COMPARATIVE EVALUATION OF THE OPERATION OF VARIOUS TYPES OF BALANCING MACHINES

P. T. MAL’TSEV, V. T. TIKHONOV

(Presented by the Department of Theoretical Mechanics)

Dynamic balancing of rotating masses in modern high-speed machines is becoming increasingly important. The accuracy and sensitivity of various types of balancing machines substantially affect the quality of dynamic balancing.

To assess the degree of accuracy of dynamic balancing, it is necessary to identify all factors associated with the balancing machine. Only with a clear understanding of the capabilities of the balancing machine can one speak of the technological process of dynamic balancing.

In the modern domestic and foreign literature there are many works dealing with questions of balancing on balancing machines; however, there are almost no comparative studies testing various types of balancing machines. Over a number of years we have observed the operation of balancing machines at plants of the electrical industry. This made it possible to compare the operation of balancing machines of various types and to determine the advantages and disadvantages inherent in one type or another.

Before proceeding to the practical part of our investigations, we shall briefly consider the classification and certain questions of the theory of balancing machines.

The classifications of balancing machines known in the literature [5, 11, 12] were based on design features or on differences in methods for measuring the quantities of unbalance and, in our view, require substantial revision. We consider it advisable to propose a classification of balancing machines based on the main principle—the principle of operation. Under such a classification, balancing machines may be divided into three types.

The first type should include resonance balancing machines, on which the magnitude and position of the balancing weight are determined at rotational speeds of the body being balanced in the resonance range or, very rarely, in the pre-resonance range. During operation, machines of this type may convert the mechanical vibrations of the supports of the body being balanced into electrical vibrations, or may use compensation of the action of the unbalanced mass. The compensation may be mechanical or electrical. Among the most widespread machines of this type are

balancing machines may include: balancing machines DB-3, DB-4, of the firm “Gisholt,” models $S$, $V$, and $C$, “General Motors,” designed by the Department of TMM of MVTU—PE-2, PE-3, 7512, 3672, and the type 9720 balancing automatic machine designed by the Experimental Scientific-Research Institute of Metal-Cutting Machine Tools.

The second type may include resonant balancing machines operating at the critical number of revolutions, i.e., at a rotational speed of the balanced body corresponding to the resonance state of the system. Balancing machines of this type may operate both on the basis of measuring the amplitude of oscillation and on the basis of compensating the effect of unbalance. This type includes machines with an oscillating frame and machines with oscillating supports.

The most widespread balancing machines of the second type include: balancing machines designed by MIIT, machines of the firms “Schenck,” “Trebel,” “Losenghausen,” “Behrat,” and others.

The third type includes balancing machines operating without the use of oscillatory processes. The balancing process in these machines is carried out by directly measuring dynamic pressures on rigid supports. These include experimental models of 77 MDU-2 machines, in which the sensitive elements are ceramic plates made of barium titanate [12, 13, 14].

General questions of balancing rotating masses have been studied by many authors [2; 3; 5; 6; 9, etc.]. The foundations of the theory of balancing machines were laid by Academician Krylov [8]. However, the question is considered by him in general form.

As is known, the determination of the magnitude of the amplitude of oscillation due to the action of an unbalanced mass on balancing machines is based on the expression

\[ A=\frac{h_1\omega^2}{\sqrt{(k^2-\omega^2)^2+4n^2\omega^2}}, \tag{1} \]

where

\[ k=\sqrt{\frac{c}{M}};\qquad h_1=\frac{H}{M};\qquad n=\frac{b}{2M};\qquad z=\frac{\omega}{k};\qquad \gamma=\frac{n}{k}, \]

$A$ — amplitude of oscillation,
$\omega$ — frequency of the disturbing force,
$H\omega^2$ — disturbing force,
$b$ — coefficient of the resistance force,
$c$ — stiffness of the elastic element,
$M$ — reduced mass of the oscillating system.

