Mechanical Engineering-Tribology In Machine Design Episode 5 pps

The bearing area curve, however, does not give the true bearing area when a rough surface is in contact with a smooth flat one.. Figure 3.1 1 shows a surface of unit nominal area consist

Elements of contact mechanics 87

where

y = [ 1 + 0 8 7 ~ - ~ ] - '

and y ranges from 0.72 at L= 5 to 0.92 at L = 100

3.8 contact between There are no topographically smooth surfaces in engineering practice Mica

rough surfaces can be cleaved along atomic planes to give an atomically smooth surface

and two such surfaces have been used to obtain perfect contact under laboratory conditions The asperities on the surface of very compliant solids such as soft rubber, if sufficiently small, may be squashed flat elastically by the contact pressure, so that perfect contact is obtained through the nominal contact area In general, however, contact between solid surfaces is discontinuous and the real area ofcontact is a small fraction

of the nominal contact area It is not easy to flatten initially rough surfaces

by plastic deformation of the asperities

The majority of real surfaces, for example those produced by grinding, are not regular, the heights and the wavelengths of the surface asperities vary in a random way A machined surface as produced by a lathe has a regular structure associated with the depth of cut and feed rate, but the heights of the ridges will still show some statistical variation Most man- made surfaces such as those produced by grinding or machining have a pronounced lay, which may be modelled, t o a first approximation, by one- dimensional roughness

It is not easy t o produce wholly isotropic roughness The usual procedure for experimental purposes is t o air-blast a metal surface with a cloud of fine particles, in the manner of shot-peening, which gives rise t o a randomly cratered surface

The topographical characteristics of random rough surfaces which are relevant t o their behaviour when pressed into contact will now be discussed briefly Surface texture is usually measured by a profilometer which draws a stylus over a sample length of the surface of the component and reproduces

a magnified trace of the surface profile This is shown schematically in Fig 3.9 It is important t o realize that the trace is a much distorted image of the actual profile because of using a larger magnification in the normal than in the tangential direction Modern profilometers digitize the trace at a suitable sampling interval and send the output t o a computer in order t o extract statistical information from the data First, a datum or centre-line is established by finding the straight line (or circular arc in the case of round components) from which the mean square deviation is at a minimum This implies that the area of the trace above the datum line is equal to that below

it The average roughness is now defined by

Figure 3.9

where z(x) is the height ofthe surface above the datum and L is the sampling length A less common but statistically more meaningful measure of average roughness is the root mean square (r.m.s.) or standard deviation a

of the height of the surface from the centre-line, i.e

The relationship between a and R, depends, to some extent, on the nature of the surface; for a regular sinusoidal profile a = (n/2 JZ)R, and for a Gaussian random profile a = (n/2)*~,

The R, value by itself gives no information about the shape of the surface profile, i.e about the distribution of the deviations from the mean The first attempt to d o this was by devising the so-called bearing area curve This curve expresses, as a function of the height z, the fraction of the nominal area lying within the surface contour at an elevation z It can be obtained from a profile trace by drawing lines parallel to the datum at varying heights, z, and measuring the fraction of the length of the line at each height which lies within the profile (Fig 3.10) The bearing area curve, however, does not give the true bearing area when a rough surface is in contact with a smooth flat one It implies that the material in the area of interpenetration vanishes and no account is taken of contact deformation

An alternative approach to the bearing area curve is through elementary statistics If we denote by $(z) the probability that the height of a particular point in the surface will lie between z and z +dz, then the probability that the height of a point on the surface is greater than z is given by the

cumulative probability function: @(z)=j: $(zl)dz' This yields an S-

shaped curve identical to the bearing area curve

It has been found that many real surfaces, notably freshly ground

'" @('' surfaces, exhibit a height distribution which is close to the normal or

Figure 3.10 Gaussian probability function:

where a is that standard (r.m.s.) deviation from the mean height The cumulative probability, given by the expression

can be found in any statistical tables When plotted on normal probability graph paper, data which follow the normal or Gaussian distribution will fall

on a straight line whose gradient gives a measure of the standard deviation

It is convenient from a mathematical point of view to use the normal probability function in the analysis of randomly rough surfaces, but it must

be remembered that few real surfaces are Gaussian For example, a ground surface which is subsequently polished so that the tips of the higher asperities are removed, departs markedly from the straight line in the upper height range A lathe turned surface is far from random; its peaks are nearly all the same height and its valleys nearly all the same depth

