Mathematics
Coefficient of Friction
The coefficient of friction is a measure of the resistance to motion between two surfaces in contact. It is a dimensionless quantity that represents the ratio of the force of friction between the surfaces to the force pressing them together. A higher coefficient of friction indicates greater resistance to motion, while a lower coefficient indicates less resistance.
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9 Key excerpts on "Coefficient of Friction"
- John B. Wachtman(Author)
- 2009(Publication Date)
- Wiley-American Ceramic Society(Publisher)
Friction may be defined as the resistance to relative motion between two solid bodies in contact, or more comprehensively, as the latent resistive force which opposes incipiative movement at and parallel to the slip plane of the two interfacing surfaces, e.g., the human foot or shoe and the walking surface and is proportional to the normal component of force pressing the two surfaces together. Expressed mathematically: FR FN FR = p FN; p = - = Tana where F, is the resistive, shear, tangential, or horizontal component of force which must be overcome; F, is the normal or vertical component of force; a is the angle at which the force must be applied for a slip to occur; and p is a constant, the Coefficient of Friction. The Coefficient of Friction is an intrinsic property of the two interfac- ing, interacting surfaces and serves as a measure of their micro- and macroroughness, inter- and intramolecular forces of attraction and 30 repulsion, and viscoelastic properties. As such, the area of contact, duration of contact before movement (dwell time), velocity of move- ment, pressure, etc., are contributing factors to the results and also to the inconsistency of Coefficient of Friction values obtained with different friction testers and/or protocols. The Coefficient of Friction is referred to as either static or dynamic (kinetic) depending on whether it is a measure of the forces at the instant that relative motion begins or after there is a continuous, uniform sliding motion, respectively. A slip may be defined as a sudden, unforeseen, unexpected, and out-of-control slide of the foot resulting from loss of footing. Normally, it is the end product of insufficient friction, that which is required (biomechanically) by the individual vs that which is available between the foot/shoe and the walking surface (static Coefficient of Friction). Slip resistance is a function of many factors, among which the Coefficient of Friction is only one, albeit probably the most important.
eBook - PDF- Kenneth G. Budinski(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
It is the Coefficient of Friction. In this era it was also learned that the friction force to initiate motion of a body at rest was usually higher than the force required to sustain motion on the body after motion started. Thus, Coefficient of Friction was recognized as a dimensionless ratio of the friction force to the normal force, F/N. It was given the Greek symbol µ, which is still in use, and early researchers identified a difference between starting friction force, F s , and friction to sus-tain motion, F k . Around the same time, the mathematics were developed to show that the tangent of the angle at which a body on an inclined plane starts to move is equal to the static Coefficient of Friction (see Figure 11.3): µ = tan θ Also about this time the concept and mathematics were developed for viscosity, which is a form of internal friction. By the time of the Industrial Revolution, the world had the benefit of many significant learnings regarding friction. Also, by that time, countless experimenters unsuccess-fully tried to develop a perpetual motion device by eliminating all friction forces. Even Leonardo da Vinci engineered a device that would pump water continuously, as shown in Figure 11.4; he conquered friction. The device may never have been built, but at any time there are many devices that claim zero friction in their mechanisms. However small it might be, there is always some type of frictional resistance to motion that will eventually w θ μ = tan θ FIGURE 11.3 The coefficient of static friction as the tangent of the angle at which slip occurs on an object on an elevating plane. FIGURE 11.4 One of Leonardo da Vinci’s machines that would operate continuously—perpetual motion. 139 Sliding Friction © 2008 Taylor & Francis Group, LLC stop a device. There are clocks in museums that will run for 100 years without attention; 100 years does not constitute perpetual motion.
