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The temperature distribution due to frictional heat generated between a stationary cylinder and a rotating cylinder
23
Wear, 42 (1977) 23 - 34 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
THE TEMPERATURE DISTRIBUTION DUE TO FRICTIONAL HEAT GENERATED BETWEEN A STATIONARY CYLINDER AND A ROTATING CYLINDER
M. EL-SHERBINY
and T. P. NEWCOMB
Department of Transport Technology, (G t. Britain)
University of Loughborough,
LEll
3TU
(Received June 1, 1976)
Summary An analytical model has been developed to find the distribution of heat between two cylinders placed at right angles to one another. One cylinder is stationary and the other rotates at high speed. The model assumes that the contact width is very small compared with the overall dimensions of the cylinders. The heat conduction problem has been solved for both the rotating and the stationary cylinder, the average temperature over the contact area has been determined and the heat distribution between them obtained by matching the average temperatures. Allowance has been made for surface heat transfer from both bodies.
Introduction Wear measurements are frequently carried out on a crossed cylinder apparatus. This consists of a stationary cylinder loaded with its curved surface against a rotating cylinder with their axes at right angles (Fig. 1). The area of contact between the cylinders is small and high localised temperatures
Fig. 1. Diagrammatic representation of the crossed cylinder apparatus.
24
can be reached at such areas under the right operating conditions. These high temperatures have a profound influence on wear. A steady build-up in temperature in the cylinders also occurs during continuous rubbing. By solving two heat problems the temperatures attained at any point on the surfaces of both cylinders can be predicted. The results obtained are used to estimate the amount of heat that enters both bodies. These are the problems considered in this paper and the results derived are useful for other mechanisms that involve rotating cylindrical components such as mechanical O-ring seals, as well as adding to our knowledge of the thermal behaviour of small areas in sliding contact.
Fig. 2. Model for the rotating
cylinder.
Heat transfer solution for the rotating cylinder It is assumed that the heat entering into the cylinder is uniformly distributed over a circumferential band of width 2bi (Fig. 2). A cylindricai coordinate system can therefore be used and the differential equation of heat conduction for the cylinder temperature Ti takes the form 1 aT, a2T1 _+--_.__+_ r
ar2
subject
ar
azTi
to the boundary
-K,
aT,/ar
=o
az2
O
O
(1)
conditions
= H,(T,
-To)
- ql(.z)
r=R,
where 41(z) =
41
= 0
O
T1 = 0 and aT,/az = 0
aT,/a2
=0
2=0
and the ambient temperature dimensional variables 77= r/R,
t = z/R,
(2) z+c-=
O
(3)
O
(4) the non-
01 = hlR,
the solution [l] to the above heat problem for the average temperature within the band can be expressed in the form
(5) T;
25
T; =- 2&q,
sin’@rs)ds
s 0
NM1
s2[s~Ms)/Ms))
=- 2Riql
+ &Hi/K,
1 &K,
s 11
(6)
where I0 and I, are the modified Bessel functions of the first kind of zero and first order, respectively. Values of the integral S1r for different values of pi and C = RHIK have been given previously [ 11.
Fig. 3. Model
for the stationary
Heat transfer
solution
cylinder.
for the stationary
cylinder
On the stationary cylinder it is assumed that heat is generated over a band of width 2bz in the z direction and extends over an arc of contact 28e as shown in Fig. 3. The temperature T2 is then dependent on f(r, 8, z) and is a solution of the differential heat equation
l12T2+ 1 ar2
a%+1 a25 + a2Tz_ 0
r
ar
r2 ae2
subject to the boundary = H,T,
-K2aT21ar
q2(,w
conditions - q2(z, e)
q2
OGz
0
elsewhere
(8)
-BeGe<&,
(9) z-+00
aT2/a2 = 0
2=0
the non-dimensional
n = r/R2
t = z/R2
the above equations a2T2 +l a772
K2
0
(10) (11)
variables 02 = b2lR2
(12)
reduce to
aT2+LaZT2 -+-= r12 ae2 71 as
aT2 W6.z -+a77
r=R2
=
T2=Oand~T2/&r=0
Introducing
(7)
az2
a2T, at2
0
(13)
(14)
26
(15) T2 =OandaTs/ag=O
t-+00
aT,/a[
O,
Applying
=0
<=O
the Fourier
Fc{T2(71,~,4.))
