Rolling Friction

1. Rolling Friction Jayadeep U.B. PhD (MED) IISc

2. Outline • Introduction • Case Studies • “Free” or “Inertial” Rolling • Accelerated Rolling • Rolling with Deformation • Mechanisms of Rolling Friction • Interfacial slip in the Contact Area • Adhesion Hysteresis • Effect of Surface Roughness • Elastic and Plastic Deformation • Electric Double Layer • Concluding Remarks

3. Introduction • Invention of Wheel – Difference between Rolling and Sliding Friction • Energy loss is much higher for sliding friction compared to rolling friction, when the components are reasonably rigid. • Highly counter-intuitive: static friction has almost no effect on rolling friction! • Combined effect of a number of energy dissipating effects • We do not generally talk about a “rolling friction force”! • Rolling friction could be a misnomer; Resistance to Rolling is much better…

4. v ω I: Free Rolling or Inertial Rolling • Continuum assumption • Rigid Cylindrical Roller • Rigid Horizontal Surface • Velocity remains constant • FBD gives: • N = W • No Frictional Force • No Rolling Friction W N Free body diagram

5. II-A: Accelerated Rolling – down an incline • Continuum assumption • Rigid Cylindrical Roller • Rigid Inclined Surface • Velocity increases • FBD gives: • N = W cosθ • f≤μ N = μW cosθ • f=μ N leads to slipping • r f= Iα • No Rolling Friction ω,α v,a θ W f N Free body diagram

6. II-B: Accelerated Rolling – on a Horizontal Surface • Continuum assumption • Rigid Cylindrical Roller • Rigid Horizontal Surface • Velocity increases • FBD gives: • N = W • F – f = m a • r f=I α • f ≤ μ N • f = μ N leads to slipping • No Rolling Friction ω,α v,a F W F f N Free body diagram

7. Continuum assumption Rigid Cylindrical Roller Deformable Horizontal Surface Velocity decreases FBD gives: N cosβ = W N sinβ = ma d N cosβ – rN sinβ = I α No Sliding Friction Rolling with Deformation ω v α a W d ~ r β N Free body diagram

8. Mechanisms of Rolling Friction: Coulomb Laws & Interfacial Slip • In case of hypothetical continuum, the only mechanism for rolling resistance is what we discussed… • Coulomb (1781): For same materials, resistance to rolling is proportional to weight and inversely proportional to diameter. • Reynolds (1876): Slipping & friction at the contact (due to deformation) is the reason for Rolling Friction (Hence so-called!) and Heathcote (1921): • Explained difference in rolling resistance in hard and soft materials • But, lubrication never significantly reduces rolling friction! • It was explained that reducing friction increases slip, and slipping area.

9. Molecular Adhesion Hysteresis • Tomlinson (1929): • Micro-slip suggested by Reynolds is extremely small to account for experimental values of rolling friction. • Reynolds type micro-slip should have produced fretting, which was not observed. • Surface atoms are pulled away from equilibrium position; exceeding a critical distance they flick back. • Hysteresis during this process leads to energy dissipation, accounting for rolling friction • Insignificant influence of lubricant films on rolling friction can not be explained

10. Adhesion Hysteresis – An illustration Hysteresis Loop F F x Motion K Magnet x r Ignoring gravity, force is given by: Steel Surface F = Kδ≤ C/r2

11. Effect of Surface Roughness • Bikerman (1949): • Based on experiments by rolling stainless steel balls on a brass plate • For the roughness values considered (0.02 – 3 microns), higher surface roughness increases rolling friction. • Balls can rest on hills, while tilting the plate. • Other effects like elastic deformation, adhesion, capillary forces etc. found to be negligible.

12. Elastic & Plastic Deformation • Eldredge & Tabor (1955), Tabor (1955): • For metals, plastic deformation is the predominant mechanism during initial traversals. • Plastic deformation, and hence rolling friction, reduces on repeated traversals on same track. • Elastic hysteresis accounts for the rolling friction in later traversals • Rolling friction is independent of presence of lubricants/greases – effect of slip is “minute” • Work hardening promotes the change of mechanism from plastic deformation to elastic hysteresis • Elastic hysteresis is the major mechanism of rolling friction of elastomers like rubber

13. Elastic Hysteresis • Greenwood, Minshall, Tabor (1961): • Hysteresis loops from more complicated stress cycles required for rolling friction • Obtained expressions, which correctly predicted dependence of rolling friction on load, diameter and elastic constants of rubber

14. Effect of Electric Double Layer • Derjaguin & Smilga (1963): • When dielectric/semi-conductor cylinder rests on a metal surface (or vice versa), electric double-layer is created. • While rolling, electric double-layer is not symmetric about mid-point. • This asymmetry leads to a moment on cylinder, leading to rolling friction. • Reduction in surface conductivity and gas pressure increases rolling friction.

15. Crack Propagation (Peel Adhesion) • Kendall (1975): • Rolling is similar to two cracks propagating in same direction & same speed – one opening and another closing – for smooth roller & surface. • Force required can be calculated from fracture mechanics. • Rolling friction shown to be connected to peel adhesion – dwell time (speed) affects friction. • Energy required for breaking bonds is much higher than that obtained while making the bond. • Explains unexpectedly high rolling friction in rolling smooth glass cylinder over a smooth rubber surface, drastic reduction due to contamination in this case & static rolling friction, and predicts stick-slip in rolling at high speeds giving noise.

16. Concluding Remarks • Rolling Friction is used to account for a number of energy dissipation mechanisms • Rolling Friction might be a misnomer; “resistance to rolling” is a better terminology • From a pure continuum mechanics perspective, rolling friction can be explained using deformations. • Physical mechanisms of rolling friction include: • Plastic deformation • Elastic Hysteresis • Adhesion Hysteresis • Electrostatic (Electric Double layer) effects • Interfacial slip and many others • Coulomb’s law is only very approximately true.

17. References • Shames, I.H.: Engineering Mechanics – Statics and Dynamics, Prentice Hall of India • Reynolds, O.: On Rolling Friction, Phil Trans 166 (1876), 155-174. • Tomlinson, G.: A Molecular Theory of Friction, Phil. Mag. 7 (1929), 905 • Eldredge K.R., Tabor, D.: Mechanism of Rolling Friction – I. The Plastic Range, Proc. R. Soc. Lond. A (1955) 229, 181-198 • Tabor, D.: Mechanism of Rolling Friction – I. The Elastic Range, Proc. R. Soc. Lond. A 229 (1955), 198-220.

18. References • Derjaguin B.V. Smilga V. P.: Prog. Surf. Sci. 45 (1994) 108. • Kendall, K.: Rolling Friction and Adhesion between Smooth Solids, Wear 33 (1975) 351-358. • Bikerman, J.J.: Effect of Surface Roughness on Rolling Friction, J. Appl. Phys.20 (1949) 285-296. • Greenwood, J.A., Minshall, H., Tabor, D.: Hysteresis Losses in Rolling and Sliding Friction, Proc. R. Soc. Lond. A 259 (1961), 480-507.

19. Thank You... Questions???