Tangential Vibrations of Brushes
Abstract Generated abstract
This paper investigates tangential self-excited vibrations of brushes in sliding electrical contacts, a problem linked to disturbances in commutation but less studied than radial brush vibrations. Using the theory of frictional vibrations, the authors develop a mechanical model of the brush, derive equations relating vibration amplitude and frequency to contact geometry, stiffness, inertia, pressure, friction characteristics, and brush-holder clearance, and compare these results with measurements on slip rings and commutators. The experiments indicate that increased pressure and pressure toward the leading edge intensify vibrations, while larger brush-holder gaps increase amplitude and reduce frequency, and that tangential rocking vibrations modulate contact resistance and voltage drop, potentially degrading commutation. The paper proposes practical mitigation measures including reduced guide clearance, smaller brush-holder gaps, harder brush materials, and reduced commutator runout.
Full Text
TANGENTIAL VIBRATIONS OF BRUSHES
L. Ya. Zinner, A. I. Skorospeleshkin
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
At present, ever greater attention is being paid to theoretical and experimental investigations of the dynamics of sliding contact. This is due to the fact that an unsatisfactory mechanical state of the sliding contact leads to serious disturbances in the commutation process of a commutator electric machine.
The radial vibrations of the commutator–brush assembly have been studied in the greatest detail to date. However, as is known, in a real sliding contact there are always also other types of brush vibration, which include tangential and axial vibrations. If the question of radial vibrations is removed by switching to end-face commutators and by improving the technology of cylindrical commutators, then the elimination of tangential vibrations is associated with great difficulties, since they are determined only by the frictional characteristics of the friction pairs.
The tangential vibrations of brushes that occur in the operation of a sliding electrical contact are, by their nature, self-excited vibrations and in this respect do not differ from the frictional self-excited vibrations that are very widespread in nature.
It should be noted that the mechanics of frictional self-excited vibrations has been studied in considerable detail by a number of researchers [1, 2, 3]. However, as applied to the sliding contact of electrical machines, which is distinguished by certain specific features, this question still remains unstudied. The available works [4, 5] only partially touch upon the question of the influence of tangential vibrations on current collection, without addressing the physics of the process and ways of eliminating them.
Numerous investigations have established that frictional contact has an elastic-viscous nature, while at the points of contact phenomena occur that are, to some extent, analogous to creep, and that the occurrence of self-excited vibrations is determined by a certain form of the static and kinetic characteristics of friction. Thus, the cause of the occurrence of self-excited vibrations of brushes on the surface of a commutator and a ring is the presence of a difference between the static and kinetic coefficients of friction. This difference may be caused both by an increase in the static friction force with the duration of immobility of the contact during the joint motion of the brush and the commutator, and by a decrease in the sliding-friction force with an increase in the relative velocity of motion of the friction pairs.
The present work is devoted to an analytical and experimental-
nominal investigation of the tangential vibrations of brushes and their influence on current collection. As the calculation model, the authors have adopted the scheme shown in Fig. 1. The mechanism of the oscillations that occur may be explained as follows.
Fig. 1. Calculation scheme
When the ring or commutator rotates, the brush, under the action of the friction force, is carried along by the surface of the ring in the direction of rotation and deviates from the vertical by a certain angle \(\varphi_0\), taking up the clearance in the guides. Subsequently the brush is deformed at the points where it bears against the brush holder, as a result of which a normal reaction arises, playing the role of a restoring force. As long as the brush moves together with the ring, the restoring force is continuously balanced by the friction force. When the restoring moment is equal to the maximum moment due to the friction force, there occurs a jump-like increase in the relative velocity of motion and a decrease in the moment of the friction force, as a result of which the brush begins to move in the opposite direction. At the point corresponding to the minimum of the restoring moment, the direction of motion of the brush again changes. The process repeats.
Thus, in addition to the moment of the friction force, the brush is acted upon by the moment due to the pressure and by a certain restoring moment proportional to the angle of deflection of the brush from the vertical. For an unchanged friction moment the brush deviates by a certain angle \(\varphi_0\), determined from the equality:
\[ M_p + M_{\varphi 0} = M_{\mathrm{tr}}, \tag{1} \]
where \(M_p\) is the moment due to the pressing force,
\(\qquad M_{\varphi 0}\) is the restoring moment,
\(\qquad M_{\mathrm{tr}}\) is the friction moment.
