SOME PROBLEMS IN THE DYNAMICS OF SLIDING BRUSH–COMMUTATOR CONTACT

L. Ya. Zinner, A. I. Skorospeleshkin

(Presented by the seminar of the Departments of Electrical Machines and General Electrical Engineering)

As is known [1, 2], the operation of a sliding contact is greatly influenced by mechanical factors, which under certain conditions lead to unsatisfactory commutation in commutator electrical machines. At present, ever greater attention is being devoted to theoretical questions in the dynamics of sliding contact; nevertheless, many uncertainties still remain concerning the oscillations of the commutator–brush assembly.

Most of the existing works [3, 4, 5] consider oscillations of brushes under the action of comparatively low-frequency harmonic disturbances from the commutator. Such an approach is correct when considering low-speed machines, where separation of the brushes from the commutator does not occur even at large values of irregularities. However, in most cases one has to deal not with harmonic, smooth disturbances of the commutator surface, but with high-frequency intermittent disturbances, which may lead to separation even at comparatively low rotational speeds. These disturbances acquire especially great significance for machines operating at high speeds. The latter is explained by the fact that the commutators of modern machines are subject to significant deformations, leading to the protrusion of individual segments or groups of segments. Obviously, in this case the interaction of the brush with an approaching irregularity of the commutator must be considered as a short-duration dynamic impact.

The previously proposed simplified schemes of the brush assembly, based on the assumption that the oscillatory system has only one degree of freedom, are in fact far from reality. The point is that brushes in an actual machine are subject to both radial and tangential oscillations. However, taking into account the fact that radial and tangential oscillations occur in mutually perpendicular planes, they may, with accuracy sufficient for practical calculations, be considered separately as independent.

In the present work, primary attention is devoted to radial oscillations of brushes during elastoplastic short-duration impacts with approaching irregularities of the commutator, taking into account the elastic properties of the brush itself.

As the computational scheme we adopt the scheme shown in Fig. 1. The differential equation of motion of the brush, taking into account all the forces acting on it, will be written in the form:

Fig. 1. Computational scheme of a simple commutator–brush unit.

\[ m\frac{d^{2}y}{dt^{2}}+\mu N_{0}\left(C-D\frac{dy}{dt}\sin\alpha\right)+ky+k'(y-h\sin\omega t)=0. \tag{1} \]

The notation in Fig. 1 and in the equation is as follows:

\(m\) — mass of the brush, g,
\(\mu\) — coefficient of friction,
\(P_{0}\) — pressure on the brush, N;
\(N_{0}=N_{1}+N\) — total reaction imposed on the brush by the brush holder;
\(C\) and \(D\) — coefficients in the resistance force characterizing the frictional properties of the electrobrush material (Table [3]);
\(k\) — stiffness of the pressure spring;
\(k'\) — stiffness of the body of the brush, characterizing its elastic properties;
\(\alpha\) — angle at which the brush encounters the incoming irregularity;
\(F\) — reaction from the commutator, N,
\(h\) — height of the irregularity, μm,
\(\omega\) — angular frequency of the disturbing force, determined from the length of the irregularity and the peripheral speed,

\[ \omega=\frac{\pi V}{l}, \tag{2} \]

where \(V\) is the peripheral speed of the commutator, m/sec,
\(l\) is the length of the irregularity, m.

From the standpoint of ensuring ideal contact between the brush and the commutator, it is necessary that the condition of synchronism of their motion be satisfied; this consists in the radial motion of the brush completely coinciding with the relief of the commutator surface. In an actual machine, however, this condition is, as a rule, not fulfilled. The main cause of the brush falling out of synchronism is the inertial masses of the oscillatory system and the presence of nonlinear resistance forces.

At first glance, the presence of resistance forces seems to play a positive role. Indeed, when there is an irregularity on the commutator surface, the resistance forces exert a damping action, preventing a jump-like increase in the brush velocity in the radial direction. But since the lifting of the brush is determined only by the height of the irregularity, the velocity and acceleration of the brush practically do not depend on the resistance forces. At the same time, damping by the resistance forces decreases with increasing velocity and becomes, in practice, very small, thus not preventing rebound of the brush.

