Tribology Lubricants and Lubrication 2012 Part 6 potx

Methodology of Calculation of Dynamics and Hydromechanical Characteristics of Heavy-Loaded Tribounits, Lubricated with Structurally-Non-Uniform and Non-Newtonian Fluids 117 of comparison of calculated results with those obtained experimentally or during operation. On the basis of experimental studies and modern methods of calculation the criteria of performance of hydrodynamic tribounits are developed: the smallest allowable film thickness p er h , maximum allowable hydrodynamic pressure p er p , minimum film thickness reduced to the diameter of the journal, the maximum unit load max f . According to calculations of crankshaft engine bearings of several dimensions the maximum permissible loading parameters listed in the table 5 are obtained. Assessment of performance of bearings is also done according to the calculated value of the relative total lengths of areas per the cycle of loading p er h α and p er p α , where the values of min inf h are less, and max sup p are bigger than acceptable values. Experience has shown that these parameters should not exceed 20% (Fig. 12). Loading parameters Maximum specific load max f , MPa Reduced to the diameter of the journal minimum film thickness, mμ/100 mm The largest hydrodynamic pressure in the lubricating film max sup p , MPa Bearing Type Crank Main Crank Main Crank Main Antifriction material: SB - stalebronzovye inserts coated with lead bronze, SA - staleallyuminievye inserts coated aluminum alloy AMO 1-20 Engine group SB SA SB SA SB SA SB SA SB SA SB SA Highly accelerated 55 49 41 37 2,0 1,2 448 397 336 305 Medium accelerated 52 46 34 30,5 2,3 1,5 420 377 275 245 Low accelerated 45 39,5 31 27 2,5 1,9 367 326 255 225 Table 5. Maximum permissible loading parameters of sliding bearings of a crankshaft of automotive internal combustion engines Fig. 12. The dependence of the hydromechanical characteristics on the rotation angle of crankshaft Tribology - Lubricants and Lubrication 118 6. Conclusion Thus, the methodology of calculating the dynamics and HMCh of heavy-loaded tribounits lubricated by structurally heterogeneous and non-Newtonian fluids, consists of three interrelated tasks: defining the field of hydrodynamic pressures in a thin lubricating film that separates the friction surfaces of a journal and a bearing with an arbitrary law of their relative motion; calculation of the trajectory of the center of the journal; the calculation of the temperature of the lubricating film. Mathematical models used in the calculation must reflect the nature of the live load, lubricant properties, geometry and elastic properties of a construction. The choice of models is built on the working conditions of tribounits in general and the properties of the lubricant. This will allow on the early stages of the design of tribounits to evaluate their bearing capacity, thermal stress and longevity. 7. Acknowledgment The presented work is executed with support of the Federal target program «Scientific and scientifically pedagogical the personnel of innovative Russia» for 2009-2013. 8. References Elrod, H. (1981). A Cavitation Algorithm. Journal of lubrication Technology, Vol.103, No.3, (July 1981), pp. 354-359, ISSN 0201-8160 Prokopiev, V., Rozhdestvensky, Y. et al. (2010). The Dynamics and Lubrication of Tribounits of Piston and Rotary Machines: Ponograph the Part 1, South Ural State University, ISBN 978-5-696-04036-3, Chelyabinsk Prokopiev, V. & Karavayev, V. (2003). The Thermohydrodynamic Lubrication Problem of Heavy-loaded Journal Bearings by Non-Newtonian Fluids, Herald of the SUSU. A series of "Engineering" , Vol.3, No 1(17), pp. 55-66 Whilkinson, U. (1964). Non-Newtonian fluids, Moscow: Mir Gecim, B. (1990). Non-Newtonian Effect of Multigrade Oils on Journal Bearing Perfomance, Tribology Transaction, Vol. 3, No 3, pp. 384-394. Mukhortov, I., Zadorozhnaya, E., Levanov, I. et al. (2010). Improved Model of the Rheological Properties of the Boundary layer of lubricant, Friction and lubrication of machines and mechanisms , No 5, pp. 8-19 Oh, K. & Genka, P. (1985). The Elastohydrodynamic