However, according to data in [6], the friction force does not remain constant and has a certain dependence on the velocity of displacement of the moving member of the friction pair. In the general case this dependence is represented by a linearized characteristic, decreasing slightly with increasing velocity.
In accordance with this, the moment of the friction force is also a certain function of the relative velocity
\[ M_{\mathrm{tr}} = |p|\left[C - D\left(V - l'\dot{\varphi}\right)\right], \tag{2} \]
where
\[ \begin{aligned} D &\text{— coefficient taking into account the slope of the friction characteristic;}\\ V &\text{— peripheral speed of the ring;}\\ l'\dot\varphi &\text{— linear velocity of the brush surface;}\\ l' &\text{— effective arm of application of the friction force;}\\ P &\text{— pressure on the brush.} \end{aligned} \]
Taking (1) and (2) into account, we write the equation of motion of the brush in the form
\[ I\ddot\varphi + mga\varphi = PlD(V-l'\dot\varphi), \tag{3} \]
where
\[ \begin{aligned} a &\text{— reduced length of the brush,}\\ I &\text{— moment of inertia of the brush,}\\ m &\text{— mass of the brush,}\\ q &\text{— acceleration of free fall.} \end{aligned} \]
Expression (3) describes only the free vibrations of the brush under the action of a variable friction force. However, in view of the fact that a real brush placed in a brush holder is subjected to certain actions from the latter, it is necessary to take into account the restoring moment \(M_\varphi\). This moment, as has already been noted earlier, is directed opposite to the action of the moment of the friction force and is completely determined by the elastic-plastic properties of the brush in its interaction with the brush holder. Taking into account that the discussion concerns very small amplitudes of vibration, measured in units of microns, the arm of application of the restoring force may be defined as \(l'-d\). Then the moment of the restoring force is determined from the expression
\[ M_\varphi = K(l'-d)^2\varphi, \tag{4} \]
where
\[ \begin{aligned} d &\text{— projection of the brush from the brush holder,}\\ K &\text{— stiffness of the connection, depending on the elastic properties of the brush.} \end{aligned} \]
At the same time it is also necessary to take into account the moment due to the pressing force, which is determined by the magnitude of the pressure and the point of its application. The following cases may occur:
-
The pressure is directed toward the entering edge of the brush. In this case the moment \(M_p\) coincides in direction with the moment of the friction force and increases the instability of the brush.
-
The pressure is directed along the axis of the brush, \(M_p=0\).
-
The pressure is directed toward the leaving edge. In this case \(M_p\) acts opposite to the friction moment. In this case, by choosing \(P\) and the arm of its application, it is possible to compensate some fraction of the friction moment.
\[ P\,\frac{b_{\mathrm{sh}}}{2}=l'PC. \tag{5} \]
Strictly speaking, the list of forces acting on the brush is not limited to those indicated above. For a rigorous calculation, the differential equation of motion would also include the moment arising from the interaction of the entering edge of the brush with the ring or commutator and accompanied by indefinitely small deformations of the brush surface. But, since this moment is small in comparison with \(M_\varphi\), we shall not take it into account in what follows.
Proceeding from the foregoing, we write the complete equation of motion of the brush without taking damping into account in the form:
\[ I\ddot\varphi + mga\varphi + M(\varphi) - Pl'D(V-l'\dot\varphi)=0. \tag{6} \]
As is seen from (6), the character of the vibrations is determined by the function \(M_\varphi\). Strictly speaking, this function is nonlinear owing to the fact that the return-
… the resisting moment, caused by elastic-plastic deformations, depends nonlinearly on the magnitude of the deformations. Nevertheless, taking into account that the amplitude of the brush vibrations is a very small quantity, it is possible, with sufficient accuracy, to regard \(M_{(\varphi)}\) as linearly dependent on \(\varphi\). Adopting this assumption, we have
\[ I\ddot{\varphi}+K(l'-d)^2\varphi-Pl'D(V-l'\varphi)=0. \tag{7} \]
It follows from (7) that the system has a single state of unstable equilibrium when \(V-l'\varphi=0\). In this case the brush is deflected through a certain angle \(\varphi_0\) and is in engagement with the ring, which occurs when the condition
\[ K(l'-d)^2\varphi_0=Pl'[C-D(V-l'\varphi)] \tag{8} \]
is satisfied.
