- Email: [email protected]
Temperature rise due to slip between wheel and rail—an analytical solution for hertzian contact
Wear, 61 (1980) 295 - 308 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
295
TEMPERATURE RISE DUE TO SLIP BETWEEN WHEEL AND RAIL AN ANALYTICAL SOLUTION FOR HERTZIAN CONTACT
M. A. TANVIR Research and Development DE2 8 UP (Ct. Britain)
Division,
Railway
Technical
Centre,
London
Road, Derby
(Received July 31,1979)
Summary The temperature rise due to slip between wheel and rail is obtained by the Laplace transform method. The pressure distribution at the wheel-rail contact is taken to be elliptical and it is assumed that the fast moving heat source can be approximated to an instantaneous static source.
1. Introduction The continuing improvements in traction and braking performance of railway vehicles demand the optimum use of the available wheel-rail adhesion. When the traction or braking forces exceed the available adhesion, gross slipping occurs which generates heat at the wheel-rail contact zone. The consequent rise in temperature probably increases wheel and rail wear, and in the extreme case of slipping the temperatures generated may be high enough to cause metallurgical transformation of the wheel or rail material. This on subsequent quenching produces a surface layer of brittle material which, even when very thin, can initiate cracks which may propagate into the bulk material leading to rail breakages and spalling of wheels [l] . Automatic wheel slip detection and control devices can be provided which, to be effective, should respond fast enough to prevent dangerously high temperatures being generated. Since it is difficult to measure the temperature of the wheel-rail contact, theoretical predictions of temperature rise for various wheel loads, vehicle speed and slip speed are necessary. The theory is essentially that of a heat flow problem involving moving sources of heat. Jaeger [ 21 has studied these problems in considerable detail, but since a vast range of parameters is involved any mathematical discussion is limited to the numerical calculations for a particular model. Blok [ 31 has studied the problem of temperature rise due to sliding of lubricated gear teeth by using a parabolic pressure distribution. Ling and Mow have given a partial analytical solution for an elastohydrodynamic contact with an arbitrarily distributed fast moving heat
296
source [4], In the case of wheel and rail contact Hertzian elastic theory gives an elliptical contact area with a semi-ellipsoid distribution of pressure [ 51. In this paper an analytical solution for the elliptical pressure dist~bution is obtained by the Laplace transform method. The analysis is basically valid only for the conditions of gross slip occurring while the wheel is moving forward. When traction and braking forces are exerted during normal rolling, the temperature at the rolling contact will rise owing to the creep of the wheel over the rail. Although slip conditions during creep are not simply defined, the calculation is extended to determine the temperature rise when a rolling wheel is just about to undergo gross slipping due to traction or braking forces.
2. Boundary conditions When two elastic cylinders of different radii are in contact under static load with their axes at right angles they make an elliptical contact patch in the tangent plane. It can be assumed that similar contact conditions are obtained with the railway wheel loaded against the rail and also that the normal pressure distribution in the contact patch is not signifi~~tly altered due to rolling or slipping of the wheel. The normal pressure at any point in the contact ellipse of a wheel rolling on the rail can therefore be given by Hertzian theory [ 61, i.e. p=
_E 1-s _f,“’ 2nab (
(1)
where a and b are the semi-axes of the contact ellipse, L is the normal load and x and y are the Cartesian coordinates as defined in Fig. 1. The magnitude of these axes is given by
b=n-
L
1
3(X-
E l/R,
v2)
v3
+ l/R,
where E is the modulus of elasticity, v Poisson’s ratio and R, and R, the radii of the wheel and rail respectively. $ and n are functions of + R,)}and their values for various values of R, and R, cos-1 t(R, - RMR, are given by Kraft [6]. The major axis of the contact ellipse lies in the axial direction of the smaller diameter cylinder; in the case of the wheel and rail it lies along the length of the rail. Consider a wheel slipping at a constant speed u,. The thermal flux per unit area per unit time at any point within the contact ellipse can be given by
297
Q=
3&J
-
(3)
2nab
where p is the friction coefficient between wheel and rail. Since the maximum flux and hence the maximum temperature occur along the major axis of the contact ellipse, the problem can be simplified by considering the flux only along this axis. Equation (3) then reduces to Q = Q, (1 - x2/a2)1’2
(4)
where Q,
=
%
2nab
u, = pnlw,
and P,,, is the maximum pressure. Consider a point on the wheel or rail moving over the-major axis with velocity u relative to the contact ellipse. Then the time t to travel a distance x is t = x/u and the total time T to traverse the major axis is T = 2a/u. Substituting these values in eqn. (4) and shifting the origin to the rear of the contact ellipse gives 112 &=2Q, $ (I-- -)I t (5) T 1 A proportion of the flux will enter the wheel tyre and the rest will go into the rail [ 71, and if
then
Q, = &S/O + 6)
Q, =Q/(1+6)
Also, if slipping is to occur the rail and wheel velocities relative to the contact ellipse must be different. This means that the contact time T of any point on the wheel or rail must be different, i.e. T, = 2a/u, where u, = V (V is the forward speed of the wheel) and T, = 2a/u, (u, is the peripheral speed of the wheel and is equal to V f us). The positive sign is for wheel spin in traction and the negative sign is for wheel slide during braking. Equation (5) for the thermal flux can therefore be modified depending upon whether it is the wheel or the rail temperature which is of interest. 3. Solution of equations To simplify the analysis of temperature rise by the Laplace transform method we make the approximations that the heat flow is linear into a semiinfinite solid and that the moving heat source can be treated as quasi-static. These approximations are reasonable provided certain assumptions can be made.
298
/ L
Fig. 1. Coordinate
system.
Fig. 2. Instantaneous thermal flux distribution along the major axis of the elliptical contract: solid curve, exact elliptical flux distribution; broken curve, polynomial approximation; chain curve, Blok’s parabolic distribution.
