MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular 24.41 Formula Is ¼ moment of inertia of the slotted cross-section ring, mm4 lco ¼ twice the diameter between center of gravity and outside diameter, mm The bending stress present in the case of slotted oil control ring of rectangular cross section in terms of tangential load, F The tangential load or force required for opening of a rectangular cross-section piston ringa bso ¼ 6ðd À hÞlco Im F wh3 Is ð24-142dÞ where bso in N/mm2 ; F in N; lco , d, h, and w in mm; Im and Is in mm4 ;max ẳ hE 1:26"T 1:84k ỵ 0:025ị dh SI 24-142eị where dỵh dh k ẳ piston ring parameter from Eq (24-142f ) and (24-142h) "T ¼ 3ðd À hÞ2 F E wh3 The piston ring parameter (k) in terms of tangential load F for rectangular cross-section rings k¼ The tangential load or force required for opening of rectangular cross-section slotted oil control rings ;max T ẳ 24-142f ị lci E I 1:26"T 1:84k ỵ 0:025ị m dh Is 24-142gị where dỵh SI dh lci ẳ twice the distance between center of gravity and inside diameter, mm "T ¼ k ¼ piston ring parameter from Eq (24-142e) and (24-142f ) The piston ring parameter (k) in terms of free ring gap (f ) for rectangular cross-section slotted oil rings for use in Eq (24-142g) k¼ f 3 d À h ð24-142hÞ The piston ring parameter (k) in terms of the constant pressure ( p) for rectangular cross-section rings also for use in Eqs (24-142f ) and (24-142g) kẳ p dd hị2 2E h3 24-142iị The radial thickness of the ring at a section which makes an angle measured from the center line of the gap of the ring a sffiffiffiffiffiffiffiffiffiffiffi 24pr sin2 h¼ E Goetze AG, Piston Ring Manual, 3rd ed., Burscheid, Germany, 1987 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-142jÞ MISCELLANEOUS MACHINE ELEMENTS 24.42 CHAPTER TWENTY-FOUR Particular Formula The maximum thickness of the ring which occur opposite the gap of the ring (i.e., at ẳ ) hmax s 24pr4 ẳ E 24-142kị For piston ring dimensional deviation, hardness, and minimum wall pressure Refer to Tables 24-7 to 24-9 For cylinder bore diameter Refer to Table 24-10 TABLE 24-5 Values of c for various inclinations of coned pistons Cone Inclination ranges, , deg c — Slightly Medium Strong 0–6 6–18 18–28 28–35 0.85–0.95 0.75–0.85 0.65–0.75 TABLE 24-6 Values of coefficient K for pistons (admissible pressures, kgf/mm2 absolute) Pressure, kgf/mm2 Diameter of cylinder, mm 0.01 to 0.02 0.02 to 0.04 0.04 to 0.06 0.06 to 0.08 0.08 to 0.10 0.10 to 0.12 0.12 to 0.14 0.14 to 0.16 380–575 575–775 775–975 975–1175 1175–1375 1375–1575 1575–1775 1775–1975 1975–2150 2150–2350 2350–2550 2550–2750 1.000 1.375 1.500 1.750 2.000 2.375 2.500 2.750 3.000 3.000 3.125 3.375 1.125 1.500 1.750 2.000 2.250 2.500 3.000 3.125 3.375 3.500 3.500 3.750 1.375 1.750 2.000 2.375 2.750 3.125 3.375 3.500 3.750 4.000 1.500 2.000 2.500 3.000 3.125 3.500 4.000 4.125 4.375 1.750 2.500 3.125 3.500 3.750 4.000 4.375 2.000 2.750 3.500 4.000 4.125 4.375 2.125 3.000 3.750 4.500 2.500 3.375 4.000 Key: kgf/mm2 ¼ 1:42247 kpsi; kpsi ¼ 6:894757 MPa Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS 24.43 MISCELLANEOUS MACHINE ELEMENTS TABLE 24-7 Recommended hardness for piston rings of IC engines Nominal diameter, d, mm 200 TABLE 24-8 Minimum wall pressure for piston rings of IC engines Hardness HRD Compression rings Oil rings MPa 95–107 93–105 90–102 Petrola Diesel a kgf/cm MPa kgf/cm2 0.059 0.013 0.60 1.05 0.137 0.196 1.40 2.00 Gasoline TABLE 24-9 Permissible deviation on the dimensions of piston rings of IC engines Dimensions Deviations, mm Axial width, b À0.010 À0.022 Radial thickness 80 mm ring diameter >80 mm with 175 mm ring diameter 175 mm ring diameter Parallelism of sides—40% of tolerance on axial width Æ0.08 Æ0.12 Æ0.15 TABLE 24-10A Preferred cylinder bore diameters for internal-combustion (IC) engines (all dimensions in mm) 30 32 34 (35) 