cosine transform
= Tc(skS)
=
W741’2
O,
(10) (17)
in [ defined j-
T2(v,o,t)
as cm
St dt
(13)
0
eqn. (13) reduces to a2T,
+
a772
1 1)
w +_f_ a%aq
and eqn. (14) is transformed aTc R2H2 -+-Tc= a17
s2Tc(q,e,s)
q2 ae2
= 0
(19)
into
o
-sinsP2
~=l
K,
Again applying the finite Fourier Tcc(q,n,s) = F,{T,(q,fJ,s)}
(20)
cosine transform
= jTc
cos
ne de
(21)
0
(22) However since aT2jae
= 0 at e = 0,71,2n
(23)
aT,lae
= 0 at e = 0,77, 2~
(24)
so that = -n2Tc,
(25)
and eqns. (19) and (20) reduce to
(26) and
aT,c -+--T,,= 377
h&z K2
II2 H&2 sin sp2 sin no0 -K-__ ____ n S 2
atq=l
(27)
27
Equation (26) is a generalised general solution T,, = AI,(w)
Bessel differential
equation
and has the (23)
+ BK,(v)
and if the temperature
is to be finite B = 0. This gives
T,, = AL(w)
(29)
aTJan
(30)
and = A&+,(w)
Applying the boundary T,,(l,
conditions
at n = 1 gives (31)
n, s) = AI,(s)
and
aTc,(l, n, s)Prl = AsZn+,(s) and substituting A=-
for T,,(l,
(32)
n, s) and ~T,,(l,n,s)/~~~
‘I2 R2q2 sin n _p_ K2 n i 71 1 2
in eqn. (27) gives 1
sin sp2
00
s
In(S)[S{ln+l(S)Iln(S)}+
Wh/Kzl
(33)
Using eqn. (29) 1/2 R2q2 sin s(js sin _ F ~ n
S
K2
1
ne0
(34)
&&l&l
[S{k+,(S)/h(S)l+
and since T,(l,
8, s) = 5 T,,(l, lr
then by substituting
0, s) + z : T,,(l, 71 n=l
n, s) cos ne
(35)
for T,, from eqn. (34) ‘I2 R2q2 sin sf12 1
- K2
-
m sin no0
+:z71 n=l
n
Finally using the inversion T,(l,
e, ,g) = (2/n)“2
S
e. ii- [s{l,(s)l~o(s)~+ W&/K21 +
cos t2e [sI~n+~(s)/L(s)l+ &Hd&l
(36)
theorem J- T,(l,
e, s) cos s.g ds
(37)
0
00 [S{Z,(SVZo(SD+
+2p-
m sin
7rn=l
nBo
12
cos
R2H2lK2
t2e
[s{Zn+l(s)/Zn(s)} + R,H,/K,]
1+
522
Fig. 4. Values for the integral
Szz when C = 0.001.
Fig. 5. Values for the integral 5’22 when C = 0.01.
522
Fig. 6. Values for the integral S22 when C = 0.1.
29
The average temperature T;
=-
TL over the friction
zone is
PI 00
1 P2~0,
ss
7'2(1,8,E)d~dE
(39)
His,2 +2s22)
(40)
0
and T’
=
2
2 n2
R2q2 a
where 1 Is{~l(s)l~o(s)I+
R2H2/K2
(41)
1 ds
and 1 [s{~,+,(sYL(s)I
sin2sP2 +
~&/&I
s2
ds
(42)
The integrals SI1 and Slz are particular cases of the more general integral Sz2 and their values were obtained by numerical integration, as described in the Appendix. The variation of S22 with p is given in Figs. 4, 5 and 6 for different values of C and Bo.
