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Paper XIV (i) A Comprehensive Analysis for Contact Geometry, Kinematics, Lubrication Performance, Bulk and Flash Temperatures in Helical Gears
383
Paper XIV (i)
A Comprehensive Analysis for Contact Geometry, Kinematics, Lubrication Performance, Bulk and Flash Temperatures in Helical Gears D. Zhu and H.S. Cheng
For several decades helical gears have been known as one of the most commonly used mechanical components in automobiles and other machines. In this paper a comprehensive numerical analysis for helical gears is presented and the corresponding computer program described. This program can be used in design or research work to calculate contact geometry and kinematics, and to predict lubricant film thickness, friction, bulk and flash temperatures as functions of gear angular location based on a static tooth load estimation. It can also be combined with a dynamic load program to include dynamic effect on the lubrication performance and the contact surface temperature increase. 1. INTRODUCTION Helical gears have long been known as one of the most commonly used mechanical components in transmitting power. However,the basic mechanisms which govern the major failure modes, especially those of lubrication, are quite complicated and not fully understood. In industrial applications long life is often attained by overdesign at the sacrifice of cost, material and compactness. Any improvements in design criteria against surface pitting, scuffing and other excessive wear must depend on a more thorough understanding of the contact pattern, kinematics, lubrication and surface temperature in the gear tooth contacts. There is a continuous need for improved methods in calculating lubrication performance and temperature increase in helical gears. Although early papers concerning the spur gear lubrication (1-3) were presented long ago, the successful estimaticn of the film thickness did not appear until Dowson and Higginson ( 4 ) developed their isothermal EHL theory. Their formula remains to date as a good method for predicting film thickness at pitch point, provided the bulk surface temperatures of the spur gear teeth are known beforehand. Later, Gu ( 5 ) extended Dowson and Higginson's approach to determine the film thickness variation along the entire line of action. However, his analysis still assumed that the surface temperatures are known. Since the lubrication failure is critically influenced by the total surface temperatures in the tooth contacts, it is necessary to combine temperature analysis with the film thickness and friction calculation. The flash temperature rise in a sliding Hertzian contact was extensively investigated in the past by Blok et a1 ( 6 - 9 ) based on simplified heat conduction analyses for a moving heat source over a semi-infinite body. Using a solution of this type, Wang and Cheng (LO) contributed the first system analysis for spur gears, in which the flash temperature was summed together with the bulk temperature calculated by finite element method, and the squeeze effect on the film thickness and a dynamic load analysis were also included. Since the film thickness, friction, heat generation in the film
and the flash and bulk temperatures are all mutually dependent, an iterative scheme was used in their work. More recently Chao (11) presented a similar analysis for spiral bevel gears, in which the squeeze effect was neglected due to its relative insignificancy, but the contact geometry was more complicated than spur gears. In this paper a comprehensive analysis for helical gears is presented and the corresponding computer program described. This program can be used in design or research work to calculate contact geometry and kinematics, and to predict film thickness, friction and both bulk and flash temperature increases on the tooth contact surfaces based on either a static load estimation or a given dynamic load variation. NOTATION central distance of pinion and gear specific heat Young's modulus = 2[ (l-~,~)/E~+(l-v,~)/E~]-~, effective elastic modulus tooth load coefficient of friction = aE', dimensionless material parameter limiting elastic shear modulus lubricant film thickness central film thickness minimum film thickness conductivity conductivity of lubricant length of contact line maximum Hertzian contact pressure heat flux time averaged heat flux =plpz/(pl+pz),effective radius of curvature tip radius radius of base cylinder T, +Tb +T,, local surface temperature ambient temperature bulk temperature increase flash temerature increase T i , J temperature influence coefficient U rolling velocity u surface velocity 0 r),U/(E'R), dimensionless speed parameter V sliding velocity
-
-
384
- F/(E'Rl,), dimensionless load parameter x,y,z coordinates
Y
0
2 a at 3/,
+ r)
P
A Y
p 7
;"
P ', o
number of teeth pressure-viscosity exponent transverse pitch angle base helix angle shear strain rate viscosity density heat partition ratio Poisson's ratio radius of curvature shear stress limiting shear stress location angle of contact line, see Fig.2 thermal reduction factor of film thickness angular velocity
l i x Involute Surface 1 6 2
Y
Subscripts: 1 2
refers to pinion or its surface refers to gear or its surface
Fig. 1. Contact Line on the Plane of Action
.-.
2. GEOMETRY AND KINEMATICS In order to develop the analysis of lubrication and temperature, it is needed beforehand to determine tooth surfaces of both pinion and gear mathematically and to calculate the variations of radius of curvature, rolling and sliding velocities at each location of engagement of a pair of teeth and along each contact line. The tooth profile on transverse plane consists of four curves: addendum circle, involute, dedendum fillet curve (equidistant line of trochoid), and dedendum circle. These curves can be determined and then helix surfaces can be generated according to engagement theory(l2),if machine settings and cutter specifications in gear making process are given. Fig.1 shows a pair of teeth being in mesh. The contact line, MM', is a straight line moving only on the plane of action, NiNZNlNi. The line of action on transverse plane is better shown in Fig.2, and the starting and ending points of engagement of a pair of teeth can be given by Fig.2. The Line of Action on Transverse Plane
where A+ is the angle difference between two sides of a tooth. A diagrammatic sketch of contact lines on a tooth contact surface is given in Fig.3. If the gear is a driver, the contact line moves from the lower corner to the upper during the engagement. If the gear is a follower, it moves in the opposite direction. Note that different contact lines correspond to the different locations of the engagement of a pair of teeth. Equations of helix involute surface can be written as
- rbcos B +- 66 cos Ahsin 8 cos Abcos
1 y - r,sin J x z
B
- pB - 6 sin
B
A,
where 0 , r, , A,, , p and 6 are shown in Fig.4. Equation of contact line: e - x + a t
- 4
where is the angle specifying the location of the contact line, as defined in Fig.2. The principal radius of curvature in the
Fig. 3. Contact Lines on the Tooth Surface
direction perpendicular be obtained from
to the contact line can
- r sin sin where r - (x + y2)Oa5, cos (r,/r). The effective radius of curvature is defined as ar/
p
2
R -
A,
at-
-1
P1 p2
P1 + P2 The entraining and sliding velocities for the calculation of lubrication performance and
385
flash temperatures can te given by U V
, - 0.5 wlcos pb[rb,4 + zZ, (a sin u t - rb14)]
- wlcos pb[rb14 - z, (a sin 1'
p
u t - rbld)l
--rbn / 2tan
)Lb Pb
3. FILM THICKNESS The following formulae contributed by Dowson, Higginson and Toyoda (13-14) are used in this analysis to predict the minimum and central film thicknesses :
4 hc
-
2.65
GO.7
3.09 ~
G
0.54
fi-0'13R
0 . 6 90 . 5 6 p-0
.I
R
where fJ, l?and , G are dimensionless speed, load and material parameters'respective1y.To consider the effect of thermal action on film thickness, results from an inlet zone analysis by Cheng (15) are employed. The thermal reduction factor of film thickness can be,expressedas OT
- fl(1
where S is the f , and f, can squeeze effect film thickness f icancy . 4.