Next the dynamic coefficient is introduced

\[ \lambda=\frac{z^2}{\sqrt{(1-z^2)+4\gamma^2z^2}}, \tag{2} \]

then expression (1) is transformed into the following:

\[ A=H\cdot\lambda. \]

In Fig. 1 a graph is given of the dependence of $\lambda$ on $z$ for different values of $\gamma$. From the graph and expression (2) it is evident that, with an increase in the angular rotational speed of the balanced body during operation of balancing machines in the beyond-resonance zone, there occurs a sharp decrease in the amplitude of oscillation, i.e., the sensitivity of the balancing machines falls.

The sensitivity of resonant-type balancing machines operating with the use of the phenomenon of resonance during the run-out of the balanced body or during the process of introducing the balanced body

into resonance reaches its maximum value at resonance, that is, at \(k=\omega\) or \(z=1\). For machines of this type, at the moment of resonance, taking into account the transmission ratio \(i\) of the amplitudometer, the maximum amplitude will be expressed as follows:

\[ A=\frac{h_1\omega}{2n}\,i. \tag{3} \]

The sensitivity of a balancing machine is directly proportional to the amplitude, that is, it will be the greater, the greater the linear or angular amplitude with which the machine responds to a given magnitude of unbalance.

Fig. 1

From expression (3) it is evident that the sensitivity of a balancing machine decreases as resistance increases. As the transmission ratio of the amplitudometer increases, sensitivity increases.

At first it seems that increasing the transmission ratio of the amplitudometer is an easy means of increasing the sensitivity of a balancing machine. However, in the amplitudometer mechanism, as the transmission ratio increases, an excessively large frictional resistance arises because of the increase in the number of joints; this indirectly leads to an increase in resistance and reduces sensitivity.

The presence of a large number of joints also leads to an increase in backlash and errors in the readings of the amplitudometer. It follows from this that increasing the sensitivity of balancing machines by increasing the transmission ratio of the amplitudometer is limited. The sensitivity of balancing machines is also affected by the magnitude of the angular speed of rotation of the body being balanced.

As is seen from (3), the most advantageous angular speed, from the standpoint of sensitivity, for the technological process of dynamic balancing is the speed corresponding to resonance of the system “balancing machine—body being balanced.” When the resistance tends to zero, the sensitivity of a resonance-type balancing machine increases without bound.

If, in expression (1), the values

\[ \frac{h_1\omega^2}{k^2}=\frac{P^1}{c}; \qquad k=\sqrt{\frac{c}{M}}; \qquad n=\frac{b}{2M}, \]

are substituted, then after the simplest transformations we obtain the expression for the amplitude in the following form:

\[ A=\frac{P^1}{\sqrt{c^2+\omega^2 M(\omega^2 M-2c)+\omega^2 b^2}} . \tag{4} \]

It follows from expression (4) that the amplitude of vibration, other conditions being equal, increases as the mass of the vibrating system, the stiffness of the elastic element \(c\), and the resistance \(b\) decrease (the vibrating system is the combination of the part being balanced plus the supports, or the part being balanced plus the pendulum frame).

Let us examine the influence of the distance between the axes of oscillation of the frame and the springs on the amplitude of vibration of resonance-type balancing machines.

Schematically, the vibrating system of the BM-19 balancing machine (Schenck type) is shown in Fig. 2.

Fig. 2.

Here

\(M_1\) — mass of the part being balanced,
\(M_2\) — mass of the pendulum frame,
\(a_1\) — distance from the axis of oscillation to the center of gravity of the part being balanced,
\(a\) — distance to the balancing plane,
\(r\) — radius of placement of the unbalanced mass.

