Frequency Response of Servomechanisms with Non-linear Friction

In this paper, the effect of non-linear friction, i.e. static and coulomb friction, on frequency response of the servomechanism, is considered in order to get the knowledge for synthesis of the servomechanism. The effect of non-linear friction does not depend on the absolute value of friction torque, but on the ratio of the static friction torque to the amplitude of sinusoidal input torque F/A. If the amplitude of sinusoidal input torque is not so large as the static friction torque, the output speed of the servomechanism has a dead zone in the low frequency range, and whether the output speed has a dead zone or not depends on the frequency of the sinusoidal input torque, the ratio F/A, and the other system parameters. The limiting condition of the output speed with a dead zone is considered. When the input torque varies sinusoidally, the waveforms of output speed of the servomechanism in which the non-linear friction characteristic is contained, are strictly analysed. From the results of the above analyses, the effect of non-linear friction on the frequency response of the servomechanism is discussed through the method of comparing the first harmonic component of the output speed with input torque. The magnitude of gain is less than that of the linear first order system and converges to a certain value determined by the characteristic of the non-linear friction in the low frequency range. The shape of gain curves is like the linear first order system and the curves approach the -20dB/decade line in the high frequency range. The break point frequency becomes a little larger than that of the linear first order system as F/A becomes larger. On the other hand, the phase angle delays more than that of the linear first order system in the low frequency range and less than that in the high frequency range. Moreover, the phase characteristic curves converge to a certain angle larger than -90°. So, in the high frequency range, the non-linear friction acts so as to stabilize the performance of the whole system

Self-excited Oscillation of Relay Servomechanism with Coulomb Friction and its Graphic Solution

In this paper, the authors treat the relay servomechanism containing Coulomb friction as one of the control systems with multiple nonlinearities, and are particularly concerned with the self-excited oscillation of this relay servomechanism, discussing the effect of the Coulomb friction on its performance. In the case where the relay has hysteresis but no dead zone in its characteristics, the self-excited oscillation occurs in the servomechanism and the waveform and the frequency of the oscillation are greatly affected by the Coulomb friction. The output waveform and the freqency in this state are analyzed exactly, using the fact that the Coulomb friction characteristic in a periodic state can be treated as the ideal relay characteristic, and the effects of the Coulomb friction on them are considered. Moreover, the authors develop the graphic method of solving the self-excited oscillation of the relay servomechanism. Using this approximation method, the frequency of the self-excited oscillation can be obtained graphically through the vector locus of the transfer function of the linear part of this servomechanism plotted on a complex plane. The result obtained by this approximation method shows a good agreement with the theoretical one