The distribution
of frictional
heat between
cylinders
If it is assumed that the average temperatures at the contact zones are the same then the distribution of heat between the cylinders can be determined. By introducing the non-dimensional variables P=
PlIP2
fi =
I-t= K,/K,
RI/R2
a=
&l/Q2
(43)
the heat sharing ratio becomes 2EF2R s 2s Q=--12+--.22 ( S 11 Gsll n
i
(44)
Values of Q have been evaluated at loads that will develop a contact semi-angle not exceeding 2” for any combination of materials and heat convection coefficients and these are given in Fig. 7. These results show that at small angles of contact the amount of heat entering the stationary cylinder is about 5.5% of that entering the rotating cylinder when the two cylinders are of the same dimensions. As the load increases and the contact angle widens to 2” the heat transferred to the stationary cylinder increases to 9.5% for the same configuration. As the stationary cylinder becomes larger than the rotating cylinder the total flow of heat into the latter body is reduced. When R = 10 the heat
R/s
---------_
---
-- -=-z J:.l
to3
10' -
ld -
10"
q:, k ld' : 3 I
Fig. 7. The heat sharing Max Temp
ratio
2.0
1.5
1-o
0.5
I
,
I
I
1
for crossed
cylinders
(
at different
loads.
_
T4C
,j: R
zRc1.0
fi,= 8, = 8. 403_
E z 2.1x ld
N/m2
R,: R, = .lm. V
~10
m/set
300-
zoo-
0
1
2
3
4
s
6
7
x z0.0011 Fig. 8. The computed load).
temperature
tb radtons
for mild steel cylinders
as a function
of 60 (i.e. of
31
transferred to the stationary cylinder approaches 55 - 60% of that entering the rotating cylinder. In a particular practical application, the maximum temperature reached by two crossed cylinders made of mild steel and of the same radius has been calculated when one cylinder is rotating at a speed of 10 m s-r. Figure 8 shows the calculated maximum temperature rise when the cylinders are loaded so that 0s increases up to 2”. This curve shows how the temperature build-up increases quite rapidly as the load increases. At loads of 10 kg the temperature rise is about 70 “C and this value compares quite well with experimental results.
Conclusions An analysis has been presented to enable the temperature to be determined when a stationary cylinder is loaded against a rotating cylinder. The results have been used to estimate the heat sharing ratio between cylinders and a numerical calculation was given of the temperature reached for the particular problem of two mild steel cylinders in contact.
Nomenclature bandwidth RH/K surface heat transfer coefficient modified Bessel function of the first kind, zero order modified Bessel function of the first kind, nth order thermal conductivity modified Bessel function of the second kind, nth order rate of heat generation per unit area radial coordinate outer radius temperature of the cylinder average steady temperature over the bandwidth axial coordinate dimensionless unit along the heat band dimensionless unit in the radial direction initial contact angle dimensionless unit along the cylinder subscripts referring to the rotating and stationary cylinders, respectively
Reference 1 P. S. Kounas and A. D. Dimarogonas, The distribution of friction heat between a stationary pin and a rotating cylinder, Wear, 19 (1972) 415.
32
Appendix Evaluation of the integrals Sll, S12 and Sz2 The integrals under consideration are of an oscillatory nature within the infinite range. The integral Szz also includes an infinite summation. Evaluation of such integrals is difficult. The problem is further complicated by the Bessel functions of high arguments and orders. The method of integrating between the zeros of the oscillatory function has been considered in ref. Al. A modified version of Euler’s transformation has been used to accelerate the convergence of the series of the finite integrals [A21 . A number of related methods for evaluating such integrals have been critically compared [ A3]. The present problem may be generalised to take the form S= 5
$(7Zi)Vi
(Al)
i=O
where s J/ (n,;
vi =
ds
(A2)
sin2nBo/n2
(A3)
S)
0 @(ni) =
1 $ (ni, s) =
2&l
cos 20s + C) - 2s2(sl+
C)
(A4)
and (A5)
‘= ‘“i+l (‘)l’,i(‘)
Several trials were made to evaluate the infinite integral Vi. Integrations over the half-cycles of the oscillating term by Gaussian quadrature and over the infinite range of the non-oscillatory term proved to be inefficient. The non-oscillatory term and the non-oscillatory part of the second term appeared to have singularities at s = 0. Therefore it was necessary to split the integration range into two parts: (0, a) and (a, -). The four resulting integrals were dealt with separately but the procedure followed in ref. A3 proved to be lengthy and inefficient. This was because too many evaluations of the Bessel functions were involved. Efficient evaluation was achieved by the following procedure. (1) The infinite integral vi was first evaluated for a specific value of n (ni). This was made by integrating over the half-cycles of the regular function + (ni,
S) =
sin2 0
S/S2(Sl +
These can be expressed Uj =
r
C)
(A6)
by
$ (ni, S) ds
(i-l)77
t-47)
33
The integration was made by using the Clenshaw-Curtis method for a finite number (N > 5) of half-cycles. The infinite integral was then computed using Shanks operators [ A41 so that Vi = C
Uj (A81
j=l
(2) The infinite summation over at was then performed by a series of finite summations over the half-cycles of the oscillatory function #(q), i.e. knie, wk=
L:
@(nibi
i=(k-l)n/eo
t-49)
and then Shanks operators were used to evaluate (M > 5) of the evaluated half-cycles. T=C
T from a finite number
Wk k=l
t-410)
The Bessel functions
were evaluated
as follows.
