Study of the Stability of Electric Brush Operation on Commutators and Slip Rings
Abstract Generated abstract
This paper analyzes the mechanical stability of electric brush operation on commutators and slip rings, focusing on brush lift-off caused by eccentricity, ellipticity, surface harmonics, and interlamellar height differences. It develops analytical models for brush response to harmonic and impact disturbances, derives conditions for continuous contact, and uses these expressions to calculate permissible surface irregularities and parameters of the brush and spring system. An experimental study with a copper ring relates contact voltage-drop modulation to proposed stability criteria, the mechanical stability margin and contact-pressure pulsation coefficient. The authors conclude that reliable sliding contact requires low values of these criteria, increased natural frequency of the brush and spring system, reduced brush mass or increased spring stiffness, and stricter surface tolerances, especially for high-speed machines.
Full Text
STUDY OF THE STABILITY OF ELECTRIC BRUSH OPERATION ON COMMUTATORS AND SLIP RINGS
L. Ya. Zinner, A. I. Skorospeshkin, E. K. Damm
(Recommended by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
Satisfactory commutation of commutator-type electrical machines is determined to a considerable extent by the quality of execution of the commutator-and-brush apparatus and, consequently, by the condition of the working surface.
One of the causes of unsatisfactory commutation is instability of the commutator surface, leading to separation of the brushes and, consequently, to sparking even at low rotational speeds.
In this connection there arises the need for an analytical and experimental study of the stability of brush operation on commutators and slip rings, and for the development of tolerances for the principal parameters of the current-collection unit.
In the present work, primary attention is given to the study of brush operation under the action on them of harmonic and impact disturbances from the commutator, to the determination of the conditions for uninterrupted brush operation, and to the development of tolerances.
Eccentricity, ellipticity, and the interlamellar difference in height.
As the computational scheme for the action of harmonic disturbances on the brush, the scheme in Fig. 1 is adopted, where \(T\) is the friction force of the brush against the brush holder, \(W\) is the force of viscous friction of the brush against the air, \(N\) is the reac-
Fig. 1. Computational diagram of the commutator-brush unit
of the commutator, arising under elastoplastic deformation of the contact layer; \(P_0\) is the pressure on the brush, \(m\) is the mass of the brush, \(K\) is the stiffness of the pressure spring, \(K'\) is the stiffness of the contact layer, \(S_1\) is the deformation zone of the brush.
In the case of a simple brush holder, when there are no damping masses for reducing external vibrations, the force is very small and is determined by the relation [1]
\[ W = C q S, \tag{1} \]
where
\(C\) is the coefficient of frontal resistance;
\[ q = -\frac{\rho}{2} V_{\text{sh}}^{2}, \]
\(S\) is the cross-sectional area of the brush;
\(\rho\) is the density of air at \(+20^\circ\text{C}\);
\(V_{\text{sh}}\) is the velocity of radial displacement of the brush.
In the first approximation, the velocity of the brush may be determined as
\[ V_{\text{sh}} = \sqrt{\frac{2h_{\max}P_0}{m}}, \tag{2} \]
where
\(h_{\max}\) is the maximum height of an irregularity;
\(P_0\) is the pressure force;
\(m\) is the mass of the brush.
The condition for uninterrupted motion of the brush may be written in the form:
\[ \frac{P_0}{m} = \frac{d^2 h(e)}{dt^2}. \tag{3} \]
Hence the pressure required to ensure continuous following of the commutator irregularities by the brush is determined from the expression:
\[ P_0 = K[\delta - \xi + h(e)], \tag{4} \]
where
\(\xi\) is the approach of the contacting surfaces due to elastic deformation of the brush under the action of the force \(P_0\),
\(\delta\) is the initial displacement of the pressure spring;
\(h(e)\) is the function describing the commutator profile.
In the general case, the function \(h(l)\) can be represented as a Fourier-series expansion, expressing the circumference of the commutator in terms of circumferential velocity and time:
\[ h(e) = a_0 + \sum_{i=1}^{\infty} A_i \sin(i\omega t + \varphi_i), \tag{5} \]
where
\(a_0\) is the constant term of the expansion,
\[ \omega = \frac{2\pi V}{e} \]
is the angular frequency of the first harmonic, corresponding to the eccentricity of the commutator;
\(A_i\) and \(\varphi_i\) are the amplitude and phase of the \(i\)-th harmonic.
