named after Academician A. N. Krylov, Sudpromgiz, 1959.

8. Aleksandrovskii V. V. “Vibrations of Electrical Machines and Methods for Their Control.” Vestnik elektropromyshlennosti, 1961, No. 12.

9. Federn K. Erfahrungswerte, Richtlinien und Gütemaßstäbe für die Beurteilungen von Maschinenschwingungen. Konstruktion, 10. Jahrgang, (1958) Heft 8, pp. 289–298.

10. VDI-Richtlinien 2056, Beurteilungsmaßstäbe für mechanische Schwingungen von Maschinen, März, 1958.

Received by the editors
in June 1962.