Self-Oscillations

Self-Oscillations

 

undamped oscillations which may exist in some system in the absence of variable external influences, whereby the amplitude and period of oscillations are determined by the properties of the system. Thus self-oscillations differ from forced oscillations, the amplitude and period of which are determined by the character of external influences (the prefix “self’ indicates that the oscillations originate in the system and not as a result of external influences). Self-oscillations differ from free oscillations (for example, oscillations of a freely suspended pendulum, current oscillations in an electric circuit) in that, first, free oscillations gradually decay and, second, their amplitude is dependent upon the initial “jolt” which started these oscillations. The oscillations of a clock pendulum, oscillations of a string in a string instrument or of the column of air in a wind instrument, and electrical oscillations in a tube oscillator are examples of self-oscillations. Systems in which self-oscillations occur are called self-oscillating systems.

In many cases self-oscillating systems can be divided into three basic elements: (1) the actual oscillating system itself; (2) the energy source maintaining the self-oscillations; and (3) the device regulating the transfer of the energy from the source into the oscillating system. These three basic elements can be clearly differentiated. For example, in watches the pendulum or balance is the oscillating system, the spring or weight drive is the energy source, and the movement is the mechanism regulating the transmission of energy from the source into the system. In a tube oscillator the oscillating system is the circuit which contains capacitance and inductance and has a small active impedance, the energy source is the rectifier (or battery) supplying the voltage to the anodes of the tube, and the device regulating the energy transfer from the source into the oscillating circuit is an electronic tube with a regenerative coupling.

In watches, for example, self-oscillations are attained as the balance wheel, moving back and forth, passes through a predetermined position (usually twice during each period), and the drive mechanism (the pallet and the escape wheel) actuates the balance wheel, transmitting to it the energy needed to make up for energy losses which occur during the half-cycle of oscillation. The balance of the watch undergoes self-oscillation with the amplitude fully defined by the characteristics of the watch mechanism. However, to have these self-oscillations start, it usually is necessary not only to wind the spring but also to tilt the watch slightly in order to transmit an initial impulse to the balance. Thus a watch is in most cases a self-oscillating system without self-excitation. In wind instruments blowing an air stream through the column of air in the tube of the instrument and in string instruments the friction force between the bow and the string maintain self-oscillations.

In order to have these oscillations undamped, the energy entering the system from the source should compensate for energy losses in the system itself. Such compensation occurs during the entire period of oscillations; however, during one part of the period the energy entering may exceed the losses within the system, while during another the losses may exceed the energy being transferred. That value of amplitude of the oscillations at which compensation of losses occurs for the entire period and which appears to be stable (no change with respect to time) is the amplitude of the self-oscillations. Such a balance of transfer and losses of energy is only feasible with predetermined values of self-oscillation amplitude (in the simplest cases only for one single value).

Usually where amplitude values of oscillations are below the stable value, the transfer of energy into the system exceeds the losses, and as a result the amplitude of the oscillations increases and reaches a stable value. Specifically, if energy exceeding that being lost in the system is being transferred to it at relatively small values of oscillation amplitude, then self-excitation of oscillations occurs. Conversely, energy losses in the system where amplitudes exceed the stable value usually exceed the energy transfer from the source, and as a result the oscillation amplitude decreases and reaches a stable value. Thus the deviations of amplitude of self-oscillations in either direction from the stable value are subject to decay, and the self-oscillations in these cases become stable.

However, in certain cases deviation of the amplitude of oscillations from the stable value and a change in compensation for energy losses within the system lead to a further increase in deviations of the amplitude of the stable value. This will occur if with a decrease of amplitude, losses begin to exceed the energy being transferred or, conversely, during an increase of amplitude if the transferred energy begins to exceed the losses. In this case self-osci’lations are unstable; with the presence in every real system of unavoidable disturbances and shocks, such self-oscillations cannot sustain themselves for any length of time.

Different shapes of self-oscillation can be distinguished. If the figure of merit of the oscillating system is high (that is, the energy losses in the oscillating system are relatively small), then in order to sustain self-oscillations the quantity of energy entering the system should be very small in comparison with the total energy within the oscillating system. In this case the character of the processes occurring in the system is almost the same as the character of the system without the transfer of energy to it. In this case the period and shape of self-oscillations will be very close to the period and shape of natural oscillations of the oscillating system; if by shape the natural oscillations in the system are close to harmonic, then the self-oscillations will also be close to harmonic.

To maintain self-oscillations in a system of low figure of merit, the input energy must be large compared to the energy of the system. This input in turn can significantly change the character of processes occurring in the system; specifically, the shape of the self-oscillations may differ significantly from the sinusoidal. If during a period of self-oscillation the entire energy stored in the system is dissipated (that is, the system is already not aperiodic and oscillating), then self-oscillations may very significantly differ in shape from the sinusoidal—that is, they will be transformed into relaxation oscillations.

The possibility of establishing an energy balance exists only with specific amplitude values of self-oscillation dependent upon the presence in the system of a so-called nonlinear element, one whose properties are dependent upon the conditions of the system (for instance, impedance dependent upon the applied voltage drop).

REFERENCES

Gorelik, G. S. Kolebaniia i volny, 2nd ed. Moscow, 1959.
Teodorchik, K. F. Avtokolebatel’nye sistemy, 3rd ed. Moscow-Leningrad, 1952.

S. E. KHAIKIN

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