36 38 40 42 44 46 48 50 52 54 56 (57) 58 (59) 60 (62) 65 (68) 70 (72) (73) 74 (76) (78) (79.4) 80 82 85 87.3 88 (88.9) 90 (91.4) (92) 95 98 (98.4) 100 (101.6) (102) (103.2) (104.8) 105 108 110 (111.1) 112 (114.3) 115 (118) 120 (120.6) (122) 125 (127) (128) (128.2) 130 (132) (133.4) 135 (138) (139.7) 140 (142) 142.9 (145) (146) (148) (149.9) 150 (152) (152.4) 155 (158) (158.8) 160 (162) 165 (165.1) (168) 170 (171.4) (172) 175 (177.8) (178) 180 (182) (184.2) 185 (188) 190 (190.5) (192) 195 (196.8) 198 200 (205) (209.6) 210 (215) (215.9) 220 (225) (228.6) 230 (235) 240 (241.3) (245) (250) (254) (255) 266 (265) 270 (273) (275) 280 (285) 290 (292.1) (295) (298.4) 300 (305) 103 315 (317.5) 320 (325) 330 (335) 340 (343) (345) 350 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 34,850 49,582 49,285 2285 4658 4928A 4928B 3.5–4.5 0.8–1.5 0.8–1.5 0.8–1.5 Cu 1.2–1.8 0.8–1.3 0.8–1.3 2.8–1.3 Mg 0.6 11.0–13.0 17.0–19.0 23.0–26.0 Si 0.7 0.8 0.7 0.7 Fe 0.2 0.2 0.2 0.2 Mn 1.1–2.3 1.5 0.8–1.3 0.1–1.3 Nic 0.2 0.35 0.2 0.2 Zn Chemical composition, % 0.23 0.2 0.2 0.2 Ti 0.05 0.05 0.05 0.05 So 0.05 0.05 0.05 0.05 Pb c b Alloys have been designated in accordance with IS 6051, 1970 Code for designation of aluminum and aluminum alloys Physical properties are attainable after suitable heat treatment The purchaser may specify nickel content, if so desired Source: Bureau of Indian Standards, New Delhi a Forging Casting Alloy designationa 0.3–0.6 Cr MPa 225–275 195–245 175–215 165–205 Hardness, HB 90–130 90–140 90–125 90–125 Al 32.7–39.8 28.5–35.6 25.6–31.3 24.2–29.7 kpsi 49.8–59.7 42.7–52.6 32.7–28.5 kpsi Forging 345–410 295–365 225–295 MPa Tensile strength Chill casting Remainder Physical propertiesb TABLE 24-10B Chemical composition of alloys and physical properties of aluminum alloy piston (values in % maximum unless shown otherwise) 23–24 20.5–21.5 18.5–19.5 17–18 mm/mm/ 8C Â 10À4 Coefficient of thermal expansion (20 to 2008C) MISCELLANEOUS MACHINE ELEMENTS 24.44 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS 24.45 TABLE 24-10C Preferred number of piston rings Differential pressure Std atm MPa psi Minimum number of rings 0–9 10–14 15–24 25–29 30–49 50–99 100–200 0–0.88 0.98–1.37 1.47–2.35 2.45–2.85 2.94–4.80 4.90–9.71 9.81–19.61 0–128 142–199 213–341 355–412 426–696 710–1406 1422–2844 Source: M J Neale, Tribology Handbook, Butterworth-Heinemann, London, 1973; reproduced with permission TABLE 24-10D Properties of typical piston ring materials Tensile strength, t Nominal modulus of elasticity, En Typical coefficient of expansion, MPa Material Metallic: Gray irons Carbide malleable irons Malleable and/or nodular irons Sintered irons Nonmetallic: Carbon-filled PTFE Graphite/MoS2 -filled PTFE Resin-bonded PTFE Carbon Resin-bonded carbon Glass-filled PTFE Bronze-filled PTFE Resin-bonded fabric kpsi GPa Mpsi Brinell hardness number, HB Bulk density, g/cm3 Â106 =8C 230–310 400–580 540–820 33.4–45.0 58.0–84.1 78.3–119.0 83–124 140–160 155–165 12.1–18.0 20.3–23.2 22.5–24.0 210/310 250/320 200/440 Good Excellent Poor 250–390 36.5–56.6 120 17.4 130/150 Good 10.3 19.6 1.49 2.85 2.05 2.20 55 115 29.4 43.4 19.6 16.7 12.8 110.8 4.27 6.30 2.85 2.42 1.85 16.07 1.75 1.8 1.9 2.26 3.90 1.36 30 43 20 80 118 22.5/87.5a a Material is anisotropic Source: M J Neale, Tribology Handbook, Butterworth-Heinemann, London, 1973, extracted with permission Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Wear rating MISCELLANEOUS MACHINE ELEMENTS 24.46 CHAPTER TWENTY-FOUR TABLE 24-10E Piston rings and piston ring elements Mechanical properties Designation Grade Hardness Tensile strength, st , MPa Modulus of elasticity, E, MPa Steel GOE 61 GOE 62 Cr steel, 17% Cr Cr-Si steel 380–450 HV 30 500–600 HV 30 1200Ã approx 1900Ã approx 230 000 approx 210 000 approx Cr-Si steel Cr steel, 11% Cr min, high C Cr steel, 11% Cr min, low C 