COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES Particular 19.17 Formula SELLERS’ CONE COUPLING (Fig 19-10) The length of the box L ẳ 3:65d to 4d 19-62ị The outside diameter of the conical sleeve D1 ẳ 1:875d to 2d ỵ 0:0125 SI ð19-63aÞ USCS ð19-63bÞ where D, d in m D1 ẳ 1:875d to 2d ỵ 0:5 where D, d in in Outside diameter of the box D2 ¼ 3d 19-64ị The length of the conical sleeve l ẳ 1:5d ð19-65Þ FIGURE 19-11 Hydraulic coupling FIGURE 19-10 Sellers, cone coupling HYDRAULIC COUPLINGS (Fig 19-11) Torque transmitted Mt ¼ Ksn2 Wðr2 r2 ị mo mi where K ẳ coefficient ẳ Percent slip between primary and secondary speeds The mean radius of inner passage (Fig 19-11) sẳ 19-66ị 1:42 approx:ị 107 np À ns  100 np ð19-67Þ where np and ns are the primary and secondary speeds of impeller, respectively, rpm r2 À r3 ð19-68aÞ rmi ¼ r2 À r2 The mean radius of outer passage (Fig 19-11) rmo ¼ The number of times the fluid circulates through the torus in one second is given by i¼ r3 À r3 r2 À r2 13;000Mt nWðr2 À r2 Þ mo mi 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 ð19-68bÞ ð19-69Þ Cast iron or steel Cast iron Steel Hard steel Hard steel, chromium plated Hard steel, chromium plated Cast iron or steel Hard steel, chromium plated Cast iron or steel Cast iron or steel Cast iron or steel Cast iron or steel Cast iron or steel Cast bronze Cast iron Cast iron Hard steel Hard steel 0.08–0.12 0.12 0.05–0.1 0.1–0.15 Hard steel, chromium plated Cast iron or steel Cast iron or steel Steel Cast iron Woven asbestos 19.18 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 0.1 0.1–0.2 0.25 0.25 0.2–0.5 0.32 0.3–0.6 0.2–0.35 0.3–0.5 0.3–0.5 0.22 0.3–0.5 632–811 422 533 533–659 533 444–533 422 363.3 363.3 411 363.3 359–538 149 260 260–386 260 171–260 149 90.3 90.3 138 90.3 538 538 260 149 316 260 260 260 8C Maximum temperature 2.0682 0.6894 0.3452–1.0346 1.0346 8.2738 0.6894–1.3788 0.03432–0.6894 0.4138–0.6208 0.0686–0.2746 0.0549–0.0981 0.0343–0.0686 0.0686–0.2746 1.0346 2.0682 1.0346 0.5521–0.8277 1.0346–1.7240 0.8277–1.3788 0.6894 1.3788 MPa 0.2109 0.0703 0.0352–0.1055 0.1055 0.8437 0.0703–0.1406 0.0350–0.0703 0.0422–0.0633 0.0070–0.0284 0.0056–0.01 0.0035–0.0070 0.3070–0280 0.1055 0.2109 0.1055 0.0563–0.0844 0.1055–0.1755 0.0844–0.1406 0.0703 0.1406 kgf/mm2 Maximum pressure, p b Conservative values should be used to allow for possible glazing of clutch surfaces in service and for adverse operating conditions Steel, where specified, should have a carbon content of approximately 0.70% Surfaces should be ground true and smooth c For a specific material within this group, the coefficient usually is maintained within plus or minus 5% Note: kpsi ¼ 6.894757 MPa or Pa ¼ 145  10À6 psi or MPa ¼ 145 psi a Carbon graphite Molded phenolic plastic, macerated cloth base Molded asbestosc Impregnated asbestos Cast iron or steel Woven asbestos 0.1–0.2 Cast iron or steel 0.16 0.12–0.15 0.15–0.25 0.18 811 811 0.1–0.4 0.1–0.3 0.05–0.1 0.05–0.1 422 589 533 533 533 K 533 0.15–0.2 Dry 0.03 0.05 0.05 0.06 0.05 0.03 Wet Friction coefficient,a Wood Leather Cork Felt Vulcanized fiber or paper Woven asbestosc Hard-drawn phosphor bronze Powder metalc Powder metalc