Tribology Research: From Model Experiment to Industrial Problem G. Dalmaz et al. (Editors) 9 2001 Elsevier Science B.V. All rights reserved.
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Frictional Behaviour of Synthetic Gear Lubricants B.R. HtShn a, K. Michaelis b and A. Doleschel c Professor, Gear Research Centre (FZG), Research group manager, Gear Research Centre (FZG), c Research Scientist, Gear Research Centre (FZG), Technical University of Munich, D85747 Garching, Germany a
b
In this experimental examination the frictional behaviour of 19 synthetic lubricants was systematically investigated in a twin disc machine as a function of load, sliding speed and oil temperature. Additional power loss measurements were made in an FZG backtoback gear test rig, varying load, speed and temperature. From these measurements the mean coefficient of friction in the twin disc contact area and in the gear mesh could be calculated and compared. The comparative investigations led to a calculation method to predict gear mesh friction and power loss from relatively simple and inexpensive twin disc results. The different test methods and the results for some selected lubricants are shown. Using a mineral oil and synthetic gear oil in comparison the measured and calculated power loss values of an industrial gear box with one bevel gear stage and two spur gear stages are discussed. 1. INTRODUCTION The frictional behaviour of a lubricant determines the load dependent power loss in bearings and gears of a transmission. Lower friction leads to higher efficiency, lower operating temperatures, better film formation and thus to higher load carrying capacity of the components. For mineral oils gear mesh friction can be calculated from experimentally based equations; for synthetic gear lubricants little information is available. The existing calculation method for synthetic gear lubricants [ 1] gives only a rough estimation for several lubricants. For a more accurate calculation of currently used gear lubricants a more detailed data basis is necessary. The calculation of the coefficient of friction based on a physical model of the contact is often discussed in literature [2  6]. All these therotically derived equations do not fit to measured values or they require many rheological lubricant parameters which are often not available. Equations based on the assumption of Newtonian fluid models [2] never fit the experimental data. The
Characteristic EHLpressure pressure ~ / acc. Hertz ~__ / EHL. //~~ A" temperature
l
Figure 1:EHL  Lubrication fluid under pressure and temperature conditions of bearing or gear contacts shows a clear nonNewtonian behaviour. The mesh zone between two gears or two discs can be described with only few parameters. A special distribution of pressure and temperature along the contact is typical for elastohydrodynamic lubrication (EHL) (Fig. 1). With the temperature and the pressure
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of the fluid the local instantaneous viscosity can be determined. With this operating viscosity and the shear rate the coefficient of friction can be calculated a c c o r d i n g to J o h n s o n / T e v a a r w e r k [3,4] or Winer/Gecim [5]. For the application of these equations the evaluation of the limiting shear stress zL is necessary. For most lubricants the value of xL is not available. Using a test rig for determining the coefficient of friction simple experimentally based equations can be derived, which are able to describe the frictional behaviour of each lubricant with sufficient accuracy for their application in gear transmissions. The disadvantage, however, is that for each lubricant a test run has to be performed. For the investigation two standard test methods were used. For all lubricants the coefficient of friction was measured in the FZG twindisc test rig. Some of the lubricants were additionally investigated in the FZG backtoback gear test rig. 2. LUBRICANTS For the investigation 19 different synthetic lubricants and, as a reference, two mineral oils were investigated. The basic types of lubricants are given in Table 1. All lubricants are of the vicosity grade ISO VG 150. Table 1 Investigated lubricants oil type
investigated in twin disc machine
backtoback gear test rig
mineral oil MIN
2
1
polyalfaolefin PAO
2
1
polyglycol PG
5
2
ester
6
3
hydrocrack XHVI
1
0
dialkylbenzol DAB
1
0
mixtures
4
2
total
21
9
3. TWIN DISC MACHINE
3.1. Test rig
skid flatspring oilinjection\~ __~~~_~~F loadactuator frame , ~ spring
