Local Stability of the Steady-Sliding State

The introduction of the new variables, yi = u and y2 = u, converts (6.5) into a system of first-order differential equations. The Jacobian matrix of this system evaluated at the origin (i.e., steady-sliding equilibrium point) is found as 

Note that c is the equivalent damping coefficient due to the velocity-dependent friction and it becomes negative if 0. The eigenvalues of the Jacobian matrix 

Assuming To 0, the steady-sliding equilibrium point becomes unstable if 

The above instability threshold can be stated alternatively in terms of the applied axial force, R. The steady-sliding equilibrium point is unstable if 

The stable/unstable regions in the space of parameters R and d are shown in Fig. 6.1. Expectedly, when negative friction damping is present (d 0), there is a limiting value of axial force, beyond which the steady-sliding equilibrium point is unstable. This limit proportionally increases with the increase of the damping in the lead screw supports. 

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