Consider a band source of heat moving along the surface of a semiinfinite solid, i.e. z > 0, at a uniform speed u. The first assumption made is that the heat source is of infinite width perpendicular to the direction of motion and of length 2a in the direction of motion. It supplies heat uniformly along its width while the heat distribution along its length is elliptical. If this heat source traverses the solid for a sufficient distance a constant temperature field will be established in the solid which advances at the same speed as the heat source. The second assumption is that the greater the speed of the heat source the more closely it will approach the conditions of thermal impact [ 31. In the case of thermal impact the heat penetrates perpendicularly into the plane surface directly beneath the source and there is no increase in temperature at any points instantaneously located in front of the advancing source. This means that the entire temperature field of the contact surface, which is displaced at the same speed u as the heat source, is fully determined when it covers a distance equal to the length 2e of the contact area. Both these assumptions are valid [ 21 only if the values of the dimensionless length A = va/2a and width B = ub/2a of the contact area are greater than 10. In the case of the railway wheel the forward velocity is usually high enough to satisfy these conditions. For example for a forward velocity as small as 1 m s-l, a = 5 mm and (Y = 0.12 X lop4 m s-l (the diffusivity of
steel), the value of A would be about 200. For the same forward speed the value of b would need to be about 0.24 mm to approximate to an infinitely wide band across which heat is uniformly distributed. The heat conduction equation for the linear flow of heat and zero initial temperature can be given by
a28
----=o a2*
1
ae
(Y
at
z>OandO<
t
(6)
where 0 is the temperature, t the time and (Y= K/pc the thermal diffusivity. K, p and c are the conductivity, density and specific heat respectively. Equation (6) has to be solved for a thermal flux boundary condition of the type (eqn. (5)).
g/l’*
2 +(l_
atz=O
(7)
For a thermal flux into the wheel
and for a thermal flux into the rail @ =aW=2Q,/K(1+6) Equation (7) is not in a convenient form since the binomial series of the expression under the square root converges slowly and requires a large number of terms as t/T approaches unity. Consequently for ease of handling the analysis the expression has been approximated to a fourth-order polynomial giving the boundary conditions
ae - = - @ {Cl + C,(t/T) a2
+ C&/T)2
+ C,(t/T)3
+
C&/zy} atz=O
(3) where the constant coefficients take the values Ci = 0.0061, C2 = 3.67, C3 = -10.65, C4 = 13.96 and C5 = -6.98. A plot of eqns. (7) and (8) is shown in Fig. 2. It shows that the error in the analysis is likely to be very small if the above polynomial is used. For comparison Blok’s parabolic distribution is also plotted on the same diagram. Applying the Laplace transform of 0, i.e. e
(e) = &= f
exp
(-pt)e(z,
t)dt
P>O
0
to eqns. (6) and (8) gives d*@ -- p @CO dz* ff and
de dz
(9) 2C3 c2 -+-+-+p*T p3T2
6C4 p4T3
24C5 p5p
(10)
300
For a semi-infinite solid the solution of eqn. (9) must be [ 81 0 = F exp (-qz)
(11)
for both 0 and G to be bounded as z -+ 00, and the differential of eqn. (11) with respect to z is de _ =dz
qJ’ exp (- 42)
(12)
F is an arbitrary constant and q = @/cx)~~. By equating eqns. (10) and (12) when z = 0 a value of F can be obtained which when substituted in eqn. (11) leads to C,
2Cs
24C5
6C4
*+p3TZ+p4T3
+p5T4
ew
(-qz)
(13)
Inverting the above equation term by term from the table of transforms [8] and defining the dimensionless parameters X = t/T = x/2a
h = z/2(0 tp2
gives ;
(aT)- ‘I2 = XU2 {C, i erfc (A) + 4C2Xi3erfc (A) + + 32C,X2i5 erfc (A) + 384C4X3i7erfc (h) + + 6144C5X4 ig erfc (A)}
(14)
where I: -l {exp (-qz)}
p-(1+n’2)=
(4t)“/2i”erfc z/(4cvt)l12
Equation (14) gives a solution for the temperature rise during wheel slip. The temperature rise at any point below the wheel or rail surface parallel to the major axis of the contact ellipse can be determined by substituting the appropriate values of z, a, T,, T, and t.
4. Surface temperatures during gross slip The most interesting temperatures occur in the contact plane. These can be calculated by putting z = 0 in eqn. (14). When z = 0, h = 0 and i” erfc (0) = where r(4+1)
Iy-$) = 7rY2
1 2”r(n/2
=;(;-1)(:-2)
+ 1)
. . . +I(+>
301
Mayor @
AXIS of Contact
= MaxImum
Surface
Ellqse
=20
Temperature
X-
Fig. 3. Surface temperature along the major axis during and after passage through the contact ellipse.
Substitution of these values of the integrals of the complementary error function and the values of the constant coefficients in eqn. (14) gives tl*=o
--
2@
n
u2
( 1 aT
= Xy2(0.0061
+ 2.448X - 5.676X2 +
(15) A plot of the surface temperature rise along the major axis as a proportion of the maximum temperature is shown in Fig. 3. The maximum temperature occurs when the differential with respect to t of the above equation vanishes giving a positive root at X = t/T = 0.8353. Substituting this value of t/T in eqn. (15) gives the maximum surface temperature rise in the contact ellipse : + 6.394X3
-3.834X4)
0 = 0.8@(r~T/#‘~
(16)
To determine the temperature of the wheel or the rail we need to apportion the thermal flux and replace T by the velocities of the wheel and the rail relative to the contact ellipse. Assume that the interface temperatures and the thermal properties for the wheel and rail are the same [ 71, i.e. 0,
=o,
Therefore 6 = (u,/u, )‘I2 where U, = V and the V is the forward speed. Substituting this in eqn. (16) gives the maximum temperature at the interface in S.I. units:
@w = 0, =
2.26P,p
aaV
K
c-177
u2
(1 - (1 -R)l’2)
(17)
302
20 r
1o-
Fig. 4. Maximum interface temperature during wheel slide for various slip ratios (slip during braking). Fig. 5. Maximum interface temperature during wheel spin for various slip ratios (slip during traction).
