On Tangential Oscillations of Carbon Brushes

E. K. Damm, L. Ya. Zinner, A. I. Skorospeshkin

(Recommended by the seminar of the Department of Electrical Machines and General Electrical Engineering)

Although several works \([1, 2, 3, 4]\) have been devoted to the theoretical and experimental study of tangential oscillations of brushes, the nature and degree of their influence on the commutation process, as well as possible measures for combating them, have still not been studied sufficiently. In the present work an attempt is made, on the basis of a more accurate equation of motion of the brush, to analyze the conditions for the occurrence of tangential oscillations, the occurrence of the well-known phenomenon of brush “whistling,” and also certain questions concerning the influence of transverse oscillations of the brushes on the commutation process.

Equation of Motion of the Brush and Its Analysis

The computational diagram of the current-collecting unit is presented in Fig. 1. We adopt the following notation:

\(I\) — moment of inertia of the brush with respect to the axis of rotation.

\(\varphi\) — angle of deflection of the brush from the equilibrium position.

(The equilibrium position corresponds to the absence of tangential oscillations during operation, when the brush is pressed by frictional forces against edge \(C\) of the brush holder).

\(P\) — pressing force of the brush-holder spring;

\(R_{\text{sh}}\) — radius of the brush face;

\(R_k\) — radius of the commutator;

\(l\) — calculated arm of application of the friction force;

\(c\) — coefficient of static friction of the sliding pair brush—commutator;

\(D\) — coefficient taking into account the decrease in the coefficient of friction in the presence of relative displacement of the brush and the commutator.

\(v\) — peripheral velocity of the commutator surface;

\(h\) — coefficient taking into account energy losses in the body of the brush during its deformation, directly related to the coefficient of restitution of the brush material;

\(M_r\) — constant component of the reaction of the brush holder at point \(C\);

\(d\) — overhang of the brush from the brush holder;

\(k\) — stiffness of the contact layer at point \(C\).

It is assumed that the pressing force is directed along the axis of the brush. The radius of the brush is always greater than the radius of the commutator.

In the system under consideration the following moments act. The moment due to the friction force between the brush and the commutator, whose variable component is proportional to the derivative of the angle \(\varphi\),

\[ M_{\mathrm{tr}}=lP[c-D(v-l\dot{\varphi})]=(lcP-lPDv)+l^{2}PD\dot{\varphi}. \tag{1} \]

The moment due to the reaction of the brush holder at point \(C\), whose variable component is proportional to \(\varphi\),

\[ M_c=M_r+k(l-d)^2\varphi. \]

The dynamic moment

\[ M_j=-I\ddot{\varphi}. \tag{3} \]

The moment caused by energy losses arising in the periodic deformation of the contact layer at point \(C\),

\[ M_{\mathrm{v}}=h(l-d)^2\dot{\varphi}. \tag{4} \]

The moment due to the pressure of the pressure finger

\[ M_p=\frac{R_{\mathrm{щ}}}{R_{\mathrm{щ}}-R_{\mathrm{к}}}\,lP\varphi . \tag{5} \]

Expression (5) takes into account the fact that, when the angle \(\varphi\) changes, the contact area of the brush with the commutator (Hertzian zone) moves along the face surface of the brush.

Taking (1)—(5) into account, the equation of equilibrium of moments about point \(O\) for the system under consideration can be written in the form

\[ -I\ddot{\varphi}-lP\frac{R_{\mathrm{щ}}}{R_{\mathrm{щ}}-R_{\mathrm{к}}}\varphi + lPc-lPDv-l^{2}PD\dot{\varphi}-M_r+k(l-d)^2\varphi - \]

\[ -\,h(l-d)^2\dot{\varphi}=0. \tag{6} \]

At a constant speed of rotation of the commutator, the brush takes up the clearance in the brush holder and is deflected through an angle \(\psi_0\) of such magnitude that the constant components of the friction moment and of the reaction of the brush holder balance one another:

\[ lPc-lPDv-M_r=0. \tag{7} \]

Taking (7) into account, expression (6) can be rewritten in the form

\[ \ddot{\varphi} + \left[ \frac{h(l-d)^2-l^2PD}{I} \right]\dot{\varphi} + \left[ \frac{ lP\frac{R_{\mathrm{щ}}}{R_{\mathrm{щ}}-R_{\mathrm{к}}}+k(l-d)^2 }{I} \right]\varphi=0. \tag{8} \]

Equation (8) is the equation of free oscillations of the system under consideration.