~oil k~,~_l
loadcell R ......FN
" "'"~/f _~.~ f"~ "~\v2 V =
~
l ~]//~ ~
pivot pivot
. . . . . . / ~I / ~ foundation/~ /
1
1
1
1
/
Figure 2: FZG twin disc machine The coefficient of friction was measured in the FZG twin disc machine (Fig. 2). The test discs are separately driven by two AC motors. For continuous variation of speed, traction drives are mounted between the motors and driving shafts. The upper disc is mounted in a skid, which is connected over two flat springs to the frame. The flame can be rotated over the pivot and thus the upper disc can be pressed against the lower disc. The load is applied by a spring which itself is loaded by the load actuator. If the two discs have different speed, a frictional force is generated which moves the upper skid. The movement is blocked with a load cell so that the frictional force F R in the contact can directly be measured. The normal force FN is measured by strain gages at the flat springs. From the measured values FR and FN the coefficient of friction g can be calculated according to equation (1). FR ILt F (1) N An oil spray device with heating and cooling possibilities and a filter system provides lubricant at the desired temperature to be injected directly between the discs. The investigated parameters normal load, temperature, sum and sliding velocitycan be adjusted separately. For best simulation of a gear set conditions of a line contact were adjusted with cylindrical discs with a width of b = 5mm and an outer diameter of da = 80mm. For constant conditions of the surface texture under the different operating conditions the surface
761
of the discs was polished to an arithmetic mean roughness of about R, ~ 0,051am to 0,1 gm.
3.2. Experimental Results For all lubricants the load, the temperature and the sum and sliding velocity are varied. Starting with a standard test condition each parameter is varied to a lower and a higher value. The test conditions are shown in Table 2. Table 2 Test conditions in the twin disc machine Low .
.
.
Standard .
.
.
.
.
High .
.
PH N/mm~
800
1000
1200
8oiJ
~
60
90
110
vz
m/s
2
4
8, 16
The sum velocity vz is calculated from equation (2) with v~ and v 2 as the velocities of discs 1 and 2. (2)
VE = V 1 + V 2
The sliding speed vg and the slip rate s are defined as Vg
V1 S =
V2
IV,
9100%
V2 [
with v 1 >
(3)
v2
(4)
vI
For all test conditions the slip rate s is varied from 1% up to 50% for constant sum velocity. The influence of the variation of the different parameters on the coefficient of friction is shown for the mineral oil.
Variation of pressure Pn In Fig. 3 the variation of pressure for the investigated slip rates s and for mineral oil is shown. For all lubricants the coefficient of friction rises with increasing load. The influence of the Hertzian pressure pn on the coefficient of friction is higher for all synthetic lubricants compared to the mineral oil.
Figure 4: Variation of sum velocity in twin disc machine Fig. 4 shows the influence of the sum velocity vz on the coefficient of friction for mineral oil. With increasing sum velocity the EHL lubrication leads to a higher film thickness and the coefficient of friction decreases.
Variation of oil temperature ~oi~ Fig. 5 shows the influence of the oil temperature 8o~ on the coefficient of friction for the mineral oil. With increasing oil temperature the viscosity of the fluid decreases. This leads under EHLlubrication conditions to a lower viscosity and thus to a lower
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Figure 5: Variation of temperature in twin disc machine
sum velocity hertzian stress
coefficient of friction. Higher temperatures lead also to a thinner film and to mixed and boundary lubrication conditions. When the higher oil temperatures cause a transition from EHL to mixed lubrication conditions the coefficient of friction tends to rise with temperature. Variation of oil type In Fig. 6 the measured values of the coefficient of friction for different lubricants are shown. In comparison to the polyglycol the mineral oil has a 3 times higher c o e f f i c i e n t of friction la. The polyalfaolefin has a coefficient of friction of about 60% of the mineral oil and mixtures with PAO lie between mineral oil and PAO. The lowest coefficient of friction occurs with the polyglycol, where the friction depends on the mixture of polyethylen to polypropylen derivates (EO:PO). The investigated esters (not figured) are in the range between mineral oil and polyglycol (1:1).