After substituting the second derivative of expression (5) into (3) and carrying out certain transformations, we obtain an inequality for the natural frequency of the oscillatory system, whose fulfillment is necessary for contact stability:
\[ p^2 \leq -\frac{\omega^2 \displaystyle\sum_{i=1}^{\infty} e^2 A_i \sin(e\omega t + \varphi_i)} {\delta - \xi + \displaystyle\sum_{i=1}^{\infty} A_i \sin(i\omega t + \varphi_i) + a_0}, \tag{6} \]
where \(p=\sqrt{\dfrac{K}{m}}\) is the natural frequency of the brush–spring system.
From (6), after a number of transformations, we obtain an expression for the limiting value of the amplitude of any harmonic component of the commutator profile:
\[ A_i \leq \frac{\delta-\xi+a_0}{1+\dfrac{\omega^2 i^2}{p^2}} . \tag{7} \]
Expressions (6, 7) are valid for the case when the resistance force \(F\) and the contact stiffness \(K'\) are not taken into account.
Taking \(F\) and \(K'\) into account, the motion of the brush is described by a differential equation of the form:
\[ m\ddot y + m\ddot h(e) - K\delta + K'\xi \pm N'(C-D\dot y)=0, \tag{8} \]
where
\(N'\) is the pressure on the brush from the brush holder at the points of contact,
\(C\) and \(D\) are coefficients in the resistance force, characterizing the frictional properties of the electrobrush material [2].
It follows from (8) that the condition for uninterrupted motion of the brush along a deformed commutator is satisfied if the following inequalities hold:
\[ m\ddot y + m\ddot h(l) - K\delta + K'\xi \pm N'(C-D\dot y) < 0. \tag{9} \]
for \(y \leq h(l)\).
Suppose that the motion of the brush differs from the motion of the irregularity on the commutator surface and occurs according to the law \(y=y\sin\Theta t\). Then, taking into account the contact stiffness and the resistance force, the solution of inequality (9) will be an expression of the form:
\[ A_i \leq \frac{ \delta+a_0-\dfrac{K'}{K}\xi -y\dfrac{\Theta^2}{p^2} \pm N'\dfrac{C}{K} }{ \dfrac{\omega^2 i^2}{p^2} }. \tag{10} \]
Analysis of expressions (6, 7, 10) makes it possible to draw the following conclusions:
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In order to reduce the probability of brush lift-off, it is necessary to increase the natural frequency of oscillation of the system at unchanged pressure; this is achieved by reducing the mass of the brushes and increasing the stiffness of the pressure spring, and, in addition, by using brushes with lower contact stiffness.
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When hard grades of brushes are used, since they have better commutating properties, it is necessary to reduce the permissible values of the irregularities of the commutator surface and the linear velocities; this is achieved by reducing the commutator diameter at a specified rotational speed.
Significantly greater difficulties in ensuring uninterrupted motion of the brushes, as a rule, arise when there are substantial interlamellar steps.
To examine the mechanism of interaction of the brushes with protruding lamellae, let us consider the diagram in Fig. 2, where \(S\) is the impulse of the impact force; \(S_0\) is the normal component of the impulse of the impact force; \(F\) is the tangential component; \(R_k\) is the commutator radius; \(R_{\mathrm{sh}}\) is the radius of curvature of the brush; \(\alpha_0\) is the impact angle; \(h\) is the height step of adjacent lamellae.
Fig. 2. Diagram of the interaction of the brush with a protruding lamella
The behavior of the brush upon impact with a protruding plate is determined by the impulse of the impact force [2], which in the general case is written in the following form:
\[ S=\lim_{\tau \to 0}(S_0 \tau), \tag{11} \]
where \(\tau\) is the duration of the impact,
\(S_0\) is the unit impulse of the impact force.
Knowing the magnitude of the impulse of force, the displacement of the brush after collision and its velocity can be determined from the expressions
\[ y=\frac{1}{mp}\sum_{j=0}^{i} \left[ S_0 \sin pt+\ldots+S_j \sin p(t-jT_c) \right], \tag{12} \]
\[ \dot{y}=\frac{1}{m}\sum_{j=0}^{i} \left[ S_0 \sin pt+\ldots+S_j \sin p(t-jT_c) \right], \tag{13} \]
where
\[ T_c=\frac{1}{\pi}\sqrt{\frac{m}{K}} \]
is the period of the natural oscillations of the brush–spring system.