(1) Small arguments (0 ,< s d 2). An auxiliary was computed based on the series expansion
function
i,(s) = s-“l,(s)
(s/2)’ mc 2” r=O r! F(n + r + 1)
i,(s) = 1
(2) Moderate arguments (2 < s < 12.5). Stegun and Abamowitz [A51 was used.
(All) The procedure
(3) Large arguments (s > 12.5). An auxiliary was computed by the phase-amplitude expressions
followed
by
function i,(s) = evsI,(.s) given by Goldstein and
Thaler [ A6]. (4) Orders zero and one. These were obtained
References Al
by Chebyshev
series [ A7].
to the Appendix
G. Balbine and J. Franklin, The calculations of Fourier integrals, Math. Comp., 20 (1966) 570. A2 I. Longman, A method for the evaluation of finite integrals of oscillatory functions, Math. Comp., 14 (1960) 53. A3 M. Blackmore, G. Evans and J. Hyslop, The evaluation of rapidly oscillating finite integrals, Math. Research, No. 79, Loughborough University of Technology, 1975. A4 D. Shanks, Non-linear transformation of divergent and slowly convergent sequences, J. Math. Phys. Cambridge, Mass., 34 (1955) 1.
34 A5 A6 A7
I. Stegun and M. Abamowitz, Generation of Bessel functions on high speed computers, Math. Tables Comp., 11 (1957) 255. M. Goldstein and R. Thaler, Bessel functions for large arguments, Math. Tables Come., 12 (1958) 18. C. Clenshaw, Mathematical Tables, Natl. Phys. Lab., London, 1962.
Wear, 42 (1977) 23 - 34 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
THE TEMPERATURE DISTRIBUTION DUE TO FRICTIONAL HEAT GENERATED BETWEEN A STATIONARY CYLINDER AND A ROTATING CYLINDER
M. EL-SHERBINY
and T. P. NEWCOMB
Department of Transport Technology, (G t. Britain)
University of Loughborough,
LEll
3TU
(Received June 1, 1976)
Summary An analytical model has been developed to find the distribution of heat between two cylinders placed at right angles to one another. One cylinder is stationary and the other rotates at high speed. The model assumes that the contact width is very small compared with the overall dimensions of the cylinders. The heat conduction problem has been solved for both the rotating and the stationary cylinder, the average temperature over the contact area has been determined and the heat distribution between them obtained by matching the average temperatures. Allowance has been made for surface heat transfer from both bodies.
Introduction Wear measurements are frequently carried out on a crossed cylinder apparatus. This consists of a stationary cylinder loaded with its curved surface against a rotating cylinder with their axes at right angles (Fig. 1). The area of contact between the cylinders is small and high localised temperatures
Fig. 1. Diagrammatic representation of the crossed cylinder apparatus.
24
can be reached at such areas under the right operating conditions. These high temperatures have a profound influence on wear. A steady build-up in temperature in the cylinders also occurs during continuous rubbing. By solving two heat problems the temperatures attained at any point on the surfaces of both cylinders can be predicted. The results obtained are used to estimate the amount of heat that enters both bodies. These are the problems considered in this paper and the results derived are useful for other mechanisms that involve rotating cylindrical components such as mechanical O-ring seals, as well as adding to our knowledge of the thermal behaviour of small areas in sliding contact.
Fig. 2. Model for the rotating
cylinder.