From these experimental results an equation for the determination of the coefficient of friction can be derived. The coefficients Cp, Cv, and cn of equation (3) represent the influence of Hertzian pressure PH, sum velocity v~ and viscosity rim at bulk temperature. The constant Cs considers the influence of the slip rate s.
/)cp()cv/
g = Co . PII
. v~:
. rIM
"tanh (Cs.s)
(5)
oil temperature viscosity
vT_" = 4 m/s Prl = 1000 N/mm 2 ,9Oil = 90~ ISOVG150
lubricants: MIN DAB
mineral oil dialkylbenzol
XHVI
hydrocrack
PAO
polyalphaolefin
PAO
mixtures
PG
polyglycol
PG
EO:PO 0:1 .. 3:1
Figure 6: Coefficient of friction of different synthetic lubricants measured in twin disc machine 4. FZG BACKTOBACK G E A R TEST RIG 4.1. Local conditions on the path of contact
Along the path of contact gears are exposed to varying speed and load conditions. In Fig. 7 the local speed conditions along the path of contact are shown. The surface speed of the pinion Vl and of the gear v2 varies between the begin of the approach path to the end of the recess path. Only at the pitch circle C the values of Vl and v2 are identical and pure rolling appears. From the values of Vl and v2 the sum velocity vz, the sliding velocity vg and the slip rate s can be calculated. For the used gear type C (Table 3) slip rates up to 80% can be found and even near to the pitch line a high sliding speed occurs.
763
1,2
1,2
Ph / PC
1,0 S
,,,,'" ,~
0,8
,I .........................
v Z / vt v2 / vt Vg / vt
, ,, .,,,,,,,, ,~,_.~,~=,
"",,,,,,,,
1,0 0,8
,9
...................... geal
0,6
............... pinton.. 0,4
,/
0,6
FN / FN,C P/Pc
0,4 0,2
0,2
Vl/vt
,~. A
B C D path of contact
E
0,0
Figure 7" Characteristic speed parameters along the path of contact for gear type C
Table 3 Geometry of gear type C pinion
gear
16
24
number of teeth z

pressure angle %
~
20
helix angle 6
o
0
gear ratio u

1.5
centre distance a
mm
91.5
normal module mn
mm
4.5
addendum modification x
0.18
0.18
20
20
pitch circle diameter dw mm
73.2
109.8
arithmetic roughness
0.4
0.4
face width b
tooth loss factor Hv

mm
Ra
~tm 
0.202
Disregarding dynamic effects, the local normal load F N varies between single and double tooth contacts (Fig. 8). At point A tooth #2 comes into contact sharing the load with tooth # 1 which stands at D (double tooth contact) at this moment. Carrying on to point B where tooth # 1 goes out of the contact and the total load has to be transmitted only by tooth #2 (single tooth contact). Point C indicates the pitch line with pure rolling. At point D tooth #3 comes into
A
B C D path of contact
0,0
Figure 8: Characteristic load distribution along the path of contact for gear type C contact at A and the double tooth contact area starts again. At point E tooth #2 goes out of the contact. Together with the local radius of curvature p the distribution of the Hertzian stress PH along the path of contact can be derived. With the local speed and the local pressure each point on the path of contact can be simulated in the twin disc machine. With equation (5) and the parameters or, 13, and Y gained from the twin disc experiments, the coefficient of friction along the path of contact of a gear can be calculated. The results of the calculation for gear type C at standard conditions of load ( P c  1100 N/mm ~) and pitch line velocity (vt.c = 8 m/s) are shown in Fig. 9. In the double tooth contact area the coefficient of friction is lower than in the single tooth contact area. In the region of low slip (s  0 at point C) ~t drops to zero. 4.2. Test rig The frictional losses of cylindrical gears are measured in a modified backtoback gear test rig [7] (Fig. 9). The test pinion and the test gear are mounted on two parallel shafts which connect them to the drive gear stage. In the drive gear stage identical gears to the test gears are mounted, so that two equal stages are closing the inner circle. The pinion shaft consists of two separate parts, which are connected by the load clutch. By twisting the load clutch using defined weights (load stages) on the load lever a
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defined static torque is applied, which is indicated at the torque measuring clutch. The motor has only to compensate the frictional losses in the load circle. For the measurement of the loss torque of the whole system a torque meter shaft is mounted between the motor and the drive gear box.