This equilibrium state is unstable if \(PDl'(V-l'\varphi)<0\), and self-oscillations are established that are close in form to sinusoidal ones. Moreover, the smaller the moment of inertia of the brush, the closer these oscillations are to relaxation oscillations. In this case the oscillations break up into two types:
-
When \(K(l'-d)^2\varphi \approx Pl'D(V-l'\varphi)\), the system has comparatively large accelerations, despite the small moment of inertia \(I\), and, consequently, the brush moves through the equilibrium state at low velocities.
-
When \(K(l'-d)^2\varphi \ne Pl'D(V-l'\varphi)\), the accelerations of the brush are large, while the change in velocity occurs rather rapidly; and the smaller the moment of inertia \(I\), the shorter the duration of the jump, the change in the coordinate \(\varphi\), and the deformation of the adjacent edge of the brush.
After some transformations, from (7) we obtain the final expression describing the motion of the brush:
\[ \frac{d^2\varphi}{dt^2}+h\frac{d\varphi}{dt}+\omega^2\varphi = -\frac{Pl'(C-DV)}{I}, \tag{9} \]
where
\[ h=\frac{P(l')^2D}{I}, \]
\[ \omega=\frac{K(l'-d)^2}{I}. \]
The solution of (9) will be an expression of the form
\[ \varphi=Ae^{-ht}\sin\omega t-\frac{Pl'(C-DV)}{I\omega^2}. \tag{10} \]
Thus, we have obtained an expression for \(\varphi\) in which the vibration frequency does not depend on the amplitude, which is not entirely correct.
Experimental investigations of tangential vibrations of brushes show that their amplitude and frequency are in a definite functional relationship. Thus, with increasing amplitude, a decrease in the vibration frequency is observed.
To obtain some ideas about this dependence, let us assume that the vibrations occur without damping and that the equality
\[ \frac{d\varphi}{dt} = \frac{K(l'-d)^2\varphi}{2I}\,t \tag{11} \]
holds.
From (11), for \(\varphi\) we have
\[ \varphi=\frac{I_{\max}}{\left[\dfrac{(l'-d)^2 K t^2}{4I}-1\right]} . \tag{12} \]
Taking in (9) \(I\ddot{\varphi}=0\), we find the maximum angle of deflection of the brush for this case from the condition \(V-l'\varphi\approx 0\).
\[ \varphi_{\max}=\frac{P l' C}{K(l'-d)^2}. \tag{13} \]
On the basis of (12) and (13), for \(\varphi\) and for the frequency of self-oscillations we have:
\[ \varphi=\frac{P l' C}{K(l'-d)^2\left[\dfrac{(l'-d)^2 K t^2}{4I}-1\right]}, \tag{14} \]
\[ f=\frac{1}{2\pi}\sqrt{\frac{K(l'-d)^4\varphi}{I P l' C}} . \tag{15} \]
Thus, from (14) and (15) it is seen that the amplitude and frequency of the tangential vibrations of brushes are directly dependent on the parameters of the current-collecting device \(l'\), \(d\), \(I\), and \(K\). Thus, as the gap between the ring and the brush holder increases, the amplitude of the vibrations increases, while the frequency decreases, which is well confirmed by experimental investigations.
To study tangential vibrations, the authors used the installation of Fig. 2 with a carefully prepared commutator and contact rings.
The measurements were carried out using a profilometer [7] and a capacitive contactless sensor specially developed for these purposes. Simultaneously with the measurement of brush vibrations, the distribution of currents in various parts of the brushes was measured, for which three-layer composite brushes were used. The experimental investigations carried out made it possible to establish the connection between tangential vibrations and the parameters of the sliding contact. Below we give a brief description of the nature of the influence of one or another parameter on the behavior of the brush.
