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Static bow of shafts due to rubbing
Wear, 25 (1973) 17-27 0 Elsevier Sequoia SA., Lausanne - Printed in The Netherlands
STATIC BOW OF SHAFTS
17
DUE TO RUBBING
ANDREW D. DIMAROGONAS* Lehigh University, Bethlehem, Pa. (U.S.A.) (Received January 23, 1973)
SUMMARY
When rotating shafts contact stationary components, part of the heat liberated due to friction enters the shaft and causes bow, if the distribution of the heat is not uniform. This situation is very common in turborotors rubbing against the steam packings and seals. In this paper, the shaft is modeled as a hollow cylinder of finite length. Uniform heat convection on both surfaces was assumed with arbitrary distribution of the frictional heat. The transient heat conduction equation was used in conjunction with Goodiers formula for the thermoelastic bow. The bow was calculated in closed form in terms of two parameters which were tabulated for a variety of boundary conditions.
NOMENCLATURE
B D E L T
Bow of the hollow cylinder External diameter Young’s modulus Length of the cylinder Temperature Outer radius of the cylinder Inner radius of the cylinder Imaginary unit (when used as a factor) Thermal conductivity Length of the heat input zone Heat input intensity Radius Time Coordinate along the cylinder axis Heating zone of the cylinder Coefficient of thermal expansion Angle in cylindrical coordinates Thermal diffusivity Angle of flexural rotation
: z 1
4 r t Z
a E
e K (5 l
Formerly with the General Electric Co., Turbine Department, Schenectady, N.Y., U.SA.
18
A. D. DIMAROGONAS
INTRODUCTION
Friction is considered, in general, as a static phenomenon. However, there are situations where friction is associated with dynamic phenomena. In rotating machinery many cases of this type exist, such as stick-slip in bearings and dry friction whirl. One interesting case is when rotating shafts are rubbed against stationary components, such as packings and seals. If the heat entering the rotor is unevenly distributed, a thermal bow is produced which might yield severe vibration’. In this case the increasing vibration might or might not affect the rate of the generated heat. The latter case is the subject of this paper. The former case is far more complicated and will be the subject of a future report. A state of the art review has been presented2. Existing analyses for the thermoelastic problem in the cylinder consider only symmetric heat inputs, such as uniformity along the z-axis3v4 or axisymmetric heat rings 5,6. Furthermore, sp ecial boundary conditions are generally assumed, such as constant surface temperatures or isolated surfaces. Here, constant heat convection is assumed in both surfaces. The heat transfer coefficients might vary along the shaft but in practical situations they are essentially constant around the friction zone and can be considered constant throughout the rotor for all practical purposes. THE THERMAL
DEFLECTION
IN GENERAL
The heat input is highly localized and it can be assumed that the length L of the rotor is large compared with the length 1 of the heating zone. Figure 1 shows the situation for a rotor extending along the z-axis of a Cartesian coordinate system with axes x, y, z and bent in the (x, z)-plane due to rubbing of its outer cylindrical surface in the zone -a/2< cp< a/2. Assuming that the portions of the rotor outside of the heating zone stay straight or bent insignificantly only, the bow B can be expressed approximately as : B = (p$
where 0 is the angle of flexural rotation of one end relative to the other. Formula (1) has an error due to substituting the triangle A, A’, A” for the exact shape of the bowed rotor ; this error is believed to be quite negligible. To apply eqn. (1) the angle 0 must be determined. Let T denote the temperature in the rotor as a function of location and time, the temperature field caused by the heat input. For a prismatic bar, extending along the z-axis and bending in the (x, z)-
Fig. 1. Geometry
of a locally bowed shaft.
19
STATIC BOW OF SHAFTS DUE TO RUBBING
plane under the influence of T, Goodier7 defines “a mean thermoelastic rotation of one end relative to the other” by the integral : a=-
1
Txdxdydz
E
IIv
flexural
(2)
over the volume I/ of the bar. Here I represents the moment of inertia of the bar cross section with respect to the y-axis and E is the thermal expansion coefficient. Goodier’s formula is based on Betti’s theorem of reciprocity. Its angle 25is not exactly the angle of eqn. (1) ; since it appears that the difference is irrelevant, it may be ignored. If bowing is not restricted to the (x, z)-plane, it is useful to let B be complex such that Re B, Im B give the bow in x-and y-direction respectively. With hollow cylinders made of isotropic material, the moment I is the same for x- and y-axis which permits the complex bow to be set up in the form: L,L,E B=-----ET(x+iy)dxdydz LJ i v
(3)
It is convenient to refer the relevant features of the temperature field to cylindrical coordinates r, 0, z, where x =T cos 19,y=r sin 8. If the inner surface has radius b and the outer radius a, then the hollow cylinder can be described by b < r d a and - L, < 26 Lz.