Heat transfer solution for the rotating cylinder It is assumed that the heat entering into the cylinder is uniformly distributed over a circumferential band of width 2bi (Fig. 2). A cylindricai coordinate system can therefore be used and the differential equation of heat conduction for the cylinder temperature Ti takes the form 1 aT, a2T1 _+--_.__+_ r
ar2
subject
ar
azTi
to the boundary
-K,
aT,/ar
=o
az2
O
O
(1)
conditions
= H,(T,
-To)
- ql(.z)
r=R,
where 41(z) =
41
= 0
O
T1 = 0 and aT,/az = 0
aT,/a2
=0
2=0
and the ambient temperature dimensional variables 77= r/R,
t = z/R,
(2) z+c-=
O
(3)
O
(4) the non-
01 = hlR,
the solution [l] to the above heat problem for the average temperature within the band can be expressed in the form
(5) T;
25
T; =- 2&q,
sin’@rs)ds
s 0
NM1
s2[s~Ms)/Ms))
=- 2Riql
+ &Hi/K,
1 &K,
s 11
(6)
where I0 and I, are the modified Bessel functions of the first kind of zero and first order, respectively. Values of the integral S1r for different values of pi and C = RHIK have been given previously [ 11.
Fig. 3. Model
for the stationary
Heat transfer
solution
cylinder.
for the stationary
cylinder
On the stationary cylinder it is assumed that heat is generated over a band of width 2bz in the z direction and extends over an arc of contact 28e as shown in Fig. 3. The temperature T2 is then dependent on f(r, 8, z) and is a solution of the differential heat equation
l12T2+ 1 ar2
a%+1 a25 + a2Tz_ 0
r
ar
r2 ae2
subject to the boundary = H,T,
-K2aT21ar
q2(,w
conditions - q2(z, e)
q2
OGz
0
elsewhere
(8)
-BeGe<&,
(9) z-+00
aT2/a2 = 0
2=0
the non-dimensional
n = r/R2
t = z/R2
the above equations a2T2 +l a772
K2
0
(10) (11)
variables 02 = b2lR2
(12)
reduce to
aT2+LaZT2 -+-= r12 ae2 71 as
aT2 W6.z -+a77
r=R2
=
T2=Oand~T2/&r=0
Introducing
(7)
az2
a2T, at2
0
(13)
(14)
26
(15) T2 =OandaTs/ag=O
t-+00
aT,/a[
O,
Applying
=0
<=O
the Fourier
Fc{T2(71,~,4.))
cosine transform
= Tc(skS)
=
W741’2
O,
(10) (17)
in [ defined j-
T2(v,o,t)
as cm
St dt
(13)
0
eqn. (13) reduces to a2T,
+
a772
1 1)
w +_f_ a%aq
and eqn. (14) is transformed aTc R2H2 -+-Tc= a17
s2Tc(q,e,s)
q2 ae2
= 0
(19)
into
o
-sinsP2
~=l
K,
Again applying the finite Fourier Tcc(q,n,s) = F,{T,(q,fJ,s)}
(20)
cosine transform
= jTc
cos
ne de
(21)
0
(22) However since aT2jae
= 0 at e = 0,71,2n
(23)
aT,lae
= 0 at e = 0,77, 2~
(24)
so that = -n2Tc,
(25)
and eqns. (19) and (20) reduce to
(26) and
aT,c -+--T,,= 377
h&z K2
II2 H&2 sin sp2 sin no0 -K-__ ____ n S 2
atq=l
(27)
27
Equation (26) is a generalised general solution T,, = AI,(w)
Bessel differential
equation
and has the (23)
+ BK,(v)
and if the temperature
is to be finite B = 0. This gives
T,, = AL(w)
(29)
aTJan
(30)
and = A&+,(w)
Applying the boundary T,,(l,
conditions
at n = 1 gives (31)
n, s) = AI,(s)
and
aTc,(l, n, s)Prl = AsZn+,(s) and substituting A=-
for T,,(l,
(32)
n, s) and ~T,,(l,n,s)/~~~
‘I2 R2q2 sin n _p_ K2 n i 71 1 2
in eqn. (27) gives 1
sin sp2
00
s
In(S)[S{ln+l(S)Iln(S)}+
Wh/Kzl
(33)
Using eqn. (29) 1/2 R2q2 sin s(js sin _ F ~ n
S
K2
1
ne0
(34)
&&l&l
[S{k+,(S)/h(S)l+
and since T,(l,
8, s) = 5 T,,(l, lr
then by substituting
0, s) + z : T,,(l, 71 n=l
n, s) cos ne
(35)
for T,, from eqn. (34) ‘I2 R2q2 sin sf12 1
- K2
-
m sin no0
+:z71 n=l
n
Finally using the inversion T,(l,
e, ,g) = (2/n)“2
S
e. ii- [s{l,(s)l~o(s)~+ W&/K21 +
cos t2e [sI~n+~(s)/L(s)l+ &Hd&l
(36)
theorem J- T,(l,
e, s) cos s.g ds
(37)
0
00 [S{Z,(SVZo(SD+
+2p-
m sin
7rn=l
nBo
12
cos
R2H2lK2
t2e
[s{Zn+l(s)/Zn(s)} + R,H,/K,]
1+
522
Fig. 4. Values for the integral
Szz when C = 0.001.
