Frictionally Induced Quasiharmonic Vibration

Sometimes the operation of machine elements such as brakes and clutches may result in audible sound, ranging from high-pitched squeals to low-pitched growls. It is often assumed that such sounds are manifestations of stick-slip sliding, but in reality they are the result of frictionally induced quasiharmonic vibration. This aspect of friction has been analyzed and studied experimentally by Brockley and Ko [16, 17], 

Typical stick-slip motion is the combination of two distinctly 

To casual inspection traces of stick-slip motion such as Fig. 8-8a seem to be composed of linear segments, but Eqns 8-21 and 8-28 show that in actuality they are functions of the cosine or the sine of the angular displacement. Quasiharmonic oscillation gives the distinctly different type of trace shown in Fig. 8-8b. Brockley and Ko [16] showed that a linear relation between kinetic friction and tractive velocity of the kind illustrated in Fig. 8-9a leads only to either stick-slip or smooth sliding for quasiharmonic oscillation a humped curve such as is seen in Fig. 8-9b is required. 

The friction function that fulfills this requirement may be represented by a polynomial or by the exponential expression 

Equation 8-18 is coupled with this expression and manipulated to yield. . X + C, 

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The solution of Eqn 8-30 is discussed by Brockley and Ko [16] it requires considerable manipulation to put it in a form useful for computation. It can be shown there is a stable, cyclically repeated maximum amplitude of displacement of the rider, the value of which can be calcu-lated if the constants of the system such as cj, C, etc. are known. In physical terms, the sliding body does execute frictionally induced harmonic vibration. Furthermore, it can be demonstrated that the curve representing kinetic friction as a function of tractive velocity (Fig. 8-9b) is consistent with the solutions for quasiharmonic vibration. Such a curve for the undamped case (X + = 0) is characterized by the... 

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