Thus T = T(r, 8, z; t)
and eqn. (3) can be given the form : B=-
a
-b&E
SIb
LI
2n
0
Lz
.i
T(r, 0, z; t)r’e”‘dzdOdr
(4)
-LI
T must satisfy the partial differential equation : a2T
1 #T
18T
a2T
1 aT
(5)
p+;Jgy+I;Ijg+jp=;z and the initial condition
TsO
for
t-c0
(6)
if heating does not begin before t = 0. Relations (5) and (6) have to be supplemented by boundary conditions. Thus.
aT 0 az= -kg,+
kg
h,T=O
+ h,T=g(&z;
t)
for
z=L,, -L,
for
r=b
for
r=a
(7)
(8)
Here K, k, h,, hb are thermodynamic constants; g(O, z; t) gives the heat rate density per unit area, generated by rubbing at the point (0, z) of the outer cylinder r=a;
A. D. DIMAROGONAS
20
g(0, z ; t) need not be specified in all detail. Introducing the integrals Z(r, 0; t) = IL1
T(r, 8, z; t)dz; ij(6; t) =
IL’
L*
s(K z; t)dz
Lz
(9
Equation (4) demands the knowledge of Z only ; on the other hand it follows from eqns. (5) and (7) that furthermore (5) is satisfied with T replaced by Z and finally : -kg+
h,Z=O
kg
+ h,Z=g(0,
t)
for
r=b
for
r=a
(11)
The bow can be expressed in the form : 2n rz
W(r,
Z(r, 8; t)eied6
t)dri W(r, t) = 0
W(r, t) represents (after division by rr) a complex Fourier component respect to 8. It satisfies
a2w law ar”+;x-F”‘=;z
w
i aw
of Z with
(13)
and the boundary conditions : 2n
kg
+ h,W=
w(t), w(t) =
o
g(8;
t)eiedO
for r=a (14)
-kg
+ h,W=O
for r=b
W also satisfies (7) i.e. W = 0 for T < 0. The function W(r, t) can be found by the method of Laplace-transform. Introducing W(r, p) =
1: e-P’W(r,
t)dt
(15)
as the transform of W allows G and B to be introduced as transforms of w(t) and of B(t) respectively. Furthermore A2=p/k then from (13) and from W=O for t< 0 it follows that : w”+
;
W’_
L$W_~2W=O
(16)
consequently
W= yl, (Ar) + 6K, (lx)
(17)
where I,, K, are the modified Bessel functions of 1 order. The coefficients y, 6 must be determined from relations (14) which stay valid if W, w are replaced by q W.The boundary conditions yield : _
a,,y + a,26 = y azly
+
a,,6 =
6
(18)
STATIC BOW OF SHAFTS DUE TO RUBBING
21
where a 11 = all;
(Aa) + ul, (Au)
a 12= a/m; (Au) + UK, (h)
(19)
a21= - bnl; (Ib) + ol, (Lb) uz2 = - b/IK’,(Ib) + uK, (Lb)
u = ah,,/k; v = bh,/k
Since zl; (z) = zZ,(z) - II (z); zKI (z) = - zK,(z) -K,
(z) it means,
a l,=naZ,(la)+(u-l)z,(la) u12= -la,K,(~a)+(u-l)K,(la) a21 = -tIbZ,(lb) + (v+ l)K,(Ab) a,,=IbK,(Ib)+(u+l)K,(~b) The solution of (18) is : Y=r;
az2aii 6 = -2aw~
(21)
where A = a11az2 - a12azl The computation of the determinant A is simplified if the well-known expression for the Wronskian is used Z, (4 K, (4 + ZI(4 Ko (4 = 5 + We obtain A =~*abD,,+la(u+l)D,,+Lb(u-l)D,,+(u-l)(v+l)D,, with
(22)
D,, = Z.(Au)K, (2b) - (- l)m+ ’ Z#b) K, (Au) where m, n = 0,l. B may he represented in a closed form. From eqn. (12) it follows:
(23) since eqn. (16) can be rewritten in the form ~*rZW2=rZWII+~‘_-=((yZW’-r~)
(24
it is found that A2 a iE.2dr=(r2w-rI)I; jb
(25)
To utiliie eqns. (17) and (19), steps of an elementary nature yield :
(26)
22
A. D. DIMAROGONAS
where
ubi2Doo+~(u+ 1)3,0,, - 2b/U),,+2(u+l)D,,
_
+ 5 (1 -u)
p = a~1~[ab/2~D,,,+a(v+1)~D,,+h(u-1)~D,,+(u-l)(u+1)D,~ 1” represents the Laplace-transform of some function P(t). More about P(t) will be found later. Then by the convolution theorem, B(t) = L1;;;4i.