Thus, in investigating the behavior of a brush striking the protruding plates of the commutator, the problem reduces to determining the impulses of the instantaneous forces arising at each impact and to summing the series for \(y\) and \(\dot{y}\) for any number of protruding plates, \(i, j\).
In the case of a single impact, we define the impulse of force as
\[ S_j=\frac{m_{\mathrm{pr}}\cdot m(V-V_j)}{m_{\mathrm{pr}}+m}, \tag{14} \]
where \(m_{\mathrm{pr}}\) is the reduced mass of the armature,
\[ m_{\mathrm{pr}}=\frac{I_x}{r^2}, \]
\(I_x\) is the moment of inertia of the armature relative to the axis of rotation;
\(m\) is the mass of the brush;
\(r\) is the impact radius;
\(V_j\) is the velocity of the brush at the instant of impact;
\(V\) is the velocity of the lamella.
From Fig. 2 we determine the radial component of the impulse of the impact force:
\[ S_0 = S_j \alpha, \tag{15} \]
where
\[ \begin{aligned} \alpha &\text{ is the impact angle,}\\ h &\text{ is the height of the lamella,}\\ b_{\mathrm{shch}} &\text{ is the width of the brush along the commutator.} \end{aligned} \]
Taking into account that at the moment of collision the radial component of the brush velocity \(V_i = 0\), after substituting the values for \(\alpha\) and \(m_{\mathrm{pr}}\) into (14) we obtain the following expression for the impulse of the impact force:
\[ S_0 = \frac{2 I_x m V h} {r^2 \left( \dfrac{I_x}{r^2} + m \right)b_{\mathrm{shch}}}. \tag{16} \]
After substituting into (16) the values \(r=(R_k+h)\approx R_k\) and
\[ \frac{m}{I_x}\approx 0 \]
and carrying out some transformations, we have for the impulse of the impact force:
\[ S_0 = \frac{y \pi R_k m h \omega}{b_{\mathrm{shch}}} (1+K_v), \tag{17} \]
where
\[ \begin{aligned} \omega &\text{ is the angular velocity,}\\ K_v &\text{ is the coefficient of restitution of the brush velocity, taking into account}\\ &\text{the ability of the brushes to damp impact loads.} \end{aligned} \]
From (12) we find:
\[ y = \frac{1}{mp}\sum_0^1 S_0 \sin pt = \frac{4\pi R_k h \omega}{b_{\mathrm{shch}}p}(1+k_v). \tag{18} \]
Expression (18) is obtained under the condition that the pressure spring is not initially strained. Taking into account the preliminary displacement of the spring, we have:
\[ y_{\max} = \sqrt{ \delta^2+ \left( \frac{4\pi R_k h \omega}{b_{\mathrm{shch}}p} \right)^2 \cdot (1+K_v)^2 }. \tag{19} \]
From (19) we find the condition for continuous motion of the brush along the commutator when it collides with a protruding lamella. It is obvious that the condition must be satisfied for \(y_{\max}\le h\), whence
\[ h_{\max}\le \delta \sqrt{ \frac{1}{ \dfrac{4\pi^2 D_k^2 \omega^2}{b_{\mathrm{shch}}^2 p^2} \cdot (1+K_v)^2 } }. \tag{20} \]
It follows from (20) that, in order to ensure continuous motion of the brushes along the commutator when interacting with protruding lamellae, the same measures are necessary as in the case of high-frequency harmonic disturbances. The most effective measure should be considered to be reducing the difference in height between lamellae and reducing the commutator diameter, especially in high-speed machines.
