Periodic Vibration Stick-Slip Motion

In the stick-slip vibrations, the mass moves with the belt for a part of one period. This intermittent motion results in a dynamical system with varying degrees-of-freedom (DOFs). Obviously, during the stick phase, the number of DOFs is zero and it is one otherwise [see (4.2) and (4.3)]. Due to this discontinuity, the averaging method used in previous section is no longer applicable. A common but intricate approach to construct an approximate solution for the stick-slip motion is to treat the stick phase and slip phase separately and then stitch the two results together [57, 62]. Here, however, we take a different approximation approach smoothing  

To avoid the discontinuity of the friction with respect to the relative velocity, the coefficient of friction - given by (4.1) - is modified to 

As shown in Fig. 4.2, the introduction of the second bracket brings the coefficient of friction rapidly to zero in a small range of relative velocities. The parameter (r) controls the steepness of the coefficient of friction function in this small range. 

This smoothing modification to the coefficient of friction function enables us to apply the method of first-order averaging to the cases where relative velocity becomes zero or even changes sign. There are, however, side effects that must be treated with caution  

Remark 4,1, The stick-slip vibration is now replaced by a quasi-stick-slip vibration. Because of smoothing, no sticking occurs however, the relative velocity remains close to zero in a portion of a period resembling that of a stick-slip cycle. Parameter r directly affects how close quasi-stick phase gets to a true stick phase.  

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It is obvious that when is negative and c< — Nd, the origin of system (4.5) is unstable. In this situation, the vibration amplitude grows until it reaches an attractive limit cycle. If trajectories reach the stick boundary, i.e., Vb — y = 0, stick-slip periodic vibration occurs. In the next two sections, periodic vibrations in cases where pure-slip and stick-slip motions occur are smdied separately. In these sections, a perturbation method (i.e., the method of averaging) is used to construct asymptotic solutions since due to nonlinearity and discontinuity of (4.4), closed-form solutions are not available. 

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