i” P(t-s)w(s)ds 0
Applying this general formula to the special case in which w(t) represents the influence of the heat flow into the hollow cylinder due to a time varying heat input q(t). B(t) = gL i’ P(t-s)q(s)ds . 0
this formula follows from eqn. (28). The constant y is L, L2a4& “I =
L’IK
Since the integrand in eqn. (29) is dimensionless (Pdt has no dimension) and since B has the dimension of ,!., q is without dimension. This function P(t) has nothing to do with the heat input in its detail but
depends exclusively on the quality of eqns. (1) and (2). In a simple approximation for it, the approximation must be compatible with the errors that afflict 0 in eqn. (2) and P must be considered in some detail. For large values of (zl and for arg lzl < $z the well-known asymptotic formulas are used :
L(z)= (1+0(l))
&
K,(Z) = (1+0(l))
(t$ ec
(31)
This implies for the coefficients D,, introduced by eqn. (22). D,,=
as 11,(~~;largJl
(1+0(l))&
<$t
so that Dmn/DOO+ 1 as 121 + w
assuming D,,/D,,
P-
= 1 then eqn. (27) yields the asymptotic formula aA-
aV(a;l+u-
1)
for large 111 and larg 11 <$t
In particular a212P+
1 as Il(+ar,
(34)
For very small values of lil eqn. (16) indicates that the function IV, due to b > 0, will be regular analytic with respect to 1’ in some neighborhood of A2= 0. This means that P approaches a limit value as A+O. To find the limit value either asymp-
totic formulas for the behavior of Bessel functions for small arguments are used or Wis
STATIC BOW OF SHAFTS
DUE TO RUBBING
23
determined directly by setting 1=0 in eqn. (16). The latter procedure is simple and leads to : W=p+!
(35)
r
The coefficients 7,s are determined by the boundary conditions for r = a&. One finds a++
ji’
l)W kN
a%‘(u-1)W kN
;6=
(36)
with N=a2(u+l)(v+1)-b2(~-l)(u-1)=(~2-b2)(~u+1)+(u2+b2)(~+~)
(37)
Since [’ m2dr
= $(a”-b2)[y(u2+b2)
.f b
+28]
(u2-P)U+u2+3b2
(39) p(“)=p =~(u2-b2)(.u+l)+(u2+b’)(u+u For a thick walled cylinder and if u % 1, u $1 the approximation
:
u2-b2 p =
4uu2
may be used. The rational function P of 1 which has the same behavior as is for A+0 and for A-+ co and which approximates P for positive 1 within an error bound of say 1% should not present great difficulties to retransform it into a function of time. Since II is in proportion to pi it would be preferable (if at all possible) to make P”a rational function of A2 i.e. of p itself. The latter seems promising since the behavior of P for small and large [A( points to A2: (a2J2+Pw2) p
=
=---
Pl
U2A26
(u2A2+&)(u2A2+b;2)
-~ 1
P2
U2A26
2
(41)
where Pl=
(Pd2
-
h2_dl
Wl
;
P2 =
(PSI- 1P2 d2_d1
; P1-P2=]
This approximation has the desired asymptotic behavior. It depends on the two parameters 6,, d2, which might be determined by collocation, to name at least one procedure. Collocation leads to two linear equations for 6,& and 6, +6,. From these quantities 6i, 6, can be derived as roots of the quadratic equation : 62-q61+62)+6162=o An example of approximation
by collocation is shown in Fig. 2. It is apparent that good agreement between P and P has been obtained. Further improvement could be obtained by optimizing the selection of the points of collocation. For the purpose of this study the accuracy of Fig. 2 appears to be good enough.
24
A. D. DIMAROGONAS
Fig. 2. Approximation
of the Laplace transform.