torque measuring clutch
•
up into the losses of the gears (PvzP and Pvz0), the losses of the bearings (PwP and PvL0) and auxiliary loss sources (Pvx0) like seals or oil pumps. They can also be split up into load dependent losses (index P, PvzP and PwP) and noload losses (index 0, Pvz0, Pwo and Pvx0) according to equation (6) [8].
Pv torque meter rive gear
~_~__~] c.lL~
= Pvzp
+
Pvzo
+ PVLP + PVLO + P v x o
(6)
4.3. Experimental Results In Fig. 10 the loss torque measured with mineral oil using gear type C is shown.
~est gear , / test pinion
~ "
load lever with weights
temperature senso load clutch
Figure 9: FZG backtoback test rig
12,5 ,~ E z >
:::3 E
For determining the loss torque at different test conditions the motor speed and the static load are varied in a wide range (Table 4). For all test conditions spray lubrication was applied with an oil injection temperature of 8o+~ 90~ In Table 4 it can be seen, that the power in the circle varies from 0 kW (without load) to P A "~ 165kW transmitted power. The highest speed (vt = 20 m/s) together with the highest load (Pc = 1700 N/ram 2) was not investigated. Table 4 Operating conditions of backtoback gear test rig Transmitted power PA in circle [kW] Load stage KS0 speed
KS5
plv
nl vt [1/min] [m/s]
KS7
KS9
KSll
m m
o
\
10 i ~ 7,5
gear type C b = 20mm mineraloil 8oil = 90~
~ ~ _ _
Hertzian
stress PH [N/mm 2] 1700
~
5
~
~
~ 1400
2,5
" ~ ~ 
~
.
,..
,,,
0~~~
0
750
1500 2250 3000 pinion speed n 1 [l/min]
1100 78O 0
3750
Figure 10" Measured loss torque with mineral oil To determine the load dependent losses PvP two m e a s u r e m e n t s are n e c e s s a r y : the n o  l o a d measurement and the load measurement. The load dependent losses Pw are the difference between these two measurements. They include the load dependent gear (PvzP) and bearing (PwP) losses. The bearing losses can be calculated according to SKF. From the measured value of Tw and Tvo the load dependent power loss of the gear mesh can be calculated with equation (7).
Pc [N/mm'] 0
780
1 1 0 0 1400
1700
522
2
0
5.14
10
16.5
24.6
1044
4
0
10.3
20
33
49.2
2087
8
0
20.6
40.1
66
98.4
3131
12
0
30.8
60.1
99
147.6
5218
20
0
51.4
100
165

1
PvzP  2
n'~
3i) (TvP  Tv~ )  PVLP
From the load dependent gear losses the mean coefficient of friction can be calculated with the transmitted power P A and the tooth loss factor Hv according to Ohlendorf [9]. P VZP [Llmz 
The total power loss of a gear box Pv can be split
(7)
PA" Hv
(8)
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For different lubricants the measured coefficient of friction is shown in Fig. 11. In Fig. 11 the load is held constant to Pc = 1100 N/mm = and the oil injection temperature is 8o, = 90~ The values of the coefficient of friction show the same order as the values measured in the twin disc machine. For PAO a coefficient of friction of about 80% of the mineral oil is measured, for PG and TMP ester a g of about 60% of the mineral oil was found.
pitch line velocity v t [m/s] 0
5
N 0,08 E "'1 tO
9
0,06
O i_
"6 0,04
10
mineral oil polyalfaolefin . TMPester polyglycol ....... ~ 9~ " ' l ....... ,,.._ ..................