Solution of Journal Bearings Under Dynamic Loading, Journal of Tribology, No 3, pp. 70-76. Bonneau, D. (1995). EHD Analysis, Including Structural Inertia Effect and Mass-Conserving Cavitation Model, Journal of Tribology, Vol. 117, (July 1995), pp. 540-547 Zakharov, S. (1996). Calculation of unsteady-loaded bearings, taking into account the deviation of the shaft and the regime of mixed lubrication, Friction and Wear, Vol.17, No 4, pp. 425-434, ISSN 0202-4977 Zakharov, S. (1996). Tribological Evaluation Criteria of Efficiency of Sliding Bearings of Crankshafts of Internal Combustion Engines, Friction and Wear, Vol.17, No 5, pp. 606 – 615, ISSN 0202-4977 4 The Bearing Friction of Compound Planetary Gears in the Early Stage Design for Cost Saving and Efficiency Attila Csobán Budapest University of Technology and Economics Hungary 1. Introduction The efficiency of planetary gearboxes mainly depends on the tooth- and bearing friction losses. This work shows the new mathematical model and the results of the calculations to compare the tooth and the bearing friction losses in order to determine the efficiency of different types of planetary gears and evaluate the influence of the construction on the bearing friction losses and through it on the efficiency of planetary gears. In order to economy of energy transportation it is very important to find the best gearbox construction for a given application and to reach the highest efficiency. In transmission system of gas turbine powered ships, power stations, wind turbines or other large machines in industry heavy-duty gearboxes are used with high gear ratio, efficiency of which is one of the most important issues. During the design of such equipment the main goal is to find the best constructions fitting to the requirements of the given application and to reduce the friction losses generated in the gearboxes. These heavy-duty tooth gearboxes are often planetary gears being able to meet the following requirements declared against the drive systems: • High specific load carrying capacity • High gear ratio • Small size • Small mass/power ratio in some application • High efficiency. There are some types of planetary gears which ensure high gear ratio, while their power flow is unbeneficial, because a large part of the rolling power (the idle power) circulate inside the planetary gearbox decreasing the efficiency. In the simple planetary gears there is no idle power circulation. Therefore heavy-duty planetary drives are set together of simple planetary gears in order to transmit megawatts or even more power, while they must be compact and efficient. 2. Planetary gearbox types The two- and three-stage planetary gears consisting of simple planetary gears are able to meet the requirements mentioned above [Fig. 1(a)–1(d).]. Tribology - Lubricants and Lubrication 120 Varying the inner gear ratio (the ratio of tooth number of the ring gear and of the sun gear) of each simple planetary gear stage KB the performance of the whole combined planetary gear can be changed and tailored to the requirements. There are special types of combined planetary gears containing simple KB units (differential planetary gears), which can divide the applied power between the planetary stages thereby increasing the specific load carrying capacity and efficiency of the whole planetary drives [Fig. 1(b)-1.(d)]. Proper connections between the elements of the stages in these differential planetary gears do not result idle power circulation. (a) (b) (c) (d) Fig. 1. (a) Gearbox KB+KB; (b). Planetary gear PKG; (c). Planetary gear PV; (d). Planetary gear GPV The efficiency of planetary gears depends on the various sources of friction losses developed in the gearboxes. The main source of energy loss is the tooth friction of meshing gears, which mainly depends on the arrangements of the gears and the power flow inside the planetary gear drives. The tooth friction loss is influenced by the applied load, the entraining speed and the geometry of gears, the roughness of mating tooth surfaces and the viscosity of lubricant. Designers of planetary gear drives can modify the