1. Influence on tangential vibrations of the pressing force and the direction of its action. As is known, the contact of friction pairs is observed not over the entire so-called “nominal” contact area, owing to the nonuniformity of the radii of curvature and the presence of waviness and roughness of the contacting surfaces. Waviness and roughness determine the discrete character of the distribution of the points of mechanical contact, as a result of which the interaction of the surfaces occurs over an insignificant part of the nominal area, called the “actual” contact area. The actual contact area depends on the pressure acting on the contacting surfaces and, consequently, has a definite influence on the increase of the friction force. In the given case, when the friction characteristic is linear, one may consider that Amontons’ law is satisfied and, consequently, an increase in pressure leads to an increase in the friction force, which in turn leads to an increase in the amplitude and a decrease in the frequency of the vibrations.
With regard to the actual contact area, the following may be noted: the larger the area over which the brush overlaps the commutator, the smaller the amplitude of the vibrations. In addition, the direction of the pressing force has a large influence on the intensity of the vibrations. In this
Fig. 2. Experimental setup
relations, as was already noted earlier, there are two limiting cases:
a) when the pressure is directed toward the leading edge of the brush, the frictional moment increases, since the moment of the static friction force remains uncompensated and the amplitude of the oscillations increases sharply;
b) when the pressure is directed toward the trailing edge, the amplitude of the oscillations is considerably smaller. In Fig. 3a, b, oscillograms are given of the tangential vibrations of an EG-14 brush for the two limiting cases, all other conditions being identical. From the oscillograms it is evident that, in the case of pressure toward the leading edge of the brush, the amplitude of the oscillations increases.
II. Influence of the ellipticity and eccentricity of the ring on the behavior of the brush. Everything stated above is valid for the case in which we are dealing with a ring that is ideal with respect to runout. Otherwise, additional factors arise that may be the cause of instability of the brush–ring contact. In the presence of significant runout, modulation of the contact pressure occurs, and consequently also of the friction force, with the amplitude and frequency of the runout. Oscillation of the friction force leads to intensification of the vibrations of the brushes.
III. Influence of the rotational speed of the ring on the tangential vibrations of the brushes. In the course of experimental
Fig. 3. Oscillograms of tangential vibrations of brushes at \(P=250\ \text{g/cm}^2\), \(n=2300\ \text{rpm}\), brush EG-14.
a — pressure on the trailing edge,
b — pressure on the leading edge
It has been established by studies that, with an increase in the rotational speed of the ring, there is some decrease in the amplitude of the brush vibrations and an increase in frequency. The frequency, however, depends only very slightly on the rotational speed.
Thus, when the rotational speed of the ring is increased from 500 to 2500 rpm, the frequency changes by 15 percent. At the same time, the vibration frequency is different for different brush grades. For EG-49 it is 750–800 Hz, EG-14—1000 Hz, EG-74—1500–2000 Hz and, therefore, depends on the stiffness of the brush.
It should be noted that the vibration frequency for one and the same brush is the same both when the brush operates on a ring and on a commutator. The only difference is that in the latter case higher-frequency bending vibrations are superimposed on the fundamental vibrations. The spectrum of bending vibrations lies in the range of the lamellar frequency and is determined with sufficient accuracy from the relation:
\[ f_{\text{bend}} = 0.162 \frac{h^2}{l}\sqrt{\frac{E}{lm}}, \tag{16} \]
where
\(h\) — height of the brush,
\(l\) — length of the brush,
\(E\) — modulus of elasticity.
In Fig. 4 (a, b, c, d) are shown oscillograms of tangential vibrations of the EG-14 brush, recorded at various rotational speeds of the ring and pressure \(P=250\ \text{g/cm}^2\). As can be seen from the oscillograms, with a change in speed over wide limits the vibration frequency changes very little, whereas their amplitude and character undergo considerable changes. At low rotational speeds, Fig. 4a (2000 rpm), the vibration amplitude is large and their character is close to relaxation. With increasing speed the amplitude decreases somewhat, the frequency increases, and the vibrations themselves become sinusoidal. The latter is explained by the action of the moment of inertia of the brush.