The inverse transform of P is readily obtained by first substituting 1’ =P/K
“‘PI P ~---_-----
K’PZ p+Jc’&
p+lc’d,
’
K’ =
,“z -
(43)
The fractions in (43) have well known inverse-transforms P(t)x”‘~~
e-K’a1t_~‘~ze-“‘62*
Relation (29) takes now the form of the approximation m=rlq;
and, (44) :
[K’PI exp[-K’81(t-s)]--‘Pzexp[-IC’8Z(t-s)]]q(s)ds
(45)
By differentiation it can be shown that (45) is equivalent with : c!ZB=Aq d2 d .z = dt2 ,+ Jc’(C51+6,)3 + K’8*6,
(46)
supplemented by the initial conditions B(O)=0
(47)
S(0) = rpdLq(O) In words : An ordinary differential equation of order 2 with constant coefficients can be used to compute B from q. In the general case of time varying heat input, eqn. (46) is nonlinear. To obtain numerical results a suitable integration method in a digital computer should be used. In this work the 4th order Runge-Kutta has been employed. For the case of constant heat input eqn. (46) becomes equivalent to a linear one and,
B+K’(~, +62)8+~‘281c5ZB = r/Ld26,b2pq.
(48)
25
STATIC BOW OF SHAFTS DUE TO RUBBING
The general solution of this equation is : B(t)=A,
exp(-rc’6,t)
+A,exp(-~‘6,t)+f&pq.
(49)
A, and A, are to be determined from the initial conditions (47). Finally :
A2= v%’
6
(51)
_*
2
1
The heat input function q is a Fourier term of order 1 of the total heat input according to eqn. (12). Equation (49) gives therefore the response to a constant heat input. The constants 6, and b2 which appear in this equation can be calculated by the method discussed. They have been plotted in Figs. 3-6 for several combinations of diameters and heat transfer coefficients. The heat transfer coefficients are given as Biot Numbers, defined as Bi = ah/k. We note that after a long time, the bow will assume the steady state value: BzqLpq.
(52)
This asymptotic value of the bow can be calculated without the values of 6, and 6, and it can be used to check the accuracy of the calculations for the transient bow.
I I
, 2
Fig. 3. Parameter 6,.
3
4
5
6
c4/6
26
A. D. DIMAROGONAS
Pi), It.03
1000.
3.60
3.15 1.70 1.25
500
LW I.15
.
.90 .45
‘.
4 2
3
Fig. 4. Parameter
5
6
4
5
s/b
6,.
I I
Fig. 5. Parameter
2
6,.
3
6
a/ b
Two values of the heat transfer coefficient for the inner surface were selected, 1.2 and 4.2, corresponding to a non-flow and high cooling cases respectively. We note that for high a/b ratios, the results are not much affected by the internal heat convection. For thin cylinders (u/b < 1.5) hand calculations based on the curves of Figs. 3-6 will not be accurate, therefore use of a computer is suggested. This method is applicable for relatively long cylinders (L/D > 3) because of the limitation introduced with Goodier’s formula and with the measure of the bow. However, in most practical applications the thermal bow for short cylinders is negligible. CONCLUSION
The problem of shaft bow due to an asymmetric distribution of the friction
27
STATIC BOW OF SHAFTS DUE TO RUBBING
I d
3
4
5
6
O/b
Fig. 6. Parameter 6,.
heat was investigated. Non uniform and time dependent heat input was assumed as well as uniform heat transfer coefficients over the surface. Solution of the transient form of the heat conduction equation yielded a nonlinear differential equation for the static bow due to a time varying heat input. This equation is reduced to a linear one for the case of heat input constant. The frictional heat input can be of a very general form and will depend on the particular geometry and the other properties at the sliding contact. All these properties have been reduced to a dimensionless quantity which characterizes the severity of the bow and can be used for similarity purposes. The solution has been obtained in closed form and in terms of two quantities which have been tabulated for a wide range of heat transfer coeffkients, thus facilitating hand computations. A computer program was also written for cases not covered by the tabulation and is available from the author on request. REFERENCES 1 A. D. Dimarogonas. An analytic study of the packing rub effect in rotating machinery, Dissertation, Rensselaer Polytechnic Inst., Troy, N.Y., 1970. 2 A. D. Dimarogonas and G. N. Sandor. Packing rub effect in rotating machinery. I. A state of the art review, Wear, 14 (1969) 153-170. 3 M. Sokolowski. The axially symmetric thermoelastic problem of the infinite cylinder, Arch. Mech. Stosowanej, 10 (6) (1958) 811. 4 H. Parkus. Instationiire Wiirmespnnnungen, Springer, Vienna, 1959. 5 C. K. Youngdahl and E. Stemberg. Transient thermal stresses in circular disks and cylinders, Brown Univ. Tech. Rept., 8, 1960. 6 G. Horvay. Transient thermal stress in circular disks and cylinders, Trans. ASME, 76 (1954) 7 G. N. Goodier. On the integration of the thermoelastic equations. Phil. Msg., 7 (1937) 1017.