-
0.1 S ) ( 1
- f3Pm/E')
slide-to-rollratio, coefficients be found in (15). Note that the is neglected in calculating the because of its relative insigni-
FiE. 4. Determination of Helix Involute Surface
The lubricant local velocity profile and temperature profile for the sliding cases were estimated by Plint (18) and later confirmed by Trachman and Cheng (17). According to their analyses one can determine the heat partition coefficient approximately and obtain
FRICTION
Bair and Winer's non-Newtonian fluid model (16) is employed here to calculate shear stress and traction in tooth contacts. The relation between shear strain rate and shear stress is
The surface temperatures can be given by using the solution of one dimensional transient heat conduction analysis for a semi-infinite plane under an arbitrarily distributed, fast-moving heat source (19). Thus one can have
where G, is the limiting shear elastic modulus, and T~ is the limiting shear stress. Assuming -iA v/h, 8 one can have
T i * j v T i - I , j 7L )---0 - -ln(lA x V TL h, This is a nonlinear equation in T~ , j , if r i -,, is given. Thus one can solve it using bisection method and obtain the local shear stress, i i , j , at each node. Friction can be readily calculated by integrating the shear stress over the whole EHL contact area.
-
+ 0.5 q(X))
(<
+
(e -
!L.T i , j G,
5.
FLASH TEMPERATURE INCREASE
The calculation of flash temperature is based on the theory of a fast moving heat source over a semi-infinite solid. According to Archard (8) the heat flow in the direction perpendicular to the velocity of the hert source is negligible. Heat is generated by viscous shearing in the film. This heat is carried away either by the lubricant through convection or by the two solids through conduction. Trachman and Cheng (17) investigated the relative importance between them and concluded that the effect of convection can be neglected, except at very high speeds which exceed those of machine elements i n current practice. So the heat generated can be divided into two parts by the ratio A/(l-A), based on Frances' analysis (9), where A is heat partition coefficient: q'7Vs
qi-qh,
qz-q(l-A)
0.5 q(X))
d X J-a
I'
d X
The above equations are recognized as Volterra's integral equation of the second kind. These can be integrated as described in (19). Since the surface temperature and the heat generation in the film are mutually dependent, an iterative procedure has to be adopted similarly to that explained by Cheng in (19). 6.
BULK TEMPERATURE
The calculation of helical gear bulk temperature distribution in the teeth and the rim normally would require a full scale three dimensional finite element analysis. This approach usually entails lengthy computations. As a compromise, a simplified method was recently developed by Zhou and Cheng (20). It combines a 2-D axisymmetric FFM analysis for gear wheel, hub and shaft and a 3-D analysis for gear teeth, and can yield an accurate prediction of bulk temperature without large amount of computation. A model of this simplified approach is shown in Fig.5, and the basic steps include: a) Determine helical gear tooth surfaces geometrically and generate finite element mesh. b) Calculate so-called temperature influ-
386
Three Dlmenslonal FEU Uodel for Gear Teeth
I
&o Dlmenslonal A x l s m a t r l c FEU U o d d for Wheel - m d Shaft
Fig. 5. FEM Models for the Calculation of Bulk Temperature
j on the tooth contact surface.
It is important to note that for a pair of gears meshing at high speeds, the main cooling is usually provided by a mixture of oil mist and air surrounding the entire gear surface. The whole system is initially at a given ambient temperature. It is gradually heated by sliding friction between gear teeth, until it reaches a steady state distribution after many cycles of rotation. The analysis of transient temperature history in gears involves a complicated 3-D time dependent heat conduction problem. Since bne is primarily concerned with the equilibrium temperature distribution, the transient temperature history before the state of equilibrium is of no interest to the present work. At each revolution the tooth is subject to the same heating flux, and the time period of each contact position is only a small fraction of the entire period of revolution, so the local temperature jump, as shown in Fig.6, decays rapidly before it enters the contact zone at the next revolution. Thus, it is justified to use an average heat input over the revolution tc,calculate the steadystate bulk temperature rise of the body. A detailed description of solution to this problem is given in (20). 7.
(a) Bulk Temperature Tb (Steady)
Dynamic load analysis requires a full scale 3-D finite element method to calculate the combined tooth stiffness and deflection, so it is quite complicated and very time consuming. However, since the dynamic load is practically unaffected by other quantities, such as friction, lubricant film thickness and temperature, it can be solved independently. It is found from the results and discussions in (10) and (11) that usually the dynamic load differs not far away from static tooth load, and the lubrication performances are relatively insensitive to the load. Therefore, it is believed that a ststic load estimation can satisfy most engineering applications, if the major interests are in the lubrication and the temperature. In this program it is allowed at user’s option to use either a static tooth load estimated in the program, or a dynamic tooth load variation obtained from a separate program of dynamic analysis when needed. It was decided not to include the dynamic analysis in the present work, so that the program can run on most personal computers. 8.