15 ' ...........  *" " .... ~ ~ _ _
9~"=". ~:=~'T'=.':..............
IC
20
gear type C b = 20 mm PH = 1100 N/mm 2 ,9,oiI = 90~ ISOVG150
~""*
9 ~="=."=.2=:E:~.:=_:~=
O
tE 0.02 O O r
The transfer from the results of the twin disc machine to the predicted gear mesh losses needs two steps. First the result of the twin disc machine has to be analyzed and an equation to determine the coefficient of friction for all load and speed conditions has to be found. Equation (5) delivers a satisfying accuracy, especially at high slip rates as they occur in gears. As a second step for each load and speed the local coefficient of friction can be determined (Fig. 12). Integration along the path of contact results in a mean coefficient of friction for each operating condition. In Fig. 13 the results from the twin disc machine are compared to the results of the gear test rig. Fig. 13 shows a good correlation for the gear and the disc results for the influence of speed. The sensitivity for the influence of pressure, however, is stronger for the disc results than for the gear results. This dependency was found for all investigated lubricants.
0
0
E
2000
4000
pitch line velocity v t [m/s]
6000
pinion speed n,[l/min]
0
Figure 11" Mean coefficient of friction of different lubricants measured in gear test rig
N
0,08
5
10
mineral oil
Pc Pc ,_ I Pc ~"'~._ Pc
E
"1 t
o 0.06
.D
.O ~ I...
15
20
= 1700 N/mm 2 2 =1400N/mm =1100N/mm 2 = 800 N/mm 2
~ 9 O o a) (I) ~.o "~~~ o ~I E . ,,
"5 0,04 t
5. T R A N S F E R F R O M DISC TO G E A R
o,.
",4
~
tt.~
~



A
O
0,02
i 0,06 PH =1100N/mm21
O O
vt = 8 rn/s I '__.~
~ r ~ ,
I
,9,oi1=90 Cl"
E
0,05
"i'
I mineraloill_ 0,04 o,ya, a I O,O3
[ 'i
A
olefin _1 polyglycol0,02 ~. 0,01 0,0
B C D E path of contact Figure 12: Coefficient of friction along the path of contact recalculated from twin disc results
polyglycol
(::::::,l
::!il:
'
0
0
2000
4000
6000
pinion speed nl[l/min]
Figure 13" Comparison of calculated values from disc measurements to gear measurements For lubricants with a generally low coefficent of friction even lower friction was determined in the twin disc machine than in the gear test rig. For this discrepancy the following reasons can be found: 9 The surface topography and roughness of discs and gears are d i f f e r e n t . G e a r s have a topographical structure transverse to the sliding direction, discs have the main topographical structure in the direction of sliding. According to [10], depending on the operating conditions, the difference can be up to 10%. The surface roughness of the investigated gears was about 4 times higher than the roughness of the discs. It is
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well known, that the roughness has an important influence on the coefficient of friction. 9 Film formation in a gear contact is discontinuous with each new tooth entering the contact zone while the engagement in discs is continuous. The tooth has to build up a new oil film with every new engagement. At the beginning of the engagement very disadvantageous conditions with a scraping edge removing the oil film from the surface lead to a thin oil film with possibly mixed lubrication conditions and thus to a high coefficient of friction. 9 In the investigation dynamic effects were not considered, but they certainly have an influence on the friction between the tooth flanks. These differences of gear and disc contact conditions can be taken into account by introducing an empirical correction to the calculation. Fig. 14 shows the comparison of measured gear values to c a l c u l a t e d and c o r r e c t e d values from disc measurements. pitch line velocity v t [m/s] 0
5
10
15
20
The exact calculation of the viscosity, the coefficient of friction and the resulting total power loss is important to decide which oil leads to the lowest power loss for the given operating conditions. For the example of a 3 stage bevelspur gear system [11] a mineral oil and a polyalfaolefin, both ISO VG 320, were investigated for different load and speed conditions. The loss torque of the gear box was measured at the equilibrium temperature of the gear box for each operating condition. In comparison to the measurement the power loss of the gear box was calculated with the FVA programme WAEPRO [ 12]. The calculation was completed with a polyalfaolefin of a lower viscosity grade ISO VG 220. Table 5 Industrial gear box Investigated lubricants oil
type
viscosity ISO VG
measured
calculated
MIN320
mineral
320
r
r
PAO320 synthetic
320
r
r
PAO220 synthetic
220

r
N
E 0,08
Pc Pc Pc Pc
mineral oil
::I. .O m
"6 0,06 . ...~
=1700N/mm 2 =1400N/mm 2 =1100N/mm 2 =800N/mm 2
~.O ~.o ,,8. o E. o
d ~m ~8
2,5
"6 0,04
o..'.;~:::: .....