eBook - ePubBasic Engineering Mechanics Explained, Volume 1
Principles and Static Forces
- Gregory Pastoll, Gregory Pastoll(Authors)
- 2019(Publication Date)
- Gregory Pastoll(Publisher)
Below each diagram is a free-body diagram of the block, from which the equation for the value of N is obtained. The weight of the block is W in each case. In the third and fourth diagrams, additional external forces have been introduced. The Coefficient of Friction, μ This coefficient, given the symbol μ (the Greek letter ‘mu’), is a number, ranging in value from zero to approximately 1, that indicates the difficulty of sliding one material relative to the other. The nature of the two materials in contact dictates what the Coefficient of Friction will be. A Coefficient of Friction of zero indicates perfect ‘smoothness’, namely no frictional resistance whatsoever. A Coefficient of Friction of 1 indicates a large amount of frictional resistance. This value is sometimes regarded as the maximum possible value for μ. However, instances have been recorded in which μ > 1. (See the table that follows.) The friction coefficient has meaning only when both materials are specified. For example, ‘the Coefficient of Friction for rubber on glass’ can be specified. One cannot speak of the Coefficient of Friction for only one material, for example, for steel. If both surfaces are of the same material, for example: steel, then one must specify ‘the Coefficient of Friction for steel on steel’. It is important to realise that the terms ‘smoothness’ and ‘roughness’ that are sometimes used to indicate the extent of a frictional resistance, are not referring directly to the physical smoothness or roughness of the surfaces. Some very approximate values for the Coefficient of Friction for various combinations are listed in the table below
eBook - PDFEngineering Science N2 Student's Book
TVET FIRST
- MJJ van Rensburg(Author)
- 2016(Publication Date)
- Troupant(Publisher)
The Coefficient of Friction is 0,3. Calculate the: a) Smallest force when the working line of the force is horizontal. b) Total reaction force. 12. A block is at rest on an incline making an angle of 12° with the horizontal. The weight of the block is 40 N and the frictional force is 10 N. Calculate the smallest force necessary to pull the block up the slope. 13. A block is at rest on an inclined plane of 15° with the horizontal. The weight component parallel to the plane is 4 N and the maximum frictional force is 7 N. Calculate the: a) Smallest force required to move the block up the incline. b) Smallest force required to move the block down the plane. 14. A block is at rest on a horizontal plane and a force of 5 N is required to bring the block into motion. The coefficient of static friction is 0,3. Calculate the: a) Static frictional force. b) Weight of the block. c) Force required to keep the block just in motion if the coefficient of kinetic friction is 0,25. Summary • Friction is a force that exists between any two surfaces that are in contact with each other. • Kinetic friction opposes the movement of a body that is already in motion. • Static friction prevents an object from moving when it is at rest. It is the friction that the object must overcome to start moving. • The five laws of friction : ❍ Law 1: If two bodies are in contact, the frictional force will act opposite to the direction of motion or possible motion. 164 ❍ Law 2: If two bodies are in contact and the forces acting on them are in equilibrium, the frictional force will be just sufficient to prevent motion or keep the body moving with a constant velocity (speed). ❍ Law 3: The ratio of the limiting friction to the normal reaction force depends on the substances the surfaces are made of. F = μ R μ = F R ❍ Law 4: The limiting friction depends on the area and shape of the two surfaces.