(b) Flash Temperature T, (Transient)’ Fig. 6.. Characteristics of Bulk and Flash Temperatures on the Tooth Contact Surfaces
ence coefficients with FEM.Influence coefficient Ti j is defined as the temperature increase at node j due to a unit heat input applied at node i. It can be obtained by superimposing the above menttioned 3-D and 2-D axisynunetric FEM analyses after heat flux between the tooth and the wheel due to the unit heat load at node i is given. c) Predict bulk temperature distribution on the tooth contact surfaces based on the influence coefficients obtained and the heat flux generated in the tooth contacts. The bulk temperature rise at node i, Ti,can be given by
where tij is the time averaged heat load at node
DYNAMIC LOAD
COMPUTATIONAL PROCEDURE
It was shown in the previous chapters that the problem of helical gear lubrication consists essentially of the solution of the following quantities during the engagement: 1. Film thickness; 2 . Friction; 3. Flash temperature ; 4. Bulk temperature. Since these quantities are mutually dependent, an iterative numerical scheme is needed. The overall computational procedure can be divided into the following steps: Step 1. Input data, then determine tooth surfaces and generate FEM meshes for both pinion and gear; Step 2. Calculate influence coefficients with FEM, then initialize bulk temperature; Step 3 . For each time step corresponding to each location of engagement of a pair of teeth, calculate the radius of curvature, rolling and
387 sliding velocities, maximum contact stress, film thickness, friction, heat generation and flash temperature increase along the contact line based on given bulk temperature and tooth load; Step 4. Calculate bulk temperature increase based on the time averaged heat generation; Step 5 . Check whether the bulk temperature is convergent or not. If it is not convergent, go back to Step 3; Step 6 . Print the results of computation. By following the above steps the computer program was developed, and several trial runs were then successfully carried out. It was seen that in most cases the converged solution can be achieved within 5 8 iterations, which usually take 40 60 minutes on IBM PC/XT computer.
-
1°h
-
.
4b
5b
' I 0 ' 20 ' 30 ' POSITION IN DIRECTION OF M O 7 W W I D T H (MU)
Fig. 7. Effective Radius of Curvature Along Each Contact Line
9. SAMPLE RESULTS AND DISCUSSION
Program HGEAR described above is applicable for both standard and non-standard, external and internal gears with a wide range of geometric, material and operating parameters. The unit system used for input and output can be either metric or English (or even mixture). Some typical results for a pair of non-standard external gears are shown in Fig.7 Fig.13, and the main input data are listed below:
-
39 Number of teeth of pinion: 86 Number of teeth of gear: Diametral pitch: 7.16667 l/in. Normal pressure angle: 22.5 degree Helix angle: 12.383 degree Facewidth:' 53.85 mm (2.12 in.) 0.0 Profile-shift coeff. for pinion: Profile-shift coeff. for gear: 0.0 Hobbing cutters used to make both pinion and gear. Material of pinion and gear: steel Viscosity of lubricant: 0.0454 Pa*s Conductivity of lubricant: 0.145 w/(m*Deg.C) Pressure-viscosity exponent: 22.75 1/GPa 5000.0 N*m Torque input: Angular speed of pinion: 5000.0 rPm 25.0 Deg.C Ambient temperature : Convection coefficient for oil-air mist cooling: 0.15 N/(cm*Deg.C) Convection coefficient for air cooling: 0.0397 N/(cm*Deg.C) Fig.7 shows the variation of effective radius of curvature along each contact line on the tooth contact surface (also see Fig.3). Note that any one of these contact lines corresponds to a certain position of engagement of a pair of teeth. At the beginning or the end of engagement the contact line is very short and in the middle o f engagement it is longer stretching from the root to the tip of the tooth. It is seen that in the beginning of engagement the effective radius o f curyature is relatively small, and at the end of engagement it becomes larger. Along each contact line the radius of curvature increases in the direction from the root to the tip of the pinion tooth. Variations of rolling and sliding velocities along each contact line are shown in Fig.8. It is observed that rolling velocity is always positive and do not change dramatically, but sliding is negative in the beginning of the engagement and positive at the end. Along any contact line sliding velocity increases linearly from the root to the tip of the pinion tooth. Above the pitch line sliding is always positive and below the pitch line it is negative. There is no sliding at the pitch line.
Fig. 8. Sliding and Rolling Velocity Along Each Contact Line 1.08
0.80
0.60
0.40
0.20
0.0% <-
Tooth Width
->
Fig. 9. Bulk Temperature Increase8 on Tooth Contact Surface of Pinion
Based on the above geometric and kinematic results the film thickness, friction, and bulk and flash temperatures are calculated, and some results are plotted in Fig.9-13. Fig.9 is a contour plot of equilibrium bulk temperature increase on the tooth contact surface of pinion. Fig.10 shows some typical results of flash and bulk temperature increases along a contact line. Since the pinion is smaller,the bulk temperature of pinion is usually higher. The flash temperature is dominated by the sliding velocity, so it becomes zero at the crossing point of the pitch line and the contact line. Fig.11-13 give the variations of rolling and sliding velocities, maximum Hertzian contact pressure, film thickness, friction, and bulk and flash temperature increases along the line of action on transverse plane. At pitch point P shown in Fig.2, there is no sliding between two surfaces, so that the flash temperatures and the friction are zero. However, the sliding does not affect the bulk temperatures and film thickness significantly, as shown in Fig.12 and 13. Effects of torque input on the lubrication and the bulk and flash temperatures are shown in
388
0-1 0.0
Fig. 10. Bulk and Flash Temperatures on the Tooth SUKfaCOS Along a Contact Line
,
-nl
-
1
h
1
(1.00 0.~2 o.h 0.b 0.~8 1 .O RELATIVE CONTACT POSITION Fig. 11. Friction and Rolling and Sliding Velocities Along the Line of Action on Transverse Plane n.'o
9
Fig. 13. Bulk and Flash TempeKatUKO Variations Along the Line of Action on Transverse Plane
r O . 10
161
I
0.'2 0.k 0.'6 0.8 1 .O RELATIVE CONTACT POSITION
0.0
.