At resonance \(\omega=k\), and expression (4) takes the form

\[ A=\frac{\dfrac{P^1}{c}\,k}{2n}, \tag{5} \]

where

\[ P^1=mr\,\frac{ca}{Ml}. \]

\[ M=M_1\frac{a_1^2}{l^2}+M_2\frac{\left(\dfrac{l+a}{2}\right)^2}{l^2}. \tag{6} \]

Substituting the values of \(P^1\), \(k\), and \(n\) into expression (5) and carrying out the transformations, we have

\[ A=\frac{mra\sqrt{c}}{b}\cdot \frac{1}{l\sqrt{M}}. \]

Denoting the constant quantity \(\dfrac{mra\sqrt{c}}{b}=B\) and substituting the values of \(M\) from expression (6), we obtain

\[ A=\frac{B}{\sqrt{M_1a_1^2+M_2\left(\frac{l+a}{2}\right)^2}} . \]

From this expression it is evident that, other conditions being equal, for constant values of \(a\) and \(a_1\), as the distance \(l\) between the axis of oscillation of the frame and the axis of the springs increases, the amplitude of oscillation decreases and, consequently, the sensitivity of the balancing machine is reduced.

In dynamic balancing of the body being balanced on a balancing machine, there is an angular phase shift between the displacement of the frame and the disturbing force. The phase-shift angle is determined by the expression

\[ \operatorname{tg}\alpha=\frac{2n\omega}{k^2-\omega^2} =\frac{2\gamma z}{1-z^2}. \]

As the angular velocity changes, the phase-shift angle \(\alpha\) changes (Fig. 3). The value of the angle \(\alpha\), equal to \(\dfrac{\pi}{2}\), occurs at the angular velocity corresponding to the resonance phenomenon. In resonance-type balancing machines, when determining the axial plane of location of the balancing mass, the body being balanced is rotated through an angle equal to \(\dfrac{\pi}{2}\). In this case an error is inevitably introduced, which is associated with fixing the plane of action of the moment from the unbalanced mass according to the maximum amplitude of oscillations. In fact, as is evident from Figs. 3 and 4, the maximum amplitude is reached at \(z>1\) and, correspondingly, at \(\alpha>\dfrac{\pi}{2}\). On the basis of (2) one can ob-

Fig. 3.

show that the maximum value of the oscillation amplitude in steady-state operation is obtained at \(z=\dfrac{1}{\sqrt{1-2\nu^2}}\).

At small resistances this displacement of the amplitude maximum is not large. As a result, the error in angular displacement on resonance-type balancing machines is practically neglected. However, a considerable reduction in resistance adversely affects the operation of resonance-type balancing machines. As resistance decreases, the steepness of the phase-angle curves increases, which significantly complicates fixing the plane of action of the moment from the unbalanced mass on the balancing machine and introduces error into the measurements.

It follows from expression (1) and Fig. 1 that, when balancing machines operate in the superresonance zone, a further increase in the rotational speed of the balanced body leads to a decrease in the oscillation amplitude. Therefore the sensitivity of balancing machines operating in the superresonance zone depends entirely on the sensitivity of the pickups recording the vibrations of the supports. Operation of balancing machines in the superresonance zone became possible only through the use of induction pickups with a high degree of sensitivity.

Fig. 4.

In a superresonance-type balancing machine, registration of the phase-shift angle is considerably more reliable. In the superresonance zone the curves of the phase-shift angle (Fig. 3) asymptotically approach an angle equal to \(180^\circ\). At \(z>2\) and \(\nu<0.1\), error in an angular displacement differing from \(180^\circ\) is practically absent.

In balancing machines of the third type there is no angular phase shift, since they operate without using oscillations.

In studying various types of balancing machines—DB-3, DB-4, “Trebel,” and BM-19 of the “Schenck” type—the authors carried out a comparative evaluation of their sensitivity.

The experiments were conducted with specially manufactured test rotors weighing from 1.5 to 20 kg, with ground surfaces, identical in shape to the rotors of electric machines produced

at the Tomsk plants. Four rotors weighing 1.5, 3, 4, and 5 kg had the same distances between supports, between balancing planes, and the same radii of the balancing circles. The latter were used to study the sensitivity of the balancing machines BM-19, “Trebel,” and DB-4. The journals of the indicated rotors were equipped with rolling bearings.

Before the experiments were set up, the test rotors were subjected to thorough dynamic balancing on a well-adjusted BM-19 machine by the method of repeated “going around with the weights.”

