Negative Damping Instability Mechanism

Vahid-Araghi and F. Golnaraghi, Friction-Induced Vibration in Lead Screw Drives, DOI 10.1007/978-l-4419-1752-2 6, Springer Science+Business Media, LLC 2011 

At steady-sliding, we have z = 0, i = 0, and z = zq. Substituting these values in (6.3) yields 

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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. 

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. 

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. 

The velocity-dependent friction model used in this work is discussed in Sect. 5.1. The dynamics of a pair of meshing lead screw and nut threads is studied in Sect. 5.2. Based on the relationships derived in this section, the basic 1-DOF lead screw drive model is developed in Sect. 5.3. This model is used in Chaps. 6 and 8 to study the negative damping and kinematic constraint instability mechanisms, respectively. A model of the lead screw with antibacklash nut is presented in Sect. 5.4, and the role of preloaded nut on the increased friction is highlighted. Additional DOFs are introduced to the basic lead screw model in Sects. 5.5 to 5.8 in order to account for the flexibility of the threads, the axial flexibihty of the lead screw supports, and the rotational flexibility of the nut. These models are used in Chaps. 7 and 8 to investigate the mode coupling and the kinematic constraint instability mechanisms, respectively. Finally, in Sect. 5.9, srane remarks are made regarding the models developed in this chapter. 

Friction can cause instability in dynamical systems through three distinct mechanisms (1) negative damping, (2) kinematic constraint, and (3) mode coupling. Chapter 4 is dedicated to the introduction of these mechanisms. Illustrative examples are worked out in this chapter to demonstrate the techniques that are applied to the lead screw drives in the later chapters. 

The mechanisms that govern pyroacoustic instability must satisfy the Rayleigh criterion [10], which states that over the cycle of an acoustic wave the flame must add more energy during the positive-pressure phase than during the negative-pressure phase. If this is accomplished then an acoustic wave will be amplified, otherwise it will die out eis a result of viscous damping. The overall... 

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