Opposingb Wearing Contact surfaces TABLE 19-4 Friction materials for clutches High Low Very low Moderate Moderate Low Low Lowest Very low Very low Low Very low High Very high High Low Very low Very low Moderate High Relative cost Prolonged slip service ratings given This rating for short infrequent engagements Used in Napier Sabre engine Wide field of applications For demanding applications For critical requirements For light special service Unsuitable at high speed Subject to glazing Cork-insert type preferred Resinent engagement Low speeds, light duty Good wearing qualities High energy absorption Good wearing qualities Subject to seizing Good at low speeds Fair at low speeds Subject to galling Durable combination Comment COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES Particular Power transmitted by torque converter 19.19 Formula Mt À Kn2 D5 ð19-70Þ where K ¼ coefficient—varies with the design n ¼ speed of driven shaft, rpm D ¼ outside diameter of vanes, m (in) 19.2 CLUTCHES POSITIVE CLUTCHES (Fig 19-12) Jaw clutch coupling a¼ c¼ f ¼ g¼ h¼ i¼ j¼ k¼ lẳ 2:2d ỵ 0:025 m 1:2d ỵ 0:03 m 1:4d þ 0:0055 m d þ 0:005 m 0:3d þ 0:0125 m 0:4d ỵ 0:005 m 0:2d ỵ 0:0375 m 1:2d þ 0:02 m 1:7d þ 0:0584 m a¼ c¼ f ẳ gẳ hẳ iẳ jẳ kẳ lẳ 2:2d ỵ 1:0 in 1:2d ỵ 1:2 in 1:4d ỵ 0:3 in d þ 0:2 in 0:3d þ 0:5 in 0:4d þ 0:2 in 0:2d ỵ 0:15 in 1:2d ỵ 0:8 in 1:7d þ 2:3 in (19-71) The shear stress assuming that only one-half the total number of jaws i is in actual contact Aẳ 0:5a bịh sin 19-72ị ẳ 4F sin ia bịh cos 19-73ị ẳ The area in shear 2:8F ia bịh 19-74ị for tan ẳ 0:7 where ¼ angle made by the shearing plane with the direction of pressure FIGURE 19-12 Square-jaw clutch 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 COUPLINGS, CLUTCHES, AND BRAKES 19.20 CHAPTER NINETEEN Particular Formula FRICTION CLUTCHES Cone clutch (Fig 19-13) The axial force in terms of the clutch dimensions Fa ẳ Dm pb sin 19-75ị where Dm ẳ D1 ỵ D2 ị (approx.) ẳ one-half the cone angle, deg ¼ ranges from 158 to 258 for industrial clutches faced with wood ¼ 12.58 for clutches faced with asbestos or leather or cork insert Axial force in terms of normal force (Fig 19-13) Fa ¼ Fn sin The tangential force due to friction F ¼ Torque transmitted through friction Mt ¼ Power transmitted P¼ Fa Dm n 19;100 sin kl P¼ Fa Dm n 126;000 sin kl Pẳ pD2 bn m 19;100kl 19-76ị Fa sin 19-77ị Fa Dm sin ð19-78Þ SI ð19-79aÞ USCS ð19-79bÞ SI ð19-79cÞ FIGURE 19-13 Cone clutch 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 COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES Particular 19.21 Formula P¼ pD2 bn m 126;000kl USCS 19-79dị where kl ẳ load factor from Table 14-7 Refer to Table 19-4 for p The force necessary to engage the clutch when one member is rotating The ratio (Dm =b) The value of Dm in commercial clutches Fa ẳ Fn sin ỵ cos Þ ð19-80Þ Dm ¼ 4:5 to b sffiffiffiffiffiffiffiffiffiffi Pkl q Dm ẳ 18:2 pn 19-81ị qẳ SI USCS s Pkl q Dm ẳ 34:2 pn 19-82aị 19-82bị Dm ẳ 5d to 10d 19-82cị DISK CLUTCHES (Fig 19-14) The axial force Fa ¼ pD1 ðD2 À D1 Þ ð19-83Þ Refer to Table 19-4 for p The torque transmitted Mt ẳ Fa Dm 19-84ị where Dm ẳ D3 D3 ị ðD2 À D2 Þ ð19-85aÞ for uniform pressure