Fig. 5. Values for the integral 5’22 when C = 0.01.
522
Fig. 6. Values for the integral S22 when C = 0.1.
29
The average temperature T;
=-
TL over the friction
zone is
PI 00
1 P2~0,
ss
7'2(1,8,E)d~dE
(39)
His,2 +2s22)
(40)
0
and T’
=
2
2 n2
R2q2 a
where 1 Is{~l(s)l~o(s)I+
R2H2/K2
(41)
1 ds
and 1 [s{~,+,(sYL(s)I
sin2sP2 +
~&/&I
s2
ds
(42)
The integrals SI1 and Slz are particular cases of the more general integral Sz2 and their values were obtained by numerical integration, as described in the Appendix. The variation of S22 with p is given in Figs. 4, 5 and 6 for different values of C and Bo.
The distribution
of frictional
heat between
cylinders
If it is assumed that the average temperatures at the contact zones are the same then the distribution of heat between the cylinders can be determined. By introducing the non-dimensional variables P=
PlIP2
fi =
I-t= K,/K,
RI/R2
a=
&l/Q2
(43)
the heat sharing ratio becomes 2EF2R s 2s Q=--12+--.22 ( S 11 Gsll n
i
(44)
Values of Q have been evaluated at loads that will develop a contact semi-angle not exceeding 2” for any combination of materials and heat convection coefficients and these are given in Fig. 7. These results show that at small angles of contact the amount of heat entering the stationary cylinder is about 5.5% of that entering the rotating cylinder when the two cylinders are of the same dimensions. As the load increases and the contact angle widens to 2” the heat transferred to the stationary cylinder increases to 9.5% for the same configuration. As the stationary cylinder becomes larger than the rotating cylinder the total flow of heat into the latter body is reduced. When R = 10 the heat
R/s
---------_
---
-- -=-z J:.l
to3
10' -
ld -
10"
q:, k ld' : 3 I
Fig. 7. The heat sharing Max Temp
ratio
2.0
1.5
1-o
0.5
I
,
I
I
1
for crossed
cylinders
(
at different
loads.
_
T4C
,j: R
zRc1.0
fi,= 8, = 8. 403_
E z 2.1x ld
N/m2
R,: R, = .lm. V
~10
m/set
300-
zoo-
0
1
2
3
4
s
6
7
x z0.0011 Fig. 8. The computed load).
temperature
tb radtons
for mild steel cylinders
as a function
of 60 (i.e. of
31
transferred to the stationary cylinder approaches 55 - 60% of that entering the rotating cylinder. In a particular practical application, the maximum temperature reached by two crossed cylinders made of mild steel and of the same radius has been calculated when one cylinder is rotating at a speed of 10 m s-r. Figure 8 shows the calculated maximum temperature rise when the cylinders are loaded so that 0s increases up to 2”. This curve shows how the temperature build-up increases quite rapidly as the load increases. At loads of 10 kg the temperature rise is about 70 “C and this value compares quite well with experimental results.
Conclusions An analysis has been presented to enable the temperature to be determined when a stationary cylinder is loaded against a rotating cylinder. The results have been used to estimate the heat sharing ratio between cylinders and a numerical calculation was given of the temperature reached for the particular problem of two mild steel cylinders in contact.
Nomenclature bandwidth RH/K surface heat transfer coefficient modified Bessel function of the first kind, zero order modified Bessel function of the first kind, nth order thermal conductivity modified Bessel function of the second kind, nth order rate of heat generation per unit area radial coordinate outer radius temperature of the cylinder average steady temperature over the bandwidth axial coordinate dimensionless unit along the heat band dimensionless unit in the radial direction initial contact angle dimensionless unit along the cylinder subscripts referring to the rotating and stationary cylinders, respectively
Reference 1 P. S. Kounas and A. D. Dimarogonas, The distribution of friction heat between a stationary pin and a rotating cylinder, Wear, 19 (1972) 415.