-1
100
TORQVdol%%UT ( N * m )
-
f
/
g-
f
-0.8
I
Fig. 14(a). Effect of Torque Input on Film Thickness and Coefficient of Friction
-1.2 -1.0
. I 0.00 10000
i
g V
LI]
-0.6
5 0.4
-
-I
-0.4
-0.2
I
I
M
5
3
2cr,
0.0 0.'2 0.2 0.'6 0.8 1.0 RELATIVE CONTACT POSITION Fig. 12. Maximum Hertzian Pressure and Film Thickness Along the Line of Action on Transverse Plane
&0.0 0.0
I
Fig.14. It is seen that as the load increases, both the bulk and flash temperatures increase but the film thickness decreases. The friction coefficient increases first, then decreases a little after reaching its maximum value. The effects of pinion angular speed are illustrated in Fig.15. It can be observed that as the speed increases, both the bulk and flash temperatures increase but the friction decreases due to the reduction of lubricant viscosity. The minimum film thickness increases because of entraining effect. At very high speeds, however, the film thickness decreases again due to thermal effect in the inlet zone of the contact.
10. CONCLUSIONS A comprehensive numerical analysis for contact geometry, kinematics, lubricant film thickness, friction, and bulk and flash temperatures in helical gears is developed. The program can be
100
TORQUEf0I%%UT
( N * m ) 10000
Fig. 14(b). Effect of Torque Input on Bulk and Flash Temperature Increases
used on most personal computers for both standard and non-standard, external and internal gears, and the input and output can be in either IS0 or English system. In most cases the converged solution can be achieved within 5-8 iterations, which may take 40-60 minutes on IBM PC/XT. Sample results show that this program is useful in helical gear design and research. REFERENCES "Lubrication of Gear Teeth, Engineering, V01.102, 1916, p.109. McEween, E., "The Effect of Variation of Viscsity with Pressure on the Load Carrying Capacity of Oil Films Between Gear Teeth," J. of Inst. Pet., Vo1.38, 1952, p.646. Daring, D . W . , and Radzimovsky, E.I., "Lubricating Film Thickness and Load Capacity of Spur Gears: Analytical Investigation," 'I
389
-
a0.6
::
r0.10
7
.d
50.5 tn
v)
0.4
Y
3
0.3
r= 0.2
E
20.1 J:
0.0 I 100
o
ANGULAR &%D
Fig. 15(a).
.
0 1000
OF PINION
0
frpm)
Effect of Angular Speed of Pinion on Film Thickness and Friction
ASME Paper No.63-WA-85,1963. Dowson, D., and Higginson, G.R., "Elastohydrodynamic Lubrication - - - - - - - The Fundamentals of Roller and Gear Lubrication," Pergamon Press, London, 1966. Gu, A., "Elastohydrodynamic Lubrication of Involute Gears," ASME Paper No.72-PTG-34. Blok, H., "Theoretical Study of Temperature Rise at Surfaces of Actual Contact Under Oiliness Lubricating Conditions," Proc. of the General Discussion on Lubrication and Lubricants, Oct., 1937, Inst. of Mech. Eng. (London), V01.2, pp 222-235. Jaeger, J.C., "Moving Surfaces of Heat and the Temperature at Sliding Contacts," J. and Proc. of the Royal SOC. of New South Wales, Vol.wNI(76), 1942, pp 203-224. Archard, J.F., "The Temperature of Rubbing Surfaces," Wear, Vo1.2,1958-59,pp 438-455. Frances, H. A., "Interfacial Temperature Distribution Within a Sliding Hertzian Contact." ASLE Trans., Vo1.14, 1971, p.41. (10) Wang, K ;L. , and Cheng] H.S . , . "A Nde>ical Solution to the Dynamic Load, Film Thickness, and Surface Temperatures in Spur Gears, Part I: Analysis," J. of Machine Design, Vo1.103, 1981, p.177. (11) Chao, H.C., "A Computer Solution for the Dynamic Load, Lubricant Film Thickness and Surface Temperatures in Spiral Bevel Gears," Ph.D. Thesis, 1982, Northwestern University, Evanston, Illinois.
100 ANGULAR SPEED 1000 OF plNtoONoPrpm)
Fig. 15(b). Effect of Angular Speed of Pinion on Bulk and Flash Temperatures
12) Litvin, F.L., "Theory of Gearing," AVSCOM Technical Report 88-C-035,1989, NASA Reference Publication 1212. 13) Dowson, D., "Elastohydrodynamics,"Proc. of Inst.Mech.Eng., Vo1.182,3A, 1967-68,p.151. 14) Dowson, D., and Toyoda, S., "A Central Film Thickness Formula for Elastohydrodynamic Line Contact," Proc. of the 5th Leeds-Lyon Symposium on Tribology, 1978, p.60. (15) McGrew,J.M.,Gu,A., Cheng,H.S., and Murray, S.F., "Elasto-hydrodynamicLubrication - - - Preliminary Design Manua1,"Technical Report AFAPL-TR-70-27,Aero Propultion Laboratory, Air Force Systems Command, Wright-Pattenson Air Force Base, Ohio, Nov., 1970. (16) Bair,S. and Winer,W.O.,"A Rheological Model for Elastohydrodynamic Contacts Based on Primary Laboratory Data," J. of Lub. Tech., V01.101, 1979, pp 258-265. (17) Trachman,E.G.,and Cheng, H.S., "Reological Effects on Friction in Elastohydrodynamic Lubrication," NASA CR-2206, 1973. (18) Plint,M.A.,"Traction in Elastohydrodynamic Contacts," Proc. of Inst. Mech. Eng., Vol. 182, Part 1, 1967, p.300. (19) Cheng, H.S., "Calculation of Elastohydrodynamic Film Thickness in High Speed Rolling and Sliding Contacts," MTI Report 67TR24. (20) Zhou, R.S., and Cheng, H.S., "A Simplified Solution for Gear Bulk Temperature Matrix," to be submitted for publication.
