Steady-sliding

Figure 12-23. Diagram of a strong junction in steady sliding.

In the following we will focus on the low temperature, small h limit where the films have entered a glassy state. Two types of SFA experiments have been done in this limit. Granick and coworkers have applied an oscillatory shear and studied the dependence on amplitude and frequency (d-72). Israelachvili and coworkers have generally studied steady sliding (i-J). Both types of motion are discussed below. 

The pin-on-plate machine sacrifices the steady sliding speed between specimens, but partially simulates the reciprocating action broadly associated with file hip joint in ambulation. 

Considering small perturbations around the steady-sliding equilibrium point where Vb — y > 0, the linearized equation of motion is found from (4.4) as... 

The averaged amplitude equations have two equilibrium points A trivial solution, that is, a = 0, corresponding to the steady-sliding equilibrium and a nontrivial solution (i.e., a hmit cycle) is given by... 

For Vb > Vb max, the steady-sliding state (i.e., trivial solution) is stable. Note that if c> p N then the trivial solution is stable for all values of the belt velocity. 

As expected, a = 0 is the trivial solution of (4.27). Similar to the case of the pure-slip motion, the stability of the steady-sliding equilibrium point (i.e., the origin) is evaluated from the sign of da /d L=o- From (4.27) one finds... 

Shifting the equilibrium point (steady-sliding state) to the origin by setting x = X — Xeq and zi = z — Zeq, whcrc... 

Fig. 4.22 Unstable steady-sliding equilibrium point, C(0eq) —3 < 0 4.3.4.3 Painleve s Paradoxes...

For the parameter values that satisfy the above inequality and A Vt < 0, Painleve s paradoxes occur and the equation of motion given by (4.59) or the linearized equation given by (4.63) are no longer valid due to violation of existence and uniqueness of the solution. However, as we will see in the numerical examples below, the steady-sliding equilibrium point is indeed unstable for such values of system parameters. 

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... 

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. 

Fig. 6.1 Region of stability of the steady-sliding equilibrium point in the space of applied axial forces, R, and gradient of friction/velocity curve...

Fig. 6.2 System trajectories for c = 2 x 10 < Qr unstable steady-sliding equilibrium point (0,0)...

From (6.47), the condition for the stability of the steady-sliding state becomes... 

Remark 6.5. Because of the smoothing of the coefficient of friction, regardless of the value of H2 and 3, the steady-sliding state is stable for very low input angular velocities (i.e., 0 < O < mt). ... 

In this chapter, the 1-DOF model of the lead screw drives developed in Sect 5.3 was used to study the instability caused by the negative gradient of the friction coefficient with respect to velocity. The local stability of the steady-sUding equilibrium point of the system was studied by examining the eigenvalues of the Jacobian matrix of the linearized system. It was shown that the steady-sliding equdibiium point of the system loses stability when the condition given by (6.9) is satisfied. 

In Sects. 5.5 and 5.6, we have introduced two 2-DOF models for the lead screw drives. In this section, the equations of motion of these models are transformed into matrix form and linearized with respect to their respective steady-sliding equilibrium point. These equations are then used in the next sections to study the local stability of the equilibrium point and the role of mode coupling instability mechanism. In this chapter, for simplicity, the coefficient of friction, n, is taken as a constant. 

The equations of motion of the 2-DOF lead screw drive with axially compliant lead screw support are given in Sect. 5.6 by (5.30) and (5.31). Assume i = 2t where Q is a constant. The following change of variables is used to transfer the steady-sliding equilibrium point to the origin ... 

Figure 7.1 shows the stability region of the undamped 2-DOF model in the kc — p parameter space. The hatched region corresponds to the parameter range where the two natural frequencies are complex, and the steady-sliding equilibrium point is unstable. The boundary of this region is the fiutter instability threshold where co = co2-... 

To study the stability of the steady-sliding equilibrium point, the eigenvalues of the Jacobian matrix (evaluated at y = 0) are calculated. The Jacobian matrix is given by (F 0, O 0)... 

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