SELF-EXCITATION OF A SELF-OSCILLATING ELECTROMECHANICAL SYSTEM TAKING INTO ACCOUNT FRICTION FORCES

V. M. Rikkonen

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

In the practice of electric drives and automatic systems it is very important to establish the cause of the occurrence of free periodic motions, which are called self-oscillations.

Self-oscillating systems are analyzed by the methods of the theory of nonlinear oscillations, and in doing so, naturally, it is necessary to take into account all the characteristic features of the conditions and properties of the system in order to obtain a correct picture of the phenomena taking place. Usually, when formulating the differential equation of motion of an electric-drive system, the static load torque on the motor shaft is assumed to be constant and, in the limiting case, is taken to be zero; in this case the equations of motion turn out to be identical, only in the first case the equation is written in deviations, while in the second case—in the full coordinates of the system (variables). Therefore in the literature [1, 2] the self-excitation of oscillations in a system is explained by the presence of residual magnetism in the machines, by the relationship of the system parameters, and the influence of dry-friction forces is not taken into account.

However, neglecting the constant static torque on the motor shaft, which has the character of “dry-friction forces,” leads to an entirely different picture of self-excitation than that which occurs in reality. Under the indicated assumptions we obtain systems that are fundamentally qualitatively different: if, without taking the static torque into account, the system proves to be self-oscillating, then when it is taken into account the system acquires the properties of potentially self-oscillating systems.

Let us take as an example a typical self-oscillating system: a series-excited generator—an independently excited motor; and, after generalizing the conclusions obtained, extend them to analogous electromechanical systems in general (Fig. 1,a).

It is known in the theory of automatic control that, with the aid of dry-friction torque, one can effectively combat oscillations of mechanical systems (Fig. 1,b). This is so-called damping by dry friction. However, the nature of this phenomenon differs substantially from ordinary damping by viscous friction.

Even if we consider the no-load case, when there is no useful load on the motor, even in this case the motor, in rotating, overcomes a no-load torque due to frictional forces in the brush contacts, bearings, friction against air, etc. Thus, in the general case the electromagnetic torque of the motor overcomes, in addition to the dynamic torque, also a static torque \(M_s\).

Most often the static torque does not depend on the speed of the motor, while its sign changes with a change in the direction of rotation; this is what characterizes the basic property of dry-friction forces (loss torques in the transmission elements of the drive, in the working parts of the mechanism, and the load of the working tool).

Fig. 1.

In this case the characteristic of the load torque may be represented by Fig. 2.

As can be seen from the figure, the sign of the torque depends on the sign of the speed. Associated with the load torque is the power which we shall conventionally call the power of losses due to friction,

\[ P_{\mathrm{tr}}=\frac{M_{\mathrm{tr}}\omega}{9.81}. \tag{1} \]

Fig. 2.

Since self-oscillation of the system occurs in the incremental region, when the system increases its store of energy, it is absolutely necessary to take into account also the losses of energy due to the presence of friction forces (apart from the losses in the active resistance of the power circuit). The equation of the system of Fig. 1 has the form

\[ L\frac{d^2 I}{dt^2}+\left(R-\alpha+3\beta I^2\right)\frac{dI}{dt} +\frac{C_e C_m}{J}(I-I_s)=0, \tag{2} \]

where \(I\) is the current of the power circuit;
\(J\) is the moment of inertia of the drive;
\(R\) is the active resistance of the generator–motor circuit;
\(L\) is the inductance of the circuit;
\(C_m C_e\) are motor constants;
\(\alpha\) and \(\beta\) are the coefficients of approximation of the generator characteristic.

To clarify the physical picture of oscillations with a load on the motor shaft, let us first assume that the dissipative term in (2) is equal to zero.

The equation will take the form

\[ L\frac{d^2 I}{dt^2}+\frac{C_e C_m}{j}(I\mp I_s)=0. \tag{3} \]

Suppose that an initial current \(I_{01}\), greater than \(I_s\), is produced in the system by an external force. Consequently, the motor will begin to move, and its equation will be:

\[ \text{a) }\quad LI_1''+kI_1=0,\quad \text{where } I_1=I-I_s \text{ for } \omega>0, \]

\[ \text{b) }\quad LI_2''+kI_2=0,\quad \text{where } I_2=I-I_s \text{ for } \omega<0. \]

The initial current in such an oscillatory system will decrease, and the origin of the currents for equation (a) will be at the point \(I_s\); as long as the current is greater than the current \(+I_s\), the motor will accelerate; from the moment when the current becomes less than \(+I_s\) and changes sign, the motor velocity will decrease in the positive direction (oscillatory motion about the point \(O_1\), Fig. 3).

Fig. 3.