- H. D. Ram, A. K. Chauhan(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
The experimental result is shown in the Figure 2.2. Figure 2.2: Relation between F t and friction force F 84 | Foundations and Applications of Engineering Mechanics Figure 2.2 shows that as long as F t is less than the limiting friction force (F l ), the friction force is equal to F t . When F t becomes larger than F l , the body starts moving and the friction force reduces to F k . Coulomb was the first to study systematically the behaviour of friction. His findings are known as Coulomb’s law of friction. The Coulomb’s law of friction is described in the following section. 2.3 Coulomb’s law of friction According to this law, the magnitude of friction force depends on the normal reaction between the two surfaces and the nature of surfaces. The magnitude of limiting friction force, F l , is proportional to normal reaction, i.e., F l ∝ N or, F l = µ s N (2.3) where, µ s is a constant of proportionality. It is called coefficient of static friction . From the equation 2.3, μ s l F N = Coefficient of static friction: It is defined as the ratio of limiting friction force and normal reaction. It depends on the nature of surfaces that are in contact. It does not depend on the amount of area in contact. Note: (1) In general, friction force is F < F l or µ s N . (2) The value of friction force is F = F l or µ s N , only when the friction reaches its limit. The condition when the limit of friction is reached is called limiting or impending condition. Coefficient of kinetic or dynamic friction: It is defined as the ratio of kinetic friction and normal reaction. It is denoted as µ k . mathematically μ k k F N = The value of µ k is approximately 75% of µ s . 2.4 Angle of friction ( φ ) The maximum angle between the directions of normal reaction and the resultant of normal reaction and friction force is called angle of friction. Figure 2.3a shows body B to be in the state of limiting condition.
eBook - PDFTheoretical Mechanics for Sixth Forms
In Two Volumes
- C. Plumpton, W. A. Tomkys(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
CHAPTER IX FRICTION 9.1. The Nature of Friction In Chapter II, § 2.6, friction was defined as the component parallel to the reacting surface of the action of one body on another in contact with it. In conformity with Newton's third law, there is an equal and opposite component of reaction of the second body on the first. If there is no friction between two surfaces in contact, an ideal concept which cannot be realised in the physical world, the surfaces in contact are said to be smooth. The amount of friction between two surfaces is related to the roughness of the surfaces and to the normal reaction between them. The quality of roughness between two surfaces is too imprecise to permit of accurate mathematical definition and the normal reaction between two surfaces can be increased to a point at which the nature of the surfaces in contact is changed by the pressure between them. It is not possible, therefore, to construct a mathematical model of this physical phenomenon in action which is both accurate and comprehensive. Experi-mental data, however, suggest a set of rules, to which the force of friction approximately conforms, which we adopt as the mathematical definition of the force and which gives an approximation to observed experimental results in cases where the normal reaction between the surfaces is not ab-normally great. 9.2. The Laws of Friction (a) Friction is a variable force between surfaces in contact which, sub-ject to an upper limit on its magnitude, is of such a magnitude and acts in such a direction as to prevent the motion of one surface over the other. So long as there is no relative motion between the surfaces, the direction of 229 230 THEORETICAL MECHANICS friction is thus opposite to that direction in which the surface of the body on which it acts would move if there were no friction, and its line of action is the line of action of the equilibrant of the forces acting on the body.
eBook - PDFFriction and Wear
Calculation Methods
- I V Kragelsky, M N Dobychin, V S Kombalov(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
156 Dry and Boundary Friction 157 In metalforming the ratio of the tangential resistance in the contact zone between two bodies to the yield point of the weaker naterial is called the Coefficient of Friction. In this case, in accordance with plasticity theory, the Coefficient of Friction cannot exceed 0.5; m }///////y }//////y wz /y/m /M α ) ( b ) ( c ) Fig. 1. Various types of friction (schematic): 1, sliding friction; b, rolling friction; c, rotating friction. Preliminary displacement Fig. 2. Friction force as a function of displacement. (2) The Coefficient of Friction on impact is the ratio of the change in the quantity of motion of an impacting body in the normal and tangential directions during the inpact time, Δ {nwt) (3) In energy assessments the Coefficient of Friction loss is defined as the ratio of the work done in overconing the friction force Wfj. to the total work done, i.e. 158 Friction and Wear w I t i s d e s i r a b l e t o use t h i s c h a r a c t e r i s t i c f o r i n t e g r a l assessment o f the f r i c t i o n l o s s e s i n machines and mechanisms. REVIEW OF RESEARCH For a long time the science o f surface f r i c t i o n was r e s t r i c t e d t o the law Τ = fN, vdiere the p r o p o r t i o n a l i t y f a c t o r / was c a l l e d the c o e f f i c i e n t o f f r i c t i o n , and was considered t o be independent o f the l o a d . I h i s law was f i r s t established by Leonardo da V i n c i in 1508, v*io assumed that / = 0.25. In 1699 the French p h y s i c i s t Amontons [466] formulated three f r i c t i o n laws. I t i s i n t e r e s t i n g that these were not established i n the l a b o r a t o r y , but under production conditions in p o l i s h i n g o p t i c a l lenses ( F i g . 3 a ) .