127.
STATIC BOW OF SHAFTS
17
DUE TO RUBBING
ANDREW D. DIMAROGONAS* Lehigh University, Bethlehem, Pa. (U.S.A.) (Received January 23, 1973)
SUMMARY
When rotating shafts contact stationary components, part of the heat liberated due to friction enters the shaft and causes bow, if the distribution of the heat is not uniform. This situation is very common in turborotors rubbing against the steam packings and seals. In this paper, the shaft is modeled as a hollow cylinder of finite length. Uniform heat convection on both surfaces was assumed with arbitrary distribution of the frictional heat. The transient heat conduction equation was used in conjunction with Goodiers formula for the thermoelastic bow. The bow was calculated in closed form in terms of two parameters which were tabulated for a variety of boundary conditions.
NOMENCLATURE
B D E L T
Bow of the hollow cylinder External diameter Young’s modulus Length of the cylinder Temperature Outer radius of the cylinder Inner radius of the cylinder Imaginary unit (when used as a factor) Thermal conductivity Length of the heat input zone Heat input intensity Radius Time Coordinate along the cylinder axis Heating zone of the cylinder Coefficient of thermal expansion Angle in cylindrical coordinates Thermal diffusivity Angle of flexural rotation
: z 1
4 r t Z
a E
e K (5 l
Formerly with the General Electric Co., Turbine Department, Schenectady, N.Y., U.SA.
18
A. D. DIMAROGONAS
INTRODUCTION
Friction is considered, in general, as a static phenomenon. However, there are situations where friction is associated with dynamic phenomena. In rotating machinery many cases of this type exist, such as stick-slip in bearings and dry friction whirl. One interesting case is when rotating shafts are rubbed against stationary components, such as packings and seals. If the heat entering the rotor is unevenly distributed, a thermal bow is produced which might yield severe vibration’. In this case the increasing vibration might or might not affect the rate of the generated heat. The latter case is the subject of this paper. The former case is far more complicated and will be the subject of a future report. A state of the art review has been presented2. Existing analyses for the thermoelastic problem in the cylinder consider only symmetric heat inputs, such as uniformity along the z-axis3v4 or axisymmetric heat rings 5,6. Furthermore, sp ecial boundary conditions are generally assumed, such as constant surface temperatures or isolated surfaces. Here, constant heat convection is assumed in both surfaces. The heat transfer coefficients might vary along the shaft but in practical situations they are essentially constant around the friction zone and can be considered constant throughout the rotor for all practical purposes. THE THERMAL
DEFLECTION
IN GENERAL
The heat input is highly localized and it can be assumed that the length L of the rotor is large compared with the length 1 of the heating zone. Figure 1 shows the situation for a rotor extending along the z-axis of a Cartesian coordinate system with axes x, y, z and bent in the (x, z)-plane due to rubbing of its outer cylindrical surface in the zone -a/2< cp< a/2. Assuming that the portions of the rotor outside of the heating zone stay straight or bent insignificantly only, the bow B can be expressed approximately as : B = (p$
where 0 is the angle of flexural rotation of one end relative to the other. Formula (1) has an error due to substituting the triangle A, A’, A” for the exact shape of the bowed rotor ; this error is believed to be quite negligible. To apply eqn. (1) the angle 0 must be determined. Let T denote the temperature in the rotor as a function of location and time, the temperature field caused by the heat input. For a prismatic bar, extending along the z-axis and bending in the (x, z)-
Fig. 1. Geometry
of a locally bowed shaft.
19
STATIC BOW OF SHAFTS DUE TO RUBBING
plane under the influence of T, Goodier7 defines “a mean thermoelastic rotation of one end relative to the other” by the integral : a=-
1
Txdxdydz
E
IIv
flexural
(2)
over the volume I/ of the bar. Here I represents the moment of inertia of the bar cross section with respect to the y-axis and E is the thermal expansion coefficient. Goodier’s formula is based on Betti’s theorem of reciprocity. Its angle 25is not exactly the angle of eqn. (1) ; since it appears that the difference is irrelevant, it may be ignored. If bowing is not restricted to the (x, z)-plane, it is useful to let B be complex such that Re B, Im B give the bow in x-and y-direction respectively. With hollow cylinders made of isotropic material, the moment I is the same for x- and y-axis which permits the complex bow to be set up in the form: L,L,E B=-----ET(x+iy)dxdydz LJ i v
(3)
It is convenient to refer the relevant features of the temperature field to cylindrical coordinates r, 0, z, where x =T cos 19,y=r sin 8. If the inner surface has radius b and the outer radius a, then the hollow cylinder can be described by b < r d a and - L, < 26 Lz.
