Limit transmissible torque in the traction drive of a concave and convex roller pair

Tribology International 33 (2000) 233–240 www.elsevier.com/locate/triboint

Limit transmissible torque in the traction drive of a concave and convex roller pair Toshiji Nonishi a

a,*

, Satoshi Oda b, Kouitsu Miyachika b, Takao Koide

b

Faculty of Engineering, Fukuyama University, Sanzou, 1 Gakuen-cho, Fukuyama 729-0251, Japan Faculty of Engineering, Tottori University, 4-101 Minami, Koyama-cho, Tottori 680-8552, Japan

b

Abstract This paper presents a study on the limit transmissible torque in the traction drive of a concave–convex roller pair. The transmitted torque, the specific sliding, the roller surface temperature and the oil film formation under different contact pressures and roller speeds were simultaneously measured by carrying out a concave–convex roller test. The effect of the lubricant on the limit transmissible torque of the concave–convex roller pair was investigated. Furthermore, the experimental results were compared with theoretical ones based on the thermal elastohydrodynamic lubrication theory, which takes account of the effects of viscous heating and Eyring viscosity. Close agreement between the theoretical and experimental results was obtained.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Elastohydrodynamic lubrication; Lubricant; Contact pressure

1. Introduction Many studies on traction drives [1–3] have been reported. Most of these studies have treated the traction characteristics of a convex–convex roller pair. There are, however, circumstances in which traction characteristics of the concave–convex roller pair need to be studied. For example, some traction drives such as the epicyclic roller transmission consist of a concave and convex roller, so it becomes necessary to determine the traction characteristics of a concave–convex roller pair for the design. The effects of the specific sliding, the contact pressure, the roller speed and the surface roughness on traction characteristics of the concave–convex roller pair were determined experimentally by Oda et al. [4]. Theoretical investigation on the traction characteristics of a concave–convex roller pair has not been carried out. In this paper, the transmitted torque, the specific sliding and the roller surface temperature under various contact pressures and roller speeds were simultaneously measured by carrying out a concave–convex roller test. Then effects of lubricants on the limit transmissible

* Corresponding author. Tel.: +81-849-36-2111; fax: +81-849-362213. E-mail address: [email protected] (T. Nonishi).

torque of the concave–convex roller pair were examined by using paraffin oils (P150, P460) and a traction oil (T22). Furthermore, the experimental results were compared with the theoretical ones based on the thermal elastohydrodynamic lubrication (EHL) theory, which takes account of the effects of viscous heating and Eyring viscosity. A close agreement between the theoretical and experimental results was obtained.

2. Experimental apparatus and test procedure 2.1. Test rollers Fig. 1 shows the shapes and dimensions of the test rollers. Table 1 shows the materials, the treatment conditions, the hardness and the surface roughness of the rollers. Table 2 shows the properties of lubricants used in this experiment. 2.2. Concave–convex roller testing machine Fig. 2 shows the concave–convex roller testing machine used in this experiment. The concave roller is supported by bearings fixed to a movable pedestal. The convex roller is inscribed inside the concave roller. Test rollers are loaded by pressing the pedestal with a

0301-679X/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 1 - 6 7 9 X ( 0 0 ) 0 0 0 3 6 - 0

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Nomenclature 2b C C0 c 1, c 2 d E⬘ G H Hm H0 kf k 1, k 2 n1 P p pmax r r1 , r 2 S 1, S 2 T1, T3 Tmax t t0 tm t1, t2 U u 1, u 2 W w X Y a g g0 gP h0 hNm h0a h0b hPa hPb q r 1, r 2 s t0 t1, t2