450–550 HV 30 300–400 HV 30 1700Ã approx 1300Ã approx 210 000 approx 210 000 approx 270–420 HV 30 1300Ã approx 220 000 approx GOE 64 GOE 65A GOE 65B Cast Iron GOE 12 GOE 13 GOE 32 GOE 44 GOE 52 GOE 56 Unalloyed non heattreated gray cast iron Unalloyed non heattreated gray cast iron Alloyed heat-treated gray cast iron with carbides Malleable cast iron Spheroidal graphite cast iron Spheroidal graphite cast iron Main application Compression rings Coil spring loaded rings Compression rings Compression rings, nitrided Coil spring loaded rings and segments, nitrided 94–106 HRB 350 85 000 typical 97–108 HRB 420 95 000–125 000 109–116 HRB 650 130 000–160 000 Compression and oil control rings Compression and oil control rings Compression rings 102–111 HRB 104–112 HRB 800 1300 150 000 150 000 Compression rings Compression rings 40–46 HRC 1300 150 000 Compression rings Source: Goetze Federal Mogul Burscheid GmbH, Piston Ring Manual, 4th ed., January 1995, Burscheid, Germany, reproduced with permission Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS 24.5 DESIGN OF SPEED REDUCTION GEARS AND VARIABLE-SPEED DRIVES SYMBOLS2,3 center distance, m (in) number of pinions or planetary pinion (Fig 24-36) center distance (also with subscripts) (Fig 24-36) area of reduction gear housing, m2 (in2 ) noncooled, i.e., ribbed, surface of housing of reduction gear drive, m2 (in2 ) cooled surface of reduction gear drive, m2 (in2 ) surface area of contact of teeth when one-fourth of all teeth of wheel in wave-type reduction gears are engaged, m2 (in2 ) width of rim, m (in) diameter of pinion, m (in) diameter of rigid immovable rim with internal teeth of wave-type reduction gears, m (in) diameter of gear, m (in) diameter of flexible movable wheel rim with external teeth of wave-type reduction gear, m (in) maximum diameter of the circumference of the belt arrangement on the V-belt of a variable-speed drive, m (in) minimum diameter of the circumference of the belt arrangement on the V-belt of a variable-speed drive, m (in) a A An Ac Aw b d1 d2 dmax dmin D¼ dmax dmin velocity control range for a V-belt drive with only one adjustable pulley velocity control range for a V-belt drive with two adjustable pulleys working height of a V-groove of the pulley, m (in) maximum load acting on the pinion, kN (lbf ) mean load acting on the pinion, kN (lbf ) height of tooth, m (in) coefficient of heat transfer, W/m2 K (Btu/ft2 h 8F) coefficient of heat transfer of noncooled surface, W/m2 K (Btu/ ft2 h 8R) coefficient of heat transfer of cooled surface, W/m2 K (Btu/ft2 h 8R) addendum of tooth, m (in) dedendum of tooth, m (in) transmission or speed ratio D1 D2 e Fmax Fm h hn hc hf i knl ¼ L m Mts n n1 , n2 q È velocity control range for a V-belt drive Fmax Fm nonuniform load distribution factor distance between the axes of the pinions (Fig 24-36d) module, m (in) torque acting on smaller wheel, N m (lbf in) speed, rpm speeds of pinion and gear, respectively, rpm a whole number heat generated, W (Btu/h) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 24.47 MISCELLANEOUS MACHINE ELEMENTS 24.48 CHAPTER TWENTY-FOUR maximum radius of the circumference of the belt arrangement on the V-belt of a variable-speed drive, m (in) minimum radius of the circumference of the belt arrangement on the V-belt of a variable-speed drive, m (in) temperature of lubricant, 8C (8F) ambient temperature, 8C (8F) number of teeth on sun pinion and planetary pinion of epicyclic gear transmission, respectively, Fig 24-36 number of teeth on pinion and gear, respectively number of teeth on ring gear (Fig 24-36a) number of teeth on smaller wheel angular speed of pinion and gear, respectively, rad/s deformation, m (in) clearance between the pinions which should be at least mm (in) half-cone angle of V-belt, deg allowable compressive stress, MPa (psi) rmax rmin