IV. Influence of contact temperature. Analysis of the influence of temperature on brush vibrations proves to be the most complicated, since temperature affects all the physical-mechanical properties of the friction pairs. With an increase in temperature to 90–100°C, the vibration amplitude increases, while the frequency falls slightly. This is explained by an increase in the coefficient of friction. With further increa-
Fig. 4. Oscillograms of tangential vibrations of the EG-14 brush at \(P = 250\ \mathrm{g/cm^2}\).
a — \(n = 200\ \mathrm{rpm}\); b — \(n = 1200\ \mathrm{rpm}\);
c — \(n = 2000\ \mathrm{rpm}\); d — \(n = 3000\ \mathrm{rpm}\).
With increasing temperature the amplitude of the vibrations decreases, which is evidently connected with softening of the brush surface and the formation of a viscous interlayer.
V. Influence of the projection of the brush from the brush holder. Of the factors listed above that affect brush vibrations, the greatest influence is exerted by the clearance between the brush holder and the ring. Increasing the clearance from 2 to 4 mm leads to an increase in the vibration amplitude by a factor of 2–2.5.
VI. In addition to the factors listed, the angle formed by the brush with the surface of the ring also has a large influence on the intensity of brush vibrations. When the brush is set at an angle of 2–4° in the direction of rotation of the ring, it is possible to eliminate tangential vibrations almost completely.
To investigate the influence of tangential vibrations on current collection, the authors carried out an experiment with three-layer brushes. The brushes were first carefully run in on a copper ring at a current density of \(20\ \mathrm{A/cm^2}\). During the experiment, the nature of the current distribution over the brush layers and the connection of these currents with tangential vibrations were observed.
As experimental investigations showed, when a brush operates on a ring or collector, redistribution of current over the brush layers constantly occurs. In this process, the currents at the leading and trailing edges of the brush undergo the greatest changes, which indicates instability in the operation of the leading and trailing edges of the brush. Table 1 gives the values of the voltage drops across the layers of the EG-14 brush, from which it may be concluded that the depth of modulation of the drop
Table 1
| Speed $n$, rpm | Running edge, $U=$ | Running edge, $U\approx$ | Middle layer, $U=$ | Middle layer, $U\approx$ | Trailing edge, $U=$ | Trailing edge, $U\approx$ |
|---|---|---|---|---|---|---|
| 2300 | 0,54 | 0,24 | 0,50 | 0,15 | 0,52 | 0,24 |
| 1800 | 0,54 | 0,23 | 0,50 | 0,15 | 0,52 | 0,23 |
| 1400 | 0,53 | 0,22 | 0,50 | 0,15 | 0,51 | 0,22 |
| 1000 | 0,51 | 0,20 | 0,49 | 0,14 | 0,51 | 0,18 |
| 500 | 0,51 | 0,14 | 0,48 | 0,12 | 0,50 | 0,16 |
| 0 | 0,46 | 0 | 0,46 | 0 | 0,48 | 0 |
the voltage reaches a considerable magnitude, taking values from 16 to 50 percent. The modulation depth was taken to be the ratio of the variable component $U\approx$ to the constant component $U=$ of the transient voltage drop, expressed as a percentage. Measurement of $U\approx$ and $U=$ was carried out by the oscillographic method. When the voltages at the trailing and running edges of the brush are oscillographed by means of a two-beam S1-16 oscilloscope, it turns out that these voltages are strictly in antiphase (Fig. 5). The latter indicates that the oscillations of the brushes that occur are, by their nature, rocking oscillations and not flexural ones, as K. Binder [4] indicates.
Fig. 5. Oscillogram of the voltage drop in the running and trailing layers of the brush
The modulation of the voltage drop on the brush that occurs is directly connected with the modulation of the actual area of contact between brush and commutator and, consequently, with the transition resistance. As is known, the instantaneous contact area that takes up the pressure constitutes only a small fraction of the nominal area. In this case the actual area depends on the magnitude of the specific pressure and on the hardness of the electrobrush material, and is related to them by the following relation [8]:
\[ F=\frac{P}{\xi H}. \tag{17} \]
On the other hand, the transition resistance between the brush and the ring consists of the constriction resistance $R_F$ and the resistance of the oxide film. The latter may be regarded as inversely proportional to the area of contact of the contacts,
\[ R_F=a\frac{1}{F}, \tag{18} \]
where $a$ is a certain coefficient of proportionality.