The nature of the oscillatory process may differ depending on the sign of the coefficient of \(\dot{\varphi}\). Suppose that it is positive, i.e.,

\[ h(l-d)^2-l^2PD>0. \tag{9} \]

Physically this means that the characteristic of the coefficient of friction does not fall too steeply as the speed of relative displacement of the brush and the commutator increases, and that the energy losses in deformation of the brush are relatively large.

Observations show that the tangential oscillations of brushes are always periodic. Oscillations in the given system, when condition (9) is satisfied, can occur only under the action of an initial impulse. In this case the damped oscillations occur according to the law:

\[ \varphi=\varphi_{0} e^{-\frac{1}{2}lat}\sin \omega t, \tag{10} \]

where

\[ \omega=\frac{1}{2}\sqrt{ 4\left[ \frac{ lP\dfrac{R_{\mathrm{sh}}}{R_{\mathrm{sh}}-R_{\mathrm{k}}}+k(l-d)^2 }{I} \right]^2 - \left[ \frac{h(l-d)^2-l^2PD}{I} \right]^2 }, \]

\[ a=\frac{h(l-d)^2-l^2PD}{I}, \]

\(\varphi_0\) is the initial displacement of the system.

If the brush experiences no disturbances from the commutator, tangential oscillations will be absent altogether. In Fig. 2, the profilogram of the commutator and the oscillogram of the tangential oscillations of the brush are compared. Both oscillograms were recorded simultaneously from the screen of a dual-beam oscilloscope. Commutator bars protruding by several microns excite oscillations, which then decay in accordance with expression (10).

The oscillations do not decay to zero only because it is impossible to obtain an ideal sliding surface. Oscillations may also be excited because of the ellipticity and eccentricity of the commutator,

Fig. 1. Calculation scheme

since in this case the friction force changes owing to the modulation of the contact pressure. This was already pointed out by W. Volkmann [3].

If the condition

\[ h(l-d)^2-l^2PD<0 \tag{11} \]

is satisfied, the system under consideration becomes unstable. The solution of equation (8) in this case is (for complex-conjugate roots of the characteristic equation) a sinusoid with an infinitely increasing amplitude.

Fig. 2. Oscillogram of the commutator profile (below), and of the tangential oscillations of the brush (above)

In practice this does not occur. Consequently, for case (11), equation (8) is not valid. The reason for this is twofold. First, the characteristic of the coefficient of friction is approximately approximated by the expression \(K_{\mathrm{tr}} = c - Dv\) only over a small segment (Fig. 3).

Fig. 3. 1 — real friction characteristic; 2 — calculated, \(K_{\mathrm{tr}} = c - Dv\)

Second, if previously, during oscillations, the brush did not separate from edge \(C\) of the brush holder, now the amplitude of the tangential oscillations increases so much that the brush separates from the brush holder, and it would be more correct to consider the system with a constraint.

When condition (11) is satisfied, the system easily passes into a self-oscillatory regime, which is described by a complex nonlinear equation.

The well-known phenomenon of “whistling” of brushes, accompanied by a sharp increase in the amplitude of tangential oscillations, corresponds precisely to condition (11).

Figure 4 presents oscillograms of the lamellar current and the transient voltage drop, obtained on the setup described in [5] of the present collection, with “whistling” brushes. It is evident that the influence of the oscillations on the magnitude of the transient voltage drop is significant.

Fig. 4. Lamellar current (top) and transient voltage drop (bottom) with “whistling” brushes

Observations show that, in the absence of brush “whistling,” the frequency of tangential oscillations is almost independent of the collector’s rotational speed, and the oscillations are close to sinusoidal in form.

With whistling brushes, as direct oscillographic recording of the oscillations shows, the frequency depends to a considerable degree on the rotational speed, and the form of the oscillations differs substantially from sinusoidal; this is a characteristic sign of a self-oscillatory process.

The fact that the phenomenon of “whistling” has a very harmful effect on the commutation process is convincingly shown by Fig. 4.

Expression (9) is the condition under which brush “whistling” will not occur. Effective measures for combating this harmful phenomenon are: selecting the minimum possible projection of the brush from the brush holder, \(d\), and installing damping gaskets in the gap between the brush and the brush holder, on the side of the incoming edge of the brush, made of a material with a low coefficient of restitution of velocity.

Tangential oscillations of brushes and commutation

When it is said that the radius of the brush face is greater than the radius of the collector, it is assumed that the brush is bedded to the radius of the collector section having the least curvature. This would correspond to reality if the brush did not undergo tangential oscillations. These oscillations may be regarded as rotation of the brush about the axis \(O\) (Fig. 1).