Thus T = T(r, 8, z; t)
and eqn. (3) can be given the form : B=-
a
-b&E
SIb
LI
2n
0
Lz
.i
T(r, 0, z; t)r’e”‘dzdOdr
(4)
-LI
T must satisfy the partial differential equation : a2T
1 #T
18T
a2T
1 aT
(5)
p+;Jgy+I;Ijg+jp=;z and the initial condition
TsO
for
t-c0
(6)
if heating does not begin before t = 0. Relations (5) and (6) have to be supplemented by boundary conditions. Thus.
aT 0 az= -kg,+
kg
h,T=O
+ h,T=g(&z;
t)
for
z=L,, -L,
for
r=b
for
r=a
(7)
(8)
Here K, k, h,, hb are thermodynamic constants; g(O, z; t) gives the heat rate density per unit area, generated by rubbing at the point (0, z) of the outer cylinder r=a;
A. D. DIMAROGONAS
20
g(0, z ; t) need not be specified in all detail. Introducing the integrals Z(r, 0; t) = IL1
T(r, 8, z; t)dz; ij(6; t) =
IL’
L*
s(K z; t)dz
Lz
(9
Equation (4) demands the knowledge of Z only ; on the other hand it follows from eqns. (5) and (7) that furthermore (5) is satisfied with T replaced by Z and finally : -kg+
h,Z=O
kg
+ h,Z=g(0,
t)
for
r=b
for
r=a
(11)
The bow can be expressed in the form : 2n rz
W(r,
Z(r, 8; t)eied6
t)dri W(r, t) = 0
W(r, t) represents (after division by rr) a complex Fourier component respect to 8. It satisfies
a2w law ar”+;x-F”‘=;z
w
i aw
of Z with
(13)
and the boundary conditions : 2n
kg
+ h,W=
w(t), w(t) =
o
g(8;
t)eiedO
for r=a (14)
-kg
+ h,W=O
for r=b
W also satisfies (7) i.e. W = 0 for T < 0. The function W(r, t) can be found by the method of Laplace-transform. Introducing W(r, p) =
1: e-P’W(r,
t)dt
(15)
as the transform of W allows G and B to be introduced as transforms of w(t) and of B(t) respectively. Furthermore A2=p/k then from (13) and from W=O for t< 0 it follows that : w”+
;
W’_
L$W_~2W=O
(16)
consequently
W= yl, (Ar) + 6K, (lx)
(17)
where I,, K, are the modified Bessel functions of 1 order. The coefficients y, 6 must be determined from relations (14) which stay valid if W, w are replaced by q W.The boundary conditions yield : _
a,,y + a,26 = y azly
+
a,,6 =
6
(18)
STATIC BOW OF SHAFTS DUE TO RUBBING
21
where a 11 = all;
(Aa) + ul, (Au)
a 12= a/m; (Au) + UK, (h)
(19)
a21= - bnl; (Ib) + ol, (Lb) uz2 = - b/IK’,(Ib) + uK, (Lb)
u = ah,,/k; v = bh,/k
Since zl; (z) = zZ,(z) - II (z); zKI (z) = - zK,(z) -K,
(z) it means,
a l,=naZ,(la)+(u-l)z,(la) u12= -la,K,(~a)+(u-l)K,(la) a21 = -tIbZ,(lb) + (v+ l)K,(Ab) a,,=IbK,(Ib)+(u+l)K,(~b) The solution of (18) is : Y=r;
az2aii 6 = -2aw~
(21)
where A = a11az2 - a12azl The computation of the determinant A is simplified if the well-known expression for the Wronskian is used Z, (4 K, (4 + ZI(4 Ko (4 = 5 + We obtain A =~*abD,,+la(u+l)D,,+Lb(u-l)D,,+(u-l)(v+l)D,, with
(22)
D,, = Z.(Au)K, (2b) - (- l)m+ ’ Z#b) K, (Au) where m, n = 0,l. B may he represented in a closed form. From eqn. (12) it follows:
(23) since eqn. (16) can be rewritten in the form ~*rZW2=rZWII+~‘_-=((yZW’-r~)
(24
it is found that A2 a iE.2dr=(r2w-rI)I; jb
(25)
To utiliie eqns. (17) and (19), steps of an elementary nature yield :
(26)
22
A. D. DIMAROGONAS
where
ubi2Doo+~(u+ 1)3,0,, - 2b/U),,+2(u+l)D,,
_
+ 5 (1 -u)
p = a~1~[ab/2~D,,,+a(v+1)~D,,+h(u-1)~D,,+(u-l)(u+1)D,~ 1” represents the Laplace-transform of some function P(t). More about P(t) will be found later. Then by the convolution theorem, B(t) = L1;;;4i.