In this connection, it should be considered necessary to develop a rigorous system of tolerances for the principal parameters of the condition of commutator surfaces. The expressions obtained from the analytical consideration of the conditions for continuous motion of the brushes make it possible to obtain simple formulas for determining the permissible values of eccentricity, ellipticity, and interlamellar differences in height. For this purpose let us consider the equation of the balance of forces acting on the brush:
\[ \delta_k-\frac{d^2h(e)}{dt^2}=0. \tag{21} \]
From (21) we determine the critical values of eccentricity \(e\) and ellipticity \(E\), above which separation of the brushes from the commutator occurs:
\[ e \leq \frac{3600\delta_k}{4\pi^2 n^2 m}, \tag{22} \]
\[ E \leq \frac{3600\delta_k}{16\pi^2 n^2 m}. \tag{23} \]
In the general case, with a known pitch of the irregularity on the commutator, the value of its permissible height is determined from the expression:
\[ A \leq \frac{3600\delta_k e^2}{4\pi^4 D_k^2 n^2 m}. \tag{24} \]
The values for \(E\) and \(e\) obtained by calculation from (22) and (23) are limiting values, above which separation of the brushes from the commutator occurs. It is obvious that the actual values of eccentricity and ellipticity must be much smaller than the limiting ones, since even at comparatively low rotational speeds the presence of eccentricity leads to such harmful phenomena as modulation of the contact pressure, of the transient resistance, of the actual arc of contact between the brush and the commutator, and intensification of tangential oscillations.
Consequently, it is necessary to establish certain criteria that would characterize the reliability of operation of the sliding contact.
Such criteria, in our opinion, are, on the one hand, the reserve of mechanical stability with respect to speed and, on the other hand, the pulsation coefficient of the contact pressure \(K_n\).
\[ \Psi=\frac{n}{n_{\text{crit}}}. \tag{25} \]
In (25) the following designations are adopted:
\(n\) — operating rotational speed; \(n_{\text{crit}}\) — critical speed at which separation of the brushes from the commutator occurs.
The expression for the pulsation coefficient of the contact pressure is obtained from the equality of the force actions on the brush (21) by substituting into it the expressions for the acceleration of the brush by the commutator and by the pressure spring. After a series of transformations we have:
\[ K_{n(e)}=\frac{4\pi^2mn^2e}{900P_0}, \tag{26} \]
\[ K_{n(E)}=\frac{16\pi^2mnE}{900P_0}, \tag{27} \]
where
\(K_{n(e)}\) and \(K_{n(E)}\) are the pulsation coefficients of the contact pressure caused by eccentricity and ellipticity.
From (25, 26, 27) it is seen that the contact of the brush with the commutator is stable when \(\Psi=K_n=0\). When \(\Psi=1\) and \(K_n=1\), the contact is completely unstable.
For the development of a substantiated system of tolerances, it is also necessary to carry out an experimental study of the values of \(\Psi\) and \(K_n\) at which brush operation is satisfactory. For this purpose we set up an experiment on an acceleration installation with a copper ring 130 mm in diameter, having an eccentricity of 100 \(\mu\)k. The tested brushes were mounted on the ring, carefully run in at a current density of 20 a/cm\(^2\).
The experimental procedure was as follows: the disk was set into rotation, and at a specified speed the constant and alternating components of the contact voltage drop, caused by the action of eccentricity, were measured.
During the experiment, the rotational speed of the ring was varied from 500 to 6000 rpm, and the brush pressure from 100 to 500 g/cm². As the criterion of normal contact operation, the magnitude of the modulation depth of the contact voltage drop was taken, equal to
\[ m=\frac{U_{\sim}}{U_=}\cdot 100 . \tag{28} \]
It was established that signs of unsatisfactory operation of the sliding contact, consisting in the appearance of visible sparking, are already observed at \(m=50\) percent, while reliable operation occurs at \(m=(5—10)\) percent. In addition, it was found that there is a direct relationship between the modulation depth of the contact voltage drop and the coefficients \(\Psi\) and \(K_n\). Thus, at \(m=15\) percent, which occurs when \(n=3000\) rpm, \(P_0=3000\), \(e=100\ \mu\text{m}\), and \(G=10\) g, the value \(K_{n(e)}\), calculated from (26) for the same values of \(n\), \(P_0\), \(e\), and \(G\), is 0.15, while the value of \(\Psi\) is 0.10.
