LXXVIII. Frictional relaxation oscillations

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Abstract
A graphical method has been developed and applied to solve the equation of motion for frictional relaxation oscillations in cases where the intrinsic frictional coefficient varies with the speed of rubbing. The solution takes the form of a curve relating a displacement parameter with the speed of vibration. A number of representative cases are examined, with frictional characteristics of types justified by experiment. It is shown that under certain frictional and velocity conditions stick-slip oscillations of a definite form will develop and persist, while under others oscillations will decay, and in some cases either stick-slip or decay can occur according to the initial displacement. As a rule slow tractive speeds lead to stick-slip oscillations and these increase in amplitude with the speed. At a certain critical speed however, stick-slip becomes unstable and the oscillations die out, leaving a steady displacement corresponding to the friction intrinsic to the tractive speed. Under certain combinations of frictional characteristic and damping, stable “quasi-sinusoidal” oscillations may occur at tractive speeds intermediate between stick-slip and decay. These oscillations are not symmetrical about the position of static equilibrium. The period o f “critical” stick-slip is the same and o f “quasi-sinusoidal” oscillations approximately the same as for free elastic oscillations. In the range of stink-slip, the lower the tractive speed and the damping factor the greater is the periodic time.

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