when the slip ratio R = us/V < 1 and V > vw, i.e. when the wheel slides during braking, and 0,
=@,=
((1 + R)“2 - 1)
(18)
when R < 00 and V < vw, i.e. during wheel spin. Figures 4 and 5 show the maximum temperatures reached for various slip and forward speeds. The maximum temperature is expressed in a form such that different material constants, loads, contact geometries and adhesion coefficients can be plotted on the same graph. Figure 6 shows the maximum interface temperatures attained as a function of slip speed at three different forward speeds when P,,., = 1100 MN, a = 7.5 mm, OL= 0.12 X 1O-4 m2 s-l, K = 46 W m-l “C-l and th e coefficient of adhesion p is 0.1. The ease with which surface temperatures greater than 1000 “C can be generated should be noted. Temperatures as high as this will readily be achieved by a stationary spinning wheel. However, the above theory is not valid for a forward speed less than 0.1 m s-l as the condition of thermal impact no longer applies. It is also not applicable to very high spinning speeds for in this case the reentry temperature of the tyre would be appreciable and the contact area would increase rapidly owing to wear. El-Sherbiny and Newcomb [9] have given a solution for the temperature distribution of a stationary cylinder loaded against a rotating cylinder with their axes at right angles.
303
I’/
’
hForward
Speed
Forward 30 mls
511~ Speed
Speed
m/s
Fig. 6. Typical maximum interface temperature for various slip and forward speeds: wheel load, 100 kN; wheel radius, 0.5 m; adhesion coefficient, 0.1; solid curve, wheel slide during braking; broken curve, wheel spin during traction.
5. Temperature after passing through the contact area As the element of area leaves the contact zone the heat input becomes zero and the surface temperature decays by conduction into the bulk material, i.e. z > 0. The temperature of a point on the wheel or rail surface for t > T could be of interest, but it cannot readily be obtained by using the boundary condition defined by the polynomial (eqn. (8)). To determine this temperature we assume that the thermal flux along the major axis for 0 < t < T is constant and that the flow of heat is normal to the surface. The boundary condition for the average thermal flux can be given by ae - = - 0.333@ {H(t) - H( t - T)} az
(W
where H(t) is a unit step function and H(t - T) is a shifting function. Applying the Laplace transform to eqn. (19) as before gives the boundary condition for the subsidiary eqn. (9) : de Z
= - 0.333@
1 - I P
exp (-pt) P
(20)
t
Solving eqns. (9) and (20) gives e = 0.333%!“2
4
-
exp (-PT)
exp (74
I P 312 The inversion of eqn. (21) from the table of transforms gives
$ (aT)-
1P
1/2 = 0.667{Xu2i
erfc (X) - (X - 1)y2i erfc (h)t_-T}
(21)
(22)
304
where At-r = Z/2(&( t - T)}l’2 and the surface temperature at t > T as before is (Xi’2 - (X - 1)1’2} The surface temperature variation of the rail or the wheel can be calculated by substituting the appropriate values of @, t and T. In dimensionless form the surface temperature for t > T can be given as a proportion of the maximum interface temperature reached (Fig. 3) : e*=o /o = 0.834 (Xi’2 - (X - l)i’2}
(24)
The assumptions which have been made are reasonably valid for short times but they will give rather higher temperatures than are likely to be obtained in practice. This is because some heat would also diffuse in the lateral direction.
6. Heat penetration perpendicular to the contact plane The most important temperatures are those obtained at the interface of the sliding surfaces, i.e. z = 0, but a knowledge of the temperatures below the surfaces, i.e. z > 0, is needed for calculating the thermal stresses. These temperatures can be obtained for various values of z provided the forward speed of the wheel is fairly high. Figure 7 shows curves of $0 (cxT)-~~@-’ plotted against X = t/T for values of 2 of 0, 50,100 and 200.2 is defined as 2 = VZ/~CY.The maxima of the curves can be seen to move towards the right as 2 increases. This is due to the continuing diffusion of heat after the contact zone has moved forward (X eventually becomes greater than unity). These results follow similar trends to those obtained by Jaeger [2] ; the differences are due to the different boundary conditions used. Figure 8 shows curves of i13 (aT)- u2 Q-l against 2 for various values of X = t/T up to X = 1. The figure shows that at most forward speeds the heat penetration into the medium is fairly shallow and the thermal gradients are fairly steep. Therefore the thermal stresses alone would cause only superficial damage to the rail or wheel. The most extensive damage is likely to be caused by surface temperatures which are high enough to change the steel structure or increase the local wear causing wheel flats.
7. Temperature rise due to creep During normal rolling with a longitudinal tractive force the contact patch beween the wheel and rail is divided into a region of adhesion and a
305
X-
Z-
Fig. 7. Temperature rise perpendicular to the contact ellipse below the major axis. Fig. 8. Temperature gradients perpendicular to the contact ellipse below the major axis.
region of slip. The slip region is located at the trailing end of the contact ellipse and its area depends upon the tractive force and the adhesion coefficient. The microslip occurring in this region will generate heat. The resulting surface temperature will be a function of the rolling velocity, the microslip velocity, the length of the slip region, the load supported by the slip region and the adhesion coefficient. In the analysis given below a simplified view is taken in which the microslip is considered to take place over the full length of the contact zone. This situation is obtained in practice when the tractive force is just equal to the maximum allowed by the adhesion, i.e. just before gross sliding occurs. The creep y is given by [lo] Y=-
Y=
2(pure rolling velocity - actual forward velocity) pure rolling velocity + actual forward velocity
2(% - VI v,
+
v
(25)
Its magnitude can be estimated. Hobbs [ 111 has surveyed various theoretical models and experimental results which show how creepage depends on the
306
tractive forces applied to a wheel. When the traction force f reaches its upper limit of PL the theoretical estimate is that the longitudinal creep is given by the expression Eab$/3pL
=D = 1
(26)
where $ is a coefficient dependent upon a/b and D is a function of f/L. If it is assumed that P = 0.2, L = 100 kN, a = 7.5 mm and b = 5 mm the creep near gross slip works out to be about 0.005. Experimental results cited by Hobbs suggest that creep values of two to three times this value can be sustained without gross slip occurring. For creep values as small as this, eqn. (25) can be approximated to the slip ratio, i.e. 2(&
-V)
v,
+ v
a- us =R V
(27)
The temperature rise due to creep alone can then be calculated in the same way as for a slipping wheel. The temperatures determined for a slipping wheel are proportional to the adhesion coefficient and the effect of changes in adhesion can therefore readily be calculated. However, this is not the case for a wheel that is creeping. The creep itself is a function of adhesion. If the tractive force is assumed to be maintained at its highest value as limited by the adhesion, i.e. pL, then eqn. (26) holds and creep is proportional to the adhesion coefficient. If the tractive force is assumed to remain constant, the value of D in eqn. (26) becomes less than unity and the creep y decreases as adhesion increases. Therefore the temperature calculated for the creep range must be looked upon as indicating the sort of value to be expected. However, at high forward speeds, say in excess of 50 m s-i, temperatures in the creep range may exceed 100 “C when the adhesion coefficient is about 0.2. This is approaching values at which boundary lubrication begins to become ineffective and this may help in improving wheel-rail adhesion.
8. Application
to wheel slip protection
Consider the simple case of an instantaneous loss of adhesion while the wheel is being braked at a constant torque. The sliding speed will increase linearly with time [l] and is given by
where pi, is the adhesion demanded by braking or traction forces and I is the rotational inertia of the wheelset. For a wheel slide protection system to be effective it should respond faster than the time t required to reach the maximum permissible speed. This is based on the consideration that the surface temperatures should not exceed the value which would be detrimental to the wheel and rail material. For typical wheel and rail steels it could be assumed that temperatures
307
greater than 600 “C may be undesirable. The corresponding slip speeds, if this temperature is not to be exceeded, depend upon the forward speed of the wheel. At a forward speed of 10 m s-l for a typical wheel load of 100 kN an interface temperature of up to 600 “C may be attained when the slip speed reaches about 4 m s- ’ (Fig. 6), while at a forward speed of 70 m s-l it is attained at 11 m s- ‘. The time to reach these speeds after encountering a step loss of adhesion would be about 0.16 and 0.44 s respectively. The calculation shows that the response times required are more sensitive at low forward speeds, and to prevent severe wheel slide at these speeds a wheel slide protection system must react within about 0.1 s. This means that the system should be able to detect and control a slide speed within about 5% of the forward speed. 9. Discussion and conclusions The surface temperatures generated due to friction beneath a steel wheel slipping while rolling on a steel rail have been calculated. Curves are given which relate the maximum temperatures attained with various values of wheel loading, wheel radius, rolling speed and adhesion coefficient. The temperatures obtained under typical conditions of wheel loading and speed were calculated. Surface temperatures of several hundred degrees Celsius are predicted. Heat penetration below the surface is shallow and the thermal gradients are steep. Therefore thermal stresses should cause only surface damage. In the creep range where the wheel is not exhibiting gross slip temperatures up to about 100 “C will be reached. Temperatures of this magnitude can have an effect on the lubricating properties of the oily contaminants on the rail head and the adhesion coefficient may itself be changed. The above analysis is applicable only to wheel speeds above about 0.1 m s-l (0.3 mile h-l). The formulae produced are similar to those produced by the theory of Jaeger and Blok for similar forward speeds. However, the temperature rises predicted by this method are about 20% higher than those predicted by their theories. The differences are entirely due to the assumption made for the thermal flux distribution along the contact length. Acknowledgments The author wishes to thank the British Railways Board for permission to publish this work. References 1 C. L. Murray, Wheelslip and APT, Railw. Eng., 2 (3) (1978) 22 - 27. 2 J. C. Jaeger, Moving sources of heat and the temperature at sliding contacts, Sot. NSW, 76 (1943) 203 - 224.
Proc. R.
308 3 H. Blok, Les temperatures de surface dans des conditions de graissage sons extreme pression, Proc. 2nd World Petroleum Congr., Paris, 1937, Vol. 3, pp. 471 - 486; English transl., Surface Temperature under E.P. Lubricating Conditions, Royal Dutch/Shell Laboratories, Delft. 4 F. F. Ling and V. C. Mow, Surface displacement of a convective elastic half-space under an arbitrary distributed fast-moving heat source, J. Basic Eng., 87 (1965) 729 734. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1951. K. Kraft, Adhesion, EZektr. Bahnen, 39 (6 - 9) (1968) 142 - 219. J. R. Barber, Distribution of heat between sliding surfaces, J. Mech. Eng. Sci., 9 (5) (1967) 351 - 354. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edn., Oxford Univ. Press: Clarendon Press, Oxford, 1959. M. El-Sherbiny and T. P. Newcomb, The temperature distribution due to frictional heat generated between a stationary cylinder and a rotating cylinder, Wear, 42 (1) (1977) 23 - 34. 10 K. L. Johnson, The effect of a tangential contact force upon the rolling motion of an elastic sphere on a plate, J. Appl. Mech., 25 (1958) 339 - 346. 11 A. E. W. Hobbs, A survey of creep, British Rail Res. Dept. Rep. DYN52, April 1967.