t1 ta z1 , z2 z3 zs !1 , !2 Á ca Particular Formula !1 n1 d2 z2 ¼ ¼ ¼ !2 n2 d1 z1 Transmission or speed ratio for single reduction gear i¼ For different types of gear reduction drives Refer to Fig 24-35 and Table 24-11 b dmax dmin h e Belt 2α Belt (a) V-belt at top position (b) V-belt at bottom position FIGURE 24-34 Dimension of V-belt variable-speed drive Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-143Þ MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Single-reduction spur gear Single-reduction bevel gear Single-reduction worm gear with worm arranged sideways H 2 H L Double-reduction coaxial gear Double-reduction bevel spur gear H L Double-reduction spur and hellcal gear I (a) (b) L H L Single-reduction worm gear with worm on top 12 Combination wormspur helical reduction gear I I H l 11 Double-reduction worm gear H 4 Triple-reduction spur and helical gear H Single-reduction worm gear with worm underneath L I L L (b) L H H (a) 10 Single-reduction worm gear with worm arranged vertically H H L I H L H L l 4 L I H L H = High Speed H L I = Intermediate Speed L = Low Speed FIGURE 24-35 Schematic diagrams of various types of spur, helical, herringbone, bevel, and worm reduction gears Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 24.49 MISCELLANEOUS MACHINE ELEMENTS 24.50 CHAPTER TWENTY-FOUR Particular Formula PLANETARY REDUCTION GEARS L ¼ 2A1;2 sin First conditionmating ẳ z2 m ỵ 2m1 ỵ ị þ Á a ð24-144Þ where The sum of the radii of the addendum circles of the mating pinions in planetary reduction gears should be smaller than the distance between their axes (Fig 24-36d) so that the top of the pinions should not touch each other a ¼ number of pinions Á ¼ clearance between the pinions, which should be at least mm A1;2 ¼ center distance as shown in Fig 24-36 2π a A 1,2 A2,3 2 A1,2 A2,3 A1,2 3 π a ∆ L (b) (a) i=8 i = 15 (c) (d) i = 20 to 100 FIGURE 24-36 Planetary reduction gears Second condition—coaxiality The center distance of each pair of wheels should be equal (Fig 24-36) A12 ¼ A23 ; A12 ¼ A23 ¼ A20 30 The relationship between teeth in corrected or uncorrected gears (Fig 24-36a) z1 ỵ z2 ẳ z3 z2 24-145ị 24-146aị or z1 ỵ 2z2 ẳ z3 24-146bị The relationship between teeth in corrected or uncorrected gears (Fig 24-36c) to ratify two conditions (i) First condition Refer to Eq (24-146) (ii) Second condition m2 z3 z2 ị ẳ m02 z03 z02 Þ ð24-147aÞ or z3 À z2 ¼ z03 À z02 since m2 ¼ m02 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-147bÞ MISCELLANEOUS MACHINE ELEMENTS 24.66 CHAPTER TWENTY-FOUR Particular Formula Driving shaft The horizontal force due to thrust Fa on bearing D The horizontal force due to the tangential force Ft on the bearing D bFt a q ẳ F2 ỵ F2 hDa hDt The horizontal force due to thrust Fa on the bearing C The horizontal force due to the tangential force Ft on the bearing C FhDt ẳ 24-205ị FRD The resultant force on the bearing D The resultant force on the bearing C d1 Ft ð24-204Þ 2a where d1 and d2 denote the minimum and maximum diameters of driving disk FhDa ẳ 24-206ị Fhca ẳ d1 Ft 2a 24-207ị a ỵ bịFt a q ẳ F2 ỵ F2 hca hct Fhct ẳ 24-208ị FRc ð24-209Þ TABLE 24-17 Design data for friction gearing Allowable pressure, p0 Material of driver kN/m lbf/in Coefficient of friction with cast iron Cast iron Cork composition Paper Rubber Wood 530 8.9 26.5 17.7 26.5 3000 50 150 100 150 0.15 0.21 0.15 0.20 0.15 Allowable pressure, p0 Material of driver kN/m Leather Leather fiber Straw fiber Sulfite fiber Tarred fiber 26.5 42.2 26.5 24.5 44.1 lbf/in Coefficient of friction with cast iron Coefficient of friction with aluminum 150 240 150 140 250 0.09 0.18 0.15 0.20 0.28 0.13 0.18 0.16 0.19 0.28 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS 24.7 MECHANICS OF VEHICLES SYMBOLS2,3 a A b B c C Dt Dw Ef Esh F Fmax G h i k l l0 l1 l2 l3 l4 l5 l6 l7 l8 m Mt Mtt Mti Mto Mtf