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1、CHINESE JOURNAL OF MECHANICAL ENGINEERING28-Vol. 20, No. 5, 2007DENG Sier TENG HongfeiSchool of Mechanical Engineering,Dalian University of Technology,Dalian 116023, ChinaWANG YanshuangLI YunfengYANG BoyuanSchool of Mechanical Engineering,Henan University of Science andTechnology,Luoyang 471003, Chi
2、naCONSTITUTIVE EQUATION OF A NEW AVIATION LUBRICATING OIL*Abstract: The traction of a new aviation lubricating oil was measured on a self-made test rig. The calculating formulae of the rheological parameters of the oil such as Erying stress, limiting shear stress and shear elastic modulus were obtai
3、ned under the condition of the high shear strain rate in elastohydrodynamic lubrication(ERL). The constitutive equation of this oil was determined and verified by test. The results of experiments show that the behavior of the new aviation lubricating oil behaves as visco-elastic fluid and the theore
4、tical value agrees fairly well with the measured data, which implies that the constitutive equation of this oil is correct and feasible. Key words: Lubricating oil Constitutive equation Rheological parameters TractionElectric spindle IDisk specimenBall specimenTraction sensorElectric spindle IIHydro
5、static shaftLoad sensor0INTRODUCTIONFor the high speed rolling bearings, the traction force exerted on the rolling element interfaces makes the rolling elements accelerate, decelerate, skid or skew. Thus cage instabilities and the longevity of a rolling bearing are associated with the traction behav
6、ior of the lubricant. Under the working condition of elastohydrodynamic lubrication (EHL), the selection of the constitutive equation affects the traction of the lubricant directly. DYSON, et al131, proposed several kinds of rheological models to compute the traction coefficient of the lubricant. Ho
7、wever, these rheological models are practically inapplicable because some parameters of the rheological models are hard to determine. BARLOW, et al471, researched the rheological characteristics of the lubricants under static state and low shear using rheological instrument. But the rheological para
8、meters of the lubricant under the condition of high pressure and high shear rate can not be determined by using rheological instrument at present. For these reasons, the traction experiments of a new aviation lubricating oil under various velocities, temperatures and loads were carried out on the se
9、lf-made traction test rig. The rheological parameters of the new aviation lubricating oil under the condition of high pressure and high shear strain rate were obtained from the traction experiments and the constitutive equation of this aviation oil was established and verified by experiments.1TRACTI
10、ON EXPERIMENTThe traction experiment rig is shown in Fig. 1. The specimens of the test rig are a steel disk and a steel ball. The structure of the test rig and the measuring method of the traction can be seen in Ref. 8. The traction experiment is to measure the traction force F at different slide-to
11、-roll ratio s=ivi-v2)/v when pressure p, rolling velocity v=(vi+V2)/2 and inlet oil temperature T0 are kept and thus the experimental curve of the traction coefficient n (jt=F/fV, where W is the normal load) against slide-to-roll ratio 5 can be obtained.The main characteristic parameters of the new
12、aviation lubricating oil are as follows. Ambient temperature is 27 C, dynamic viscosity in ambient temperature r$ = 0.027(Pa s), viscosity- pressure coefficient a=1.55xlO*(l/Pa), viscosity-temperature coefficient P= 0.029 (l/C), thermal conductivity K = 0.096 6(N/(s V).The working condition is as fo
13、llows. Rolling velocity 10, 20, 30, 40 m/s respectively, inlet oil temperature 27, 60, 90, 125 C, maximum Hertz pressure 0.8, 1.0, 1.1, 1.2, 1.35, 1.5 GPa.* This project is supported by National Key Projects of China (MKPT-2001-004). Received October 25. 2006; received in revised form July 9, 2007;
14、accepted July II, 2007Fig. 1 Construction of the experiment rigIn the experiment, the slide-to-roll ratio s is in the range of 0 to 0.2, thus 96 curves described in the form of traction coefficient fj against the slide-to-roll ratio s can be obtained in a state of full EHL.2 CONSTITUTIVE EQUATION OF
15、 THE NEW AVIATION LUBRICATING OILNon-Newtonian effects become significant when the mean shear stress is higher than l%-5% of the mean pressure in elastohydrodynamic lubrication. At low shear strain rates the lubricant behaves as linear viscous or elastic fluid, and with a subsequent increase in the