295
TEMPERATURE RISE DUE TO SLIP BETWEEN WHEEL AND RAIL AN ANALYTICAL SOLUTION FOR HERTZIAN CONTACT
M. A. TANVIR Research and Development DE2 8 UP (Ct. Britain)
Division,
Railway
Technical
Centre,
London
Road, Derby
(Received July 31,1979)
Summary The temperature rise due to slip between wheel and rail is obtained by the Laplace transform method. The pressure distribution at the wheel-rail contact is taken to be elliptical and it is assumed that the fast moving heat source can be approximated to an instantaneous static source.
1. Introduction The continuing improvements in traction and braking performance of railway vehicles demand the optimum use of the available wheel-rail adhesion. When the traction or braking forces exceed the available adhesion, gross slipping occurs which generates heat at the wheel-rail contact zone. The consequent rise in temperature probably increases wheel and rail wear, and in the extreme case of slipping the temperatures generated may be high enough to cause metallurgical transformation of the wheel or rail material. This on subsequent quenching produces a surface layer of brittle material which, even when very thin, can initiate cracks which may propagate into the bulk material leading to rail breakages and spalling of wheels [l] . Automatic wheel slip detection and control devices can be provided which, to be effective, should respond fast enough to prevent dangerously high temperatures being generated. Since it is difficult to measure the temperature of the wheel-rail contact, theoretical predictions of temperature rise for various wheel loads, vehicle speed and slip speed are necessary. The theory is essentially that of a heat flow problem involving moving sources of heat. Jaeger [ 21 has studied these problems in considerable detail, but since a vast range of parameters is involved any mathematical discussion is limited to the numerical calculations for a particular model. Blok [ 31 has studied the problem of temperature rise due to sliding of lubricated gear teeth by using a parabolic pressure distribution. Ling and Mow have given a partial analytical solution for an elastohydrodynamic contact with an arbitrarily distributed fast moving heat
296
source [4], In the case of wheel and rail contact Hertzian elastic theory gives an elliptical contact area with a semi-ellipsoid distribution of pressure [ 51. In this paper an analytical solution for the elliptical pressure dist~bution is obtained by the Laplace transform method. The analysis is basically valid only for the conditions of gross slip occurring while the wheel is moving forward. When traction and braking forces are exerted during normal rolling, the temperature at the rolling contact will rise owing to the creep of the wheel over the rail. Although slip conditions during creep are not simply defined, the calculation is extended to determine the temperature rise when a rolling wheel is just about to undergo gross slipping due to traction or braking forces.
2. Boundary conditions When two elastic cylinders of different radii are in contact under static load with their axes at right angles they make an elliptical contact patch in the tangent plane. It can be assumed that similar contact conditions are obtained with the railway wheel loaded against the rail and also that the normal pressure distribution in the contact patch is not signifi~~tly altered due to rolling or slipping of the wheel. The normal pressure at any point in the contact ellipse of a wheel rolling on the rail can therefore be given by Hertzian theory [ 61, i.e. p=
_E 1-s _f,“’ 2nab (
(1)
where a and b are the semi-axes of the contact ellipse, L is the normal load and x and y are the Cartesian coordinates as defined in Fig. 1. The magnitude of these axes is given by
b=n-
L
1
3(X-
E l/R,
v2)
v3
+ l/R,
where E is the modulus of elasticity, v Poisson’s ratio and R, and R, the radii of the wheel and rail respectively. $ and n are functions of + R,)}and their values for various values of R, and R, cos-1 t(R, - RMR, are given by Kraft [6]. The major axis of the contact ellipse lies in the axial direction of the smaller diameter cylinder; in the case of the wheel and rail it lies along the length of the rail. Consider a wheel slipping at a constant speed u,. The thermal flux per unit area per unit time at any point within the contact ellipse can be given by
297
Q=
3&J
-
(3)
2nab
where p is the friction coefficient between wheel and rail. Since the maximum flux and hence the maximum temperature occur along the major axis of the contact ellipse, the problem can be simplified by considering the flux only along this axis. Equation (3) then reduces to Q = Q, (1 - x2/a2)1’2
(4)
where Q,
=
%
2nab
u, = pnlw,
and P,,, is the maximum pressure. Consider a point on the wheel or rail moving over the-major axis with velocity u relative to the contact ellipse. Then the time t to travel a distance x is t = x/u and the total time T to traverse the major axis is T = 2a/u. Substituting these values in eqn. (4) and shifting the origin to the rear of the contact ellipse gives 112 &=2Q, $ (I-- -)I t (5) T 1 A proportion of the flux will enter the wheel tyre and the rest will go into the rail [ 71, and if
then
Q, = &S/O + 6)
Q, =Q/(1+6)
Also, if slipping is to occur the rail and wheel velocities relative to the contact ellipse must be different. This means that the contact time T of any point on the wheel or rail must be different, i.e. T, = 2a/u, where u, = V (V is the forward speed of the wheel) and T, = 2a/u, (u, is the peripheral speed of the wheel and is equal to V f us). The positive sign is for wheel spin in traction and the negative sign is for wheel slide during braking. Equation (5) for the thermal flux can therefore be modified depending upon whether it is the wheel or the rail temperature which is of interest. 3. Solution of equations To simplify the analysis of temperature rise by the Laplace transform method we make the approximations that the heat flow is linear into a semiinfinite solid and that the moving heat source can be treated as quasi-static. These approximations are reasonable provided certain assumptions can be made.
298
/ L
Fig. 1. Coordinate
system.
Fig. 2. Instantaneous thermal flux distribution along the major axis of the elliptical contract: solid curve, exact elliptical flux distribution; broken curve, polynomial approximation; chain curve, Blok’s parabolic distribution.