.B r
1,5 ................
o 0,02
~. . . . . . .
09 g0 O
I_~
fD I'
,
polyglycol E
industrial gear box 1 x bevel gear stage ..................... i"ii' 2 x spur gear stages dip lubrication ...............i"ii nominal speed: .., r .... ; . . . nA = 2000 l/min ., .......::::":i
2
O
0
0
2000
4000
1
O
6000
" 0,5
Figure 14: Corrigated calculation (transfer form disc to gear)
0
pinion speed nl[1/min]
~. .....:.::::!::~~/:2~:SS~ :2~=;:.... mineraloil polyalfaolefin polyalfaolefin 0
100 2O0 input torque [Nm]
visc. meas. calc. 320 9 320 9 220 300
6. INDUSTRIAL GEAR BOX
Figure 15" Power loss of a industrial gear box depending on the load
The investigation led to a more accurate calculation method for the prediction of gear box losses lubricated with a synthetic oil. For the use in a gear box the total power loss has to be considered. Synthetic lubricants mostly have a lower coefficient of friction, but higher noload losses. The viscosity at operating temperature is higher in comparison to mineral oil, if the same viscosity grade is used.
In Fig. 15 the influence of load for a constant speed is shown. For all loads a higher power loss for the mineral oil versus the polyalfaolefin was measured. The calculation shows, however, that for low load the mineral oil has a lower power loss than the polyalfaolefin of the same viscosity grade. For these operating conditions the noload losses are dominating over the load dependent losses. With
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2,5 industrial gear box 1 x bevel gear stage
.,~
2 2 x spur gear stages O~
dip lubrication 1,5 nominal torque
,o o
......t:::= >':':': ;>'" ..............i"iis>>'
TA = 200 Nm
o
............. .
.........: :
1
it:".":':;)/" mineraloil
visc. meas. calc.
320 9 polyalfaolefin 320 9 polyalfaolefin 220
Q 0,5 0 0
500
1000 1500 input speed [l/min]
2000
2500
Figure 16: Power loss o f an industrial gear box
topography and different engagement behaviour of the disc and gear contact, the results are in good agreement. Power loss measurements of a mineral and a synthetic oil used in an industrial gear box are shown and compared to the calculation. For all examined loads and speeds a good correlation between measurement and calculation was found. With a more accurate calculation method the design of gear boxes can be optimized and energy and environmental resources can be saved. The results of the project will be documented in detail in a DGMKReport [ 13].