On the other hand, when the brush is raised to the maximum height, its velocity at the point of inflection of the trajectory becomes equal to zero, while the action of the resistance forces is maximal, thereby hindering the return of the brush to the initial level. An analogous action of the resistance forces is also manifested in the case of sinking portions of the profile.

For a clear understanding of the character of each of the components in equation (1), let us consider them in more detail.

The second term in (1) is the resistance force

\[ F_c=\mu N_0(C-VD\sin\alpha). \tag{3} \]

From (3) it is seen that the resistance force is a function of the peripheral velocity of the commutator and of the angle of encounter with the oncoming irregularity. When \(\sin\alpha=0\), which occurs on an ideal commutator, the radial displacement of the brush, and consequently also its velocity, are equal to zero. The resistance force has its maximum value and is equal to the force of static friction of the brush in the guides

\[ F_c=\mu N_0 C. \tag{4} \]

When \(\alpha=\max\), the coefficient of friction and, consequently, the resistance force are minimal.

The normal reaction \(N_0\), strictly speaking, is nonlinear and is determined primarily by the angle of skew of the brush in the guides and by its deformation at the points of contact. However, when considering radial oscillations, it may with sufficient accuracy be regarded as constant, especially since the magnitudes of the brush skews in the guides are insignificant. The latter applies equally to the pressing force of the spring, where the nonlinearity of stiffness appears at displacements of the turns by amounts considerably larger than the displacements of the commutator surface.

The elastic properties of the brush–commutator contact layer are, obviously, completely determined by the compliance of the brush body and, in a rigorous analysis, are a nonlinear function of the magnitude of the deformation \((y-h\sin\omega t)\).

Making in expression (1) the substitution

\[ 2\eta=\frac{\mu N_0D}{m} \quad \text{and} \quad p^2=\frac{\kappa+\kappa'}{m}, \]

we obtain

\[ \ddot y+2\eta\dot y+p^2y=\frac{\kappa'}{m}h\sin\omega t. \tag{5} \]

In (5) it is assumed that the disturbances coming from the commutator are groups of protruding plates, which may be approximated by a half-wave sinusoid. It should be noted that such a representation of the disturbing force is a particular case. In the general case the disturbing force must be represented in the form \(h(t)\), and the equation of motion in the form”】【

\[ \ddot{y}+2\eta \dot{y}+p^2 y=\frac{K'}{m}ht. \tag{6} \]

To obtain a clearer idea of how, and to what extent, individual factors affect the behavior of the oscillatory system, let us consider certain special cases of equation (5) that may occur in the operation of a sliding contact.

1. Let us consider the behavior of the oscillatory system for the case in which a low-frequency sinusoidal disturbing force, caused by commutator unbalance, acts on the brush. For this case, equation (6) is written in the form:

\[ \ddot{y}+2\eta \dot{y}+p^2y=\frac{P_0-F}{m}\sin \omega t. \tag{7} \]

Solving (7) with respect to \(y\), we obtain for the displacement of the brush

\[ y= \frac{\delta}{ \sqrt{\left(1-\frac{\omega^2}{p^2}\right)^2+\frac{4\eta^2\omega^2}{p^4}} } \sin(\omega t-\alpha), \tag{8} \]

where \(\delta\) is the initial compression of the spring,

\[ \alpha=\operatorname{arctg}\frac{2\eta\omega}{p^2-\omega^2}, \]

\[ \omega=\frac{\pi D_k}{l}\cdot \frac{n}{60}, \tag{9} \]

where

\(D_k\) is the commutator diameter, m,
\(l\) is the wavelength of the irregularity, m,
\(n\) is the rotational speed of the commutator, rpm.