Paper XIV (i)
A Comprehensive Analysis for Contact Geometry, Kinematics, Lubrication Performance, Bulk and Flash Temperatures in Helical Gears D. Zhu and H.S. Cheng
For several decades helical gears have been known as one of the most commonly used mechanical components in automobiles and other machines. In this paper a comprehensive numerical analysis for helical gears is presented and the corresponding computer program described. This program can be used in design or research work to calculate contact geometry and kinematics, and to predict lubricant film thickness, friction, bulk and flash temperatures as functions of gear angular location based on a static tooth load estimation. It can also be combined with a dynamic load program to include dynamic effect on the lubrication performance and the contact surface temperature increase. 1. INTRODUCTION Helical gears have long been known as one of the most commonly used mechanical components in transmitting power. However,the basic mechanisms which govern the major failure modes, especially those of lubrication, are quite complicated and not fully understood. In industrial applications long life is often attained by overdesign at the sacrifice of cost, material and compactness. Any improvements in design criteria against surface pitting, scuffing and other excessive wear must depend on a more thorough understanding of the contact pattern, kinematics, lubrication and surface temperature in the gear tooth contacts. There is a continuous need for improved methods in calculating lubrication performance and temperature increase in helical gears. Although early papers concerning the spur gear lubrication (1-3) were presented long ago, the successful estimaticn of the film thickness did not appear until Dowson and Higginson ( 4 ) developed their isothermal EHL theory. Their formula remains to date as a good method for predicting film thickness at pitch point, provided the bulk surface temperatures of the spur gear teeth are known beforehand. Later, Gu ( 5 ) extended Dowson and Higginson's approach to determine the film thickness variation along the entire line of action. However, his analysis still assumed that the surface temperatures are known. Since the lubrication failure is critically influenced by the total surface temperatures in the tooth contacts, it is necessary to combine temperature analysis with the film thickness and friction calculation. The flash temperature rise in a sliding Hertzian contact was extensively investigated in the past by Blok et a1 ( 6 - 9 ) based on simplified heat conduction analyses for a moving heat source over a semi-infinite body. Using a solution of this type, Wang and Cheng (LO) contributed the first system analysis for spur gears, in which the flash temperature was summed together with the bulk temperature calculated by finite element method, and the squeeze effect on the film thickness and a dynamic load analysis were also included. Since the film thickness, friction, heat generation in the film
and the flash and bulk temperatures are all mutually dependent, an iterative scheme was used in their work. More recently Chao (11) presented a similar analysis for spiral bevel gears, in which the squeeze effect was neglected due to its relative insignificancy, but the contact geometry was more complicated than spur gears. In this paper a comprehensive analysis for helical gears is presented and the corresponding computer program described. This program can be used in design or research work to calculate contact geometry and kinematics, and to predict film thickness, friction and both bulk and flash temperature increases on the tooth contact surfaces based on either a static load estimation or a given dynamic load variation. NOTATION central distance of pinion and gear specific heat Young's modulus = 2[ (l-~,~)/E~+(l-v,~)/E~]-~, effective elastic modulus tooth load coefficient of friction = aE', dimensionless material parameter limiting elastic shear modulus lubricant film thickness central film thickness minimum film thickness conductivity conductivity of lubricant length of contact line maximum Hertzian contact pressure heat flux time averaged heat flux =plpz/(pl+pz),effective radius of curvature tip radius radius of base cylinder T, +Tb +T,, local surface temperature ambient temperature bulk temperature increase flash temerature increase T i , J temperature influence coefficient U rolling velocity u surface velocity 0 r),U/(E'R), dimensionless speed parameter V sliding velocity
-
-
384
- F/(E'Rl,), dimensionless load parameter x,y,z coordinates
Y
0
2 a at 3/,
+ r)
P
A Y
p 7
;"
P ', o
number of teeth pressure-viscosity exponent transverse pitch angle base helix angle shear strain rate viscosity density heat partition ratio Poisson's ratio radius of curvature shear stress limiting shear stress location angle of contact line, see Fig.2 thermal reduction factor of film thickness angular velocity
l i x Involute Surface 1 6 2
Y
Subscripts: 1 2
refers to pinion or its surface refers to gear or its surface
Fig. 1. Contact Line on the Plane of Action
.-.
2. GEOMETRY AND KINEMATICS In order to develop the analysis of lubrication and temperature, it is needed beforehand to determine tooth surfaces of both pinion and gear mathematically and to calculate the variations of radius of curvature, rolling and sliding velocities at each location of engagement of a pair of teeth and along each contact line. The tooth profile on transverse plane consists of four curves: addendum circle, involute, dedendum fillet curve (equidistant line of trochoid), and dedendum circle. These curves can be determined and then helix surfaces can be generated according to engagement theory(l2),if machine settings and cutter specifications in gear making process are given. Fig.1 shows a pair of teeth being in mesh. The contact line, MM', is a straight line moving only on the plane of action, NiNZNlNi. The line of action on transverse plane is better shown in Fig.2, and the starting and ending points of engagement of a pair of teeth can be given by Fig.2. The Line of Action on Transverse Plane
where A+ is the angle difference between two sides of a tooth. A diagrammatic sketch of contact lines on a tooth contact surface is given in Fig.3. If the gear is a driver, the contact line moves from the lower corner to the upper during the engagement. If the gear is a follower, it moves in the opposite direction. Note that different contact lines correspond to the different locations of the engagement of a pair of teeth. Equations of helix involute surface can be written as
- rbcos B +- 66 cos Ahsin 8 cos Abcos
1 y - r,sin J x z
B
- pB - 6 sin
B
A,
where 0 , r, , A,, , p and 6 are shown in Fig.4. Equation of contact line: e - x + a t
- 4
where is the angle specifying the location of the contact line, as defined in Fig.2. The principal radius of curvature in the
Fig. 3. Contact Lines on the Tooth Surface
direction perpendicular be obtained from
to the contact line can
- r sin sin where r - (x + y2)Oa5, cos (r,/r). The effective radius of curvature is defined as ar/
p
2
R -
A,
at-
-1
P1 p2
P1 + P2 The entraining and sliding velocities for the calculation of lubrication performance and
385
flash temperatures can te given by U V
, - 0.5 wlcos pb[rb,4 + zZ, (a sin u t - rb14)]
- wlcos pb[rb14 - z, (a sin 1'
p
u t - rbld)l
--rbn / 2tan
)Lb Pb
3. FILM THICKNESS The following formulae contributed by Dowson, Higginson and Toyoda (13-14) are used in this analysis to predict the minimum and central film thicknesses :
4 hc
-
2.65
GO.7
3.09 ~
G
0.54
fi-0'13R
0 . 6 90 . 5 6 p-0
.I
R
where fJ, l?and , G are dimensionless speed, load and material parameters'respective1y.To consider the effect of thermal action on film thickness, results from an inlet zone analysis by Cheng (15) are employed. The thermal reduction factor of film thickness can be,expressedas OT
- fl(1
where S is the f , and f, can squeeze effect film thickness f icancy . 4.
