Pure-slip

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. 

Thus, when the origin is unstable, trajectories are attracted by a stable limit cycle. There is, however, one more step needed before accepting the above nontrivial pure-slip solution The condition v>0 (i.e., pure-slip motion) must be checked. In terms of the nondimensional system parameters, this condition is satisfied when Lfflmax(M ) < Vb. The maximum velocity of the mass according to the first-order averaged solution is max(i< ) = a, thus from (4.19) we must have... 

Using (4.8) and (4.10), the lower limit of belt velocity for the existence of pure-slip periodic vibrations is found as... 

Based on the above findings, the pure-slip periodic vibration can only occur if the belt velocity is within certain limits Vb min < Vb < Vb max- For lower belt velocities, i.e., Vb < Vb min7 stick-slip vibration occurs which is characterized by periodic sticking of the mass to the conveyor belt. This case is considered in the next section. 

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... 

Thomsen and Fidlin [62] also used averaging techniques to derive approximate expressions for the amplitude of stick-slip and pure-slip (when no sticking occurs) vibrations in a model similar to Fig. 4.1. They used a third-order polynomial to describe the velocity-dependent coefficient of friction. Other researchers have shown that in cases where the coefficient of friction is a nonlinear function of sliding velocity (e.g., humped friction model), the presence of one or more sections of negative slope in the friction-sliding velocity curve can lead to self-excited vibration without sticking [4, 70, 71]. 

In the first step of the parameter identification approach, the steady-state pure-slip conditions are considered (i.e., no rotational vibrations and constant lead screw angular velocity). The vibration-free operation of the system may be achieved through feedback control. Based on the mathematical model of the system developed in the preceding section, steady-state relationships are derived and by relating the measurable system inputs and states to the internal friction and damping parameters, these parameters are estimated. Table 9.1 lists the measured (or calculated) quantities and the main parameters to be identified in this step. 

Klein and co-workers have documented the remarkable lubricating attributes of polymer brushes tethered to surfaces by one end only [56], Studying zwitterionic polystyrene-X attached to mica by the zwitterion end group in a surface forces apparatus, they found /i < 0.001 for loads of 100 and speeds of 15-450 nm/sec. They attributed the low friction to strong repulsions existing between such polymer layers. At higher compression, stick-slip motion was observed. In a related study, they compared the friction between polymer brushes in toluene (ji < 0.005) to that of mica in pure toluene /t = 0.7 [57]. 

The basis for the familiar non-slip boundary condition is a kinetic theory argument originally presented by Maxwell [23]. For a pure gas Maxwell showed that the tangential velocity v and its derivative nornial to a plane solid surface should be related by... 

It is partly because of the variable effect of hydrogen (giving both softening and hardening, according to the nature of the slip) that the extrapolation of model experiments on very pure iron to predict the behaviour of commercial materials is so difficult. It is further hindered by the ability of dissolved hydrogen to modify the dislocation structure of a straining material. 

Fig. 20.33 (top) Transmission electronmicrograph showing dislocation tangles associated with precipitates in an Al-Cu-Mg-Si alloy (x 24 000, courtesy S. Blain) and (bottom) light micrograph showing slip lines in pure lead (x 100)... 

It is not possible to calculate the in-line concentrations and slip velocity from purely external measurements on the pipe, i.e. a knowledge of the rates at which the two components are delivered from the end of the pipe provides no evidence for what is happening within the pipe, It is thus necessary to measure one or more of the following variables ... 

Slip is not always a purely dissipative process, and some energy can be stored at the solid-liquid interface. In the case that storage and dissipation at the interface are independent processes, a two-parameter slip model can be used. This can occur for a surface oscillating in the shear direction. Such a situation involves bulk-mode acoustic wave devices operating in liquid, which is where our interest in hydrodynamic couphng effects stems from. This type of sensor, an example of which is the transverse-shear mode acoustic wave device, the oft-quoted quartz crystal microbalance (QCM), measures changes in acoustic properties, such as resonant frequency and dissipation, in response to perturbations at the surface-liquid interface of the device. 

The authors noted that when their friction parameter M= (pG/,) 8/G is real, it is equivalent to the real slip parameter s = fe used by McHale et al. [14]. From this analysis, a real interfacial energy G /8 is related to the slip length b, for a purely viscous fluid, by... 

The presence of a low-viscosity interfacial layer makes the determination of the boundary condition even more difficult because the location of a slip plane becomes blurred. Transitional layers have been discussed in the previous section, but this is an approximate picture, since it stiU requires the definition of boundary conditions between the interfacial layers. A more accurate picture, at least from a mesoscopic standpoint, would include a continuous gradient of material properties, in the form of a viscoelastic transition from the sohd surface to the purely viscous liquid. Due to limitations of time and space, models of transitional gradient layers will be left for a future article. 

The definition of friction factor using mean fluid properties has been most widely used because it reduces to the correct single-phase value for both pure liquid and pure gas flow. This technique is very similar to the so-called homogeneous model, because it has a clear physical significance only if the gas and liquid have equal velocities, i.e., without slip. Variations of this approach have also been used, particularly the plotting of a ratio of a two-phase friction factor to a single-phase factor against other variables. This approach is then very similar to the Lockhart-Martinelli method, since it can be seen that (G4)... 

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