geometry of tooth profile in order to lower the tooth friction loss and to reach a higher efficiency [Csobán, 2009]. The Bearing Friction of Compound Planetary Gears in the Early Stage Design for Cost Saving and Efficiency 121 3. Friction loss model of roller bearings It is important to find the parameters (such as inner gear ratio, optimal power flow) of a compound planetary gear drive which result its highest performance for a given application. The power flow and the power distribution between the stages of a compound gearbox is also a function of the power losses generated mainly by the friction of mashing teeth and the bearing friction. This is why it is beneficial, when, during the design of a planetary gear beside the tooth friction loss also the friction of rolling bearings is taken into consideration even in the early stage of design. In this work a new method is suggested for calculate the rolling bearing friction losses without knowing the exact sizes and types of the bearings. In this model first the torque and applied loads (loading forces and, if possible, bending moments) originated from the tooth forces between the mating teeth have to be determined. Thereafter the average diameter of the bearing d m can be calculated as a function of the applied load and the prescribed bearing lifetime. Knowing the average diameter d m , the friction loss of bearings can be counted using the methods suggested by the bearing manufacturers based on the Palmgren model [SKF 1989]. For determining the functions between the bearing average diameter and between the basic dynamic, static load, inner and outer diameter [Fig. 2-6.], the data were collected from SKF catalog [SKF 2005]. The functions between the bearing parameters (inner diameter d b , dynamic basic load C) and the average diameters d m being necessary for calculation of the friction moment and the load can be searched in the following form: d m Ycd = ⋅    (1) The equations of the diagrams [Fig. 2-6.] give the values of c and d for the inner diameter of the bearings d b and for the basic dynamic loads C of the bearings. Knowing the torque M 24 and the strength of the materials of the shafts ( τ m , σ m ) the mean diameter of the bearing for central gears (sun gear, ring gear) necessary to carry the load can be calculated using the following formula: 2;4 2;4 3 16 () d m m M dd c τ π ⋅ ⋅ =   (2) Calculating the tangential components of the tooth forces the applied radial loads of the planet gear shafts F r can be determined (which are the resultant forces of the two tangential components F t2 and F t4 ). The shafts of the planet gears are sheared and bended by the heavy radial forces, this is why, in this analysis, at the calculation of shaft diameter, once the shear stresses, then the bending stresses are considered. Calculating the maximal bending moment M hmax of the planet gear shafts, and the allowable equivalent stress σ m of planet gear pins, the bearing inner diameter d b necessary to carry the applied load of the planet gear shaft and the average bearing diameter d m3 can be calculated: max 3 3 32 () h d m m M dd c σπ ⋅ ⋅ =   (3) Tribology - Lubricants and Lubrication 122 The functions between the bearing geometry and load carrying capacity for deep groove ball bearings [Fig. 2(a)-2(d)]. The points are the average data of the bearings taken from SKF Catalog [SKF 2005] and the continuous lines are the developed functions between the parameters. d - d m y = 0,3984x 1,1179 R 2 = 0,9996 0 400 800 1200 1600 0 200 400 600 800 1000 1200 1400 1600 d m [mm] d [mm] D - d m y = 1,7374x 0,9364 R 2 = 0,9998 0 400 800 1200 1600 0 200 400 600 800 1000 1200 1400 1600 d m [mm] D [m m ] (a) (b) C - d m y = 109,09x 1,3236 R 2 = 0,9682 0 400000 800000 1200000 0 200 400 600 800 1000 1200 1400 1600 d m [mm] C [N] C 0 - d m y = 12,603x 1,7564 R 2 = 0,9912 0 1000000 2000000 3000000 4000000 0 200 400 600 800 1000 1200 1400 1600 d m [mm] C 0 [N] (c) (d) Fig. 2. (a) The average inner diameter of the deep groove ball bearing as a function of its average diameter. (b). The average outer diameter of deep groove ball bearing as a function