-
0.1 S ) ( 1
- f3Pm/E')
slide-to-rollratio, coefficients be found in (15). Note that the is neglected in calculating the because of its relative insigni-
FiE. 4. Determination of Helix Involute Surface
The lubricant local velocity profile and temperature profile for the sliding cases were estimated by Plint (18) and later confirmed by Trachman and Cheng (17). According to their analyses one can determine the heat partition coefficient approximately and obtain
FRICTION
Bair and Winer's non-Newtonian fluid model (16) is employed here to calculate shear stress and traction in tooth contacts. The relation between shear strain rate and shear stress is
The surface temperatures can be given by using the solution of one dimensional transient heat conduction analysis for a semi-infinite plane under an arbitrarily distributed, fast-moving heat source (19). Thus one can have
where G, is the limiting shear elastic modulus, and T~ is the limiting shear stress. Assuming -iA v/h, 8 one can have
T i * j v T i - I , j 7L )---0 - -ln(lA x V TL h, This is a nonlinear equation in T~ , j , if r i -,, is given. Thus one can solve it using bisection method and obtain the local shear stress, i i , j , at each node. Friction can be readily calculated by integrating the shear stress over the whole EHL contact area.
-
+ 0.5 q(X))
(<
+
(e -
!L.T i , j G,
5.
FLASH TEMPERATURE INCREASE
The calculation of flash temperature is based on the theory of a fast moving heat source over a semi-infinite solid. According to Archard (8) the heat flow in the direction perpendicular to the velocity of the hert source is negligible. Heat is generated by viscous shearing in the film. This heat is carried away either by the lubricant through convection or by the two solids through conduction. Trachman and Cheng (17) investigated the relative importance between them and concluded that the effect of convection can be neglected, except at very high speeds which exceed those of machine elements i n current practice. So the heat generated can be divided into two parts by the ratio A/(l-A), based on Frances' analysis (9), where A is heat partition coefficient: q'7Vs
qi-qh,
qz-q(l-A)
0.5 q(X))
d X J-a
I'
d X
The above equations are recognized as Volterra's integral equation of the second kind. These can be integrated as described in (19). Since the surface temperature and the heat generation in the film are mutually dependent, an iterative procedure has to be adopted similarly to that explained by Cheng in (19). 6.
BULK TEMPERATURE
The calculation of helical gear bulk temperature distribution in the teeth and the rim normally would require a full scale three dimensional finite element analysis. This approach usually entails lengthy computations. As a compromise, a simplified method was recently developed by Zhou and Cheng (20). It combines a 2-D axisymmetric FFM analysis for gear wheel, hub and shaft and a 3-D analysis for gear teeth, and can yield an accurate prediction of bulk temperature without large amount of computation. A model of this simplified approach is shown in Fig.5, and the basic steps include: a) Determine helical gear tooth surfaces geometrically and generate finite element mesh. b) Calculate so-called temperature influ-
386
Three Dlmenslonal FEU Uodel for Gear Teeth
I
&o Dlmenslonal A x l s m a t r l c FEU U o d d for Wheel - m d Shaft
Fig. 5. FEM Models for the Calculation of Bulk Temperature
j on the tooth contact surface.
It is important to note that for a pair of gears meshing at high speeds, the main cooling is usually provided by a mixture of oil mist and air surrounding the entire gear surface. The whole system is initially at a given ambient temperature. It is gradually heated by sliding friction between gear teeth, until it reaches a steady state distribution after many cycles of rotation. The analysis of transient temperature history in gears involves a complicated 3-D time dependent heat conduction problem. Since bne is primarily concerned with the equilibrium temperature distribution, the transient temperature history before the state of equilibrium is of no interest to the present work. At each revolution the tooth is subject to the same heating flux, and the time period of each contact position is only a small fraction of the entire period of revolution, so the local temperature jump, as shown in Fig.6, decays rapidly before it enters the contact zone at the next revolution. Thus, it is justified to use an average heat input over the revolution tc,calculate the steadystate bulk temperature rise of the body. A detailed description of solution to this problem is given in (20). 7.
(a) Bulk Temperature Tb (Steady)
Dynamic load analysis requires a full scale 3-D finite element method to calculate the combined tooth stiffness and deflection, so it is quite complicated and very time consuming. However, since the dynamic load is practically unaffected by other quantities, such as friction, lubricant film thickness and temperature, it can be solved independently. It is found from the results and discussions in (10) and (11) that usually the dynamic load differs not far away from static tooth load, and the lubrication performances are relatively insensitive to the load. Therefore, it is believed that a ststic load estimation can satisfy most engineering applications, if the major interests are in the lubrication and the temperature. In this program it is allowed at user’s option to use either a static tooth load estimated in the program, or a dynamic tooth load variation obtained from a separate program of dynamic analysis when needed. It was decided not to include the dynamic analysis in the present work, so that the program can run on most personal computers. 8.