Mtr n n0 ni no P r rmi rmo center distance, m (in) a constant in Eq (24-216b) frontal projected area of vehicle, m2 (ft2 ) face width of gear, m (in) a constant in Eq (24-216b) width of bearing, m (in) distance between adjacent rotating parts, m (in) constant (also with suffixes) maximum diameter of torus, m (in) diameter of wheel, m (in) flow loss in each member of hydraulic torque converter, N m (lbf in) shock loss in each member of hydraulic torque converter, N m (lbf in) driving force at the tire, kN (lbf ) maximum permissible load on the pitch circle of any particular pair of gears, kN (lbf ) gradient thickness of housing, m (in) gear ratio (total) a constant distance between support bearings on a shaft in gearbox, m (in) distance between bearings of overhanging shaft, m (in) distance of rotating part from the bearing, m (in) distance of bearing from the wall, m (in) cap height from bolt to end, m (in) distance of rotating parts from the bearing cap, m (in) width of boss of rotating parts, m (in) distance of coupling to cap, m (in) distance between gear and shaft, m (in) distance of rotating parts from inner wall of housing, m (in) module, m (in) output torque of the engine, N m (lbf in) torque at the tire surface, N m (lbf in) the input torque, N m (lbf in) the reaction to the output torque, which is opposite in direction to output torque, N m (lbf in) the torque that must be applied to transmission housing to balance the moments of internal friction, oil churning, etc., N m (lbf in) the torque reaction of the transmission housing due to the gear reduction in transmission, N m (lbf in) speed, rpm speed, rps speed of driving shaft, rpm speed of driven shaft, rpm power, kW (hp) radius of the driving wheel, m (in) mean radius of inflow to the runner, m (in) mean radius of outflow from the runner, m (in) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 24.67 MISCELLANEOUS MACHINE ELEMENTS 24.68 CHAPTER TWENTY-FOUR air resistance, kN (lbf ) rolling resistance, kN (lbf ) road resistance, kN/tf (lbf/ton) gradient resistance, kN (lbf ) total resistance, kN (lbf ) tonne, t tonne force, tf velocity, m/s (ft/min) speed of vehicle, km/h (ft/s) velocity of fluid relative to the vane, m/s (ft/min) shock velocity, m/s (ft/min) weight of the vehicle, kN (Tonf ) number of teeth angle of inclination of road, deg angle of repose, deg minimum clearance between gears and inner wall of housing, m (in) transmission efficiency Ra Rr R00 r Rg Rt t tf v V Vf Vsh W z Á SUFFIXES b t max pinion gear brake tonne maximum minimum Other factors in performance or in special aspects which are included from time to time in this section and, being applicable only in their immediate context, are not given at this stage Particular Formula CALCULATION OF POWER 1000P SI ð24-210aÞ ! where P in kW, ! in rad/s, and Mt in N m Torque Mt ¼ 9550P SI ð24-210bÞ n where P in kW, n in rpm, and Mt in N m φ Mt ¼ Rg 63,000P USCS ð24-210cÞ n where P in hp, n in rpm, and Mt in lbf in Mt ¼ + α α + φ w FIGURE 24-43 Forces on the vehicle moving up the gradient Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular 24.69 Formula Mtt ¼ Mt Torque at the tire surface 24-211ị where ẳ 0.90 at top gear ¼ 0.80 at other gears Mt r The driving force at the tire F¼ Tractive factor ftr ¼ Mtt 1000W ftr ẳ Mtt 2240W Rg ẳ 1000W sin ỵ ị cos Rg ẳ 2240W sin ỵ ị cos Force required to pull the vehicle of weight W up the slope (Fig 24-43) ð24-212Þ SI ð24-213aÞ USCS ð24-213bÞ SI 24-214aị USCS 24-214bị Gradient Gẳ W Rg 24-215ị The air resistance Ra ẳ kAV2 24-216aị where k ẳ constant obtained from Table 24-18 For values of air resistance at different speeds of vehicle Refer to Table 24-19 The rolling resistance Rr ẳ a ỵ bVịW 24-216bị where a ẳ constant varies from 15 to 600 b ¼ constant varies from 0.1 to 3.5 For rolling or road resistance surfaces R0r