distribution and Dm ẳ D2 ỵ D1 ị ð19-85bÞ for uniform wear FIGURE 19-14 Multidisk clutch Power transmitted P¼ iFa n 28;650kl P¼ iFa n 189;000kl D3 À D3 2 D2 À d1 D3 À D3 2 D2 À d1 SI ð19-86aÞ USCS ð19-86bÞ for uniform pressure 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 COUPLINGS, CLUTCHES, AND BRAKES 19.22 CHAPTER NINETEEN Particular Formula where Fa ¼ p D2 À D2 Pẳ ipnD1 D2 D2 ị 76;400kl SI 19-87aị Pẳ ipnD1 D2 D2 ị 504;000kl SI ð19-87aÞ for uniform wear The clutch capacity at speed n1 P1 ẳ Pn1 nks 19-88ị where P ẳ design power at speed, n ks ¼ speed factor obtained from Eq (19-89) The speed factor ks ẳ 0:1 ỵ 0:001n 19-89ị where n ẳ speed at which the capacity of clutch to be determined, rpm DIMENSIONS OF DISKS (Fig 19-15) The maximum diameter of disk D2 ¼ 2:5 to 3:6D1 19-90ị The minimum diameter of disk D1 ẳ 4d 19-91ị The thickness of disk h ¼ to mm 19-92ị The number of friction surfaces i ẳ i1 ỵ i2 À ð19-93Þ The number of driving disks i1 ¼ i ð19-94Þ The number of driven disks i2 ẳ i ỵ1 19-95ị FIGURE 19-15 Dimensions of disks 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 COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES Particular 19.23 Formula DESIGN OF A TYPICAL CLUTCH OPERATING LEVER (Fig 19-16) The total axial force on i number of clutch disk or plates Fa ¼ ip0 D1 ðD1 D2 ị 19-96ị where p ẳ actual pressure between disks Fa ẳ 4Mta ; iD1 DịD2 m MPa psiị Mta ẳ allowable torque, N m (lbf in) FIGURE 19-16 A typical clutch operating lever The force acting on disks of one operating lever of the clutch (Fig 19-16) F1 ẳ Fa i0 19-97ị where i0 ¼ number of operating levers The total force acting from the side of the bushing (Fig 19-16) P ¼ i0 p1 ð19-98Þ The force acting from the side of the bushing on one operating lever (Fig 19-16) d L cot ỵ ị e1 P ẳ F1 d e2 ỵ e3 þ ð19-99Þ The thickness of the !ever very close to the pin (Fig 19-16) 31=3 6Fa e h ¼ 6 b i db h 19-100ị where db ẳ design bending stress for the material of the levers, MPa (psi) The diameter of the pin (Fig 19-16) Ratio of b=h ¼ 0:75 to s 2Fr dẳ d 19-101ị where Fr ẳ resultant force due to F1 and P1 cot ỵ ị on the pin, kN (lbf ) d ¼ design shear stress of the material of the pin, MPa (psi) 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 COUPLINGS, CLUTCHES, AND BRAKES 19.24 CHAPTER NINETEEN Particular Formula EXPANDING-RING CLUTCHES (Fig 19-17) Torque transmitted [Fig 19-17(a)] Mt ¼ 2pwr2 19-102ị where ẳ one half the total arc of contact, rad w ¼ width of ring, m (in) FIGURE 19-17 Expanding-ring clutch The moment of the normal force for each half of the band [Fig 19-17(a)] The force applied to the ends of the split ring to expand the ring [Fig 19-17(a)] If the ring is made in one piece (Fig 19-7(b)] an additional force required to expand the inner ring before contact is made with inner surface of the shell Mo ẳ pwrL 19-103ị when % rad Fs ¼ pwr Fe ¼ Ewt3 6L ð19-104Þ 1 À d1 d ð19-105Þ where d1 ¼ original diameter of ring, m (in) d ¼ inner diameter of drum, m (in) w ¼ width of ring, m (in) t ¼ thickness of ring, m (in) F ẳ Fs ỵ Fe The total force required to expand the ring and to produce the necessary pressure between the contact surfaces F ẳ pwr ỵ Ewt3 6L 19-106ị 1 À d1 d ð19-107Þ Fn ẳ Fn sin ỵ cos ị 19-108ị Fn ¼ Fn sin ð19-109Þ Mt ¼ i1 i2 F D ẳ i1 i2 D2 bp 19-110ị RIM CLUTCHES (Fig 19-18) When the grooved rim clutch being engaged, the equation of equilibrium of forces along the vertical axis After the block is pressed on firmly the equation of equilibrium of forces along the vertical axis Torque transmitted where i1 ¼ number of grooves in the rim i2 ¼ number of shoes b ¼ inclined face, m (in) ¼ angle of contact, rad 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 COUPLINGS, CLUTCHES, AND BRAKES 19.38 CHAPTER NINETEEN Particular The actuating force The horizontal reaction at the hinge-pin Formula Mtn ỵ Mt c ð p br Rx ¼ a sin cos d sin a 1 2 ỵ sin2 d Fx Fẳ 19-185ị 19-186ị 1 The vertical reaction at the hinge-pin Ry ¼ ð pa br sin cos d sin a 1 2 sin2 d ỵ Fy 19-187ị 1 FOR COUNTERCLOCKWISE ROTATION Mtn À Mt c ð p br Rx ¼ a sin cos d sin a 1 ð 2 sin2 d Fx 19-188ị Fẳ 19-189ị 1 Ry ẳ pa br sin a À À ð 2 ð 2 1 1 sin cos d sin2 d ỵ Fy 19-190ị HEATING OF BRAKES Heat generated from work of friction Hg ẳ pAc v J joulesị Hg ¼ Heat to be radiated for a brake lowering the load pAc v 778 H ẳ Wh J joulesị SI ð19-191aÞ USCS ð19-191bÞ SI ð19-192aÞ Wh USCS ð19-192bÞ 778 where h ¼ total height or distance, m (ft) H¼ The heat generated is also given by the equation Hg ¼ 754kl P SI ð19-193aÞ USCS ð19-193bÞ where P in kW and Hg in J/s Hg ¼ 42:4kl P where P in hp 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 COUPLINGS, CLUTCHES, AND BRAKES 19.39 COUPLINGS, CLUTCHES, AND BRAKES Particular The rise in temperature in 8C of the brake drum or clutch plates Formula ÁT ¼ H mC 19-194ị where m ẳ mass of brake drum or clutch plates, kg C ¼ specific heat capacity ¼ 500 J/kg 8C for cast iron or steel for cast iron ¼ 0:13 Btu/lbm 8F ¼ 0:116 Btu/lbm 8F for steel The rate of heat dissipation Hd ¼ C2 ÁTAr SI 19-195aị where Hd in J Hd ẳ 0:25C2 TAr Metric 19-195bị where C2 ẳ radiating factor from Table 19-13 Hd in kcal The required area of radiating surface Ar ¼ 754kl N C2 ÁT SI ð19-196aÞ where Ar in m2 Ar ẳ 0:18kl N C2 T SI 19-196bị where Ar in mm2 Approximate time required for the brake to cool Gagne’s formula for heat generated during a single operation tc ẳ Wr C2 ln T KAr 19-197ị where K ¼ a constant varying from 0.4 to 0.8 " # AC nt Nt Hg ẳ Tav Ta ị c ỵ 1:5 c nc 3600 3600 19-198ị where Tav Ta ị ẳ temperature dierence between the brake surface and the atmosphere, 8C Refer to Table 19-15 for values of C For additional design data for brakes Refer to Tables 19-11 to 19-17 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 COUPLINGS, CLUTCHES, AND BRAKES 19.40 CHAPTER NINETEEN TABLE 19-10 Working pressure for brake blocks Pressure Rubbing velocity, m/s Wood blocks Asbestos fabric Asbestos blocks MPa kgf/mm2 MPa kgf/mm2 MPa kgf/mm2 10 0.5521 0.4482 0.3452 0.2412 0.1726 0.1726 0.0563 0.0457 0.0352 0.0246 0.0176 0.0176 0.6894 0.5521 0.4138 0.2756 0.2069 0.2069 0.0703 0.0563 0.0422 0.0281 0.0211 0.0211 1.1032 1.0346 0.8963 0.6894 0.4825 0.2756 0.1125 0.1055 0.0914 0.0703 0.0492 0.0281 Note: kpsi ¼ 6.894754 MPa or MPa ¼ 145 psi TABLE 19-11 Comparison of hoist brakes Block brakes Band brakes