(b) Flash Temperature T, (Transient)’ Fig. 6.. Characteristics of Bulk and Flash Temperatures on the Tooth Contact Surfaces
ence coefficients with FEM.Influence coefficient Ti j is defined as the temperature increase at node j due to a unit heat input applied at node i. It can be obtained by superimposing the above menttioned 3-D and 2-D axisynunetric FEM analyses after heat flux between the tooth and the wheel due to the unit heat load at node i is given. c) Predict bulk temperature distribution on the tooth contact surfaces based on the influence coefficients obtained and the heat flux generated in the tooth contacts. The bulk temperature rise at node i, Ti,can be given by
where tij is the time averaged heat load at node
DYNAMIC LOAD
COMPUTATIONAL PROCEDURE
It was shown in the previous chapters that the problem of helical gear lubrication consists essentially of the solution of the following quantities during the engagement: 1. Film thickness; 2 . Friction; 3. Flash temperature ; 4. Bulk temperature. Since these quantities are mutually dependent, an iterative numerical scheme is needed. The overall computational procedure can be divided into the following steps: Step 1. Input data, then determine tooth surfaces and generate FEM meshes for both pinion and gear; Step 2. Calculate influence coefficients with FEM, then initialize bulk temperature; Step 3 . For each time step corresponding to each location of engagement of a pair of teeth, calculate the radius of curvature, rolling and
387 sliding velocities, maximum contact stress, film thickness, friction, heat generation and flash temperature increase along the contact line based on given bulk temperature and tooth load; Step 4. Calculate bulk temperature increase based on the time averaged heat generation; Step 5 . Check whether the bulk temperature is convergent or not. If it is not convergent, go back to Step 3; Step 6 . Print the results of computation. By following the above steps the computer program was developed, and several trial runs were then successfully carried out. It was seen that in most cases the converged solution can be achieved within 5 8 iterations, which usually take 40 60 minutes on IBM PC/XT computer.
-
1°h
-
.
4b
5b
' I 0 ' 20 ' 30 ' POSITION IN DIRECTION OF M O 7 W W I D T H (MU)
Fig. 7. Effective Radius of Curvature Along Each Contact Line
9. SAMPLE RESULTS AND DISCUSSION
Program HGEAR described above is applicable for both standard and non-standard, external and internal gears with a wide range of geometric, material and operating parameters. The unit system used for input and output can be either metric or English (or even mixture). Some typical results for a pair of non-standard external gears are shown in Fig.7 Fig.13, and the main input data are listed below:
-
39 Number of teeth of pinion: 86 Number of teeth of gear: Diametral pitch: 7.16667 l/in. Normal pressure angle: 22.5 degree Helix angle: 12.383 degree Facewidth:' 53.85 mm (2.12 in.) 0.0 Profile-shift coeff. for pinion: Profile-shift coeff. for gear: 0.0 Hobbing cutters used to make both pinion and gear. Material of pinion and gear: steel Viscosity of lubricant: 0.0454 Pa*s Conductivity of lubricant: 0.145 w/(m*Deg.C) Pressure-viscosity exponent: 22.75 1/GPa 5000.0 N*m Torque input: Angular speed of pinion: 5000.0 rPm 25.0 Deg.C Ambient temperature : Convection coefficient for oil-air mist cooling: 0.15 N/(cm*Deg.C) Convection coefficient for air cooling: 0.0397 N/(cm*Deg.C) Fig.7 shows the variation of effective radius of curvature along each contact line on the tooth contact surface (also see Fig.3). Note that any one of these contact lines corresponds to a certain position of engagement of a pair of teeth. At the beginning or the end of engagement the contact line is very short and in the middle o f engagement it is longer stretching from the root to the tip of the tooth. It is seen that in the beginning of engagement the effective radius o f curyature is relatively small, and at the end of engagement it becomes larger. Along each contact line the radius of curvature increases in the direction from the root to the tip of the pinion tooth. Variations of rolling and sliding velocities along each contact line are shown in Fig.8. It is observed that rolling velocity is always positive and do not change dramatically, but sliding is negative in the beginning of the engagement and positive at the end. Along any contact line sliding velocity increases linearly from the root to the tip of the pinion tooth. Above the pitch line sliding is always positive and below the pitch line it is negative. There is no sliding at the pitch line.
Fig. 8. Sliding and Rolling Velocity Along Each Contact Line 1.08
0.80
0.60
0.40
0.20
0.0% <-
Tooth Width
->
Fig. 9. Bulk Temperature Increase8 on Tooth Contact Surface of Pinion
Based on the above geometric and kinematic results the film thickness, friction, and bulk and flash temperatures are calculated, and some results are plotted in Fig.9-13. Fig.9 is a contour plot of equilibrium bulk temperature increase on the tooth contact surface of pinion. Fig.10 shows some typical results of flash and bulk temperature increases along a contact line. Since the pinion is smaller,the bulk temperature of pinion is usually higher. The flash temperature is dominated by the sliding velocity, so it becomes zero at the crossing point of the pitch line and the contact line. Fig.11-13 give the variations of rolling and sliding velocities, maximum Hertzian contact pressure, film thickness, friction, and bulk and flash temperature increases along the line of action on transverse plane. At pitch point P shown in Fig.2, there is no sliding between two surfaces, so that the flash temperatures and the friction are zero. However, the sliding does not affect the bulk temperatures and film thickness significantly, as shown in Fig.12 and 13. Effects of torque input on the lubrication and the bulk and flash temperatures are shown in
388
0-1 0.0
Fig. 10. Bulk and Flash Temperatures on the Tooth SUKfaCOS Along a Contact Line
,
-nl
-
1
h
1
(1.00 0.~2 o.h 0.b 0.~8 1 .O RELATIVE CONTACT POSITION Fig. 11. Friction and Rolling and Sliding Velocities Along the Line of Action on Transverse Plane n.'o
9
Fig. 13. Bulk and Flash TempeKatUKO Variations Along the Line of Action on Transverse Plane
r O . 10
161
I
0.'2 0.k 0.'6 0.8 1 .O RELATIVE CONTACT POSITION
0.0
.