Consider a band source of heat moving along the surface of a semiinfinite solid, i.e. z > 0, at a uniform speed u. The first assumption made is that the heat source is of infinite width perpendicular to the direction of motion and of length 2a in the direction of motion. It supplies heat uniformly along its width while the heat distribution along its length is elliptical. If this heat source traverses the solid for a sufficient distance a constant temperature field will be established in the solid which advances at the same speed as the heat source. The second assumption is that the greater the speed of the heat source the more closely it will approach the conditions of thermal impact [ 31. In the case of thermal impact the heat penetrates perpendicularly into the plane surface directly beneath the source and there is no increase in temperature at any points instantaneously located in front of the advancing source. This means that the entire temperature field of the contact surface, which is displaced at the same speed u as the heat source, is fully determined when it covers a distance equal to the length 2e of the contact area. Both these assumptions are valid [ 21 only if the values of the dimensionless length A = va/2a and width B = ub/2a of the contact area are greater than 10. In the case of the railway wheel the forward velocity is usually high enough to satisfy these conditions. For example for a forward velocity as small as 1 m s-l, a = 5 mm and (Y = 0.12 X lop4 m s-l (the diffusivity of
steel), the value of A would be about 200. For the same forward speed the value of b would need to be about 0.24 mm to approximate to an infinitely wide band across which heat is uniformly distributed. The heat conduction equation for the linear flow of heat and zero initial temperature can be given by
a28
----=o a2*
1
ae
(Y
at
z>OandO<
t
(6)
where 0 is the temperature, t the time and (Y= K/pc the thermal diffusivity. K, p and c are the conductivity, density and specific heat respectively. Equation (6) has to be solved for a thermal flux boundary condition of the type (eqn. (5)).
g/l’*
2 +(l_
atz=O
(7)
For a thermal flux into the wheel
and for a thermal flux into the rail @ =aW=2Q,/K(1+6) Equation (7) is not in a convenient form since the binomial series of the expression under the square root converges slowly and requires a large number of terms as t/T approaches unity. Consequently for ease of handling the analysis the expression has been approximated to a fourth-order polynomial giving the boundary conditions
ae - = - @ {Cl + C,(t/T) a2
+ C&/T)2
+ C,(t/T)3
+
C&/zy} atz=O
(3) where the constant coefficients take the values Ci = 0.0061, C2 = 3.67, C3 = -10.65, C4 = 13.96 and C5 = -6.98. A plot of eqns. (7) and (8) is shown in Fig. 2. It shows that the error in the analysis is likely to be very small if the above polynomial is used. For comparison Blok’s parabolic distribution is also plotted on the same diagram. Applying the Laplace transform of 0, i.e. e
(e) = &= f
exp
(-pt)e(z,
t)dt
P>O
0
to eqns. (6) and (8) gives d*@ -- p @CO dz* ff and
de dz
(9) 2C3 c2 -+-+-+p*T p3T2
6C4 p4T3
24C5 p5p
(10)
300
For a semi-infinite solid the solution of eqn. (9) must be [ 81 0 = F exp (-qz)
(11)
for both 0 and G to be bounded as z -+ 00, and the differential of eqn. (11) with respect to z is de _ =dz
qJ’ exp (- 42)
(12)
F is an arbitrary constant and q = @/cx)~~. By equating eqns. (10) and (12) when z = 0 a value of F can be obtained which when substituted in eqn. (11) leads to C,
2Cs
24C5
6C4
*+p3TZ+p4T3
+p5T4
ew
(-qz)
(13)
Inverting the above equation term by term from the table of transforms [8] and defining the dimensionless parameters X = t/T = x/2a
h = z/2(0 tp2
gives ;
(aT)- ‘I2 = XU2 {C, i erfc (A) + 4C2Xi3erfc (A) + + 32C,X2i5 erfc (A) + 384C4X3i7erfc (h) + + 6144C5X4 ig erfc (A)}
(14)
where I: -l {exp (-qz)}
p-(1+n’2)=
(4t)“/2i”erfc z/(4cvt)l12
Equation (14) gives a solution for the temperature rise during wheel slip. The temperature rise at any point below the wheel or rail surface parallel to the major axis of the contact ellipse can be determined by substituting the appropriate values of z, a, T,, T, and t.
4. Surface temperatures during gross slip The most interesting temperatures occur in the contact plane. These can be calculated by putting z = 0 in eqn. (14). When z = 0, h = 0 and i” erfc (0) = where r(4+1)
Iy-$) = 7rY2
1 2”r(n/2
=;(;-1)(:-2)
+ 1)
. . . +I(+>
301
Mayor @
AXIS of Contact
= MaxImum
Surface
Ellqse
=20
Temperature
X-
Fig. 3. Surface temperature along the major axis during and after passage through the contact ellipse.
Substitution of these values of the integrals of the complementary error function and the values of the constant coefficients in eqn. (14) gives tl*=o
--
2@
n
u2
( 1 aT
= Xy2(0.0061
+ 2.448X - 5.676X2 +
(15) A plot of the surface temperature rise along the major axis as a proportion of the maximum temperature is shown in Fig. 3. The maximum temperature occurs when the differential with respect to t of the above equation vanishes giving a positive root at X = t/T = 0.8353. Substituting this value of t/T in eqn. (15) gives the maximum surface temperature rise in the contact ellipse : + 6.394X3
-3.834X4)
0 = 0.8@(r~T/#‘~
(16)
To determine the temperature of the wheel or the rail we need to apportion the thermal flux and replace T by the velocities of the wheel and the rail relative to the contact ellipse. Assume that the interface temperatures and the thermal properties for the wheel and rail are the same [ 71, i.e. 0,
=o,
Therefore 6 = (u,/u, )‘I2 where U, = V and the V is the forward speed. Substituting this in eqn. (16) gives the maximum temperature at the interface in S.I. units:
@w = 0, =
2.26P,p
aaV
K
c-177
u2
(1 - (1 -R)l’2)
(17)
302
20 r
1o-
Fig. 4. Maximum interface temperature during wheel slide for various slip ratios (slip during braking). Fig. 5. Maximum interface temperature during wheel spin for various slip ratios (slip during traction).
