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When Castigliano's theorem is applied, the de¯ection of the helical spring is d dU dF 8FD 3 N d 4 G 4FDN d 2 G X 3X17 If we use the spring index C Dad, the de¯ection becomes d 8FD 3 N d 4 G 1 1 2C 2 % 8FD 3 N d 4 G X 3X18 3.4.4 SPRING RATE The general relationship between force and de¯ection can be written as F F dX 3X19 Then the spring rate is de®ned as kd lim Dd30 DF Dd dF dd Y 3X20 where d must be measured in the direction of the load F and at the point of application of F. Because most of the force±de¯ection equations that treat the springs are linear, k is constant and is named the spring constant. For this reason Eq. (3.20) may be written as k F d X 3X21 From Eq. (3.18), with the substitution C Dad, the spring rate for a helical spring under an axial load is k Gd 8C 3 N X 3X22 For springs in parallel having individual spring rates k i (Fig. 3.4a), the spring rate k is k k 1 k 2 k 3 X 3X23 For springs in series, with individual spring rates k i (Fig. 3.4b), the spring rate k is k 1 1 k 1 1 k 2 1 k 3 X 3X24 Figure 3.4 3. Springs 287 Machine Components 3.4.5 SPRING ENDS For helical springs the ends can be speci®ed as (Fig. 3.5) (a) plain ends; (b) plain and ground ends; (c) squared ends; (d) squared and ground ends. A spring with plain ends (Fig. 3.5a) has a noninterrupted helicoid, and the ends are the same as if a long spring had been cut into sections. A spring with plain and ground ends (Fig. 3.5b) or squared ends (Fig. 3.5c) is obtained by deforming the ends to a zero-degree helix angle. Springs should always be both squared and ground (Fig. 3.5d) because a better transfer of the load is obtained. Table 3.2 presents the type of ends and how that affects the number of coils and the spring length. In Table 3.2, N a is the number of active coils, and d is the wire diameter. Figure 3.5 Table 3.2 Types of Spring Ends Term End coils, Total coil, Free length, Solid length, Pitch, N e N t L 0 L s p Plain 0 N a pN a ddN t 1L 0 À daN a Plain and ground 1 N a 1 pN a 1 dN t L 0 aN a 1 Squared or closed 2 N a 2 pN a 3ddN t 1L o À 3daN a Squared and ground 2 N a 2 pN a 2ddN t L o À 2daN a Source: J. E. Shigley and C. R. Mischke, Mechanical Engineering Design. McGraw-Hill, New York, 1989. Used with permission. 288 Machine Components Machine Components EXAMPLE An oil-tempered wire is used for a helical compression spring. The wire diameter is d 0X025 in, and the outside diameter of the spring is D 0 0X375 in. The ends are plain and the number of total turns is 10.5. Find: The torsional yield strength The static load corresponding to the yield strength The rate of the spring The de¯ection that would be caused by the static load found The solid length of the spring The length of the spring so that no permanent change of the free length occurs when the spring is compressed solid and then released The pitch of the spring for the free length Solution From Eq. (3.3) the torsional yield strength, for hardened and tempered carbon and low-alloy steel, is S sy 0X50S ut X The minimum tensile strength given from Eq. (3.1) is S ut A d m Y where, from Table 3.1, the constant A 146 kpsi and the exponent m 0X193. The minimum tensile strength is S ut A d m 146 0X025 0X193 297X543 kpsiX The torsional yield strength is S sy À 0X50S ut 0X50297X543148X772 kpsiX To calculate the static load F corresponding to the yields strength, it is necessary to ®nd the spring index, C, and the shear stress correction factor, K s . The mean diameter D is the difference between the outside diameter and the wire diameter d, D D 0 À d 0X375 À 0X025 0X350 inX The spring index is C D d 0X350 0X025 14X From Eq. (3.9), the shear stress correction factor is K s 2C 1 2C 2141 214 1X035X With the use of Eq. (3.7), the static load is calculated with F pd 3 S sy 8K s D p0X025 3 148X77210 3 81X0350X350 2X520 lbX From Table 3.2, the number of active coils is N a N t 10X5. 