Elements o f contact mechanics 89

S o far only variations in the height of the surface have been discussed However, spatial variations must also be taken into account There are several ways in which the spatial variation can be represented One of them uses the r.m.s slope a, and r.m.s curvature ak F o r example, if the sample length L of the surface is traversed by a stylus profilometer and the height z

is sampled at discrete intervals of length h, and if zi- and zi+ are three consecutive heights, the slope is then defined as

and the curvature by

The r.m.s slope and r.m.s curvature are then found from

where n = L/h is the total number of heights sampled

It would be convenient t o think of the parameters a , a, and a k as properties of the surface which they describe Unfortunately their values in practice depend upon both the sample length L and the sampling interval h used in their measurements If a random surface is thought of as having a continuous spectrum of wavelengths, neither wavelengths which are longer than the sample length nor wavelengths which are shorter than the sampling interval will be recorded faithfully by a profilometer A practical upper limit for the sample length is imposed by the size of the specimen and

a lower limit t o the meaningful sampling interval by the radius of the profilometer stylus The mean square roughness, a , is virtually independent

of the sampling interval h, provided that h is small compared with the sample length L The parameters a, and a,, however, are very sensitive t o sampling interval; their values tend t o increase without limit a s h is made smaller and shorter, and shorter wavelengths are included This fact has led

t o the concept of function filtering When rough surfaces are pressed into contact they touch a t the high spots of the two surfaces, which deform t o bring more spots into contact T o quantify this behaviour it is necessary to know the standard deviation of the asperity heights, a,, the mean curvature

of their peaks, &, and the asperity density, q,, i.e the number of asperities per unit area of the surface These quantities have t o be deduced from the information contained in a profilometer trace It must be kept in mind that

a maximum in the profilometer trace, referred t o as a peak does not necessarily correspond t o a true maximum in the surface, referred to as a summit since the trace is only a one-dimensional section of a two- dimensional surface

T h e discussion presented above can be summarized briefly as follows: (i) for a n isotropic surface having a Gaussian height distribution with

standard deviation, a, the distribution of summit heights is very nearly Gaussian with a standard deviation

The mean height of the summits lies between 0.50 and 1 5 ~ above the mean level of the surface The same result is true for peak heights in a profilometer trace A peak in the profilometer trace is identified when,

of three adjacent sample heights, zi-, and zi+ ,, the middle one zi is greater than both the outer two

(ii) the mean summit curvature is of the same order as the r.m.s curvature

of the surface, i.e

(iii) by identifying peaks in the profiIe trace as explained above, the number

of peaks per unit length of trace q, can be counted If the wavy surface were regular, the number ofsummits per unit area q, would be q i Over

a wide range of finite sampling intervals

Although the sampling interval has only a second-order effect on the relationship between summit and profile properties it must be emphasized that the profile properties themselves, i.e ak and a, are both very sensitive to the size of the sampling interval

3.8.2 Contact of nominally flat rough surfaces

Although in general all surfaces have roughness, some simplification can be achieved if the contact of a single rough surface with a perfectly smooth surface is considered The results from such an argument are then reasonably indicative of the effects to be expected from real surfaces Moreover, the problem will be simplified further by introducing a theoretical model for the rough surface in which the asperities are considered as spherical cups so that their elastic deformation charac- teristics may be defined by the Hertz theory It is further assumed that there

is no interaction between separate asperities, that is, the displacement due

to a load on one asperity does not affect the heights of the neighbouring asperities

Figure 3.1 1 shows a surface of unit nominal area consisting of an array of identical spherical asperities all of the same height z with respect to some reference plane XX' As the smooth surface approaches, due to the

smooth

Figure 3.1 1

'reference plane on rough surf a c e

Elements of contact mechanics 9 1

application of a load, it is seen that the normal approach will be given by

( z - d ) , where d is the current separation between the smooth surface and the reference plane Clearly, each asperity is deformed equally and carries the same load Wi so that for q asperities per unit area the total load W will

be equal to q Wi For each asperity, the load Wi and the area of contact A i

are known from the Hertz theory

and

where 6 is the normal approach and R is the radius of the sphere in contact with the plane Thus if p is the asperity radius, then

and

and the total load will be given by

that is the load is related to the total real area of contact, A =qAi, by

This result indicates that the real area ofcontact is related to the two-thirds power of the load, when the deformation is elastic

If the load is such that the asperities are deformed plastically under a constant flow pressure H, which is closely related to the hardness, it is assumed that the displaced material moves vertically down and does not spread horizontally so that the area of contact A' will be equal to the geometrical area 2nPS The individual load, Wi, will be given by

Thus

that is, the real area of contact is linearly related to the load

It must be pointed out at this stage that the contact of rough surfaces should be expected to give a linear relationship between the real area of contact and the load, a result which is basic to the laws of friction From the simple model of rough surface contact, presented here, it is seen that while a plastic mode of asperity deformation gives this linear relationship, the elastic mode does not This is primarily due to an oversimplified and hence

unrealistic model ofthe rough surface When a more realistic surface model

is considered, the proportionality between load and real contact area can in fact be obtained with an elastic mode of deformation