As is evident from (8), the amplitude of the brush oscillations caused by commutator runout is obtained by multiplying the static compression of the spring by a certain dynamic coefficient, whose value is determined by the ratio of the frequency of the oscillatory system to the frequency of the disturbing force. In addition, the amplitude depends on the coefficient \(\eta\), which characterizes the damping action of the resistance force. In the case when the system enters resonance \((\omega \approx p)\), the dynamic coefficient is maximal, and the amplitude of the brush oscillations is maximal:

\[ y_{\max}=\frac{\delta p}{2\eta}. \tag{10} \]

However, as noted earlier in [4], even at small pressures the brush is capable, without separation, of following the surface of the commutator at frequencies of up to several hundred hertz. Nevertheless, it is necessary to achieve a minimum value of commutator runout even in machines operating at low speeds. This is due to the fact that, in the presence of significant commutator runout, modulation of the contact pressure and of the pressure on the brush occurs.

Indeed, assuming that the brush slides without separation along the wave-shaped profile of the commutator, it can be shown that the vertical displacement of the brush increases in proportion to the square of the peripheral speed of the commutator, and the pressure on the brush does not remain constant.

Let us represent the cross-section of the commutator surface by the function \(y=f(x)\) and define the path traveled by the brush in time \(t\) as \(x=Vt\). Then the velocity and acceleration of the brush in the vertical direction are determined from the following expressions:

\[ \begin{gathered} \frac{dy}{dt}=V\frac{dy}{d(Vt)}=V\frac{dy}{dx},\\ \frac{d^2y}{dt^2}=V\frac{d}{dx}\left(V\frac{dy}{dx}\right)=V^2\frac{d^2y}{dx^2}, \end{gathered} \tag{11} \]

Taking (11) into account, the expression for the pressure on the brush in the presence of commutator eccentricity is obtained in the form

\[ P=P_0+mV^2\frac{d^2y}{dx^2}, \tag{12} \]

where

\[ \begin{aligned} P_0&\text{— force of the pressing spring, N,}\\ m&\text{— mass of the brush, g,}\\ V&\text{— peripheral speed of the commutator, m/s,}\\ x,y&\text{— current coordinates of the profile.} \end{aligned} \]

It follows from (12) that the magnitude of the pressure on the brush is a function of the peripheral speed, the mass of the brush, and the amount of commutator runout.

Instability of the contact pressure leads to an intensification of the tangential oscillations of the brushes as a result of modulation of the friction force between the commutator surface and the brush:

\[ F_{\mathrm{fr}}=\mu P(C-DV), \tag{13} \]

1. Let us consider the particular case in which the disturbing force from the commutator has a high-frequency character and is represented in the form of a constant step on its surface in the shape of a half-wave of a sinusoid. We neglect the action of resistance forces. For this case we bring equation (5) to the form:

\[ \ddot y+p^2y=\frac{k'}{m}h\sin\omega t, \tag{14} \]

and determine its solution with respect to \(y\) by the expression:

\[ y=\frac{k'h}{m(p^2-\omega^2)}\left(\sin\omega t-\frac{\omega}{p}\sin pt\right). \tag{15} \]

It follows from (15) that the elastic properties of the brush itself play a significant role in the interaction of the brush with the protruding commutator bars, since the amplitude of the oscillations is directly proportional to the stiffness of the brush and to the height of the protrusion. If the brush precisely follows the surface of the commutator, its motion at any instant must be determined by the expression:

\[ y=\delta+\frac{k'h}{k+k'}\sin\omega t. \tag{16} \]

From the analysis of profilograms of real commutators [4] it is seen that a commutator, having a profile in the form of a superposed sinusoid, acts on the brush only during a half-period of the variation of the irregularity, i.e., in the time interval

\[ 0\leq \omega t\leq \frac{\pi}{2}. \]

Expression (15) describes the transient process of brush oscillation, which is represented as forced oscillations on which free oscillations are superposed. Taking into account that the free oscillations decay rapidly, for the motion of the brush during interaction with an approaching irregularity we obtain an equation of the form:

\[ y=\frac{\kappa' h}{(\kappa+\kappa')\left(1-\frac{\omega^2}{p^2}\right)}\sin\omega t. \tag{17} \]