3. Springs 289 Machine Components The spring rate, Eq. (3.22) for N N a is k Gd 8C 3 N a 11X510 6 0X025 814 3 10X5 1X24 lbainX The de¯ection of the spring is d F k 2X520 1X24 2X019 inX The solid length, L s , is calculated using Table 3.2: L s dN t 10X02510X5 10X287 inX To avoid yielding, the spring can be no longer than the sold length plus the defection caused by the load. The free length is L 0 d L s 2X019 0X287 2X306 inX From Table 3.2, the pitch p is calculated using the relation p L 0 À d N a 2X306 À 0X025 10X5 0X217 inX 3.5 Torsion Springs Helical torsion springs (Fig. 3.6) are used in door hinges, in automobile starters, and for any application where torque is required. Torsion springs are of two general types: helical (Fig. 3.7) and spiral (Fig. 3.8). The primary stress in torsion springs is bending. The bending moment Fa is applied to each end of the wire. The highest stress acting inside of the wire is s i K i Mc I Y 3X25 where the factor for inner surface stress concentration K i is given in Fig. 3.9, and I is the moment of inertia. The distance from the neutral axis to the extreme ®ber for round solid bar is c da2, and c ha2 for rectangular bar. For a solid round bar section, I pd 4 a64, and for a rectangular bar, I bh 3 a12. Substituting the product Fa for bending moment and the equations for section properties of round and rectangular wire, one may write for round wire I c pd 3 32 Y s i 32Fa pd 3 K iYround Y 3X26 and for rectangular wire I c bh 2 6 Y s i 6Fa bh 2 K iYrectangular X 3X27 The angular de¯ection of a beam subjected to bending is y ML EI Y 3X28 290 Machine Components Machine Components where M is the bending moment, L is the beam length, E is the modulus of elasticity, and I is the momentum of inertia. Equation (3.28) can be used for helical and spiral torsion springs. Helical torsion springs and spiral springs can be made from thin rectangular wire. Round wire is often used in noncritical applications. Figure 3.6 Figure 3.7 3. Springs 291 Machine Components 3.6 Torsion Bar Springs The torsion bar spring, shown in Fig. 3.10, is used in automotive suspension. The stress, angular de¯ection, and spring rate equation are t Tr J 3X29 y Tl JG 3X30 k JG l Y 3X31 Figure 3.8 Figure 3.9 292 Machine Components Machine Components where T is the torque, r da2 is the bar radius, l is the length of the spring, G is the modulus of rigidity, and J is the second polar moment of area. For a solid round section, J is J pd 4 32 X 3X32 For a solid rectangular section, J bh 3 12 X 3X33 For a solid round rod of diameter d, Eqs. (3.29), (3.30), and (3.31) become t 16T pd 3 3X34 y 32Tl pd 4 G 3X35 k pd 4 G 32l X 3X36 3.7 Multileaf Springs The multileaf spring can be a simple cantilever (Fig. 3.11a) or a semielliptic leaf (Fig. 3.11b). The design of multileaf springs is based on force F, length L, de¯ection, and stress relationships. The multileaf spring may be considered as a triangular plate (Fig. 3.12a) cut into n strips of width b or stacked in a graduated manner (Fig. 3.12b). Figure 3.10 Figure 3.11 3. Springs 