Thus, it may be considered established that, for good operation of the sliding contact, it is necessary that \(\Psi\) and \(K_n\) have the values
\[ \Psi \leqslant 0.10, \]
\[ K_n \leqslant 0.10—0.15 . \]
Taking into account the values for \(\Psi\), the expressions for the permissible values of the harmonic amplitudes are written in the form:
\[ e \leqslant \frac{3600\delta_k\psi}{4\pi^2 n^2 m}, \tag{29} \]
\[ E \leqslant \frac{3600\delta_k\psi}{16\pi^2 n^2 m}. \tag{30} \]
Figures 3 and 4 show the dependences \(e=f(n,\delta k,m)\) and \(E=f(n,\delta k,m)\), obtained by calculation using (29, 30).
Fig. 3. Dependence of the permissible values of \(E\) and \(e\) on pressure and rotational speed at \(G=10\) g.
Fig. 4. Dependence of the permissible values of \(e\) and \(E'\) on the mass of the brushes at \(n=3000,\ 6000,\ 10\,000\) rpm and \(\delta_k=300\) g.
It is evident from the curves that the permissible values of eccentricity and ellipticity are strongly dependent on the rotational speed, the mass of the brushes, and the pressure. Thus, increasing the pressure stabilizes brush operation and increases the permissible values of \(e\) and \(E\); increasing \(n\) and \(m\) leads to a decrease in the permissible \(E\) and \(e\) and increases contact instability. It is also evident from Figs. 3 and 4 that the maximum permissible values of \(e\) and \(E\) are somewhat greater for low-speed machines and considerably smaller for high-speed machines in comparison with those existing [3].
Figure 5 shows the dependences of the permissible interlamellar height differences on the peripheral speed of the commutator, obtained by calculation.
Fig. 5. Dependence of the permissible values of interlamellar height differences on the peripheral speed of the commutator at \(\delta=1\) cm, \(b_{\text{sh}}=1\) cm, \(P^2=300\ \text{g}/\text{s}^2\), \(D_k=80\) mm.
1 — for a brush of grade EG-4E, 2 — for a brush of grade EG-14, 3 — for a brush of grade EG-74.
from (20), and in Figs. 6, 7—the dependences of the permissible values of interlamellar height differences on the natural frequency of the brush–spring oscillatory system and on the width of the brush.
From the dependences presented it is evident that increasing the natural frequency of the oscillatory system increases the permissible value of the interlamellar height differences. Increasing the natural frequency, while keeping the pressing force unchanged, can be achieved by increasing the stiffness of the spring or by reducing the mass of the brushes.
The investigations carried out into the stability of brush motion over the commutator surface make it possible to draw the following conclusions:
Fig. 6. Dependence of the permissible values of interlamellar height differences on the natural frequency of the brush–spring system at \(K=300\ \text{g/cm}\), \(m=1, 2, 3, 4, 5, 6\ \text{g}\), \(D_k=80\ \text{mm}\), \(n=3000\ \text{rpm}\).
Fig. 7. Dependence of the permissible values of interlamellar height differences on the width of the brushes at \(D_k=80\ \text{mm}\), \(p^2=300\ \text{s}^{-2}\), \(\delta_k=10\ \text{mm}\), \(K=300\ \text{g/cm}\), \(n=3000\ \text{rpm}\).
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Expressions have been obtained that make it possible to calculate both the permissible values of eccentricity, ellipticity, and lamella offset, and the principal parameters of the brush–spring oscillatory system.
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On the basis of the analytical expressions obtained, which determine the conditions for stable operation of brushes on commutators, tolerances have been developed for the principal parameters of the surface condition of commutators.
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To increase the reliability of operation of the brush–commutator sliding contact, it is necessary:
a) to increase the natural frequency of the brush–spring oscillatory system at unchanged pressure;
b) to reduce the permissible values of commutator-surface irregularities and the linear speeds, which is achieved by reducing the commutator diameter at a specified rotational speed.
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It is proposed that the reliability of operation of the sliding contact be evaluated by the criteria $\Psi$ and $K_{\mathrm{p}}$, which characterize the margin of mechanical stability and the degree of modulation of the contact pressure.
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It has been determined experimentally that, for values $\Psi \leq 0.10$ and $K_{\mathrm{p}} \leq 10$—$0.15$, reliable operation of the sliding contact takes place.
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Ya. G. Panovko. Fundamentals of the Applied Theory of Elastic Vibrations. “Mashinostroenie,” 1967.
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P. A. Babadzhanyan, B. N. Lyusin. Design and Manufacture of Commutators for Electrical Machines. GEI, 1960.