16、shear strain rate the lubricant undergoes the transition from viscoelasticity to plasticity until the shear stress arrives at the limiting shear strength91. At the present time, the constitutive model that can describe the above property well is Evans-Johnson model1121, in which not only the influen
17、ce of q, Ge, To is considered but the influence of limiting shear stress rL is also considered. In the traction experiment of this paper, the shear stress is only in the rolling direction, so the expression of the con-,112stitutive equation is1(1)y = (l/GeXdr/df) + r0sinh(r/r0)/7 tt,where G n, r0 an
18、d rL are shear elastic modulus, viscosity, Eyring stress and limiting shear stress respectively.When the rheological parameters Ge, q, t0 and rL are determined, the constitutive equation of the new aviation lubricating oil can be determined. The value of shear stress x at a given operatingCHINESE JO
19、URNAL OF MECHANICAL ENGINEERING29*(3)(5) thereforecondition can be obtained from the solution of Eq. (1).3 RHEOLOGICAL PARAMETERS OF THE NEW AVIATION LUBRICATING OIL3.1 Erying stress r0 and viscosity t/When the product of the pressure-viscosity coefficient and the mean Hertzian pressure dp is betwee
20、n 13 and 25, the oil behavior within the oil film is viscoelastic . In this paper, the new aviation lubricating oil behaves as viscoelastic fluid due to dp -/pwhere, a is equivalent viscosity-pressure coefficient, T is oil film temperature, T0 is inlet cil temperature, So and Z are Roelands paramete
21、rs,/? is contact pressure.If the contact pressure is assumed to follow a Hertzian distribution , and the center coordinate of the contact circle is (a, 0), contact circle radius is a, the following can be obtained from Eqs. (2), (3)T = T0hiaicsnTj!eplap0-lJl-(x-a)2/a2-y2/a2/T0 (4)That the shear stre
22、ss distribution approximates Hertzian is verified by experimental results of CANN, et al1151. The assumption of Hertzian shear stress distribution is reasonably valid under median to heavy loading conditions of Hertz peak pressure above 0.8 GPal61. So the shear stress is assumed to be of a Hertzian
23、distribution in the EHL contact. Then, by measuring the EHL traction force in the experiment, the shear stress may be calculated using the following equation3Fl-(x-a)/a2 -ylafm/(2na2)where F is the measured traction force and a = 3WR/(2E)Vi is radius of the contact circle. The shear stress at the co
24、ntact center can be obtained by letting x=a and y=0 in Eqs. (4), (5). The error vector ET is defined as the difference between the shear stress r calculated by Eq. (4) and the shear stress v calculated by measured traction forces together with Eq. (5); Both of the shear stresses are calculated at th
25、e centre of the contact. The norm of the error vector is given byonly the data satisfying the condition T0y exp(a*p) r0 should be taken for the regression analysis with Eq. (6) in this case. According to above method, 96 groups of the values of Erying stress can be obtained at various pressures, rol
26、ling velocities and inlet oil temperatures.For the convenience in engineering applications, the 96 group values of Erying stress ro can be curve fitted to the following formula1.4910.0370.24(7)r0= 9.501 x10za/vuuj70where p0 = p0 / E is dimensionless parameter of Hertz peakpressure, v=tj0v/ER is dime
27、nsionless parameter of velocity,and T0 = T0JK/t0/(ER) is dimensionless parameter of temperature. The coefficient of correlation is 0.997 5, and average relative error is 1.95% by adopting Eq. (7) for fit.3.2 Shear elastic modulus Gt and limiting shear stress TlThe measurement of shear elastic modulu
28、s must be based on the condition that the lubricant is a mixed response of both elastic and viscous behavior to the imposed small shear strain rate in the linear region. Under the experimental condition of this paper, Deborah number is larger than 1, so the lub bating oil behaves as linear viscoelas
29、tic fluid when the shear strain is small. Thus the relationship of shear stress xii shea,- strain rate in the linear region cai: be expressed with iineai Max;vel! model/ = (l/GeXd/U/) + r/7(8)Even though the traction test does not differentiate between elastic aod visco-elastic responses to the shea
30、r in the linear region, with Eq. (8) it is possible to extract the effective shear modulus from the linear part of the traction curves. If the fluid flows through the contact with a steady velocity v parallel to the rolling direction, due to the rolling motion, the convective derivative reduces todx