i” P(t-s)w(s)ds 0
Applying this general formula to the special case in which w(t) represents the influence of the heat flow into the hollow cylinder due to a time varying heat input q(t). B(t) = gL i’ P(t-s)q(s)ds . 0
this formula follows from eqn. (28). The constant y is L, L2a4& “I =
L’IK
Since the integrand in eqn. (29) is dimensionless (Pdt has no dimension) and since B has the dimension of ,!., q is without dimension. This function P(t) has nothing to do with the heat input in its detail but
depends exclusively on the quality of eqns. (1) and (2). In a simple approximation for it, the approximation must be compatible with the errors that afflict 0 in eqn. (2) and P must be considered in some detail. For large values of (zl and for arg lzl < $z the well-known asymptotic formulas are used :
L(z)= (1+0(l))
&
K,(Z) = (1+0(l))
(t$ ec
(31)
This implies for the coefficients D,, introduced by eqn. (22). D,,=
as 11,(~~;largJl
(1+0(l))&
<$t
so that Dmn/DOO+ 1 as 121 + w
assuming D,,/D,,
P-
= 1 then eqn. (27) yields the asymptotic formula aA-
aV(a;l+u-
1)
for large 111 and larg 11 <$t
In particular a212P+
1 as Il(+ar,
(34)
For very small values of lil eqn. (16) indicates that the function IV, due to b > 0, will be regular analytic with respect to 1’ in some neighborhood of A2= 0. This means that P approaches a limit value as A+O. To find the limit value either asymp-
totic formulas for the behavior of Bessel functions for small arguments are used or Wis
STATIC BOW OF SHAFTS
DUE TO RUBBING
23
determined directly by setting 1=0 in eqn. (16). The latter procedure is simple and leads to : W=p+!
(35)
r
The coefficients 7,s are determined by the boundary conditions for r = a&. One finds a++
ji’
l)W kN
a%‘(u-1)W kN
;6=
(36)
with N=a2(u+l)(v+1)-b2(~-l)(u-1)=(~2-b2)(~u+1)+(u2+b2)(~+~)
(37)
Since [’ m2dr
= $(a”-b2)[y(u2+b2)
.f b
+28]
(u2-P)U+u2+3b2
(39) p(“)=p =~(u2-b2)(.u+l)+(u2+b’)(u+u For a thick walled cylinder and if u % 1, u $1 the approximation
:
u2-b2 p =
4uu2
may be used. The rational function P of 1 which has the same behavior as is for A+0 and for A-+ co and which approximates P for positive 1 within an error bound of say 1% should not present great difficulties to retransform it into a function of time. Since II is in proportion to pi it would be preferable (if at all possible) to make P”a rational function of A2 i.e. of p itself. The latter seems promising since the behavior of P for small and large [A( points to A2: (a2J2+Pw2) p
=
=---
Pl
U2A26
(u2A2+&)(u2A2+b;2)
-~ 1
P2
U2A26
2
(41)
where Pl=
(Pd2
-
h2_dl
Wl
;
P2 =
(PSI- 1P2 d2_d1
; P1-P2=]
This approximation has the desired asymptotic behavior. It depends on the two parameters 6,, d2, which might be determined by collocation, to name at least one procedure. Collocation leads to two linear equations for 6,& and 6, +6,. From these quantities 6i, 6, can be derived as roots of the quadratic equation : 62-q61+62)+6162=o An example of approximation
by collocation is shown in Fig. 2. It is apparent that good agreement between P and P has been obtained. Further improvement could be obtained by optimizing the selection of the points of collocation. For the purpose of this study the accuracy of Fig. 2 appears to be good enough.
24
A. D. DIMAROGONAS
Fig. 2. Approximation
of the Laplace transform.