when the slip ratio R = us/V < 1 and V > vw, i.e. when the wheel slides during braking, and 0,
=@,=
((1 + R)“2 - 1)
(18)
when R < 00 and V < vw, i.e. during wheel spin. Figures 4 and 5 show the maximum temperatures reached for various slip and forward speeds. The maximum temperature is expressed in a form such that different material constants, loads, contact geometries and adhesion coefficients can be plotted on the same graph. Figure 6 shows the maximum interface temperatures attained as a function of slip speed at three different forward speeds when P,,., = 1100 MN, a = 7.5 mm, OL= 0.12 X 1O-4 m2 s-l, K = 46 W m-l “C-l and th e coefficient of adhesion p is 0.1. The ease with which surface temperatures greater than 1000 “C can be generated should be noted. Temperatures as high as this will readily be achieved by a stationary spinning wheel. However, the above theory is not valid for a forward speed less than 0.1 m s-l as the condition of thermal impact no longer applies. It is also not applicable to very high spinning speeds for in this case the reentry temperature of the tyre would be appreciable and the contact area would increase rapidly owing to wear. El-Sherbiny and Newcomb [9] have given a solution for the temperature distribution of a stationary cylinder loaded against a rotating cylinder with their axes at right angles.
303
I’/
’
hForward
Speed
Forward 30 mls
511~ Speed
Speed
m/s
Fig. 6. Typical maximum interface temperature for various slip and forward speeds: wheel load, 100 kN; wheel radius, 0.5 m; adhesion coefficient, 0.1; solid curve, wheel slide during braking; broken curve, wheel spin during traction.
5. Temperature after passing through the contact area As the element of area leaves the contact zone the heat input becomes zero and the surface temperature decays by conduction into the bulk material, i.e. z > 0. The temperature of a point on the wheel or rail surface for t > T could be of interest, but it cannot readily be obtained by using the boundary condition defined by the polynomial (eqn. (8)). To determine this temperature we assume that the thermal flux along the major axis for 0 < t < T is constant and that the flow of heat is normal to the surface. The boundary condition for the average thermal flux can be given by ae - = - 0.333@ {H(t) - H( t - T)} az
(W
where H(t) is a unit step function and H(t - T) is a shifting function. Applying the Laplace transform to eqn. (19) as before gives the boundary condition for the subsidiary eqn. (9) : de Z
= - 0.333@
1 - I P
exp (-pt) P
(20)
t
Solving eqns. (9) and (20) gives e = 0.333%!“2
4
-
exp (-PT)
exp (74
I P 312 The inversion of eqn. (21) from the table of transforms gives
$ (aT)-
1P
1/2 = 0.667{Xu2i
erfc (X) - (X - 1)y2i erfc (h)t_-T}
(21)
(22)
304
where At-r = Z/2(&( t - T)}l’2 and the surface temperature at t > T as before is (Xi’2 - (X - 1)1’2} The surface temperature variation of the rail or the wheel can be calculated by substituting the appropriate values of @, t and T. In dimensionless form the surface temperature for t > T can be given as a proportion of the maximum interface temperature reached (Fig. 3) : e*=o /o = 0.834 (Xi’2 - (X - l)i’2}
(24)
The assumptions which have been made are reasonably valid for short times but they will give rather higher temperatures than are likely to be obtained in practice. This is because some heat would also diffuse in the lateral direction.
6. Heat penetration perpendicular to the contact plane The most important temperatures are those obtained at the interface of the sliding surfaces, i.e. z = 0, but a knowledge of the temperatures below the surfaces, i.e. z > 0, is needed for calculating the thermal stresses. These temperatures can be obtained for various values of z provided the forward speed of the wheel is fairly high. Figure 7 shows curves of $0 (cxT)-~~@-’ plotted against X = t/T for values of 2 of 0, 50,100 and 200.2 is defined as 2 = VZ/~CY.The maxima of the curves can be seen to move towards the right as 2 increases. This is due to the continuing diffusion of heat after the contact zone has moved forward (X eventually becomes greater than unity). These results follow similar trends to those obtained by Jaeger [2] ; the differences are due to the different boundary conditions used. Figure 8 shows curves of i13 (aT)- u2 Q-l against 2 for various values of X = t/T up to X = 1. The figure shows that at most forward speeds the heat penetration into the medium is fairly shallow and the thermal gradients are fairly steep. Therefore the thermal stresses alone would cause only superficial damage to the rail or wheel. The most extensive damage is likely to be caused by surface temperatures which are high enough to change the steel structure or increase the local wear causing wheel flats.
7. Temperature rise due to creep During normal rolling with a longitudinal tractive force the contact patch beween the wheel and rail is divided into a region of adhesion and a
305
X-
Z-
Fig. 7. Temperature rise perpendicular to the contact ellipse below the major axis. Fig. 8. Temperature gradients perpendicular to the contact ellipse below the major axis.