Axial brakes Brake characteristics Double block V-grooved sheave b a Average numerical value Relative value Force ratio F F Simple Both directions of rotation Cone Multidisk b sin a b ae 1ị be ỵ 1ị ae 1Þ b sin a b na 0.667 0.282 0.0323 0.165 0.161 0.097 20.6 8.7 5.1 5.0 h1 a b sin h1 a 2b h1 a 4b h1 a b sin a 3.00 b ih01 a b Travel at lever end h1 a b Average travel, mm (in) 8.0 (0.313) 18.8 (0.74) 74.5 (2.943) 37.36 (1.471) 32.8 (1.292) 5.56 (0.219) 1512.7 (2000) 18.9 (25) 227.0 (300) 75.6 (100) 37.8 (50) 90.8 (120) Maximum capacity P, kW (hp) a b h1 ¼ the normal distance between the sheave and the stationary braking surface to prevent dragging h ¼ b in Fig 19-21 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 COUPLINGS, CLUTCHES, AND BRAKES TABLE 19-12 Service factors for typical machines Service factors for prime movers Electric motor steam or water turbine High-speed steam or gas engine !4 Cyla Alternators and generators (excluding welding generators), induced-draft fans, printing machinery, rotary pumps, compressors, and exhausters, conveyors 1.5 2.0 2.5 3.0 3.5 5.0 Woodworking machinery, machine tools (cutting) excluding planing machines, calenders, mixers, and elevators 2.0 2.5 3.0 3.5 3.0 5.5 Forced-draft fans, high-speed reciprocating compressors, high speed crushers and pulverizers, machine tools (forming) 2.5 3.0 3.5 4.0 4.5 6.0 Rotary screens, rod mills, tube, cable and wire machinery, vacuum pumps 3.0 3.5 4.0 4.5 5.0 6.5 Low-speed reciprocating compressors, haulage gears, metal planing machines, brick and tile machinery, rubber machinery, tube mills, generators(welding) 3.5 4.0 4.5 5.0 5.5 7.0 Type of driven machine TABLE 19-13 Radiating factors for brakes Petrol engine Cyl Oil engine !6 Cyl Cyl TABLE 19-14 pv values as recommended by Hutte for brakes Radiating factor, C2 C2 ÁT pv Temperature difference, ÁT W/m2 K cal/m2 s 8C W/m2 cal/m2 s Service SI Metric 55.5 111.5 166.5 222.6 12.26 15.33 16.97 18.40 2.93 3.66 4.05 4.39 162.73 406.83 675.34 976.40 Intermittent operations with long rest periods and poor heat radiation, as with wood blocks Continuous service with short rest periods and with poor radiation Continuous operation with good radiation as with an oil bath 26.97 2.75 13.73 1.40 40.70 4.15 681.36 1703.41 2827.66 4088.19 TABLE 19-15 Values of beat transfer coefficient C for rough block surfaces Heat-transfer coefficient, C Velocity, v, m/s W/m2 K kcal/m2 h 8C 0.0 6.1 12.2 18.3 24.4 30.5 8.5 14.1 18.8 22.5 25.6 29.0 7.31 12.13 16.20 19.30 22.00 24.90 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 COUPLINGS, CLUTCHES, AND BRAKES 19.42 CHAPTER NINETEEN TABLE 19-16 Values of e Leather belt on Wood Cast iron Proportion of contact to whole circumference Steel band on cast iron ¼ 0:18 Slightly greasy ¼ 0:47 Very greasy ¼ 0:12 Slightly greasy ¼ 0:28 0.1 0.2 0.3 0.4 0.425 0.45 0.475 0.500 0.525 0.6 0.7 0.8 0.9 1.0 1.12 1.25 1.40 1.57 1.62 1.66 1.71 1.76 1.81 1.97 2.21 2.47 2.77 3.10 1.34 1.81 2.43 3.26 3.51 3.78 4.07 4.38 4.71 5.88 7.90 10.60 14.30 19.20 1.08 1.16 1.25 1.35 1.38 1.40 1.43 1.46 1.49 1.57 1.66 1.83 1.97 2.12 1.19 1.42 1.69 2.02 2.11 2.21 2.31 2.41 2.52 2.81 3.43 4.09 4.87 5.81 