The inverse transform of P is readily obtained by first substituting 1’ =P/K
“‘PI P ~---_-----
K’PZ p+Jc’&
p+lc’d,
’
K’ =
,“z -
(43)
The fractions in (43) have well known inverse-transforms P(t)x”‘~~
e-K’a1t_~‘~ze-“‘62*
Relation (29) takes now the form of the approximation m=rlq;
and, (44) :
[K’PI exp[-K’81(t-s)]--‘Pzexp[-IC’8Z(t-s)]]q(s)ds
(45)
By differentiation it can be shown that (45) is equivalent with : c!ZB=Aq d2 d .z = dt2 ,+ Jc’(C51+6,)3 + K’8*6,
(46)
supplemented by the initial conditions B(O)=0
(47)
S(0) = rpdLq(O) In words : An ordinary differential equation of order 2 with constant coefficients can be used to compute B from q. In the general case of time varying heat input, eqn. (46) is nonlinear. To obtain numerical results a suitable integration method in a digital computer should be used. In this work the 4th order Runge-Kutta has been employed. For the case of constant heat input eqn. (46) becomes equivalent to a linear one and,
B+K’(~, +62)8+~‘281c5ZB = r/Ld26,b2pq.
(48)
25
STATIC BOW OF SHAFTS DUE TO RUBBING
The general solution of this equation is : B(t)=A,
exp(-rc’6,t)
+A,exp(-~‘6,t)+f&pq.
(49)
A, and A, are to be determined from the initial conditions (47). Finally :
A2= v%’
6
(51)
_*
2
1
The heat input function q is a Fourier term of order 1 of the total heat input according to eqn. (12). Equation (49) gives therefore the response to a constant heat input. The constants 6, and b2 which appear in this equation can be calculated by the method discussed. They have been plotted in Figs. 3-6 for several combinations of diameters and heat transfer coefficients. The heat transfer coefficients are given as Biot Numbers, defined as Bi = ah/k. We note that after a long time, the bow will assume the steady state value: BzqLpq.
(52)
This asymptotic value of the bow can be calculated without the values of 6, and 6, and it can be used to check the accuracy of the calculations for the transient bow.
I I
, 2
Fig. 3. Parameter 6,.
3
4
5
6
c4/6
26
A. D. DIMAROGONAS
Pi), It.03
1000.
3.60
3.15 1.70 1.25
500
LW I.15
.
.90 .45
‘.
4 2
3
Fig. 4. Parameter
5
6
4
5
s/b
6,.
I I
Fig. 5. Parameter
2
6,.
3
6
a/ b
Two values of the heat transfer coefficient for the inner surface were selected, 1.2 and 4.2, corresponding to a non-flow and high cooling cases respectively. We note that for high a/b ratios, the results are not much affected by the internal heat convection. For thin cylinders (u/b < 1.5) hand calculations based on the curves of Figs. 3-6 will not be accurate, therefore use of a computer is suggested. This method is applicable for relatively long cylinders (L/D > 3) because of the limitation introduced with Goodier’s formula and with the measure of the bow. However, in most practical applications the thermal bow for short cylinders is negligible. CONCLUSION
The problem of shaft bow due to an asymmetric distribution of the friction
27
STATIC BOW OF SHAFTS DUE TO RUBBING
I d
3
4
5
6
O/b
Fig. 6. Parameter 6,.
heat was investigated. Non uniform and time dependent heat input was assumed as well as uniform heat transfer coefficients over the surface. Solution of the transient form of the heat conduction equation yielded a nonlinear differential equation for the static bow due to a time varying heat input. This equation is reduced to a linear one for the case of heat input constant. The frictional heat input can be of a very general form and will depend on the particular geometry and the other properties at the sliding contact. All these properties have been reduced to a dimensionless quantity which characterizes the severity of the bow and can be used for similarity purposes. The solution has been obtained in closed form and in terms of two quantities which have been tabulated for a wide range of heat transfer coeffkients, thus facilitating hand computations. A computer program was also written for cases not covered by the tabulation and is available from the author on request. REFERENCES 1 A. D. Dimarogonas. An analytic study of the packing rub effect in rotating machinery, Dissertation, Rensselaer Polytechnic Inst., Troy, N.Y., 1970. 2 A. D. Dimarogonas and G. N. Sandor. Packing rub effect in rotating machinery. I. A state of the art review, Wear, 14 (1969) 153-170. 3 M. Sokolowski. The axially symmetric thermoelastic problem of the infinite cylinder, Arch. Mech. Stosowanej, 10 (6) (1958) 811. 4 H. Parkus. Instationiire Wiirmespnnnungen, Springer, Vienna, 1959. 5 C. K. Youngdahl and E. Stemberg. Transient thermal stresses in circular disks and cylinders, Brown Univ. Tech. Rept., 8, 1960. 6 G. Horvay. Transient thermal stress in circular disks and cylinders, Trans. ASME, 76 (1954) 7 G. N. Goodier. On the integration of the thermoelastic equations. Phil. Msg., 7 (1937) 1017.
127.