region of slip. The slip region is located at the trailing end of the contact ellipse and its area depends upon the tractive force and the adhesion coefficient. The microslip occurring in this region will generate heat. The resulting surface temperature will be a function of the rolling velocity, the microslip velocity, the length of the slip region, the load supported by the slip region and the adhesion coefficient. In the analysis given below a simplified view is taken in which the microslip is considered to take place over the full length of the contact zone. This situation is obtained in practice when the tractive force is just equal to the maximum allowed by the adhesion, i.e. just before gross sliding occurs. The creep y is given by [lo] Y=-
Y=
2(pure rolling velocity - actual forward velocity) pure rolling velocity + actual forward velocity
2(% - VI v,
+
v
(25)
Its magnitude can be estimated. Hobbs [ 111 has surveyed various theoretical models and experimental results which show how creepage depends on the
306
tractive forces applied to a wheel. When the traction force f reaches its upper limit of PL the theoretical estimate is that the longitudinal creep is given by the expression Eab$/3pL
=D = 1
(26)
where $ is a coefficient dependent upon a/b and D is a function of f/L. If it is assumed that P = 0.2, L = 100 kN, a = 7.5 mm and b = 5 mm the creep near gross slip works out to be about 0.005. Experimental results cited by Hobbs suggest that creep values of two to three times this value can be sustained without gross slip occurring. For creep values as small as this, eqn. (25) can be approximated to the slip ratio, i.e. 2(&
-V)
v,
+ v
a- us =R V
(27)
The temperature rise due to creep alone can then be calculated in the same way as for a slipping wheel. The temperatures determined for a slipping wheel are proportional to the adhesion coefficient and the effect of changes in adhesion can therefore readily be calculated. However, this is not the case for a wheel that is creeping. The creep itself is a function of adhesion. If the tractive force is assumed to be maintained at its highest value as limited by the adhesion, i.e. pL, then eqn. (26) holds and creep is proportional to the adhesion coefficient. If the tractive force is assumed to remain constant, the value of D in eqn. (26) becomes less than unity and the creep y decreases as adhesion increases. Therefore the temperature calculated for the creep range must be looked upon as indicating the sort of value to be expected. However, at high forward speeds, say in excess of 50 m s-i, temperatures in the creep range may exceed 100 “C when the adhesion coefficient is about 0.2. This is approaching values at which boundary lubrication begins to become ineffective and this may help in improving wheel-rail adhesion.
8. Application
to wheel slip protection
Consider the simple case of an instantaneous loss of adhesion while the wheel is being braked at a constant torque. The sliding speed will increase linearly with time [l] and is given by
where pi, is the adhesion demanded by braking or traction forces and I is the rotational inertia of the wheelset. For a wheel slide protection system to be effective it should respond faster than the time t required to reach the maximum permissible speed. This is based on the consideration that the surface temperatures should not exceed the value which would be detrimental to the wheel and rail material. For typical wheel and rail steels it could be assumed that temperatures
307
greater than 600 “C may be undesirable. The corresponding slip speeds, if this temperature is not to be exceeded, depend upon the forward speed of the wheel. At a forward speed of 10 m s-l for a typical wheel load of 100 kN an interface temperature of up to 600 “C may be attained when the slip speed reaches about 4 m s- ’ (Fig. 6), while at a forward speed of 70 m s-l it is attained at 11 m s- ‘. The time to reach these speeds after encountering a step loss of adhesion would be about 0.16 and 0.44 s respectively. The calculation shows that the response times required are more sensitive at low forward speeds, and to prevent severe wheel slide at these speeds a wheel slide protection system must react within about 0.1 s. This means that the system should be able to detect and control a slide speed within about 5% of the forward speed. 9. Discussion and conclusions The surface temperatures generated due to friction beneath a steel wheel slipping while rolling on a steel rail have been calculated. Curves are given which relate the maximum temperatures attained with various values of wheel loading, wheel radius, rolling speed and adhesion coefficient. The temperatures obtained under typical conditions of wheel loading and speed were calculated. Surface temperatures of several hundred degrees Celsius are predicted. Heat penetration below the surface is shallow and the thermal gradients are steep. Therefore thermal stresses should cause only surface damage. In the creep range where the wheel is not exhibiting gross slip temperatures up to about 100 “C will be reached. Temperatures of this magnitude can have an effect on the lubricating properties of the oily contaminants on the rail head and the adhesion coefficient may itself be changed. The above analysis is applicable only to wheel speeds above about 0.1 m s-l (0.3 mile h-l). The formulae produced are similar to those produced by the theory of Jaeger and Blok for similar forward speeds. However, the temperature rises predicted by this method are about 20% higher than those predicted by their theories. The differences are entirely due to the assumption made for the thermal flux distribution along the contact length. Acknowledgments The author wishes to thank the British Railways Board for permission to publish this work. References 1 C. L. Murray, Wheelslip and APT, Railw. Eng., 2 (3) (1978) 22 - 27. 2 J. C. Jaeger, Moving sources of heat and the temperature at sliding contacts, Sot. NSW, 76 (1943) 203 - 224.
Proc. R.
308 3 H. Blok, Les temperatures de surface dans des conditions de graissage sons extreme pression, Proc. 2nd World Petroleum Congr., Paris, 1937, Vol. 3, pp. 471 - 486; English transl., Surface Temperature under E.P. Lubricating Conditions, Royal Dutch/Shell Laboratories, Delft. 4 F. F. Ling and V. C. Mow, Surface displacement of a convective elastic half-space under an arbitrary distributed fast-moving heat source, J. Basic Eng., 87 (1965) 729 734. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1951. K. Kraft, Adhesion, EZektr. Bahnen, 39 (6 - 9) (1968) 142 - 219. J. R. Barber, Distribution of heat between sliding surfaces, J. Mech. Eng. Sci., 9 (5) (1967) 351 - 354. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edn., Oxford Univ. Press: Clarendon Press, Oxford, 1959. M. El-Sherbiny and T. P. Newcomb, The temperature distribution due to frictional heat generated between a stationary cylinder and a rotating cylinder, Wear, 42 (1) (1977) 23 - 34. 10 K. L. Johnson, The effect of a tangential contact force upon the rolling motion of an elastic sphere on a plate, J. Appl. Mech., 25 (1958) 339 - 346. 11 A. E. W. Hobbs, A survey of creep, British Rail Res. Dept. Rep. DYN52, April 1967.