A rotor was considered prepared for the experiment if the deflection of the light beam of the amplitude meter from the zero position did not exceed ±0.5 scale division.

To determine the sensitivity of the balancing machines, pieces of plasticine weighing 0.1, 0.2, 0.3 g, etc., were fastened to the test rotor separately in the right and then in the left balancing plane, increasing the weight successively by 0.1 g.

The experiments were carried out at various distances \(l\) between the axis of oscillation of the frame and the axis of the springs, and at a constant distance \(a\) between the balancing planes.

The results of the experiments on the BM-19 balancing machine are given in Table 1.

Table 1

It is clear from the table that, with a decrease in the weight of the part being balanced and in the distance between the axis of oscillation of the frame and the axis of the springs, all other conditions being equal, the sensitivity of the balancing machine increases.

The results of the experiments do not contradict the theoretical propositions given above. An increase in the sensitivity of the machine can be obtained by reducing the mass of the pendulum frame and of the oscillating supports.

Tests of the “Trebel” balancing machine were carried out with the compensating device switched on and switched off.

The results of the experiments are summarized in Table 2.

The sensitivity of the “Trebel” balancing machine is very low even with the compensating device switched off, which may be explained by the small amplification factor of the oscillations of the amplitude meter and by the influence of the drive.

With the compensating device switched on, the sensitivity of the machine decreases still further. This depends on the introduction of additional transmission mechanisms.

Tests of the DB-4 balancing machines showed that their sensitivity threshold lies within the limits guaranteed by the manufacturer and is approximately 0.6–0.7 g·cm. However, when balancing—

Table 2

for rotors weighing up to 1 kg, the presence of drive action reduces the sensitivity of the machine.

An increase in the sensitivity of DB-4 balancing machines is possible through the use of a lightweight insert design (Fig. 5), as is shown in Fig. 6. The use of lightweight inserts on machines proves especially expedient when balancing rotors weighing up to 1 kg.

In the drawing: “Along AB.” “Material—light alloy.”

Fig. 5.

When balancing small rotors (1–2 kg and less), the weight of an insert of non-lightened design constitutes a significant part in relation to the weight of the part being balanced. Thus, for example, a cast-iron insert of non-lightened design, when balancing a rotor of 1 kg, was equal to 796 g; a lightened insert made of light alloy was only 38 g.

In studying the operating features of various types of balancing machines, we consider it possible to note the high service qualities of resonance-type balancing machines of frame construction. Machines of this type are simple in design, reliable, and trouble-free in operation. Thus, for example, a balancing machine manufactured by the training-and-production workshops of the Moscow Electromechanical Institute of Railway Transport Engineers has been operating without failure at the Tomsk Electromechanical Plant for more than seven years and requires no repair.

The DB-3 and DB-4 balancing machines are considerably worse in operation. Of the eight machines observed by us in the course of operation, all

during two years of operation, the electrical part of the machine control panel repeatedly required repair.

As our observations have shown, dynamic balancing on various types of balancing machines requires approximately the same amount of time—5–10 minutes. When the quality of balancing is increased, the balancing time on DB-4 machines increases sharply. The reason for this is the poor performance of the neon lamps.

An analysis of the results of investigating various types of balancing machines (operating at plants in the city of Tomsk) makes it possible to draw the following conclusions:

1. The most sensitive balancing machines are resonance-type machines of frame construction.

2. On the BM-19 machine, balancing should be carried out with a minimum distance between the axis of oscillation of the frame and the axis of the spring.

Magnitude of unbalance, in g·cm
○—factory insert; □—lightened-type insert

Fig. 6.

1. To increase the sensitivity and accuracy of balancing on BM-19 and DB-4 machines, it is advisable to reduce the weight of the pendulum frame and supports.

2. The sensitivity of DB-4 balancing machines is somewhat lower than the sensitivity of BM-19 balancing machines. However, the balancing quality on DB-4 machines satisfies the requirements imposed on asynchronous motors.

3. “Trebel” balancing machines, which have low sensitivity, may be used only for rough balancing of rotating parts of slow-speed machines.

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