32
Appendix Evaluation of the integrals Sll, S12 and Sz2 The integrals under consideration are of an oscillatory nature within the infinite range. The integral Szz also includes an infinite summation. Evaluation of such integrals is difficult. The problem is further complicated by the Bessel functions of high arguments and orders. The method of integrating between the zeros of the oscillatory function has been considered in ref. Al. A modified version of Euler’s transformation has been used to accelerate the convergence of the series of the finite integrals [A21 . A number of related methods for evaluating such integrals have been critically compared [ A3]. The present problem may be generalised to take the form S= 5
$(7Zi)Vi
(Al)
i=O
where s J/ (n,;
vi =
ds
(A2)
sin2nBo/n2
(A3)
S)
0 @(ni) =
1 $ (ni, s) =
2&l
cos 20s + C) - 2s2(sl+
C)
(A4)
and (A5)
‘= ‘“i+l (‘)l’,i(‘)
Several trials were made to evaluate the infinite integral Vi. Integrations over the half-cycles of the oscillating term by Gaussian quadrature and over the infinite range of the non-oscillatory term proved to be inefficient. The non-oscillatory term and the non-oscillatory part of the second term appeared to have singularities at s = 0. Therefore it was necessary to split the integration range into two parts: (0, a) and (a, -). The four resulting integrals were dealt with separately but the procedure followed in ref. A3 proved to be lengthy and inefficient. This was because too many evaluations of the Bessel functions were involved. Efficient evaluation was achieved by the following procedure. (1) The infinite integral vi was first evaluated for a specific value of n (ni). This was made by integrating over the half-cycles of the regular function + (ni,
S) =
sin2 0
S/S2(Sl +
These can be expressed Uj =
r
C)
(A6)
by
$ (ni, S) ds
(i-l)77
t-47)
33
The integration was made by using the Clenshaw-Curtis method for a finite number (N > 5) of half-cycles. The infinite integral was then computed using Shanks operators [ A41 so that Vi = C
Uj (A81
j=l
(2) The infinite summation over at was then performed by a series of finite summations over the half-cycles of the oscillatory function #(q), i.e. knie, wk=
L:
@(nibi
i=(k-l)n/eo
t-49)
and then Shanks operators were used to evaluate (M > 5) of the evaluated half-cycles. T=C
T from a finite number
Wk k=l
t-410)
The Bessel functions
were evaluated
as follows.
(1) Small arguments (0 ,< s d 2). An auxiliary was computed based on the series expansion
function
i,(s) = s-“l,(s)
(s/2)’ mc 2” r=O r! F(n + r + 1)
i,(s) = 1
(2) Moderate arguments (2 < s < 12.5). Stegun and Abamowitz [A51 was used.
(All) The procedure
(3) Large arguments (s > 12.5). An auxiliary was computed by the phase-amplitude expressions
followed
by
function i,(s) = evsI,(.s) given by Goldstein and
Thaler [ A6]. (4) Orders zero and one. These were obtained
References Al
by Chebyshev
series [ A7].
to the Appendix
G. Balbine and J. Franklin, The calculations of Fourier integrals, Math. Comp., 20 (1966) 570. A2 I. Longman, A method for the evaluation of finite integrals of oscillatory functions, Math. Comp., 14 (1960) 53. A3 M. Blackmore, G. Evans and J. Hyslop, The evaluation of rapidly oscillating finite integrals, Math. Research, No. 79, Loughborough University of Technology, 1975. A4 D. Shanks, Non-linear transformation of divergent and slowly convergent sequences, J. Math. Phys. Cambridge, Mass., 34 (1955) 1.
34 A5 A6 A7
I. Stegun and M. Abamowitz, Generation of Bessel functions on high speed computers, Math. Tables Comp., 11 (1957) 255. M. Goldstein and R. Thaler, Bessel functions for large arguments, Math. Tables Come., 12 (1958) 18. C. Clenshaw, Mathematical Tables, Natl. Phys. Lab., London, 1962.