It is well known that on real surfaces the asperities have different heights indicated by a probability distribution of their peak heights Therefore, the simple surface model must be modified accordingly and the analysis of its contact must now include a probability statement as to the number of the asperities in contact If the separation between the smooth surface and that reference plane is d, then there will be a contact at any asperity whose height was originally greater than d (Fig 3.12) If 4(z) is the probabilitydensity of the asperity peak height distribution, then the probability that a particular asperity has a height between z and z +dz above the reference plane will be 4(z)dz Thus, the probability of contact for any asperity of height z is

Figure 3.12 o f peak heights ~ ( Z I

If we consider a unit nominal area of the surface containing asperities, the number of contacts n will be given by

Since the normal approach is (z-d) for any asperity and N i and A i are known from eqns (3.48) and (3.49), the total area of contact and the expected load will be given by

n =qF,(h),

A = xqBgFl(h),

N = 9 q ~ f o t ~ l ~ + (h),

Elements of contact mechanics 93 where

+*(s) being the probability density standardized by scaling it to give a unit standard deviation Using these equations one may evaluate the total real area, load and number of contact spots for any given height distribution

An interesting case arises where such a distribution is exponential, that is,

a simple relationship will not apply, but for distributions approaching an exponential shape it will be substantially true For many practical surfaces the distribution of asperity peak heights is near to a Gaussian shape Where the asperities obey a plastic deformation law, eqns (3.53) and (3.54) are modified to become

m A' = 2nqP J ( Z - d)+(z) dz,

d

It is immedately seen that the load is linearly related to the real area of contact by N ' = HA' and this result is totally independent of the height distribution +(z), see eqn (3.51)

The analysis presented has so far been based on a theoretical model of the rough surface An alternative approach to the problem is to apply the concept of profilometry using the surface bearing-area curve discussed in Section 3.8.1 In the absence of the asperity interaction, the bearing-area curve provides a direct method for determining the area of contact at any given normal approach Thus, if the bearing-area curve or the all-ordinate distribution curve is denoted by $(z) and the current separation between the smooth surface and the reference plane is d, then for a unit nominal

surface area the real area of contact will be given by

so that for an ideal plastic deformation of the surface, the total load will be given by

To summarize the foregoing it can be said that the relationship between the real area of contact and the load will be dependent on both the mode of deformation and the distribution of the surface profile When the asperities deform plastically, the load is linearly related to the real area of contact for any distribution of asperity heights When the asperities deform elastically the linearity between the load and the real area ofcontact occurs only where the distribution approaches an exponential form and this is very often true for many practical engineering surfaces

3.9 Representation of Many contacts between machine components can be represented by

machine element cylinders which provide good geometrical agreement with the profile of the

contacts undeformed solids in the immediate vicinity of the contact The geometrical

errors at some distance from the contact are of little importance For roller-bearings the solids are already cylindrical as shown in Fig 3.13 On the inner race or track the contact is formed by two convex

contact (1) - contact ( 2 )

e q u ~ v a l e n t cylinders

equivalent cylinders and planes

r R, R = - r(R,+2r) R,+ r

Figure 3.13 = - R,+ r

Elements of contact mechanics 95

Figure 3.14

Figure 3.1 5

cylinders of radii r and R , , and on the outer race the contact is between the

roller of radius r and the concave surface of radius ( R , + 2r)

For involute gears it can readily be shown that the contact at a distances from the pitch point can be represented by two cylinders of radii,

R , , , sin$? s, rotating with the angular velocity of the wheels In this

expression R represents the pitch radius of the wheels and $ is the pressure angle The geometry of an involute gear contact is shown in Fig 3.14 This form of representation explains the use of disc machines to simulate gear tooth contacts and facilitate measurements ofthe force components and the film thickness

From the point of view of a mathematical analysis the contact between two cylinders can be adequately described by an equivalent cylinder near a plane as shown in Fig 3.15 The geometrical requirement is that the separation of the cylinders in the initial and equivalent contact should be the same at equal values of x This simple equivalence can be adequately satisfied in the important region of small x, but it fails as x approaches the radii of the cylinders The radius of the equivalent cylinder is determined as follows :

Using approximations

and

For the equivalent cylinder

Hence, the separation of the solids at any given value of x will be equal if

The radius of the equivalent cylinder is then

If the centres of the cylinders lie on the same side of the common tangent at

the contact point and R , > Rb, the radius of the equivalent cylinder takes the form