Having reached the maximum value of the opposite sign, the current will then begin to decrease in magnitude, while the velocity will change its sign—becoming negative—and \(M_s\) will also change its sign. The motion must proceed according to equation (b) about the center \(O_2\), analogously to the motion according to equation (a).

As we see, the forces of dry friction do not change the frequency of the oscillations, which in this case is determined by the frequency of the harmonic oscillator

\[ h_0=\sqrt{\frac{C_e C_m}{jL}}. \tag{4} \]

Let us consider the energy balance at the extreme positions of the generalized coordinate \(I\) [3]: in the position \(I_{01}\), the energy is only “potential” and is equal to

\[ V_1=\frac{kI_{01}^2}{2}, \]

in the opposite extreme position

\[ V_2=\frac{kI_{02}^2}{2}. \]

Obviously, upon traversing the “dry-friction” zone, the losses due to friction forces are

\[ \frac{k}{2}\left(I_{01}^{2}-I_{02}^{2}\right)=(I_{01}+I_{02})\cdot f_0 . \]

The forces of dry friction perform the work of energy loss proportional to the coordinate, since they have the nature of an elastic force.

Hence

\[ I_{01}-I_{02}=\frac{2f_0}{k}=2I_s . \tag{5} \]

From the ideal case considered, the following conclusions may be drawn:

1. In the presence of an initial current of sufficient magnitude, the system becomes capable of executing some number of oscillations until the energy imparted to the system is expended in the work of the dry-friction forces.

2. In traversing the “dry-friction” zone, the system loses in current amplitude an amount equal to the magnitude of this zone.

3. In the theory of oscillations, in the presence of viscous-friction forces, systems are distinguished as oscillatory and aperiodic.

In the presence of dry-friction forces, one can choose a sufficiently large deflection for the system to execute a series of oscillations; therefore, in this case there is no need to distinguish systems as oscillatory or non-oscillatory.

It is important for us to note here that, in the presence on the motor shaft of a constant torque \(Ms\), the system under consideration is potentially self-oscillatory, i.e., the occurrence of oscillations in the system depends on the initial impulse.

Let us consider the solution of equation (3) in the phase plane.

\[ I''+h_0^2(I\mp I_s)=0 . \tag{3} \]

The equation of the integral curves is

\[ \frac{dy}{dI}=\frac{-h_0^2(I\mp I_s)}{y}. \tag{6} \]

After integration we have:

\[ \begin{aligned} &\text{for } \omega>0:\quad \frac{y^2}{2R^2}+\frac{(I-I_s)^2}{\dfrac{2R^2}{h_0^2}}=1,\\[6pt] &\text{for } \omega<0:\quad \frac{y^2}{2R^2}+\frac{(I+I_s)^2}{\dfrac{2R^2}{h_0^2}}=1 . \end{aligned} \tag{7} \]

As can be seen, this is the equation of ellipses shifted relative to the origin by \(\pm I_s\) (Fig. 4).

It is evident from the figure that the segment \(O_1—O_2\) is the locus of equilibrium states of the system, to which it comes after several oscillations with a sufficiently large initial current. For nearly harmonic oscillations (quasilinear ones), the influence of dry-friction forces is analogous: under any impulse the system has the ability to execute some number of oscillations, but the self-oscillations will always die out. The system possesses absolute stability. Let us now consider equation (2), taking into account the second term, but only in the initial part of the characteristic.

of the generator, since self-excitation occurs precisely in this zone (Fig. 5).

For values \(R>\alpha\) the system has periodically decaying motions. When the static moment \(M_s\) is added, we should expect an intensification of the damping phenomenon.

Fig. 4.

Fig. 5.

The larger \(M_s\) is in magnitude, the greater, naturally, must be the supply of energy to the system for the development of oscillations; and this means a greater difference \((\alpha-R)\), or, according to equation (2), a greater negative resistance. Consequently, in order to maintain oscillatory decaying stability of the system at larger \(M_s\), there exists a larger zone of resistance variation.

However, under the inequality \(\alpha-R>0\), one cannot speak of absolute stability of the system for different initial deviations and different loads.

Let us rewrite equation (2) for the initial part of the characteristics

\[ I''-2\delta I' + h_0^2(I \mp I_s)=0, \tag{8} \]

where

\[ 2\delta=\frac{\alpha-R}{L}. \]

Let us note that, under the condition \(\delta\ne 0\), the system has a frequency reduced in comparison with \(h_0\),

\[ P^2=h_0^2-\delta^2. \]

In the first part of the article, for \(\delta=0\), we saw that the decrease in the current amplitude occurred by the magnitude of the dry-friction zone, i.e.,

\[ I_{01}-I_{02}=2I_s. \]

If \(\delta>0\), which means \(\alpha>R\), then in the system, during motion, partial compensation of friction losses takes place, so that the indicated change in amplitude decreases, i.e.,

\[ I_{01}-I_{02}<2I_s. \]

For analysis of the system in this case, let us apply the method of point mappings of A. A. Andronov [3].