Damp ¼ 0:38 1.27 1.61 2.05 2.60 2.76 2.93 3.11 3.30 3.50 4.19 5.32 6.75 8.57 10.90 TABLE 19-17 Coefficient of friction and permissible variations on dimensions for automotive brakes lining Type and class of brake lining Type I—rigid molded sets or flexible molded rolls or sets Class A—medium friction Class B—high friction Type II—rigid woven sets or flexible woven rolls or sets Class A—medium friction Class B—high friction Tolerance on width for sizes, mm Tolerance on thickness for sizes, mm Range of coefficient of friction, Permissible variation in , % mm thickness >5 mm thickness mm thickness >5 mm thickness 0.280.40 0.360.45 ỵ30, 20 ỵ30, 20 ỵ0 0.2 ỵ0 0.3 ỵ0 0.8 ỵ0 0.8 0.330.43 0.430.53 ỵ20, 30 ỵ20, 30 REFERENCES Shigley, J E., Machine Design, McGraw-Hill Book Company, New York, 1962 Maleev, V L and J B Hartman, Machine Design, International Textbook Company, Scranton, Pennsylvania, 1954 Black, P H., and O E Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1968 Norman, C A., E S Ault, and I F Zarobsky, Fundamentals off Machine Design, The Macmillan Company, New York, 1951 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 COUPLINGS, CLUTCHES, AND BRAKES COUPLINGS, CLUTCHES, AND BRAKES 19.43 Spotts, M F., Machine Design Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1964 Spotts, M F., Design of Machine Elements, Prentice-Hall of India Ltd., New Delhi, 1969 Vallance, A., and V L Doughtie, Design of Machine Members, McGraw-Hill Book Company, New York, 1951 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986 10 Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986 11 Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994 12 Bureau of Indian Standards, New Delhi, India 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 Source: MACHINE DESIGN DATABOOK CHAPTER 20 SPRINGS SYMBOLS A b b0 c c1 , c2 C1 , C2 d d1 , d2 D D1 D2 esz e0sr E F F Fmax ko Fcr g G h i i0 area of loading, m2 (in2 ) width of rectangular spring, m (in) width of laminated spring, m (in) width of each strip in a laminated spring, m (in) spring index constants taken from Table 20-1 and to be used in Eqs (20-1) to (20-36) constants to be used in Eqs (20-20) and (20-21) and taken from Fig 20-3 diameter of spring wire, m (in) diameter of torsion bar, m (in) diameter of outer and inner wires of concentric spring, m (in) mean or pitch diameter of spring, m (mm) overall diameter of the absorber, m (in) mean or pitch diameter of outer concentric spring, m (in) smallest mean diameter of conical spring, m (in) mean or pitch diameter of inner concentric spring, m (in) largest mean diameter of conical spring, m (in) size coefficient surface influence coefficient modulus of elasticity, GPa (psi) frequency, cycles per minute, Hz load, kN (lbf ) steady-state load [Eq (20-84)] maximum force that can be imposed on the housing, kN (lbf ) force to compress the spring one meter (in) N/m (lbf/in) [spring rate, N/m (lbf/in)] critical load, kN (lbf ) acceleration due to gravity, 9.8066 m/s2 9806.06 mm/s2 (32.2 ft/s2 ; 386.4 in/s2 ) modulus of rigidity, GPa (psi) height (thickness) of laminated spring, m (in) axial height of a rectangular spring wire, m (in) total