of the average diameter. (c). The average basic dynamic load of deep groove ball bearing as a function of the average diameter. (d). The average static load of deep groove ball bearing as a function of the average diameter The Bearing Friction of Compound Planetary Gears in the Early Stage Design for Cost Saving and Efficiency 123 The functions between the bearing geometry and load carrying capacity for cylindrical roller bearings [Fig. 3(a)-3(d)]. d - d m y = 0,4167x 1,0966 R 2 = 0,9993 0 400 800 1200 0 200 400 600 800 1000 d m [mm] d [m m ] D - d m y = 1,7114x 0,9481 R 2 = 0,9997 0 400 800 1200 0 200 400 600 800 1000 d m [mm] D [mm] (a) (b) C - d m y = 69,121x 1,6675 R 2 = 0,9808 0 2000000 4000000 6000000 8000000 0 200 400 600 800 1000 d m [mm] C [N] C 0 - d m y = 22,552x 1,9273 R 2 = 0,9863 0 4000000 8000000 12000000 16000000 0 200 400 600 800 1000 d m [mm] C 0 [N] (c) (d) Fig. 3. (a) The average inner diameter of the cylindrical roller bearing as a function of its average diameter. (b). The average outer diameter of cylindrical roller bearing as a function of the average diameter. (c). The average basic dynamic load of the cylindrical roller bearing as a function of its average diameter. (d). The average static load of different types of cylindrical roller bearing as a function of the average diameter Tribology - Lubricants and Lubrication 124 The functions between the bearing geometry and load carrying capacity for full complement cylindrical roller bearings [Fig. 4(a)-4(d)]. d - d m y = 0,4595x 1,0951 R 2 = 0,9987 0 400 800 1200 0 200 400 600 800 1000 1200 d m [mm] d [mm] D - d m y = 1,6961x 0,9399 R 2 = 0,9993 0 400 800 1200 0 200 400 600 800 1000 1200 d m [mm] D [mm] (a) (b) C - d m y = 444,04x 1,3465 R 2 = 0,9621 0 2000000 4000000 6000000 8000000 0 200 400 600 800 1000 1200 d m [mm] C [N] C 0 - d m y = 164,13x 1,6125 R 2 = 0,9845 0 4000000 8000000 12000000 16000000 0 200 400 600 800 1000 1200 d m [mm] C 0 [N] (c) (d) Fig. 4. (a) The average inner diameter of the full complement cylindrical roller bearing as a function of its average diameter. (b). The average outer diameter of the full complement cylindrical roller bearing as a function of the average diameter. (c). The average basic dynamic load of the full complement cylindrical roller bearing as a function of its average diameter. (d). The average static load of different types of full complement cylindrical roller bearing as a function of the average diameter The Bearing Friction of Compound Planetary Gears in the Early Stage Design for Cost Saving and Efficiency 125 The functions between the bearing geometry and load carrying capacity for spherical roller bearings [Fig. 5(a)-5(d)]. d - d m y = 0,4331x 1,0936 R 2 = 0,9995 0 400 800 1200 1600 2000 0 500 1000 1500 2000 d m [mm] d [m m ] D - d m y = 1,72x 0,9448 R 2 = 0,9997 0 400 800 1200 1600 2000 0 500 1000 1500 2000 dm [mm] D [mm] (a) (b) C - d m y = 188,21x 1,5827 R 2 = 0,987 0 7000000 14000000 21000000 28000000 0 500 1000 1500 2000 d m [mm] C [N] C 0 - d m y = 45,547x 1,9064 R 2 = 0,9965 0 14000000 28000000 42000000 56000000 0 500 1000 1500 2000 d m [mm] C 0 [N] (c) (d) Fig. 5. (a) The average inner diameter of the spherical roller bearing as a function of its average diameter. (b). The average outer diameter of spherical roller bearing as a function of the average diameter. (c). The average basic dynamic load of spherical roller bearing as a function of its average diameter. (d). The average static load of spherical roller bearing as a function of its average diameter [...]... 