A steady nondecaying motion arises in the case when the maximum deviations of the currents are identical, i.e.,

\[ I_{01}=I_{02}. \]

If the amplitude of the current relative to \(0_1\) is \(I_{01}-I_s\), then after a half-period \(\frac{1}{2}T\) (where \(T=\frac{2\pi}{P}\)) it will become equal to

\[ (I_{01}-I_s)e^{\frac{1}{2}\delta T}. \tag{9} \]

Relative to the origin \(0\), the deviation will be

\[ I_{02}=(I_{01}-I_s)e^{\frac{1}{2}\delta T}-I_s. \tag{10} \]

The conditions for increase of the amplitudes of the currents will be written as follows:

\[ (I_{01}-I_s)e^{\frac{1}{2}\delta T}-I_s>I_{01}. \tag{11} \]

From (11) we obtain the conditions of “hard” excitation of self-oscillations for a potential self-oscillatory system:

\[ I_{01}>\frac{1+e^{\frac{1}{2}\delta T}}{e^{\frac{1}{2}\delta T}-1}\cdot I_s=k_1I_s. \tag{12} \]

For stable motions to arise in the system, caused by the presence of a limit cycle, it is necessary that the initial impulse in the system be greater than the right-hand side of equation (12).

The conditions of hard excitation of self-oscillations can be represented graphically.

Equation (10) can be rewritten in the following form:

\[ I_{02}=I_{01}\cdot e^{\frac{1}{2}\delta T}-I_s(1+e^{\frac{1}{2}\delta T}). \]

The coefficient \(e^{\frac{1}{2}\delta T}\) gives the tangent of the angle of inclination of the straight line \(I_{02}=f(I_{01})\) (Fig. 6).

The line \(OB\) is drawn at an angle of \(45^\circ\) and gives the values of \(I_{01}\) equal to \(I_{02}\). The line \(MN\) gives the dependence \(I_{02}=f(I_{01})\). With a current impulse in the system \(I_{01}<I_A\), we obtain current amplitudes decreasing with time.

With a current impulse or disturbances \(I_{01}>I_A\), the oscillations increase. At point \(A\) we have the value of an unstable equilibrium or an unstable limit cycle. The value of the current \(I_A\) is found from equation (12):

\[ I_A=\frac{1+e^{\frac{1}{2}\delta T}}{e^{\frac{1}{2}\delta T}-1}I_{sA}. \]

Fig. 6.

For values \(e^{\frac{1}{2}\delta T}=1\), the line \(MN\parallel OB\), which means the absence of an intersection point, and, consequently, stable limit cycles cannot arise in the system. Since

\[ \frac{1}{2}\delta T = \frac{\alpha - R}{2L}\,\frac{2\pi}{2h_0} = \frac{\alpha - R}{2L}\, \frac{2\pi}{2\sqrt{\frac{1}{LC}}} = \frac{\alpha - R}{\sqrt{\frac{L}{C}}}\,\frac{\pi}{2} = \frac{\pi}{2}\,\frac{\alpha - R}{\rho}, \tag{13} \]

then the absence of stable self-oscillations means either \(\alpha = R\) or \(\rho=\infty\); \(\rho\) is the characteristic impedance of the circuit.

With an increase in the ratio \(\dfrac{\pi}{2}\cdot\dfrac{\alpha-R}{\rho}\), the point of intersection of the lines approaches the origin of coordinates, which means smaller initial impulses in the system for stable self-oscillations to arise at the former values of the load current \(I_s\). It is very important to note the fact that, at one and the same accelerating voltage (i.e., \(\alpha-R=\mathrm{const}\)), the system behaves differently with respect to the onset of self-oscillations depending on the characteristic impedance of the circuit. This conclusion is new for the conditions of hard excitation of self-oscillations in an electromechanical system and is important for understanding the physical picture of self-buildup.

Figure 7 shows damped periodic motions with an insufficient initial current impulse in the system of Fig. 1. In the same system, with a smaller active resistance of the power circuit and even a smaller initial impulse, self-buildup occurred in the system, bringing the system to a regime of self-oscillations with a stable limit cycle (Fig. 8).

Fig. 7.

Fig. 8.

References

1. M. V. Shuleikin. Self-excitation of electrical oscillations under conditions of collector dynamo machines. Proceedings of the St. Petersburg Polytechnic Institute, vol. XX, 1913.

2. M. N. Kostenko. Special Course on Electrical Machines, 1949.

3. N. A. Andronov, A. A. Vitt, S. E. Khaikin. Theory of Oscillations. GITFML, 1959.