Whence, for the velocity and acceleration of the brush, we have

\[ \dot y=\frac{\pi\kappa' h V}{l(\kappa+\kappa')\left(1-\frac{\omega^2}{p^2}\right)}\cos\omega t. \tag{18} \]

\[ \ddot y=-\frac{\pi^2\kappa' h V^2}{l^2(\kappa+\kappa')\left(1-\frac{\omega^2}{p^2}\right)}\sin\omega t. \tag{19} \]

Continuous motion of the brush over the commutator irregularity occurs when

\[ |a_{\mathrm{b}}|>|a'_{\mathrm{b}}|, \tag{20a} \]

the limiting condition is when

\[ |a_{\mathrm{b}}|=|a'_{\mathrm{b}}|, \tag{20б} \]

and the condition for separate motion of the brush is when

\[ |a_{\mathrm{b}}|<|a'_{\mathrm{b}}|, \tag{20в} \]

where

\(a_{\mathrm{b}}\) is the acceleration of the brush due to the spring,
\(a'_{\mathrm{b}}\) is the acceleration of the brush due to the commutator.

From condition (20б) we determine the phase at which separation of the system occurs:

\[ \sin\omega t_0=\frac{\delta\kappa l^2p^2}{\pi\kappa' h v^2}\left(1-\frac{\omega^2}{p^2}\right)^2. \tag{21} \]

Analysis of expression (21) makes it possible to draw the following conclusions: the phase of separation of the system is directly proportional to the pressure on the brush, the stiffness of the pressure spring, and the square of the length of the irregularity, and inversely proportional to the stiffness of the brush, the height of the irregularity, and the square of the peripheral speed.

Thus, the greater the preliminary pressure, the larger the values of \(\sin\omega t_0\) at which brush separation occurs; the steeper (for one and the same height of irregularity) its leading front, the more rapidly separation occurs.

It is interesting to note the influence of the elastic properties of the brush itself. It follows from (21) that the lower the stiffness of the brush, the later its separation from the commutator surface occurs. The latter is explained by the fact that soft brushes are subjected to the action of significantly smaller forces during collision than are hard ones. Increasing the stiffness of the pressure spring also contributes to improved contact during short-time impacts.

After certain transformations, we shall put expression (21) into a form more convenient for practical calculations:

\[ \sin\omega t_0=\frac{\delta\kappa}{\kappa' h}\left(\frac{p}{\omega}\right)^2. \tag{22} \]

From (22) one can determine the moment of separation as a function of the speed, the length of the irregularity, the spring stiffness, and the initial compression

\[ t_0=\frac{1}{\pi V}\left[\frac{\pi}{2}-\arcsin\frac{\delta\kappa}{\kappa' h}\left(\frac{p}{\omega}\right)^2\right]. \tag{23} \]

When conditions (20б) and (20в) are satisfied, separation of the system occurs, and the subsequent motion of the brush is determined by the initial conditions at the instant of detachment. We shall write the equation of motion of the brush after the end of impact in the form:

\[ y = y_0 \cos pT + \frac{V_0}{p}\sin pT, \tag{24} \]

\[ y_0 = \delta + \frac{k'h}{k+k'}\sin \omega t_0, \tag{25} \]

\[ V_0 = \frac{\pi V h k_{\mathrm{v}}}{l}\cos \alpha \cos \omega t_0, \tag{26} \]

where

\[ \begin{aligned} y_0 &\text{— displacement of the brush at the instant of detachment, m,}\\ V_0 &\text{— velocity of the brush at the instant of detachment, m/sec,}\\ \alpha &\text{— angle of encounter of the brush with the approaching irregularity, deg.} \end{aligned} \]

From (24) we obtain an expression for the maximum value of the displacement of the brush after collision with a projecting plate or group of plates:

\[ y_{\max} = \sqrt{y_0^2 + \left(\frac{V_0}{P}\right)^2}. \tag{27} \]

Or, after substituting into (27), in place of \(V_0\) and \(y_0\), their values from (25) and (26), we have

\[ y_{\max} = \sqrt{ \left( \delta + \frac{k'h}{k+k'}\sin \omega t_0 \right)^2 + \left( \frac{\omega}{p} h k_{\mathrm{v}}\cos \alpha \cos \omega t_0 \right)^2 }. \tag{28} \]

As is seen from (28), the magnitude of the brush rebound upon collision with a projecting commutator plate is determined by the height of the projection, the peripheral speed of the commutator, the coefficient of restitution of velocity, the length of the projection along the commutator development, the stiffness of the pressure spring, the amount of preliminary spring compression, and the angle of encounter.