depending on the speed 8. N O M E N C L A T U R E higher transmitted power, the synthetic oils lead to a lower mesh friction in comparison to the mineral oil. In Fig. 16 the measured and the calculated power loss is shown as a function of the speed of the bevel gear shaft. For constant load (Y a = 200 Nm) the power loss is higher for mineral oil than for the polyalfaolefin. In general the calculated values are in good accordance with the measured values. Using a lower viscosity synthetic gear oil as e.g. the PAO220 can reduce the power loss compared to the MIN320 and the PAO320. The noload losses decrease with lower viscosity. When optimizing a gear box for the highest efficiency careful selection of the lubricant concerning scuffing, wear and pitting capacity is necessary. 7. SUMMARY
a
b C cq Cp Cv
dw FN FR Hv mn
n nA
P. PA
Pv PvLo PVLP
Pvzo In this project the coefficient of friction of some 20 different lubricants was measured in a twin disc machine and an FZG backtoback gear test rig. In both test rigs the influence of pressure and speed was investigated, in the twin disc machine additionally the temperature was varied. The results of the measurements lead to an empirical equation which fits well the measured values depending on pressure, speed and temperature. With this equation the coefficient in the gear mesh zone can be calculated. In comparison to the measured mean coefficient of friction in the gear mesh the twin disc experiments lead to lower values, especially for low friction fluids. If a correction factor is introduced, considering the different surface
Pvzp PVXO Ra
s
TA Tv0
Tvp u v Vg vt vE x z
mm mm mm N N mm 1/min 1/min N/mm 2 kW kW kW kW kW kW kW gm 
Nm Nm Nm 
m/s m/s m/s m/s 

centre distance face width constant viscosity exponent pressure exponent speed exponent pitch circle diameter normal force friction force tooth loss factor normal module speed nominal speed hertzian stress nominal power power loss load independent bearing losses load dependent bearing losses load independent mesh losses load dependent mesh losses other losses arithmetic roughness slip rate nominal torque load independent torque losses load dependent torque losses gear ratio velocity sliding velocity pitch line velocity sum velocity addendum modification number of teeth
768
13
o
rl p
mPas mm
~oil
~
la Indices 1 2 C R M
pressure angle helix angle dynamic viscosity radius of curvature oil temperature coefficient of friction
pinion, disc 1 gear, disc 2 pitch line reference bulk temperature
9. ACKNOWLEDGEMENT The authors would like to thank the German Society for Petroleum and Coal Sciences and Technology (DGMK) and the German Federal Ministry of Economics and Technology (BMWi, AiFNo. 11220N) for their kind sponsorship of this project. REFERENCES
1. Schlenk, L. : Untersuchungen zur FreBtragf~ihigkeit von GroBzahnr~idern. Diss. TU Mtinchen, 1994. 2. Schouten, M.W.: Die Elastohydrodynamik: Geschichte und Neuentwicklungen. VDIBerichte Nr. 1207, 1995.
3. Johnson, K.; Tevaarwerk, J.L.: Shear behaviour of elastohydrodynamic oil films. Proc. R. Soc. Lond. A. 356, 215236, 1977 4. Tevaarwerk, J.L.: A Simple Thermal Correction for Large Spin Traction Curves. Trans ASME, Vol 103, 1981 pp 440445 5. Winer, W.O.; Gecim, B." Lubricant Limiting Shear Stress Effect on EHD Film Thickness. Journal of Lubrication Technology, April 1980 6. Hirst, W.; Moore, A.J.: NonNewtonian behaviour in elastohydrodynamic lubrication. Proc. R. Soc. Lond. A. 337, 101121, 1974 7. DIN 51354; FZGZahnradVerspannungsprtifmaschine. 8. Niemann, G. und Winter, H.: Maschinenelemente Band 2. Springer, Berlin 1985. 9. Ohlendorf H.: Verlustleistung und Erw/armung von Stirnr~idern. Diss TU Mtinchen, 1958 10. Prexler, F.: EinfluB der W~ilzfl~ichenrauheit auf die Grtibchenbildung vergtiteter Scheiben im EHDKontakt. Diss TU Mtinchen 1990. 11. ISO 14179. Thermal Load Capacity of Gear Units, 2000 12. Funck, G.: EDVProgramm W~irmeabftihrung (WAEPRO). FVA Forschungsheft Nr. 243, 1987 13. Doleschel, A.: Einfluss verschiedener GrundOle auf Reibung und V e r l u s t l e i s t u n g von Zahnradgetrieben. DGMK Forschungsbericht Nr. 526, Hamburg, 2001