width of Hertzian contact ratio of half Hertzian width to equivalent radius constant specific heats of rollers 1 and 2 specific gravity of lubricant equivalent modulus of elasticity material parameter [G=aE⬘] dimensionless film thickness constant dimensionless central film thickness without elastic deformation thermal conductivity of lubricant thermal conductivities of rollers 1 and 2 rotational speed of convex roller dimensionless pressure pressure maximum Hertzian contact pressure equivalent radius [r=r1r2/(r2⫺r1)] radii of rollers 1 and 2 boundary positions of lubricating area limit transmissible torque gained for s=1 and 3% maximum limit transmissible torque temperature of lubricant temperature of lubricant in inlet mean temperature of lubricant across film temperatures at surfaces of rollers 1 and 2 speed parameter [U=h0(u1+u2)/(2E⬘r)] velocities of rollers 1 and 2 in x-direction load parameter [W=w/(E⬘r)] load per unit contact length dimensionless abscissa on contact point of roller pair dimensionless coordinate across oil film pressure–viscosity coefficient temperature–viscosity coefficient [g=g0+gPp] temperature–viscosity coefficient at atmospheric pressure P0 increment of temperature–viscosity coefficient per pressure viscosity at P0 and t0 viscosity at t0 viscosity at P0 and ta (=40°C) viscosity at P0 and tb (=60°C) viscosity at P and ta viscosity at P and tb dimensionless temperature densities of rollers 1 and 2 specific sliding [=(u1⫺u2)×100/u1] Eyring stress of lubricant shear stress at surfaces of rollers 1 and 2

hydraulic cylinder. The convex roller is coupled to a VS (variable speed) motor which has a capacity of 7.5 kW, and the concave roller to a direct current (DC) electrical machine (maximum torque 23.5 Nm). The DC electrical machine is able to drive or brake the concave roller,

according to its use as a power source or as a power absorber. By varying the field current of the DC electrical machine, it is possible to control the rotational speed of the concave roller and also the amount of sliding between the concave and convex rollers.

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roller speeds were simultaneously measured. Fig. 3 shows the block diagram for the measurements.

3. Calculation method of traction characteristics based on EHL theory Many studies on the traction characteristics of the roller pair were based on the nonlinear Maxwell model proposed by Johnson and Tevaarwerk [1]. Johnson and Tevaarwerk also showed that the traction is dominated by the nonlinear viscosity and that the elastic effect is negligible at large shear rate. In this study, the specific sliding becomes so large that the elastic behaviour of the nonlinear Maxwell model is considered to be negligible. Therefore, the theoretical calculation of the EHL considering only the effects of viscous heating and Eyring viscosity was carried out and the calculated results were compared with the experimental results. The equations [5,6] used are as follows:

Fig. 1.

冫冤

dP 12CUnm(H−Hm) ⫽ dX H3

Shapes and dimensions of test rollers.

Table 1 Test rollers Roller

Convex roller

Concave roller

0.250

SCM415 Case-hardened Fine ground 703 0.438

Materials Treatment conditions Vickers hardness Hv Surface roughness Rmax µm ⌺Rmax µm

3(z cosh z−sinh z) z3

冪1+再H(2−s)sinh z冎 2snmUxz

⫻

冥

2

nm⫽exp{GP⫺e(qm⫺1)}

0.688

z⫽ 2.3. Test procedure The transmitted torque, the specific sliding and the surface temperature under various contact pressures and

xH dP 2C dx

(2) (3)

再

冎

再

∂2q bH2 2snmUxz 2snmUxz ⫹ ⫻sinh−1 ∂Y2 nmx2 H(2−s)sinh z H(2−s)sinh z ⫽0

Table 2 Properties of lubricants Lubricating oil

Specific gravity Viscosity cSt Viscosity index Flash point °C Pour point °C

40°C 100°C

(1)

Paraffin base P150

P460

Traction oil T22

0.884 150 14.6 96 250 ⫺12.5

0.896 460 29.9 97 292 ⫺12.5

0.863 21.96 3.645 ⬍0 122 ⬍⫺50

冎

(4)

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T. Nonishi et al. / Tribology International 33 (2000) 233–240

Fig. 2.

Concave–convex roller testing machine.

冕

S2

C ⌫ dX m⫽ zW 1

(8)

S1

where the dimensionless variables are defined as follows: Fig. 3.

p x h w P⫽ , X⫽ , H⫽ , W⫽ , E⬘ b r E⬘r

Block diagram for measurements.