The angle of encounter can be determined by considering the calculation scheme of Fig. 1 statically, taking into account all the forces acting on the brush. With sufficient accuracy, the angle of encounter can be computed from the following expressions:

\[ \alpha = \operatorname{arc\,tg}\frac{hd}{lB_{\mathrm{sh}}\mu(2d+d')}; \tag{29} \]

for the case when the commutator surface is deformed according to a sinusoidal law,

\[ \alpha = \arccos \frac{2h}{lB_{\mathrm{sh}}} \tag{30} \]

for the case when the brush interacts with an individual plate or group of plates.

In (29) and (30) the following designations are adopted:

\[ \begin{aligned} d &\text{— height of the brush holder, m,}\\ d' &\text{— distance from the brush holder to the commutator, m,}\\ B_{\mathrm{sh}} &\text{— width of the brush, m.} \end{aligned} \]

Expression (28) makes it possible, from the known law of variation of the commutator profile, to calculate all necessary parameters of the oscillatory system for which the most favorable operation of the current-collecting unit occurs and, by varying the values of the variables \(\omega\), \(\alpha\), \(h\), to construct graphical dependences \(y = f(\delta, p, \omega, h, \alpha)\) that determine the zones of uninterrupted operation of the brushes. From (22) one can construct graphical dependences

...dependence of the phase of separation on the peripheral speed of the commutator at constant values of \(h\) and \(\delta\).

Fig. 2.

In Fig. 2, curve 1 represents the dependence \(\sin \omega t_0=f(V)\), and curve 2 represents the dependence of the magnitude of the brush rebound on the peripheral speed for three brush grades, EG-4, EG-14, and EG-74. The plots were constructed from the data of the profilogram of the commutator of machine PBS-32 (Fig. 5г [4]) for the maximum value of the height of the deformed section of the commutator. The following were taken as initial data: \(h=44\cdot10^{-6}\ \mathrm{m}\), \(l=0.02\ \mathrm{m}\), \(\delta=0.02\ \mathrm{m}\), \(p=200\ \mathrm{rad/sec}\). As is evident from Fig. 2, with an increase in rotational speed the phase of brush separation decreases, while the rebound amplitude increases. At the same time, softer brushes rebound by a considerably smaller amount than harder ones.

Fig. 3.

By (22) and (28), the permissible values of the total commutator runout and the protrusion of individual bars or of a group of bars can be calculated for any commutator machine operating in a specified range of operating speeds, if the pressure \(P_0\) and the stiffness of the pressure spring are known.

Figure 3 presents the calculated dependences \(y=f(p)\) for \(V=\mathrm{const}\), \(\delta=\mathrm{const}\), according to the data of the profilogram in Fig. 5g [4], for three grades of brushes (EG-4, EG-14, EG-74), while Fig. 4 gives the dependences \(y=f(\delta)\) for various values of the natural frequency of the oscillatory brush-spring system.

Figure 4. Plot of \(y\), microns, versus \(\delta\), cm. Curves are labeled \(p=60\), \(p=70\), \(p=90\), \(p=100\), \(p=200\).

Fig. 4.

From Figs. 3 and 4 it is clear that, in order to improve the operating conditions of sliding contact, it is necessary to increase the stiffness of the spring and its preliminary compression.