After prolonged operation, the radius of the brush face becomes such that, for any position of the brush, the area of the face surface receiving the pressure is the same. Knowing the geometrical dimensions of the current-collecting assembly, one can obtain an expression relating the radius of curvature of the brush face to the amplitude of the tangential oscillations.

Let us assume that the brush, while oscillating, moves parallel to itself. Exact determination of the position of point \(O\) is difficult; however, since the amplitude of the oscillations is several orders of magnitude smaller than the quantity \(l\), the position of the center of oscillations plays no essential role. Let us further assume that, as it wears, the brush acquires a curvature with constant radius \(R_{\text{щ}}\). The elastic deformation of the brush and collector is not taken into account.

If the double amplitude (peak-to-peak value) of the tangential oscillations of the brush is denoted by \(A_{m}\), then the gap \(\delta_{k}\) under the edge of the brush, when the latter occupies the mean position, will be equal to

\[ z_k=A_m\frac{b_{\mathrm{sh}}}{2R_k}. \tag{12} \]

where \(b_{\mathrm{sh}}\) is the width of the brush.

The radius of curvature of the brush is

\[ R_{\mathrm{sh}}=R_k+ \frac{A_m b_{\mathrm{sh}}} { 2\sqrt{R_k^2-\dfrac{b_{\mathrm{sh}}^2}{4}} - \dfrac{2R_k^2-\dfrac{b_{\mathrm{sh}}^2}{2}}{R_k} }. \tag{13} \]

If the brush occupies the middle position, the gap between the brush along the width of the brush varies according to the law

\[ z=(R_{\mathrm{sh}}-R_k)+\left(\sqrt{R_k^2-a^2}-\sqrt{R_{\mathrm{sh}}^2-a^2}\right), \tag{14} \]

where \(a\) is the distance from the axis of the brush.

To obtain an idea of the degree of influence of tangential oscillations, let us carry out a calculation. Let \(R_k=4\ \mathrm{cm}\), \(b_{\mathrm{sh}}=1\ \mathrm{cm}\), \(A_m=10^{-3}\ \mathrm{cm}=10\ \mu\mathrm{m}\). Then from (4) we have \(R_{\mathrm{sh}}=4.0166\ \mathrm{cm}\). The radius of the brush will increase by \(166\ \mu\mathrm{m}\). From this example it is clear that the principal role in increasing the radius of curvature of the face surface of the brush is played by tangential oscillations.

Direct measurements of the gap between the brush and the commutator with the aid of a capacitive sensor placed in the interlamellar slot show that, in the absence of brush “whistling,” there is no gap (Fig. 5a). If there were a gap of variable magnitude under the brush, the pulse from the capacitive sensor would not have a rectangular shape.

Fig. 5a. Gap between the brush and the commutator during steady operation

With “whistling” brushes, the edges of the brush can for a short time break away from the surface of the commutator (Fig. 5b).

Fig. 5b. Gap between the brush and the commutator with “whistling” brushes

The clearance under the brush can be completely taken up owing to elastic deformation of the brush and commutator materials, but the distribution of specific pressure in the contact layer becomes nonuniform, which directly affects the magnitude of the transient voltage drop between the lamella and the brush in multiple brush overlap.

Conclusions

1. The equation of tangential oscillations of brushes has been obtained and analyzed. Under normal brush operation, the amplitude of tangential oscillations depends on the shape of the commutator profile, and the frequency depends on the parameters of the brush–commutator assembly.

2. The conditions under which brush “whistling” arises, accompanied by a sharp increase in the amplitude of tangential oscillations, have been established, and measures for combating this phenomenon have been indicated.

3. The radius of curvature of the face surface of the brush is determined mainly by the tangential oscillations of the brushes.

References

1. W. Volkmann. Über Kohlenbürsten—Schwingungen mit kleiner Amplitude. ETZ—A, No. 5, 1961.

2. K. Binder. Der Einfluß mechanischer Eigenschwingung von Kohlenbürsten auf Kommutierung und Stromabnahm. ETZ—B, Bd. 13, H. 10, 1961.

3. W. Volkmann. Über den Einfluß der Reibungskräfte auf das Dynamische Laufverhalten der Kohlenbürste. ETZ—A, No. 6, 1964.

4. L. Ya. Zinner, A. I. Skorospeshkin. Tangential oscillations of brushes. Izvestiya TPI, vol. 190, 1968.

5. A. I. Skorospeshkin, L. Ya. Zinner, E. K. Damm, V. P. Karataev. Apparatus for the experimental investigation of the sliding contact brush–commutator. Article in the present collection.