for various road The general formula for total resistance or tractive resistance (Fig 24-44) Refer to Table 24-20 Rt ẳ kAV2 ỵ W sin ỵ ị cos ỵ a ỵ bVịW Another formula for total resistance Rt ẳ Ra ỵ Rg ỵ Rr 1000 ỵ kAV2 ẳ W R0r ỵ G 24-217aị 24-217bị where k and R0r are obtained from Tables 24-19 and 24-20 where R0r in N/tf, W in tf, A in m2 , V in m/s Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS 24.70 CHAPTER TWENTY-FOUR Particular Formula Rt ẳ Ra ỵ Rg ỵ Rr 2240 ẳ W Rr ỵ ỵ kAV2 G USCS 24-217cị where R0r in lbf/t, W ẳ weight of vehicle, tonf A ¼ projected frontal area of vehicle, ft2 V ¼ speed of vehicle, ft/s Tractive effort at the tire surface Ftr ẳ iMt r 24-218ị where i ẳ gear ratio obtained from Table 24-21 The speed of the vehicle V ẳ 0:00297 nDw i USCS 24-219aị where V in mph (miles per hour), Dw in in, and n in rpm V ¼ 0:052 Power AIR RESISTANCE Ra = kAV2 ROLLING RESISTANCE Rr = (a+bv) W nDw i SI ð24-219bÞ where V in m/s, Dw in m, and n in rpm Pẳ 0:002VMtt Dw SI 24-220aị where V in m/s, Mtt in N m, Dw in m, and P in kW W SIN (α+φ) GRADIENT RESISTANCE Rg= COS φ Pẳ USCS 24-220bị where V in mph, Mtt in lbf ft, Dw in in, and P in hp P¼ FIGURE 24-44 Various resistances on the moving vehicle 0:00163VMtt Dw 5:5VMtt Dw Customary Metric ð24-220cÞ where V in km/h, Mtt in kgf m, Dw in mm, and P in kW TRANSMISSION GEARBOX (Fig 24-45) The equation for center distance between main and countershafts for the case of three-speed passenger car a ¼ 0:5 pffiffiffiffiffiffiffi Mt USCS ð24-221aÞ SI ð24-221bÞ where a in in and Mt in lbf ft a ¼ 0:0106 pffiffiffiffiffiffiffi Mt where a in m and Mt in N m Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular 24.71 Formula F G A Clutch I D H II B C E FIGURE 24-45 A typical four-speed gearbox The distance between support bearings of shaft l ¼ 0:0254 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 3 Mt to 0:0318 Mt SI ð24-222aÞ where l in m and Mt in N m l ¼ 1:2 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 3 Mt to 1:5 Mt USCS ð24-222bÞ where l in in and Mt in lbf ft The maximum permissible load at the pitch circle of any pair of gears Fmax ¼ c1 bm ¼ c1 b Pm 24-223ị where c1 ẳ constant obtained from Table 24-22 The face width of gear tooth b¼ The expression for center distance for the case of fourspeed truck transmission a ¼ 0:017 Fmax Fmax Pn ¼ mc1 c1 pffiffiffiffiffiffiffi Mt SI ð24-225aÞ where a in m and Mt in N m a ¼ 0:8 The distance between support of bearings of shaft ð24-224Þ pffiffiffiffiffiffiffi Mt USCS ð24-225bÞ where a in in and Mt in lbf ft pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 3 l ¼ 0:0254 Mt to 0:0318 Mt where l in m, and Mt in N m pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 3 l ¼ 1:2 Mt to 1:5 Mt SI ð24-226aÞ USCS ð24-226bÞ where l in in and Mt in lbf ft The face width of gear tooth b¼ Fmax Fmax Pn ¼ mc1 c1 For values of c1 , refer to Table 24-22 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-227Þ MISCELLANEOUS MACHINE ELEMENTS 24.72 CHAPTER TWENTY-FOUR Particular The expression for center distance for the case of five-speed and reverse truck transmission Formula a ¼ 0:0170 pffiffiffiffiffiffiffi Mt where a in m and Mt in N m pffiffiffiffiffiffiffi a ¼ 0:8 Mt The distance between support of bearings of shaft where a in in and Mt in lbf ft pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 3 l ¼ 0:0254 Mt to 0:0318 Mt where l in m and Mt in N m pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 3 l ¼ 1:2 Mt to 1:5 Mt The face width of gear tooth The expression for center distance for a farm tractor transmission Effective face width of gear tooth The efficiency of transmission SI ð24-228aÞ USCS ð24-228bÞ SI ð24-229aÞ USCS ð24-229bÞ where l in in and Mt in lbf ft F F P b ¼ max ¼ max n mc1 c1 ð24-230Þ For values of c1 , refer to Table 24-22 pffiffiffiffiffiffiffi SI a ¼ 0:021 Mt