Velocity-Dependent Coefficient of Friction

As mentioned in Sect. 1.1, numerous models for the velocity-dependent coefficient of friction can be found in the literature [5,11,12]. These models generally include the following three parts  

the following model for the friction coefficient is considered  

---

In a case study (Chap. 9), an actual product is studied where friction-induced vibration in the lead screw drives resulted in unacceptable levels of audible noise. An important part of this study is the development of a novel approach to identify system parameters including the velocity-dependent coefficient of friction between lead screw and nut. 

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]. 

The rubbing action of the contacting lead screw threads against the nut threads is assumed to be the main source of friction in the systems considered in this monograph. We start this chapter by presenting a velocity-dependent coefficient of friction model for the lead screw and nut interface. 

Table 9.5 and the resulting velocity-dependent coefficient of friction is plotted in Fig. 9.17. 

The new parameters in the cost function in (9.32) are defined by the following modified velocity-dependent coefficient of friction... 

A practical case study is presented in Chap. 9 where friction-induced vibration in a lead screw drive is the cause of excessive audible noise. Using a complete dynamical model of this drive, a two-stage system parameter identification and fine-tuning method is developed to estimate the parameters of the velocity-dependent coefficient of friction model. The verified mathematical model is then used to study the role of various system parameters on the stability of the system and on the amplitude of vibrations. These smdies lead to possible design modifications that can solve the system s excessive noise problem. 

In this chapter, a two-step identification approach is developed to estimate various parameters of a lead screw drive system. In the first step, using the steady-sliding test results, the velocity effects (i.e., damping and velocity-dependent parts of friction) were separated from the force effects (i.e., coulomb coefficient of friction) and appropriate parameters were estimated using the least squares technique. 

The coefficient of friction may also depend on the relative velocity of the two surfaces. This will, for example, affect the local temperature, the extent... 

In this equation, r is the cyclone radius and n is dependent on the coefficient of friction. Theoretically, in the absence of wall friction, n should equal 1.0. Actual measurements, however, indicate that n ranges from 0.5 to 0.7 over a large portion of the cyclone radius. The spiral velocity in a cyclone may reach a value several times the average inlet-gas velocity. 

For a specific resin, the shear stress at the interface depends on the temperature of the interface, pressure, and the sliding velocity, it also depends on resin type, additives and additive levels, and the rheological properties of the resin. Stresses at the interface and the coefficients of friction for numerous resins have been published previously from two sources, and the data can be found in the references [15-31]. Additional stress data are provided in Appendix A4 and in several of the case studies in Chapter 12. 

The dependence of rate on the discharge pressure was considerably less for HIPS resin as compared to HDPE resin. As shown in Fig 5.13, the solids conveying rate for HIPS resin decreased to a lesser extent with increasing discharge pressure, and the rate did not seem to be dependent on temperature in this temperature range. This result is consistent with the dynamic coefficient of friction for HIPS resin, as shown by Fig. 12.17. As shown in Fig. 12.17, the coefficient did not depend to a high level on temperature or velocity in this low temperature range. 

The coefficients of friction,and f are dependent on temperature, pressure, and velocity (Hyun-Spalding model [2]). 

The distinction is sometimes made between static and dynamic friction, implying that there is one level of the coefficient of friction just at the point when movement between the surfaces starts and another level when the surfaces are steadily separating. There can of course be no measure of friction without movement so that static friction is actually friction at an extremely low velocity and thereafter the coefficient of friction of rubbers may vary markedly with velocity. Hence, it is necessary to measure friction over the range of velocities of interest. Friction is also dependent on temperature, which can lead to inaccuracies at high velocities because of heat build-up at the contacting surfaces. 

The properties of the stochastic forces in the system of equations (3.31)-(3.35) are determined by the corresponding correlation functions which, usually (Chandrasekhar 1943), are found from the requirement that, at equilibrium, the set of equations must lead to well-known results. This condition leads to connection of the coefficients of friction with random-force correlation functions - the dissipation-fluctuation theorem. In the case under consideration, when matrixes f7 -7 and G 7 depend on the co-ordinates but not on the velocities of particles, the correlation functions of the stochastic forces in the system of equations (3.31) can be easily determined, according to the general rule (Diinweg 2003), as... 

The coefficient of friction is by no means a constant, since it still depends on the load, the contact area, the surface structure, the velocity of sliding, the temperature and, above all, on lubricants. 

The wafer contour determines the area of contact between the wafer and the pad along with the abrasives. Thus, the amount of surface asperity interaction and the particle-wafer interaction also depends on the wafer contour. The fluid film that is in contact with the wafer surface is also dependent on the wafer contour. Thus, the pressure experienced by the wafer at different applied pressures and velocities changes with the shape of the surface. Scarfo et al. [20] conducted polishing tests on wafer samples with concave, convex, and intermediate surface contours and noted that the shape of the wafer affects the coefficient of friction. 

In denudation all of the dust deposit is removed in approximately 0.5 sec. Hence the denudation velocity is the basic parameter defining this process. If - ad 0 aut the removal of large particles (diameter 2-4 mm) will depend solely on the air-flow velocity and will be observed along the boundary between the dust layer and the surface since the coefficient of friction will be greater for movement of particles along a surface of similar particles than for movement of particles along a solid (hard) surface [281]. 

For SRR = 50%, apparently higher values of friction were observed in pure buffer solution than the polymer-containing solution. Furthermore, monotonically decreasing frictional properties (from p, = 0.43 to p = 0.22) as a function of increasing velocity were observed. The polymer-containing solution, on the other hand, showed similar velocity-dependent frictional behavior, but with clearly lower coefficients of friction (p = 0.25 to p = 0.12) within the same velocity range. The frictional properties at SRR = 100% were observed to be similar to those at SRR = 50%. The coefficient of friction reduced from 0.42 to 0.26 in buffer solution to 0.26 to 0.09 in the presence of the polymers. It is noted that all measurements were performed on the same disk track, although different balls and disks were used for measurements with or without polymers. 

Finally, the load and velocity dependence for pure sliding of the FeOx/FeO tribo-pair has been investigated. The coefficient of friction obtained from the slope of the friction versus load plots (AF/AL (L = 0.5 to 5.0 N)) after 5 rotations decreased from 0.30 to 0.14 upon addition of polymer. However, the coefficient of friction, dF/dN (N = 2.0 N) that was obtained after 100 rotations for each velocity, which remained virtually constant over the range of 1 to 100 mm/s, was not significantly reduced upon addition of polymer (fj, = 0.31 0.05 without polymer to /r = 0.27 0.03... 

---