22,552 c 0,4595 1 ,69 61 444,04 164 ,13 c 0,4331 1,72 188,21 45,547 c 0, 569 1 1,4795 135,35 59, 869 1,1179 0,9 364 1,32 36 1,7 564 d 1,0 966 0,9481 1 ,66 75 1,9273 d 1,0951 0,9399 1,3 465 1 ,61 25 d 1,09 36 0,9448 1,5827 1,9 064 d 1,0529 0, 967 4 1 ,64 35 1, 863 7 Table 1 Parameters for the bearing selection Fig 7 The bearing selecting and efficiency calculation algorithm 130 Tribology - Lubricants and Lubrication 4 Comparing...1 26 Tribology - Lubricants and Lubrication The functions between the bearing geometry and load carrying capacity for of CARB toroidal roller bearings [Fig 6( a) -6( d)] d - dm 1200 D - dm 160 0 1200 D d [m m ] [m m ] 800 800 0, 967 4 y = 1,4795x 1,0529 y = 0, 569 1x 2 R = 0,9993 400 2 R = 0,9997 400 0 0 0 200 400 60 0 800 1000 1200 1400 0 200 400 60 0 800 1000 1200 dm [mm] dm [mm]... statement and solutions of the problem of determining three-dimensional stress-strain state of the models of pipes with corrosion defects under the action of internal pressure, friction caused by oil flow and temperature discussed in the present chapter are 140 Tribology - Lubricants and Lubrication important for pipeline systems and such related disciplines as solid mechanics, fluid mechanics and tribology. .. planetary gear PV as a function of gear ratio Prescribed gearbox lifetime=50000[h] 134 Tribology - Lubricants and Lubrication 100 90 80 v i/Σ v [%] 70 v tooth (ib"=2) v bearing (ib"=4) v bearing (ib" =6) 40 v tooth (ib"=8) v bearing (ib"=2) 50 v tooth (ib"=4) v tooth (ib" =6) 60 v bearing (ib"=8) 30 20 10 0 0 20 40 i PKG 60 80 100 Fig 12 The power loss ratio of planetary gear PKG as a function of gear ratio... Gépészet 20 06 ISBN 963 593 465 3 Csobán Attila, Kozma Mihály: Influence of the Power Flow and the Inner Gear Ratios on the Efficiency of Heavy-Duty Differential Planetary Gears, 16th International Colloquium Tribology, Technische Akademie Esslingen, 2008 Csobán Attila, Kozma Mihály: A model for calculating the Oil Churning, the Bearing and the Tooth Friction Generated in Planetary Gears, World Tribology. .. parameters of Table 2 and 3 were used σF ηM [MPa] 500 [mPas] 63 Ra23 [μm] 0 ,63 Ra34 [μm] 1,25 Pin [kW] 2000 β [°] 0 nin [1/min] 1500 x2 [-] 0 N [-] 3 b/dw [-] 0,8 Table 2 Other important parameters for the analyses a b f0 f1 c(db) d(db) c(L1h) d(L1h) 1 1 7,5 0,00055 0,4595 1,0951 444,04 1,3 465 Table 3 Parameters for calculate the bearing friction losses 132 Tribology - Lubricants and Lubrication The results... gear ratio, is the ratio of the number of teeth of sun gear and ring gear at the third stage, is the ratio of the number of teeth of sun gear and ring gear at the second stage, 138 ib” k L1h M0 M1 M2;4 M3 n nin ν P1 Pin Ra σF σm, τm Σv v V vBearing vtooth ω3g Y Tribology - Lubricants and Lubrication is the ratio of the number of teeth of sun gear and ring gear at the first stage, planetary carrier, prescribed... tooth (ib" =6) 100 v bearing (ib"=8) v i/Σ v [%] 70 60 50 40 30 20 10 0 0 20 40 60 i PV 80 100 Fig 10 The power loss ratio of planetary gear PV as a function of gear ratio Prescribed gearbox lifetime=5000[h] v tooth (ib"=2) v bearing (ib"=4) v bearing (ib" =6) 80 v tooth (ib"=8) v bearing (ib"=2) 90 v tooth (ib"=4) v tooth (ib" =6) 100 v bearing (ib"=8) v i/Σ v [%] 70 60 50 40 30 20 10 0 0 20 40 60 i PV... ratio of the first stage while the inner gear ratios of the second and third stage were changed and combined (Figure 16- 17) 100 v i/Σ v [% ] 90 80 70 60 v tooth (ib"=8, ib'=2) v tooth (ib'=4) 50 40 v tooth (ib' =6) v tooth (ib'=8) v bearing (ib"=8, ib'=2) v bearing (ib'=4) v bearing (ib' =6) v bearing (ib'=8) 30 20 10 0 0 50 i GPV 100 Fig 16 The power loss ratio of planetary gear GPV as a function of gear... undercut or too thin top land 100 90 80 v i/Σ v [% ] 70 v tooth (ib"=2) v bearing (ib"=4) v bearing (ib" =6) 40 v tooth (ib"=8) v bearing (ib"=2) 50 v tooth (ib"=4) v tooth (ib" =6) 60 v bearing (ib"=8) 30 20 10 0 0 20 40 60 i KB+KB 80 100 Fig 8 The power loss ratio of planetary gear KB+KB as a function of gear ratio Prescribed gearbox lifetime=5000[h] 100 90 80 v i/Σ v [% ] 70 60 50 40 30 v tooth (ib"=2) . – d m 1,7374 0,9 364 C – d m 109,09 1,32 36 C 0 – d m 12 ,60 3 1,7 564 Cylindrical roller bearings c  d  d – d m 0,4 167 1,0 966 D – d m 1,7114 0,9481 C – d m 69 ,121 1 ,66 75 C 0 – d m 22,552. diameter Tribology - Lubricants and Lubrication 1 26 The functions between the bearing geometry and load carrying capacity for of CARB toroidal roller bearings [Fig. 6( a) -6( d)]. d. 444,04x 1,3 465 R 2 = 0, 962 1 0 2000000 4000000 60 00000 8000000 0 200 400 60 0 800 1000 1200 d m [mm] C [N] C 0 - d m y = 164 ,13x 1 ,61 25 R 2 = 0,9845 0 4000000 8000000 12000000 160 00000 0 200 400 60 0 800 1000