eBook - ePubMetal Forming Science and Practice
A State-of-the-Art Volume in Honour of Professor J.A. Schey's 80th Birthday
- J.G. Lenard(Author)
- 2002(Publication Date)
- Elsevier Science(Publisher)
Chapter 6An Examination of the Coefficient of Friction
John g. Lenard Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada6.1 INTRODUCTION
Product and process design, needed for the development of draft schedules in the flat rolling industry, make use of off-line mathematical models of the process. These models vary broadly in their complexities and sophistication. They include simple, one-dimensional descriptions, which may be based on the original Orowan – von Karman approach. Alternatively, they may be especially written, dedicated finite-element or finite-difference descriptions of the process. All of them may be used to gain insight into the behaviour of the three components of the metal rolling system - the mill, the rolled metal and their interface - or they may be employed to predict the significant rolling parameters - the roll force, torque, forward slip and the resulting attributes - the yield strength and the hardness - of the product; the mill stretch and vibrations of the rolling mill, roll flattening, bending and the thermal camber of the rolls. When hot rolling is modeled, the resulting grain distribution, the retained strain, the amount of recrystallization and precipitation may also be calculated. To a large extent, the success of either endeavour depends on the appropriate formulation of the boundary conditions, which should be as sophisticated as the model itself. The boundary conditions are often expressed in terms of the Coefficient of Friction and the coefficient of heat transfer. In the cold rolling process with adequate application of lubrication the Coefficient of Friction is, arguably, the more important of the two, while in hot rolling both coefficients are of significant importance.Yuen et al (1996) write that since the trend in modem strip rolling is to produce thinner gauges of higher strength metals, the control of friction in the roll bite is most important. It is also useful to recall the comments of Roberts (1997)
eBook - PDFFriction Science and Technology
From Concepts to Applications, Second Edition
- Peter J. Blau(Author)
- 2008(Publication Date)
- CRC Press(Publisher)
If, for example, the mean value of the friction coefficient for N = 9 was 0.60 and the standard deviation was 0.03, then the 95% confidence interval is I 0 60 2 262 0 03 9 0 60 0 023 . . . . . (3.21) where the value 2.262 was obtained from a table of 95% confidence limits with a degree of freedom of 8. Then, with 95% confidence, the true mean of the sample is between 0.577 and 0.623. In designing friction tests, it is extremely difficult to vary just one thing and hold everything else constant. For example, let us assume sliding velocity is a variable. As sliding velocity is increased, the frictional temperature rise increases and tribochemical reactions may be promoted. Thus, changing the velocity also 108 Friction Science and Technology: From Concepts to Applications changes temperature, and perhaps surface film formation kinetics. As another exam-ple, assume normal force is intended to be a variable. At low normal forces the con-tact pressure may not change surface roughness very much, but as load is increased, the surface may begin to wear more, thereby introducing the variable of surface roughness into the experiment. It is important to recognize that when plotting either friction force or friction coefficients versus a single test variable, the concomitant changes in other variables may be concealed within those results. In summary, friction measurement requires careful consideration of the reasons why friction data are needed and how they might be used. The next chapter discusses the fundamentals of friction processes, especially with respect to the issue of contact size relative to the dominant frictional processes. It will support the contention that to simulate certain devices friction test methods should be selected with an under-standing of the relative scale of operative phenomena.
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