ð24-231aÞ where a in m and Mt in N m p a ẳ Mt 24-231bị USCS where a in in and Mt in lbf ft Fmax v SI ð24-232aÞ b¼ 28 Â 105 m where b in m, Fmax in N, m in m, and v in m/s F vP b ẳ max n USCS 24-232bị 8,000,000 where b in in, Fmax in lbf, Pn in inÀ1 , and v in ft/min n M Mto 24-233ị ẳ o to ẳ ni Mti ir ẵMto Mtr Mtf Þ where ir ¼ reduction ratio of transmission ¼ ðni =no Þ Distance of rotating parts from the inner wall of housing l8 ¼ 10 to 15 mm or more for high-power and heavyduty operation l8 ¼ 0:4 to 0:6 in USCS 24-234ị Distance between adjacent rotating parts c ẳ 10 to 15 mm (0.4 to 0.6 in) ð24-235Þ Minimum clearance between gears and inner wall of housing Á ! 1:2h 24-236ị Distance between bearings of overhanging shaft l0 ẳ 1:2d to 3d Distance of bearing from the wall l2 ¼ to 10 mm (0.2 to 0.4 in) where h ẳ thickness of housing 24-237ị where d ẳ diameter of shaft 24-238ị Cap height from bolt end l3 ẳ depends on the design by empirical formula Distance of rotating parts from the bearing cap l4 ¼ 15 to 20 mm (0.6 to 0.8 in) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð24-239Þ MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular 24.73 Formula Width of boss of rotating part l5 ¼ 1:2d to 1:5d Distance of coupling to cap (depends on the type of coupling) Distance between gear and shaft l7 ! 20 mm (0.8 in) ð24-241Þ Distance of rotating part from the bearing l1 ¼ B l ỵ l3 ỵ l4 ỵ 2 Refer to Chapter 24, Section 24.5 ð24-242Þ For planetary gear transmission For detail design equations of spur, helical, bevel, crossed-helical and worm gears ð24-240Þ Refer to Chapter 23 HYDRAULIC COUPLING (Fig 24-46) Torque transmitted by the coupling Mt ¼ ksn2 Wðr2 r2 ị mo mi 24-243ị where k ẳ coefficient ¼ 1:42 Â 10 Percent slip between primary and secondary speeds r3 where np and ns are primary and secondary speeds of impeller, respectively, rpm r1 C DA ð24-244Þ r2 r4 B (approx.) ðnp À ns Þ Â 100 s¼ np FIGURE 24-46 Hydraulic coupling The mean radius of the inner passage (Fig 24-46) The mean radius of the outer passage (Fig 24-46) rmi ¼ rmo ¼ r3 À r3 r2 À r2 r3 À r3 r2 À r2 The expression for number of times the fluid circulates through the torus in one second if ¼ The torque capacity of hydraulic coupling at a given slip 24-245ị 24-246ị Mt ẳ Kn2 D5 t 13,000Mt nWðr2 À r2 Þ mo mi ð24-247Þ ð24-248Þ where K ¼ coefficient varying from 0:166 Â 108 to 0:244 Â 108 ¼ 1.56 to 2.28 SI USCS Dt ¼ diameter of torus, m (ft) Mt ¼ torque capacity, N m (lbf ft); n in rpm Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS 24.74 CHAPTER TWENTY-FOUR Particular Formula HYDRODYNAMIC TORQUE CONVERTER (Fig 24-47) Mti ¼ Kn2 D5 i t The equation for input torque 24-249ị where Impeller Runner K ẳ coecient depending on design ni ¼ speed of input shaft, rpm Dt ¼ any linear dimension such as maximum diameter of impeller Reactor FIGURE 24-47 Hydrodynamic torque converter The equation for the input power 24-250ị P ẳ Cn3 D5 i t where C ¼ coefficient depending on design The expression for flow loss or friction loss in each member of the torque converter under any particular operating conditions in energy unit per kilogram of fluid circulated Ef ¼ Cf V2 f 2g 24-251ị where Cf ẳ coecient whose value depends mainly on the Reynolds number and the relative smoothness of the metallic surface ¼ 0.445 to 0.890 SI (where Ef in N m and Vf in m/s) ¼ 0.328 to 0.656 USCS (where Vf in ft/s and Ef in lbf ft) The expression for shock loss per kg fluid circulated in the impeller of a torque converter Esh ¼ Csh V2 sh 2g 24-252ị where Csh ẳ coefficient The maximum inside diameter of torus Dt ¼ 0:00135C qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mt =n02 SI ð24-253aÞ where Dt in m, Mt in N m, and n in rps Dt ¼ 0:00168C rffiffiffiffiffiffiffi Mt n2 USCS ð24-253bÞ where Dt in in, Mt in lbf in, and n in rpm C ¼ coefficient ¼ 14 for a ratio