293 Machine Components To support transverse shear N e , extra full-length leaves are added on the graduated stack, as shown in Fig. 3.13. The number N e is always one less than the total number of full length leaves N. The prestressed leaves have a different radius of curvature than the graduated leaves. This will leave a gap h between the extra full-length leaves and the graduated leaves before assembly (Fig. 3.14). Figure 3.12 Figure 3.13 294 Machine Components Machine Components 3.7.1 BENDING STRESS s e The bending stress in the extra full-length leaves installed without initial prestress is s e 18FL bt 2 3N e 2N g Y 3X37 where F is the total applied load at the end of the spring (lb), L is the length of the cantilever or half the length of the semielliptic spring (in), b is the width of each spring leaf (in), t is the thickness of each spring leaf (in), N e is the number of extra full-length leaves, and N g is the number of graduated leaves. 3.7.2 BENDING STRESS s g For graduated leaves assembled with extra full-length leaves without initial prestress, the bending stress is s g 12FL bt 2 3N e 2N g 2s e 3 X 3X38 3.7.3 DEFLECTION OF A MULTILEAF SPRING, d The de¯ection of a multileaf spring with graduated and extra full-length leaves is d 12Fl 3 bt 2 E 3N e 2N g Y 3X39 where E is the modulus of elasticity (psi). 3.7.4 BENDING STRESS, s The bending stress of multileaf springs without extra leaves or with extra full length prestressed leaves that have the same stress after the full load has been applied is s 6Fl Nbt 2 Y 3X40 where N is the total number of leaves. Figure 3.14 3. Springs 295 Machine Components 3.7.5 GAP The gap between preassembled graduated leaves and extra full-length leaves (Fig. 3.14) is h 2FL 3 Nbt 3 E Y 3X41 3.8 Belleville Springs Belleville springs are made from tapered washers (Fig. 3.15a) stacked in series, parallel, or a combination of parallel±series, as shown in Fig. 3.15b. The load±de¯ection and stress±de¯ection are F E d 1 À m 2 d o a2 2 M h Àda2h Àdt t 3 3X42 s E d 1 À m 2 d o a2 2 M C 1 h À da2C 2 tY 3X43 Figure 3.15 296 Machine Components Machine Components [...]... 3 26 130 280 58 3.05 160 .0 253.0 280 58 6. 35 160 .3 254.5 L28 140 210 33 2.03 153.7 195.3 210 33 228 140 250 42 2.54 161 .5 228 .6 250 42 4.75 161 .5 232.4 328 140 300 62 7.92 172.0 271.3 L30 150 225 35 2.03 164 .3 209.8 225 35 3. 96 164 .3 212.3 230 150 270 45 2.54 173.0 247 .6 270 45 6. 35 174.2 251.0 L32 160 240 38 2.03 175.8 223.0 232 160 6. 35 185.7 269 .5 L 36 180 280 46 4.75 199 .6 262 .9 2 36 180 320 52 6. 35... 0 .64 1.02 23.9 25.9 38.1 41.7 42 47 12 14 0 .64 1.02 24.4 25.9 36. 8 42.7 304 Machine Components L00 20 52 15 1.02 27.7 27.7 52 15 1.02 25.9 46. 2 L05 25 47 12 0 .64 29.0 42.9 47 12 0 .64 29.2 43.4 205 25 52 15 1.02 30.5 46. 7 52 15 1.02 30.5 47.0 305 25 62 17 1.02 33.0 54.9 62 17 1.02 31.5 55.9 L 06 30 55 13 1.02 34.8 49.3 47 9 0.38 2 06 30 62 16 1.02 36. 8 55.4 62 16 1.02 36. 1 56. 4 3 06 30 72 19 1.02 38.4 64 .8... 180 34 2.03 1 16. 1 164 .1 180 34 3. 96 1 16. 1 167 .1 320 95 215 47 2.54 122.9 194.1 215 47 4.75 122.4 194 .6 L21 100 160 26 2.03 1 16. 1 1 46. 8 160 26 221 100 190 36 2.03 121.9 173.5 190 36 3. 96 121.4 175.3 321 100 225 49 2.54 128.8 203.5 225 49 4.75 128.0 203.5 L22 105 170 28 2.03 122.7 1 56. 5 170 28 2.54 121.9 159.3 222 322 105 105 200 240 38 50 2.03 2.54 127.8 134.4 182 .6 218.2 200 240 38 50 3. 96 4.75 127.3... 309 45 100 25 1.52 57.2 88.9 100 25 2.03 55.9 90.4 L10 50 80 16 1.02 55 .6 73.7 72 12 0 .64 54.1 68 .1 210 310 50 50 90 110 20 27 1.02 2.03 57.7 64 .3 82.3 96. 