In addition to knowing the magnitude of the brush rebound from the commutator surface, it is also important to know the duration of the contact interruption and, moreover, the behavior of the system after the first impact with the commutator irregularity. After impact with the approaching irregularity, the brush undergoes oscillations on the spring, whose amplitude and frequency depend on the parameters of the oscillatory system and on the conditions imposed on the brush by the commutator during subsequent impacts. We determine the duration of the break in contact after the brush strikes the irregularity from (24), setting \(y=y_0\):

\[ y_0 = y_0 \cos pT' + \frac{V_0}{p}\sin pT', \tag{31} \]

whence, for the time of contact interruption, we obtain an expression of the form

\[ T'=\frac{1}{p}\left(\pi - 2\arctg \frac{V_0}{p y_0}\right). \tag{32} \]

At time \(T_1\) a new collision of the brush with the commutator surface occurs. The magnitude of the rebound in the repeated collision is determined by the position and velocity of the brush at the instant of collision, by the parameters of the oscillatory system, and by the elastic properties of the brush, and may be calculated from the expression

\[ y'=\sqrt{y^2+\frac{P_0}{p^2G}g\left(y_{\max}-y\right)\cos\alpha_1^2}, \tag{33} \]

where

\[ \begin{aligned} y&\text{— ordinate of the commutator surface at the instant of the repeated collision, m,}\\ P_0&\text{— pressure on the brush, N,}\\ G&\text{— weight of the brush, N,}\\ y_{\max}&\text{— ordinate of the brush after the first collision,}\\ g&\text{— acceleration of gravity, m/sec}^2,\\ \alpha_1&\text{— angle at which the brush meets the commutator profile in the repeated collision.} \end{aligned} \]

Thus, using expressions (22), (28), (30), (32), and (33), one can calculate the trajectory of motion of the brush during collision with a commutator irregularity. In practice, however, repeated collisions beyond the second do not occur because of the damping action of the resistance forces.

1. In the foregoing we considered certain special cases of the behavior of an oscillatory system under the action of disturbances from the commutator.

These special cases were based, in one case, on the assumption that the disturbances from the commutator have a low-frequency harmonic character, and, in the other, on the assumption that the interaction of the brush with the commutator has the character of a brief impact in the absence of resistance forces. In a real system, however, the resistance forces may play a perceptible role; it is therefore reasonable to consider the system under the action of all the forces. In doing so, we shall consider the behavior of the oscillatory system under the action of an arbitrary disturbing force of the type of a force impulse \(F(t)\).

Then equation (5) is written in the form

\[ \ddot{y}+2\eta\dot{y}+p^2y=F(t); \tag{34} \]

as is known from the theory of oscillations [5], the solution of this equation has the form

\[ y(t)= \sqrt{ y_0+\frac{y_0^2\eta+\dot{y}_0^{\,2}}{p^2-\eta^2} } \,e^{-\eta t} \sin\left[ \sqrt{p^2-\eta^2} -\operatorname{arctg} \frac{y_0\eta+\dot{y}_0}{y_0\sqrt{p^2-\eta^2}} \right], \tag{35} \]

where \(y(0)=y_0\) and \(\dot{y}(0)=\dot{y}_0\) are the initial conditions corresponding to the instant of time at which the action of the force impulse ends. After certain transformations, we obtain for the displacement of the brush after the end of the impact an expression of the form

\[ y(t)= \frac{V_0}{\sqrt{p^2-\eta^2}}\, e^{-\eta t}\, \sin\sqrt{p^2-\eta^2}\,t. \tag{36} \]

To calculate the displacement of the brush from the instant at which the action of the impulse ends, it is necessary to know the magnitude of the force impulse, from which the velocity of the brush at the instant the impact ends can be determined. For practical calculations it is convenient to use the expression for the impulse of the impact force proposed in [6]:

\[ F_{\mathrm{ud}}= m_{\mathrm{sh}}V \sqrt{\mu^2\sin\alpha+\left(1+k_{\mathrm{v}}\right)^2\cos\alpha^2}, \tag{37} \]

where

\[ \mu \text{ is the instantaneous coefficient of friction,} \]

\[ \alpha \text{ is the angle of encounter,} \]

\[ k_{\mathrm{в}} \text{ is the coefficient of restitution of velocity.} \]