31、/d/ = v(9)Eq. (8) is then integrated and further rearranged asy=jydt = J(l/Ce)dr+ (T/rjv)dx(10)So, the average shear strain in the contact region can be expressed as(11)where a is the half-width of the contact region along the rolling direction. Since(12)y =r/Ge +8r a/(37i77v)r=npr=j/d/=j;(v,-v2)/Ad
32、/(13)f = (l/l)|rd = (l/()|jj(v1-v2)/AdrJL4 =8a(v,-v2)/(37t/7v)(14)where A is the area of contact circle, a is the radius of contact circle. Substitute Eqs. (12) and (14) into Eq. (11), then the mean shear modulus takes the form of(15)Ce = 37tpw/i/8al - pmh/(vrf)(6)K*-rf1*1 =r0 can be determined by s
33、earching for the minimum value of the error function which can be obtained using Powell optimization method. It should be noted that Eq. (4) is applicable only when the non-linear region of a traction curve is clearly distinguished. Sowhere m=/s is the slope of the traction curve in the linear regio
34、n. Under the operating condition of high pressure, when the sliding velocity increases to a certain degree, the lubricating oil becomes plastic, which causes the shear stress to approach a limit value, and the traction force F to approach a constant value F,x = %(16)30*DENG Sier, et al: Constitutive
35、 equation of a new aviation lubricating oilThe maximum traction coefficient of the lubricating oil ist=FW(17)From Eqs. (16) and (17)TL=fnmW/na2(18)where W=2npyOtrS is the normal load, and /, can be obtained from experimental data.Average shear elastic modulus Ge and average limiting shear stress rL
36、in the contact region under the given experimental conditions can be obtained by Eqs. (IS) and (18). For the convenience in engineering applications, the shear modulus Gc and limiting shear stress FL can be related to the dimensionless parameters p0, v and T0. Suppose local limiting shear stress rL
37、and shear elastic modulus Ge at all points are assumed to beconstant at their average value rL and Gc respectively over the contact, then the following equations can be obtainedrL -rL = 1.856 6xlO-,p02,lVMMf0-20,44(19)GtGt= 4.331 4 x 10p025M V,04446f0-5 5637 (20)The correlation coefficient is 0.979
38、1 for rL and 0.982 5 for Ge, and the average relative error is less than 4%.4 COMPARATIVE ANALYSIS BETWEEN THEORETICAL COMPUTATION AND EXPERIMENTAL RESULTSFig. 2 Comparison between experimental and calculating values of traction coefficientTo illustrate the reasonableness of the constitutive equatio
39、n of the new aviation lubricating oil, three testing conditions were selected to carry out the comparison between model prediction and experimental observation. These three testing conditions are 0.8, 1.2 and 1.5 GPa maximum Hertzian pressure respectively at 25 m/s rolling velocity and 90 V inlet oi
40、l temperature. Fig. 2 shows the results of the comparison including the experimental traction curves obtained with the new aviation lubricating oil and the corresponding theoretical curves calculated with Eq. (1) in the form of traction coefficients ft versus slide-to-roll ratio s. Table shows the r
41、esults of comparison between the experimental values and calculating values of traction coefficient at rolling velocity v=25 m/s, inlet oil temperature 7*0=90 V, maximum Hertz pressure p0=l-5 GPa. It can be demonstrated from Fig. 2 and Table that there exists a good agreement in the overall range. T
42、he relative errors between theoretical computing values and experimental measuring values are less than 11% and the errors at low shear strain rates are larger. So the constitutive equation of the new aviation lubricating oil is correct and feasible, which can be used for the theoretical analysis of
43、 the dynamics of rolling bearings.Table Comparison between experimental and calculating values of traction coefficientSlide-to-roll ratio sExperimentaltraction coefficient /j.Calculatingtractioncoefficient aRelative error el%0.003 480.005 420.005 125.500.005 150.007 860.007 741.530.007 670.010 200.0
44、09 0910.890.009 340.012 950.011 798.960.011020.014 540.012 9311.070.013 530.016 210.014 4910.610.016 880.018 500.016 4710.970.026 100.021 280.019 4210.150.034 480.022 930.021 008.800.047 880.023 660.022 660.100.064 640.024 060.023 960.010.081 390.024 260.024 441.580.098 IS0.024 380.024 590.860.119 1
45、00.024 470.024 700.940.140 040.024 520.024 441.710.160 990.024 560.024 101.870.181930.024 590.023 783.290.198 690.024 610.023 633.985 CONCLUSIONS(1) The new aviation lubricating oil behaves as visco-elastic fluid when the maximum Hertz pressure is from 0.8 GPa to 1.5 GPa, rolling velocity is from 10 m/s to 40 m/s and inlet oil temperature is from 27 TJ to 125 C. The traction coefficient / of this oil rises linearly with the increase of slide-to-roll ratio s. When the sl