5 90 110 20 27 1.52 2.03 57.7 61 .0 82.8 99.1 L11 55 90 18 1.02 61 .7 83.1 90 18 1.52 62 .0 83 .6 211 55 100 21 1.52 65 .0 90.2 100 21 2.03 64 .0 91.4 311 55 120 29 2.03 69 .8 1 06. 2 120 29 2.03 66 .5 108.7 3.33 43.9 (continued) 307 4 Rolling Bearings Table... 24 2.54 81.8 115 .6 314 70 150 35 2.03 86. 9 134.4 150 35 3.18 84.3 135 .6 L15 75 115 20 1.02 82.3 107.2 115 20 215 75 130 25 1.52 86. 1 118.9 130 25 2.54 85 .6 120.1 215 75 160 37 2.03 92.7 143.8 160 37 3.18 90.4 145.8 L 16 2 16 80 80 125 140 22 26 1.02 2.03 88.1 93.2 1 16. 3 1 26. 7 125 140 22 26 2.03 2.54 88.4 91.2 117 .6 129.3 3 16 80 170 39 2.03 98 .6 152.9 170 39 3.18 96. 0 154.4 L17 85 130 22 1.02 93.2 121.4... 4.50 5.80 6. 10 8.20 8.70 9.10 11 .6 12.2 14.2 15.0 17.2 18.0 18.0 21.0 23.5 24.5 29.5 30.5 34.5 1.42 1.42 1. 56 2.70 3.35 3 .65 5.40 8.50 9.40 9.10 9.70 12.0 13 .6 16. 0 17.0 17.0 18.4 22.5 25.0 27.5 30.5 32.0 35.0 37.5 41.0 47.5 1.90 2. 46 3.05 3.75 5.30 5.90 8.80 10 .6 12 .6 14.8 15.8 18.0 20.0 22.0 24.5 25.5 28.0 30.0 32.5 38.0 40.5 43.5 46. 0 1.02 1.10 1.28 1. 36 2.20 2 .65 3 .60 4.75 4.95 6. 30 6. 60 9.00 9.70... (mm) 60 95 18 1.02 66 .8 87.9 95 18 1.52 67 .1 88 .6 212 60 110 22 1.52 70 .6 99.3 110 22 2.03 69 .3 101.3 312 60 130 31 2.03 75.4 115 .6 130 31 2.54 72.9 117.9 L13 65 100 18 1.02 71.9 92.7 100 18 1.52 72.1 93.7 213 65 120 23 1.52 76. 5 108.7 120 23 2.54 77.0 110.0 313 65 140 33 2.03 81.3 125.0 140 33 2.54 78.7 127.0 L14 70 110 20 1.02 77.7 102.1 110 20 214 70 125 24 1.52 81.0 114.0 125 24 2.54 81.8 115 .6 314... (mm) dS (mm) dH (mm) 10 26 8 0.30 12.7 23.4 200 10 30 9 0 .64 13.8 26. 7 300 10 35 11 0 .64 14.8 31.2 L01 12 28 8 0.30 14.5 25.4 201 12 32 10 0 .64 16. 2 28.4 301 12 37 12 1.02 17.7 32.0 L02 15 32 9 0.30 17.5 29.2 202 15 35 11 0 .64 19.0 31.2 302 15 42 13 1.02 21.2 36. 6 L03 17 35 10 0.30 19.8 32.3 35 10 0 .64 20.8 32.0 203 17 40 12 0 .64 22.4 34.8 40 12 0 .64 20.8 36. 3 303 17 47 14 1.02 23 .6 41.1 47 14 1.02 22.9... 12.5 14.9 18.9 21.1 23 .6 23 .6 26. 2 30.7 37.4 44.0 48.0 49.8 54.3 61 .4 69 .4 77.4 83 .6 113.4 140.1 162 .4 211.3 258.0 4.90 6. 20 8.5 10.0 13.1 16. 5 20.9 24.5 27.1 32.5 38.3 44.0 45.4 51 .6 55.2 65 .8 65 .8 72.9 84.5 85.4 100.1 120.1 131.2 7.20 7.40 11.3 12.0 12.2 17.3 18.0 20.9 55.0 29.4 71.0 48.9 58.7 97.9 Source : New Departure-Hyatt Bearing Division, General Motors Corporation 4 .6. 2 RELIABILITY REQUIREMENT... 1.02 38.4 64 .8 72 19 1.52 37.8 64 .0 L07 207 35 35 62 72 14 17 1.02 1.02 40.1 42.4 56. 1 65 .0 55 72 10 17 0 .64 1.02 39.4 41.7 50.8 65 .3 307 35 80 21 1.52 45.2 70.4 80 21 1.52 43.7 71.4 L08 40 68 15 1.02 45.2 62 .0 68 15 1.02 45.7 62 .7 208 40 80 18 1.02 48.0 72.4 80 18 1.52 47.2 72.9 308 40 90 23 1.52 50.8 80.0 90 23 1.52 49.0 81.3 L09 45 75 16 1.02 50.8 52.8 75 16 1.02 50.8 69 .3 209 45 85 19 1.02 52.8 77.5 . 1.02 30.5 46. 7 52 15 1.02 30.5 47.0 305 25 62 17 1.02 33.0 54.9 62 17 1.02 31.5 55.9 L 06 30 55 13 1.02 34.8 49.3 47 9 0.38 3.33 43.9 2 06 30 62 16 1.02 36. 8 55.4 62 16 1.02 36. 1 56. 4 3 06 30 72 19. 80 16 1.02 55 .6 73.7 72 12 0 .64 54.1 68 .1 210 50 90 20 1.02 57.7 82.3 90 20 1.52 57.7 82.8 310 50 110 27 2.03 64 .3 96. 5 110 27 2.03 61 .0 99.1 L11 55 90 18 1.02 61 .7 83.1 90 18 1.52 62 .0 83 .6 211. 38.4 64 .8 72 19 1.52 37.8 64 .0 L07 35 62 14 1.02 40.1 56. 1 55 10 0 .64 39.4 50.8 207 35 72 17 1.02 42.4 65 .0 72 17 1.02 41.7 65 .3 307 35 80 21 1.52 45.2 70.4 80 21 1.52 43.7 71.4 L08 40 68 15
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