of minimum inside diameter to maximum diameter of torus of one-third n ¼ speed in hundreds of rpm Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS Particular Formula TRACTIVE EFFORT CURVES FOR CARS, TRUCKS, AND CITY BUSES For finding the diameter of tire of vehicles for a particular wheel speed Refer to Fig 24-48 For tractive effort of a passenger car Refer to Fig 24-49 For tractive effort of trucks, tractors, and city buses Refer to Fig 24-50 850 800 750 700 Pa ss 600 en i Tra ge s, rc ler ars 550 tr u ck s, an 500 dc ity Wheel speed, rpm 650 es ss bu 450 400 350 300 250 50 60 70 80 90 100 Tyre diameter, cm 110 120 130 FIGURE 24-48 Wheel speed vs tire diameter of vehicles Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website 24.75 MISCELLANEOUS MACHINE ELEMENTS 24.76 CHAPTER TWENTY-FOUR Tractive effort, kgf 200 100 50 500 1000 1500 Gross vehicle weight, kgf 2000 2500 FIGURE 24-49 Tractive effort curve for passenger cars (1 kgf ¼ 9.8066 N ¼ 2.2046 lbf ) 950 900 850 800 750 700 650 Tractive effort, kgf 600 550 500 450 400 350 300 250 200 150 100 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 Gross vehicle weight, kgf FIGURE 24-50 Tractive effort curve for trucks, tractors, and city buses Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website MISCELLANEOUS MACHINE ELEMENTS MISCELLANEOUS MACHINE ELEMENTS 24.77 TABLE 24-18 Values of k for use in Eq (24-216) and (24-217) Particular k in USCSUa k in SI Average automobile of modern design Streamlined racing car Truck or omnibus 0.0017 0.20 0.0006 0.0024 0.07 0.28 a US Customary System units TABLE 24-19A Air resistancea TABLE 24-19B Air resistance in SI units Speed of vehicle, mph Velocity of wind, V, ft/s 0.0024V2 0.0017V2 0.0006V2 Speed of vehicle km/h Velocity of wind V, m/h 0.28V2 0.20V2 0.07V2 10 20 30 40 50 60 90 150 14.67 29.35 44.00 58.60 73.30 88.00 132.00 220.00 0.516 2.060 4.650 8.240 12.900 18.600 — — 0.366 1.460 3.300 5.830 9.130 13.160 29.650 — — 0.516 1.160 2.060 3.220 4.650 10.450 29.000 10 20 30 40 50 60 70 80 90 100 2.78 5.56 8.34 11.12 13.90 16.68 19.46 22.21 25.02 27.80 2.17 8.68 19.50 34.72 54.25 78.00 106.40 139.00 176.00 217.00 1.55 6.20 13.92 24.80 38.80 57.60 76.00 99.20 125.60 155.00 0.54 2.17 4.88 8.68 13.56 19.50 26.60 34.75 44.00 54.25 a Values given in this table are in US Customary System units TABLE 24-20 Road resistance, R0r Solid Surface N/tf Polished marble Concrete Asphalt Stone Good quality Poor quality Vitrified bricks Good macadam (metal road) Good Fair Rough Clay Sand lbf/ton Pneumatic % N/tf lbf/ton % 29.3 62.4 67.4 12.12 14.25 15.40 0.541 0.636 0.687 35.3 41.5 448.2 8.08 9.50 10.25 0.360 0.423 0.457 71.8 153.2 85.0 16.40 35.00 19.45 0.732 1.562 0.866 477.6 102.0 56.7 10.92 23.30 12.95 0.487 1.040 0.578 146.2 220.0 307.0 438.4 1314.0 33.50 50.00 70.00 100.00 300.00 1.491 2.240 3.130 4.470 13.400 97.9 186.4 389.3 389.3 874.8 22.20 33.20 46.60 66.60 200.00 0.998 1.900 2.100 3.970 8.920 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website Ratio SI Â 106 USCSU SI Â 106 USCSU SI Â 106 USCSU III I SI Â 106 USCSU IV SI Â 106 USCSU V Low speed reduction Three-speed passenger 124.6–145.2 18,000–21,000 145.2–153.5 21,000–24,000 193.7–221.2 28,000–32,000 car transmission Four-speed truck 76.0–90.0 11,000–13,000 128.0–149.0 18,500–21,500 96.8–110.9 14,000–16,000 175.2–207.5 26,000–30,000 transmission Five-speed reverse 76.0–90.0 11,000–13,000 138.2–152.0 20,000–22,000 105.0–117.7 15,000–17,000 90.0–105.0 13,000–15,000 175.2–207.5 26,000–30,000 truck transmission No of speeds and type of transmission Intermediate speed reduction Gear wheels belonging to High speed reduction TABLE 24-22 Value of coefficient c1 in SI and USCSU II Final drive (rear-axle differential) Second Low Reverse gear Overdrive : to : 1:6 : (total or : 1) 2:5 : (total 10 : 1) Same as or higher than low gear ratio