number of strips or leaves in a leaf spring number of coils in a helical spring total number of full-length blunt-ended leaves in a leaf spring 20.1 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 SPRINGS 20.2 CHAPTER TWENTY moment of inertia, area, m4 , cm4 (in4 ) stress factor (Wahl factor) correction factor factor depends on the ratio lo =D as shown in Fig 20-8 reduced stress correction factor or Wahl stress factor or fatigue stress correction factor shear stress correction factor length, m (in) free length of helical spring, m (in) iD length of the coil part of torsion spring, m (in) effective length of bushing, m (in) overall length of the absorber (Fig 20-15), m (in) constant depends on =di as indicated in Fig 20-3 twisting moment, N m (lbf in) factor of safety actual factor of safety or reliability factor resilience, N m (lbf in) energy to be absorbed by a rubber spring, N m (lbf in) volume of spring, m3 , mm3 (in3 ) specific weight of the spring material, N/m3 (lbf/in3 ) weight of spring, kN (lbf ) weight of effective number of coils i involved in the operation of the spring [Eq (20-77)], kN (lbf ) deflection, m (in) critical deflection, m (in) section modulus, m3 , cm3 (in3 ) polar section modulus, m3 , cm3 (in3 ) stress, normal, MPa (psi) shear stress, MPa (psi) constant from Table 20-3 constants from Table 20-3 angular deflection, rad Poisson’s ratio I k, k1 , k2 k4 Kl kr l lf or lo L M Mt n na U V W y ycr Z Zo , , SUFFIXES a m max f o outside inside amplitude mean maximum minimum endurance lirnit (also used for reversed cycle) endurance limit for repeated cycle Particular Formula LEAF SPRINGS (Table 20-1)1;2;3 The general equation for the maximum stress in springs ¼ c1 Fl bh2 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 ð20-1Þ SPRINGS SPRINGS Particular 20.3 Formula The general equation for the maximum deflection springs y¼ 20-2ị hẳ The thickness of spring c2 Fl Ebh3 c2 l c1 yE ð20-3Þ Refer to Table 20-1 for values of c1 and c2 For sizes and tolerances for leaf springs for motor vehicle suspension [4] Refer to Tables 20-2 to 20-5 TABLE 20-1 Deflection formula for beams of rectangular cross section and constants in beam Eqs (20-1) to (20-3) Particular Maximum deflection, ymax c1 , for the stress c2 , for the deflection Unit resilience, Nm/m3 (kgf mm/mm3 ) ymax ¼ 2F bE 3 h 2 18E ymax ¼ 4F bE 3 h 2 6E ymax ¼ 3F bE 3 h 3 2 6E ymax ¼ 4F bE 3 h 2 18E ymax ¼ 8F bE 3 h 2 6E ymax ¼ 6F bE 3 h 6 2 6E Source: K Lingaiah and B R Narayana Iyengar, Machine Design Data Handbook, Engineering College Cooperative Society, Bangalore, India, 1962; K Lingaiah and B R Narayana Iyengar, Machine Design Data Handbook, Vol I, Suma Publishers, Bangalore, India, 1986; and K Lingaiah, Machine Design Data Handbook, Vol II, Suma Publishers, Bangalore, India, 1986 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 SPRINGS 20.4 CHAPTER TWENTY Particular Formula LAMINATED SPRING (Fig 20-1)5 FIGURE 20-1 Laminated springs for automobiles The load on the spring ib0 h2 c1 l Fẳ 20-4ị where ib0 ẳ b The maximum deflection The maximum deflection in case of laminated semielliptical spring for heavy loads The correction factor to be