冉冊冕

S2

H⫽H0⫹(4W/p)X2⫺

8 W p 2p

U⫽

1/2

P(S)⫻ln(X⫺S)2 dS

(5)

S1

⫹C0

冕

W⫽C P(S) dS

⌫1,⌫2⫽⫾z⫹sinh−1

u1−u2 s⫽ , e⫽gt0⫽e0⫹ePP, e0⫽g0t0, u1

冕 1

S2

S1

h0(u1+u2) b E⬘ , G⫽aE⬘, C⫽ , x⫽ , 2E⬘r r t0

(6)

tm t y eP⫽gPt0E⬘, qm⫽ ⫽ q dY, q⫽ , Y⫽ , t0 t0 h 0

2

再

2snmUxz H(2−s)sinh z

冎

b⫽ (7)

⬘2

rE hNm t1 t2 , nm⫽ , ⌫1⫽ , ⌫2⫽ t0kfh0 h0 t0 t0

T. Nonishi et al. / Tribology International 33 (2000) 233–240

Eqs. (1) and (2) represent the modified Reynolds equation involving the effect of Eyring viscosity. Eq. (4) is the energy equation of the lubricating fluid. Eqs. (7) and (8) represent the dimensionless shear stress on the roller surface and the traction coefficient, respectively. As shown in Eq. (2), constant viscosity across the film is assumed in this study. The viscosity is obtained on the basis of the mean oil temperature (tm) calculated by averaging the oil temperature t across the film. The following conditions are used to solve Eq. (4); At Y=0,

冉 冊 冉 冊冕 X

l2C1/2 2−s q⫽qr2⫽1⫹ 1/2 U 2−2s

1/2

∂q ∂Y Y=0 dS H(X−S)1/2

(9)

S1

and at Y=1,

冉 冊 冉 冊冕 X

1/2

l1C 2−s q⫽qr1⫽1⫹ 1/2 U 2

1/2

∂q − ∂Y Y=1 dS H(X−S)1/2

(10)

S1

where li and qri are defined as follows: li⫽(k 2fh0/pkiriciE⬘r2)1/2, (i⫽1,2) qri⫽ti/t0, (i⫽1,2) The thermal conductivity of the lubricant [2] is determined from: kf⫽1.65⫻

0.117 (1⫺0.00054t) d

(11)

237

4. Experimental results Fig. 4 shows one set of experimental curves of the transmitted torque, the specific sliding and the roller surface temperature measured with P150. These curves were obtained for pmax=486 MPa as specific sliding s varied from 0 to 30% under constant speed of the convex roller (n1=1000 rpm). The transmitted torque increases with s and reaches the maximum value near s=25%. Though the increase of the transmitted torque ceases for sⱖ25%, the roller surface temperature continues to increase. Fig. 5 shows the relationship between the contact pressure and the limit transmissible torques (T1, T3 and Tmax) measured with P150 for n1=1000, 1500 and 2000 rpm. The limit transmissible torque increases with contact pressure. Fig. 6 shows the relationship between the roller speed and the limit transmissible torques measured with P150 and P460 under pmax=687 MPa. The limit transmissible torques decrease with increase of the roller speed. The rates of decrease of T1 and T3 for P460 are larger than those for P150. T1 and T3 for P460 are larger in the range n1⬍1500 rpm, and smaller in the range n1ⱖ1500 rpm than those for P150. Tmax for P150 is larger than that for P460 within the range shown in Fig. 6, and the difference between these values increases with roller speed. Fig. 7 shows the relationship between the contact pressure and the limit transmissible torques measured with P150 and T22 for n1=1500 rpm. The limit transmissible torques for T22 are much larger than those for P150, and increase with contact pressure, and the rates of increase for T22 are much greater than those for P150.

The temperature–viscosity coefficient [2], which is considered to be affected by the pressure, can be written as: g⫽g0⫹gPp, where g0 is determined from the ASTM viscosity–temperature chart and gP, which means the increment of the temperature–viscosity coefficient per the pressure, is determined from

冉 冊

hPah0b hPbh0a gP⫽ p(tb−ta) ln

(12)

Eq. (12) is derived from Eq. (2). In this study, ta and tb are taken as 40 and 60°C, respectively, to calculate gP from Eq. (12). Hertzian mean contact pressure is substituted into p of Eq. (12). Values of hpa and hpb are determined by the Chu-Cameron formula [7] for P150 and by the Wooster formula [7] for T22.

Fig. 4. Simultaneously measured results of transmitted torque, specific sliding and roller surface temperature.

238

Fig. 5. torque.

T. Nonishi et al. / Tribology International 33 (2000) 233–240

Relation between contact pressure and limit transmissible Fig. 7. torque.

Fig. 6.

Relation between roller speed and limit transmissible torque.

5. Comparison between theoretical and experimental results The pressure–viscosity coefficient and the Eyring stress of the lubricant, which are necessary for the theoretical calculation [Eqs. (1)–(12)], are determined from the traction behaviour of the lubricant in the nonlinear region of the traction curve [3]. Fig. 8 shows the pressure–viscosity coefficient and the Eyring stress obtained

Relation between contact pressure and limit transmissible

Fig. 8.