Thus, the modulation of the voltage drop observed in practice can be explained by a change in the contact area and in the transient resistance in the presence of tangential vibrations of the brushes.
\[ R=f(\varphi, P, t). \tag{19} \]
From (19) it is clear that, in order to take into account the magnitude of the modulation of the transient resistance, it is necessary to know the dependence of the actual contact area on the vibration parameters.
As is known from practice, the voltage drop in a sliding contact exceeds the voltage drop at a stationary contact by a certain amount \(\Delta U\):
\[ U=RI+\Delta U. \tag{20} \]
If one assumes that the current through the contact is not modulated (which in most cases is indeed the case), then for the contact voltage drop one may write
\[ U=\frac{a\xi H}{P}\cdot I+\frac{ma\xi H}{P}\cdot I\sin 2\pi f t. \tag{21} \]
From (21) we obtain for \(U\) and \(R\):
\[ U=\frac{a\xi H I}{P}(1+m\sin 2\pi f t), \tag{22} \]
\[ R=\frac{a\xi H}{P}(1+m\sin 2\pi f t). \tag{23} \]
In the case where intense brush vibrations are observed, accompanied by crackling and whistling, modulation of the current through the contact also takes place, which leads to sparking.
Thus, there is no doubt that transverse vibrations of the brushes under certain conditions are extremely harmful, leading to serious disturbances of commutation. On the basis of (23) it can be shown that the transient resistance of the brush–lamella contact also does not remain constant during the short-circuiting of the commutating section, and varies according to a certain law determined by the depth of modulation of the contact resistance and by the phase of the brush position:
\[ R=R_0\left(1+\frac{m\sin 2\pi f t}{T-t}\right), \tag{24} \]
where
\(T\) — the theoretical commutation period,
\(t\) — the current time coordinate,
\(f\) — the frequency of brush vibrations,
\(m\) — the depth of modulation of the contact voltage drop.
From the foregoing it is clear that tangential vibrations, in addition to everything else, exert a significant influence on the nonidentity of commutation cycles with respect to the commutation period. The latter follows from the fact that tangential vibrations lead to instability of the entering and leaving edges of the brush.
Conclusions
- The paper presents the results of investigations of tangential vibrations of brushes on a slip ring and a short-circuited commutator, on the basis of the general theory of frictional vibrations [1, 2, 3].
-
It has been shown that the nature and intensity of tangential vibrations depend on the following parameters of the collector–brush assembly: the mass and moment of inertia of the brush, the stiffness of the brush body, the projection of the brush from the brush holder, the height of the brush holder, the pressure, the temperature, and the clearance in the guides.
-
On the basis of the studies carried out, the following recommendations are given for the partial elimination of tangential vibrations:
a) reducing the clearance in the guides to a minimum;
b) reducing the gap between the brush holder and the commutator;
c) using harder brushes;
d) reducing the overall runout of commutators.
- A relationship is established between tangential vibrations and the electrical characteristics of the sliding contact. Thus, tangential vibrations are always accompanied by modulation of the transition resistance and of the contact voltage drop, which may lead to modulation of the current of the commutating section and, consequently, to deterioration of commutation.
References
-
B. V. Deryagin, V. E. Push, D. I. Tolstoi. Theory of sliding of solids with periodic bonds. Zh. T. F., vol. XXXVI, issue 6, 1956.
-
S. P. Strelkov. The Froude Pendulum. Zh. T. F., vol. III, issue 4, 1933.
-
O. A. Toropov. On nonlinear oscillations of frictional systems with one-sided coupling. Trudy NPI, vol. 163, 1966.
-
K. Binder. Influence of mechanical natural vibrations of carbon brushes on commutation and current collection. ETZ—B, Bd. 13, H. 10, 1961.
-
W. Wolkman. On the influence of frictional forces on the dynamic behavior of carbon brushes. ETZ-A, No. 6, 1964.
-
P. S. Livshits. Operation of electric brushes in the metallurgical industry. Metallurgiya, 1966.
-
L. Ya. Zinner, A. I. Skorospelshkin. Instrument for measuring small displacements. Izvestiya TPI, vol. 172, 1967.
-
R. Holm. Electrical Contacts. IIL, 1961.