used in Eq (20-6) yẳ c2 Fl Eib0 h3 20-5ị yẳ c2 Fl k4 Eib0 h3 20-6ị k4 ẳ 4r ỵ 2r2 1:5 ln rị rị3 where r ẳ 20-7ị i0 l For standard sections of steel plates for laminated springs Refer to Tables 20-2 to 20-6 The correction factor k4 can also be obtained from k4 ¼ Size coefficient 0:73r0:1 for < r < 20 for r > 20 0:0025 h where h in mm esz ẳ 0:8 ỵ 0:1 h where h in in esz ẳ 0:8 ỵ 2:5 h where h in mm esz ẳ 0:8 ỵ 20-8ị SI ð20-9aÞ USCS ð20-9bÞ SI ð20-9cÞ 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 SPRINGS SPRINGS Particular 20.5 Formula TABLE 20-2 Cross section tolerances for leaf springs for motor vehicle suspension—metric bar sizes—SAE J1123a Tolerance, mm in thickness (ỵ)a and in atness ()b Maximum dierence in thicknessc For thickness Width tolerance, mm For thickness, mm Width, mm Minus 0.00 5.00–9.50 10.00–21.20 22.40–37.50 5.00–9.50 10.00–21.20 22.40–37.50 40.0 45.0 50.0 56.0 63.0 75.0 90.0 100.0 125.0 150.0 ỵ0.75 þ0.75 þ0.75 þ0.75 þ0.75 þ1.15 þ1.15 þ1.15 þ1.65 þ2.30 0.13 0.13 0.13 0.13 0.13 0.15 0.15 0.15 0.18 — 0.15 0.15 0.15 0.15 0.15 0.20 0.20 0.20 0.25 0.30 — — — — — 0.30 0.30 0.30 0.40 0.50 0.05 0.05 0.05 0.05 0.05 0.08 0.08 0.08 0.10 — 0.05 0.05 0.05 0.05 0.05 0.10 0.10 0.10 0.13 0.15 — — — — — 0.15 0.15 0.15 0.20 0.25 a Thickness measurements shall be taken at the edge of the bar where the flat surfaces intersect the rounded edge This tolerance represents the maximum amount by which the thickness of the center of the bar may be less than the thickness at the edges Thickness of the center may never exceed the thickness at the edges c Maximum difference in thickness between the two edges of each bar Source: Reproduced from SAE Handbook, Vol I, 1981, courtesy SAE b Size factor ksz ẳ ẳ 4:66h0:35 esz SI 20-10aị USCS 20-10bị SI 20-10cị where k in m ksz ẳ 1:27h0:35 where h in in ksz ¼ 0:415h0:35 where h in mm LAMINATED SPRINGS WITH INITIAL CURVATURE Fg ¼ 2ð1 À rị F 2ỵr 20-11ị The load shared by full-length leaves of the spring Ff ẳ 3r F 2ỵr 20-12ị The maximum stress in the graduated leaves g ¼ The load shared by graduated leaves of the spring The maximum stress in the full-length leaves Fl lb0 h2 1:5 Fl f ¼ ib 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 ð20-13Þ ð20-14Þ SPRINGS 20.6 CHAPTER TWENTY Particular Formula The maximum deflection of the leaves (in both graduated and full-length leaves) y¼ Fl Eib0 h3 ð20-15Þ The camber to be provided for equalization of stress in both graduated and full-length leaves c¼ ... 0.3–0.5 6 32? ??811 422 533 533? ?65 9 533 444–533 422 363 .3 363 .3 411 363 .3 359–538 149 26 0 26 0–3 86 26 0 171? ? 26 0 149 90.3 90.3 138 90.3 538 538 26 0 149 3 16 26 0 26 0 26 0 8C Maximum temperature 2. 06 82 0 .68 94... kgf/mm2 MPa kgf/mm2 MPa kgf/mm2 10 0.5 521 0.44 82 0.34 52 0 .24 12 0.1 7 26 0.1 7 26 0.0 563 0.0457 0.03 52 0. 024 6 0.01 76 0.01 76 0 .68 94 0.5 521 0.4138 0 .27 56 0 .20 69 0 .20 69 0.0703 0.0 563 0.0 422 0. 028 1 0. 021 1... 0.34 52? ??1.03 46 1.03 46 8 .27 38 0 .68 94–1.3788 0.034 32? ??0 .68 94 0.4138–0. 62 0 8 0. 068 6–0 .27 46 0.0549–0.0981 0.0343–0. 068 6 0. 068 6–0 .27 46 1.03 46 2. 06 82 1.03 46 0.5 521 –0. 827 7 1.03 46? ??1. 724 0 0. 827 7–1.3788 0 .68 94