From the lubrication point of view the representation of a contact by an

equivalent cylinder near a plane is adequate when pressure generation is considered, but care must be exercised in relating the force components on the original cylinders t o the force components on the equivalent cylinder The normal force components along the centre-lines as shown in Fig 3.15 are directly equivalent since, by definition

The normal force components in the direction of sliding are defined as

Hence

and

F o r the friction force components it can also be seen that

where To,., represents the tangential surface stresses acting o n the solids

References to Chapter 3 1 S Timoshenko and J N Goodier Theory of Elasticity New York: McGraw-

Hill, 1951

2 D Tabor The Hardness of Metals Oxford: Oxford University Press, 1951

3 J A Greenwood and J B P Williamson Contact of nominally flat surfaces Proc Roy Soc., A295 (1966), 300

4 J F Archard The temperature of rubbing surfaces Wear, 2 (1958-9), 438

5 K L Johnson Contact Mechanics Cambridge: Cambridge University Press,

4,1, Introduction

lower kinema tic pairs

Every machine consists of a system of pieces or lines connected together in such a manner that, if one is made t o move, they all receive a motion, the relation of which t o that of the first motion, depends upon the nature of the connections The geometric forms of the elements are directly related t o the nature of the motion between them This may be either:

(i) sliding of the moving element upon the surface of the fixed element in directions tangential t o the points of restraint;

(ii) rolling of the moving element upon the surface of the fixed element; or (iii) a combination of both sliding and rolling

If the two profiles have identical geometric forms, so that one element encloses the other completely, they are referred t o as a closed or lower pair

It follows directly that the elements are then in contact over their surfaces, and that motion will result in sliding, which may be either in curved or rectilinear paths This sliding may be due t o either turning or translation of the moving element, so that the lower pairs may be subdivided t o give three kinds of constrained motion:

(a) a turning pair in which the profiles are circular, so that the surfaces of the elements form solids of revolution;

(b) a translating pair represented by two prisms having such profiles as t o prevent any turning about their axes;

(c) a twisting pair represented by a simple screw and nut In this case the sliding of the screw thread, or moving element, follows the helical path

of the thread in the fixed element or nut

All three types of constrained motion in the lower pairs might be regarded

as particular modifications of the screw; thus, if the pitch of the thread is reduced indefinitely so that it ultimately disappears, the motion becomes pure turning Alternatively, if the pitch is increased indefinitely so that the threads ultimately become parallel t o the axis, the motion becomes a pure translation In all cases the relative motion between the surfaces of the elements is by sliding only

It is known that if the normals t o three points of restraint of any plane figure have a common point of intersection, motion is reduced t o turning about that point For a simple turning pair in which the profile is circular, the common point of intersection is fixed relatively to either element, and continuous turning is possible

4.2 The concept of Figure 4.1 represents a body A supporting a load W and free to slide on a

friction angle body B bounded by the stationary horizontal surface X - Y Suppose the

motion of A is produced by a horizontal force P so that the forces exerted by

A on B are P and the load W Conversely, the forces exerted by B on A are the frictional resistance F opposing motion and the normal reaction R Then, at the instant when sliding begins, we have by definition

We now combine F with R, and P with,W, and then, since F = P and R = W, the inclination of the resultant force exerted by A and B, or vice versa, to the common normal NN is given by

The angle 4 = t a n ' f is called the angle of friction or more correctly the

limiting angle offriction, since it represents the maximum possible value of 4

at the commencement of motion T o maintain motion at a constant velocity, V, the force P will be less than the value when sliding begins, and for lubricated surfaces such as a crosshead slipper block and guide, the minimum possible value of 4 will be determined by the relation

= tan- ' fmin

In assessing a value for f , and also 4 , for a particular problem, careful distinction must be made between kinetic and static values An example of dry friction in which the kinetic value is important is the brake block and

-/ t o , '

Figure 4.2

R =the normal force exerted by the block on the drum,

F = the tangential friction force opposing motion of the drum,

Q = Flsin 4 =the resultant of F and R,

D = the diameter of the brake drum

The retarding or braking is then given by

The coefficient of friction, f , usually decreases with increasing sliding

velocity, which suggests a change in the mechanism of lubrication In the case of cast-iron blocks on steel tyres, the graphitic carbon in the cast-iron may give rise to adsorbed films of graphite which adhere to the surface with considerable tenacity The same effect is produced by the addition of colloidal graphite to a lubricating oil and the films, once developed, are

Y I

I d 1 I generally resistant to conditions of extreme pressure and temperature

I

$ 3 1 7 ' 4.2.1 Friction in slideways

F * I ' ? Q I guides G A force F applied by the lead screw will tend to produce clockwise

rotation of the moving element and, assuming a small side clearance,

Figure 4.3 rotation will be limited by contact with the guide surfaces at A and B Let P