Taking into account the fact that, under the action of the friction force, the brush takes up the clearance in the guides in the direction of motion of the commutator, one may make the not very strict assumption that the brush is worn in to the commutator by its leading edge, while the air wedge occurs at the trailing edge. In this case the velocity of the brush after impact will be determined from (37) as \(V_0 = V(1+k_{\mathrm{в}})\cos\alpha\), and the displacement, velocity, and acceleration of the brush after impact as

\[ y=\frac{V(1+k_{\mathrm{в}})\cos\alpha}{\sqrt{p^2-\eta^2}}\,e^{-\eta t}\sin\sqrt{p^2-\eta^2}\,t, \tag{38} \]

\[ \dot y=\eta V(1+k_{\mathrm{в}})\cos\alpha\, e^{-\eta t}\cdot \cos\sqrt{p^2-\eta^2}\,t, \tag{39} \]

\[ \ddot y=\eta^2 V(1+k_{\mathrm{в}})\cos\alpha\sqrt{p^2-\eta^2}\, e^{-\eta t}\sin\sqrt{p^2-\eta^2}\,t. \tag{40} \]

The time during which the brush will return to the initial level and again meet the surface of the commutator is determined by the expression

\[ t=\frac{2}{\sqrt{p^2-\eta^2}}\operatorname{arc\,tg}\frac{\sqrt{p^2-\eta^2}}{\eta}. \tag{41} \]

The time during which the brush will deviate by the maximum amount is determined from the expression:

\[ t_1=\frac{1}{\sqrt{p^2-\eta^2}}\operatorname{arctg}\frac{\sqrt{p^2-\eta^2}}{\eta}. \tag{42} \]

Then the magnitude of the maximum rebound of the brush, taking into account the resistance forces, may be determined as

\[ y_{\max}=\frac{U(1+k_{\mathrm{в}})\cos\alpha}{\sqrt{p^2-\eta^2}}\,e^{-\eta t_1}\sin\varphi, \tag{43} \]

where

\[ \text{where}\quad \varphi=\operatorname{arctg}\frac{\sqrt{p^2-\eta^2}}{\eta}. \]

Thus, when an impact force acts on the brush, the amplitude of the first deflection of the brush, with damping of the system taken into account, is proportional to the impulse of the force or, in other words, to the circumferential velocity of the commutator, to the coefficient of restitution of the brush velocity, and, through the angle of encounter, depends on the height of the irregularity and its length along the developed surface of the commutator. In practical calculations it is necessary to take into account that expression (43) gives the value of the brush deflection from the equilibrium position \(\delta=0\). The value of the rebound from the surface of the commutator is determined as \(\Delta=y_{\max}-\delta\).

On the basis of the foregoing, the following conclusions may be drawn:

1. The expressions obtained, (22), (28), (30), (32), (33), (42), and (43), make it possible to calculate completely the trajectory of motion of the brush over the commutator, the law of variation of whose surface is specified, and, in addition, to calculate the maximum permissible values of commutator runout and protrusion both of individual bars and of groups of bars for a specified range of operating speeds of the machine.

2. Using the analytical expressions obtained, it is possible to calculate the required pressure and stiffness of the pressure spring.

1. For more favorable operation of the current-collection unit, all other conditions being equal, it is necessary to use stiffer springs with increased preliminary compression and, if possible, softer brushes.

2. It is necessary to reduce the total runout of the commutator to a minimum, since the presence of significant runout may lead to pulsation of the contact pressure and to a sharp increase in tangential oscillations.

References

1. M. F. Karasev. Commutation of DC commutator machines. GEI, 1961.

2. I. Neikirchen. Carbon brushes and the causes of instability in the commutation conditions of DC machines. ONTI, 1937.

3. P. S. Livshits. Operation of electric brushes in the metallurgical industry. Metallurgiya, 1966.

4. L. Ya. Zinner, A. I. Skorospelkin, A. I. Proshin. Dynamic properties of commutators of electrical machines. Izvestiya TPI, vol. 160, 1966.

5. A. A. Andronov, S. E. Khaikin. Theory of Oscillations. ONTI, 1937.

6. A. G. Loitsyanskii, A. I. Lur’e. Course of Theoretical Mechanics. Vol. 2, GEI, 1948.