-1
100
TORQVdol%%UT ( N * m )
-
f
/
g-
f
-0.8
I
Fig. 14(a). Effect of Torque Input on Film Thickness and Coefficient of Friction
-1.2 -1.0
. I 0.00 10000
i
g V
LI]
-0.6
5 0.4
-
-I
-0.4
-0.2
I
I
M
5
3
2cr,
0.0 0.'2 0.2 0.'6 0.8 1.0 RELATIVE CONTACT POSITION Fig. 12. Maximum Hertzian Pressure and Film Thickness Along the Line of Action on Transverse Plane
&0.0 0.0
I
Fig.14. It is seen that as the load increases, both the bulk and flash temperatures increase but the film thickness decreases. The friction coefficient increases first, then decreases a little after reaching its maximum value. The effects of pinion angular speed are illustrated in Fig.15. It can be observed that as the speed increases, both the bulk and flash temperatures increase but the friction decreases due to the reduction of lubricant viscosity. The minimum film thickness increases because of entraining effect. At very high speeds, however, the film thickness decreases again due to thermal effect in the inlet zone of the contact.
10. CONCLUSIONS A comprehensive numerical analysis for contact geometry, kinematics, lubricant film thickness, friction, and bulk and flash temperatures in helical gears is developed. The program can be
100
TORQUEf0I%%UT
( N * m ) 10000
Fig. 14(b). Effect of Torque Input on Bulk and Flash Temperature Increases
used on most personal computers for both standard and non-standard, external and internal gears, and the input and output can be in either IS0 or English system. In most cases the converged solution can be achieved within 5-8 iterations, which may take 40-60 minutes on IBM PC/XT. Sample results show that this program is useful in helical gear design and research. REFERENCES "Lubrication of Gear Teeth, Engineering, V01.102, 1916, p.109. McEween, E., "The Effect of Variation of Viscsity with Pressure on the Load Carrying Capacity of Oil Films Between Gear Teeth," J. of Inst. Pet., Vo1.38, 1952, p.646. Daring, D . W . , and Radzimovsky, E.I., "Lubricating Film Thickness and Load Capacity of Spur Gears: Analytical Investigation," 'I
389
-
a0.6
::
r0.10
7
.d
50.5 tn
v)
0.4
Y
3
0.3
r= 0.2
E
20.1 J:
0.0 I 100
o
ANGULAR &%D
Fig. 15(a).
.
0 1000
OF PINION
0
frpm)
Effect of Angular Speed of Pinion on Film Thickness and Friction
ASME Paper No.63-WA-85,1963. Dowson, D., and Higginson, G.R., "Elastohydrodynamic Lubrication - - - - - - - The Fundamentals of Roller and Gear Lubrication," Pergamon Press, London, 1966. Gu, A., "Elastohydrodynamic Lubrication of Involute Gears," ASME Paper No.72-PTG-34. Blok, H., "Theoretical Study of Temperature Rise at Surfaces of Actual Contact Under Oiliness Lubricating Conditions," Proc. of the General Discussion on Lubrication and Lubricants, Oct., 1937, Inst. of Mech. Eng. (London), V01.2, pp 222-235. Jaeger, J.C., "Moving Surfaces of Heat and the Temperature at Sliding Contacts," J. and Proc. of the Royal SOC. of New South Wales, Vol.wNI(76), 1942, pp 203-224. Archard, J.F., "The Temperature of Rubbing Surfaces," Wear, Vo1.2,1958-59,pp 438-455. Frances, H. A., "Interfacial Temperature Distribution Within a Sliding Hertzian Contact." ASLE Trans., Vo1.14, 1971, p.41. (10) Wang, K ;L. , and Cheng] H.S . , . "A Nde>ical Solution to the Dynamic Load, Film Thickness, and Surface Temperatures in Spur Gears, Part I: Analysis," J. of Machine Design, Vo1.103, 1981, p.177. (11) Chao, H.C., "A Computer Solution for the Dynamic Load, Lubricant Film Thickness and Surface Temperatures in Spiral Bevel Gears," Ph.D. Thesis, 1982, Northwestern University, Evanston, Illinois.
100 ANGULAR SPEED 1000 OF plNtoONoPrpm)
Fig. 15(b). Effect of Angular Speed of Pinion on Bulk and Flash Temperatures
12) Litvin, F.L., "Theory of Gearing," AVSCOM Technical Report 88-C-035,1989, NASA Reference Publication 1212. 13) Dowson, D., "Elastohydrodynamics,"Proc. of Inst.Mech.Eng., Vo1.182,3A, 1967-68,p.151. 14) Dowson, D., and Toyoda, S., "A Central Film Thickness Formula for Elastohydrodynamic Line Contact," Proc. of the 5th Leeds-Lyon Symposium on Tribology, 1978, p.60. (15) McGrew,J.M.,Gu,A., Cheng,H.S., and Murray, S.F., "Elasto-hydrodynamicLubrication - - - Preliminary Design Manua1,"Technical Report AFAPL-TR-70-27,Aero Propultion Laboratory, Air Force Systems Command, Wright-Pattenson Air Force Base, Ohio, Nov., 1970. (16) Bair,S. and Winer,W.O.,"A Rheological Model for Elastohydrodynamic Contacts Based on Primary Laboratory Data," J. of Lub. Tech., V01.101, 1979, pp 258-265. (17) Trachman,E.G.,and Cheng, H.S., "Reological Effects on Friction in Elastohydrodynamic Lubrication," NASA CR-2206, 1973. (18) Plint,M.A.,"Traction in Elastohydrodynamic Contacts," Proc. of Inst. Mech. Eng., Vol. 182, Part 1, 1967, p.300. (19) Cheng, H.S., "Calculation of Elastohydrodynamic Film Thickness in High Speed Rolling and Sliding Contacts," MTI Report 67TR24. (20) Zhou, R.S., and Cheng, H.S., "A Simplified Solution for Gear Bulk Temperature Matrix," to be submitted for publication.