Eyring stress and pressure–viscosity coefficient.

from the traction curves. The Eyring stress increases and the pressure–viscosity coefficient decreases with increase of pressure. The theoretical and experimental results of traction curves for pmax=583 MPa and n1=1500 rpm are shown in Figs. 9 and 10. The traction coefficient for P150 continues to increase with s. The traction coefficient for T22 increases to s=10%, reaches a maximum at s=10%, and then decreases with increase of s. It is found from Figs.

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9 and 10 that the close agreements between the theoretical and experimental results are obtained. The pressure, the film thickness and the lubricant temperature for s=1, 10 and 20% under pmax=583 MPa and n1=1500 rpm calculated by Eqs. (1)–(12) are shown in Fig. 11. The pressure spike and the minimum film thickness decrease, and the lubricant temperature increases with increase of s. Fig. 12 shows the comparison of the theoretical and experimental results of the limit transmissible torques. Close agreements between the theoretical and experimental results are obtained for both P150 and T22. Consequently it is found that the limit transmissible torque of the concave–convex roller pair can be predicted according to the EHL theory considering the viscous heating and Eyring viscosity effect. Fig. 9.

Traction curve (P150).

6. Conclusions The limit transmissible torque was studied investigated by carrying out the concave–convex roller tests with paraffin oils (P150, P460) and traction oil (T22). Furthermore, the experimental results were compared with the theoretical ones based on the thermal elastohydrodynamic lubrication theory. The main results obtained from this investigation are summarized as follows:

Fig. 10. Traction curve (T22).

1. The theoretical traction curve of the concave–convex roller pair shows a close agreement with the measured one. The theoretical results of maximum limit transmissible torque also show close agreement with the measured ones. 2. The limit transmissible torque in the traction drive of the concave–convex roller pair can be predicted by the theoretical calculation method derived in this investigation. 3. The limit transmissible torque of the concave–convex

Fig. 11. Pressure film thickness and temperature (T22, pmax=583 MPa, n1=1500 rpm) h0=0.019 Pas, t0=3.58 MPa, a =26.8 GPa⫺1, k1,2=50 W/mK, r1,2=7.8×103 kg/m3, c1,2=4801 J/kgK, E⬘=230 GPa, g0=0.041 K ⫺1, gP=0.165 GPa⫺1K ⫺1, kf=0.219 W/mK.

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pressure for the traction oil is much greater than that for the paraffin oil.

Acknowledgements The authors would like to thank Mr C. Namba of Tottori University for his assistance in carrying out this investigation. References

Fig. 12. Relation between limit transmissible torque and contact pressure.

roller pair increases with increase of contact pressure and with decrease of the roller speed. 4. The limit transmissible torque of the concave–convex roller pair for the traction oil is much larger than that for the paraffin oil at similar conditions. The rate of increase of the limit transmissible torque with contact

[1] Johnson KL, Tevaarwerk JL. Shear behavior of elastohydrodynamic oil films. Proc Roy Soc Lond 1977;A356:215–36. [2] Muraki M, Kimura Y. Traction characteristics of lubricating oils (3rd Report). J Jpn Soc Lubr Eng 1984;29(3):216–23 (in Japanese). [3] Terauchi Y, Nagamura K, Kamitani S. Behavior of lubricants in elastohydrodynamic lubrication (2nd Report). J Jpn Soc Lubr Eng 1987;32(11):818–24 (in Japanese). [4] Oda S, Miyachika K. Traction characteristics in concave and convex roller pair contacts. Trans Jpn Soc Mech Eng 1987;53(492C):1869–76 (in Japanese). [5] Nonishi T, Terauchi Y. Relation between oil temperature increase and EHL film thickness reduction caused by sliding velocity. Trans Jpn Soc Mech Eng 1996;62(593C):264–9 (in Japanese). [6] Wang S, Cusano C, Conry TF. Thermal analysis of elastohydrodynamic lubrication of line contacts using the Ree–Eyring fluid model. Trans ASME, J Tribol 1991;113(2):232–44. [7] Konishi S, Kamita T. Principle and Application to Lubricating Oil (in Japanese). Corona Publishing Co., Ltd, 1992.