Negative Damping Instability

In Sect. 4.2, the mode coupling instability mechanism is considered. In Chap. 3, we have seen the effect of nonconservative forces in creating circulatory systems capable of exhibiting flutter instability. Examples are presented in this section to study the flutter instability with or without friction. Material presented in this section is a prelude to Chap. 7 where we study the mode coupling instability in the lead screw drives. 

Finally, in Sect. 4.3, we turn our attention to the kinematic constraint instability. This section begins by the introduction of the Painleve s paradoxes which play an important role in the kinematic constraint instability mechanism. The self-locking property which is another consequence of MctiOTi in the rigid body dynamics is also discussed in this section. This effect has a prominent presence in the study of the lead screws in Chaps. 7 and 8. The concepts presented here form the basis for the study of the kinematic constraint instability in the lead screws in Chap. 8. 

The negative slope in the friction-sliding velocity curve or the difference between static and kinematic coefficients of friction can lead to the so-called stick-slip vibratiOTis (see, e.g., [14, 59]). In most instances, researchers adopted the well-known mass-on-a-conveyor model to study the stick-slip vibrations (see, e.g., [17, 60, 61]). In this section, we will also consider this simple model - as shown in Fig. 4.1 - to investigate the effects of the negative damping instability mechanism. 

Vahid-Araghi and F. Golnaraghi, Friction-Induced Vibration in Lead Screw Drives, DOI 10.1007/978-1-4419-1752-2 4, Springer Science- -Business Media, LLC 2011 

As shown in Fig. 4.1, a block of mass m is held by a linear spring k and a linear damper c. The block slides against a moving conveyor that has a constant velocity Vb 0. Here, following [57, 62], the coefficient of friction is assumed to be a cubic function of relative velocity  

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The method of first-order averaging is introduced in Sect. 3.6. This method is utilized in Sect. 4.1 and Chap. 6 to expand the results of the eigenvalue analysis in the study of negative damping instability mechanism. 

Since AT(0eq) >0, instabilities occur if either A/(0eq) <0 (Painleve s Paradox) or C (0eq) < 0. (Negative damping). Starting with the negative damping instability, we can see that if the following conditions are satisfied, the origin of the system (4.63) becomes unstable. 

In chapter 8, it will be shown that in systems with constant coefficient of friction, there are situations where a different instability mechanism can lead to negative damping instability. 

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