Chapter 17: Springs It must be confessed that the inventors of the mechanical arts have been much more useful to men than the inventors of syllogisms. Voltaire A collection of helical compression springs. (Courtesy of Danly Die) Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Stress Stress Cycle 6U U Strain Figure 17.1: Stress-­‐‑strain curve for one complete cycle. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Spring Materials Elastic modulus, E, GPa (Mpsi) Shear modulus, G, GPa (MPsi) Density, Ε, kg/m3 (lbm/in3 ) Maximum service temperature, Ȥ C ( Ȥ F) 207 (30.0) 207 (30.0) 79.3 (11.5) 79.3 (11.5) 7840 (0.283) 7840 (0.283) 120 (248) 120 (248) High strength; excellent fatigue life General purpose use; poor fatigue life 200 (29.0) 193 (28.0) 75.8 (11.0) 68.9 (9.99) 7750 (0.280) 7840 (0.283) 250 (482) 315 (600) Unsatisfactory for subzero applications Good strength at moderate temperatures; low stress relaxation 110 (15.9) 41.4 (6.00) 8520 (0.308) 90 (194) Phosphor bronze (ASTM B159) 103 (14.9) 43.4 (6.29) 8860 (0.320) 90 (194) Beryllium copper (ASTM B197) 131 (19.0) 44.8 (6.50) 8220 (0.297) 200 (392) Low cost; high conductivity; poor mechanical properties Ability to withstand repeated Ě¡res; popular alloy High yield and fatigue strength; hardenable 214 (31.0) 214 (31.0) 75.8 (11.0) 75.8 (11.0) 8500 (0.307) 8250 (0.298) 315 (600) 600 (1110) 186 (27.0) 66.2 (9.60) 8140 (0.294) 90 (194) Ȧęn High-­‐‑carbon steels Music wire (ASTM A228) Hard drawn (ASTM A227) Stainless steels Martensitic (AISI 410, 420) Austenitic (AISI 301, 302) Copper-­‐‑based alloys Spring brass (ASTM B134) Nickel-­‐‑based alloys Inconel 600 Inconel X-­‐‑750 Ni-­‐‑Span C Principal characteristics Good strength; high corrosion resistance Precipitation hardening; for high temperatures Constant modulus over a wide temperature range Table 17.1: Typical properties of common spring materials. Source: Adapted from Relvas [1996]. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Spring Material Properties Size range Exponent, Constant, A p Material in. mm m ksi MPa a Music wire 0.004-0.250 0.10-6.5 0.146 196 2170 b 0.020-0.500 0.50-12 0.186 149 1880 Oil-tempered wire c 0.028-0.500 0.70-12 0.192 136 1750 Hard-drawn wire d 0.032-0.437 0.80-12 0.167 169 2000 Chromium vanadium e Chromium silicon 0.063-0.375 1.6-10 0.112 202 2000 302 stainless steel 0.013-0.10 0.33-2.5 0.146 169 1867 0.10-0.20 2.5-5 0.263 128 2065 0.20-0.40 5-10 0.478 90 2911 f Phosphor-bronze 0.004-0.022 0.1-0.6 0 145 1000 0.022-0.075 0.6-2 0.028 121 913 0.075-0.30 2-7.5 0.064 110 932 a Surface is smooth and free from defects and has a bright, lustrous ÀQLVK b Surface has a slight heat-treating scale that must be removed before plating. c Surface is smooth and bright with no visible marks. d Aircraft-quality tempered wire; can also be obtained annealed. e Tempered to Rockwell C49 but may also be obtained untempered. f SAE CA510, tempered to Rockwell B92-B98. Table 17.2: Coefficients used in Eq. (17.2) for selected spring materials. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Helical Coil R P D P T = PR P R P (a) (b) Figure 17.2: Helical coil. (a) Coiled wire showing applied force; (b) coiled wire with section showing torsional and direct (vertical) shear acting on the wire. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Wire Stresses and Correction 1.5 d 1.4 (a) (b) Spring axis Spring axis Spring factor d Kw 1.3 Kb 1.2 1.1 d Kd d 1.0 D/2 (c) D/2 3 6 9 12 Spring Index, C (d) Figure 17.3: Shear stresses acting on wire and coil. (a) Pure torsional loading; (b) transverse loading; (c) torsional and transverse loading with no curvature effects; (d) torsional and transverse loading with curvature effects. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson Figure 17.4: Comparison of the Wahl and Bergstraesser curvature correction factors used for helical springs. The transverse shear factor is also shown. © 2014 CRC Press Compression Spring Ends (a) (b) (c) (d) Figure 17.5: Four end types commonly used in compression springs. (a) Plain; (b) plain and ground; (c) squared; (d) squared and ground. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Deflection (P = 0) Pr Po Ps lf li lo ga (a) (b) (c) ls (d) Figure 17.6: Various lengths and forces applicable to helical compression springs. (a) Unloaded; (b) under initial load; (c) under operating load; (d) under solid load. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Spring Equations Term Number of end coils, N e Total number of active coils, N a Free length, l f Solid length, l s Pitch at free length, p Plain 0 Nt pN a + d d(N t + 1) (l f ƺ d)/N a Type of spring end Plain and ground Squared or closed 1 2 Nt ƺ 1 Nt ƺ 2 p(N a + 1) pN a + 3 d dN t d(N t + 1) l f / (N a + 1) ( l f ƺ 3d)/N a Squared and ground 2 Nt ƺ 2 pN a + 2 d dN t (l f ƺ 2d)/N a Table 17.3: Useful formulas for compression springs with four end conditions. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Deflection – Graphical Representation lf li lo ls Length, l 0 Spring force, P Ps Po Pi 0 0 bi bo bs Deflection, b Figure 17.7: Graphical representation of deflection, force and length for four spring positions. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Spring Buckling Ratio of deflection to free length, b/l f 0.80 Stable Unstable 0.60 Unstable 0.40 Stable Pa rall el en d 0.20 0 3 s Nonpa rallel e nds 4 5 6 7 8 9 Ratio of free length to mean coil diameter, lf /D 10 Figure 17.8: Critical buckling conditions for parallel and nonparallel ends of compression springs. Source: Engineering Guide to Spring Design, Barnes Group, Inc., [1987]. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Design Procedure 17.1: Design Synthesis of Helical Springs The following are important considerations for synthesis of springs. The considerations are strictly applicable to helical compression springs, but will have utility elsewhere as well. 1. The application should provide some information regarding the required force and spring rate or total deflection for the spring. It is possible that the solid and free lengths are also prescribed. Usually, there is significant freedom for the designer, and not all of these quantities are known beforehand. 2. Select a spring index in the range of 4 to 12. A spring index lower than 4 will be difficult to manufacture, while a spring index higher than 12 will result in springs that are flimsy and tangle easily. Higher forces will require a smaller spring index. A value between 8 and 10 is suitable for most design applications. 3. The number of active coils should be greater than 2 in order to avoid manufacturing difficulties. The number of active coils can be estimated from a spring stiffness design constraint. 4. For initial design purposes, the solid height should be specified as a maximum dimension. Usually, applications will allow a spring to have a smaller solid height than the geometry allows, so the solid height should not be considered a strict constraint. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Design Procedure 17.1 (concluded) 5. When a spring will operate in a cage or with a central rod, a clearance of roughly 10% of the spring diameter must be specified. This is also useful in compensating for a coating thickness from an electroplating process, for example. 6. At the free height, the spring has no restraining force, and therefore a spring should have at least some preload. 7. To avoid compressing a spring to its solid length, and the impact and plastic deformation that often result, a clash allowance of at least 10% of the maximum working deflection should be required before the spring is compressed solid. 8. Consider the application when designing the spring and the amount of force variation that is required. Sometimes, such as in a garage door counterbalance spring, it is useful to have the force vary significantly, because the load changes with position. For such applications, a high spring rate is useful. However, it is often the case that only small variations in force over the spring'ʹs range of motion are desired, which suggests that low spring rates are preferable. In such circumstances, a preloaded spring with a low stiffness will represent a befer design. Fundamentals of Machine Elements, 3rd ed. © 2014 CRC Press Schmid, Hamrock and Jacobson P Extension Spring Ends P d r3 d A r1 r2 r4 B (a) (b) P P d d r3 r1 A r2 r4 B (c) Figure 17.9: Ends for extension springs. (a) Conventional design; (b) side view of Fig. 16.8a; (c) improved design over Fig. 16.8a; (d) side view of Fig. 16.8c. (d) Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Dimensions and Preload do 200 28 175 lb lf Preload stress, MPa ll 150 20 125 Preferred range 16 100 12 75 8 50 ga lh 25 Figure 17.10: Important dimensions of a helical extension spring. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson Preload stress, ksi 24 di 4 6 8 10 12 Spring index 14 4 16 Figure 17.11: Preferred range of preload stress for various spring indexes. © 2014 CRC Press Torsion Springs P P d a D Figure 17.12: Helical torsion spring. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Leaf Spring Rear shock absorber Spring shackle Brake drum Spring eye Leaf spring Figure 17.13: Illustration of a leaf spring used in an automotive application. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson Figure 17.14: Leaf spring. (a) Triangular plate, cantilever spring; (b) equivalent multiple-­‐‑leaf spring. © 2014 CRC Press Gas Springs High pressure nitrogen gas chamber Metering orifice Integral grease chamber Seals Oil zone for end position damping and lubrication (a) Polished steel rod (b) Figure 17.15: Gas springs. (a) A collection of gas springs. Note that the springs are available with a wide variety of end afachments and strut lengths. Source: Courtesy of Newport Engineering Associates, Inc. (b) Schematic illustration of a typical gas spring. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Belleville Springs 200 Height-tothickness ratio, 2.828 Di h t Do (a) (b) Figure 17.16: Typical Belleville spring. (a) Isometric view of Belleville spring; (b) cross section, with key dimensions identified. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson Percent force to flat 160 2.275 1.000 120 1.414 80 40 0.400 0 0 40 80 120 Percent deflection to flat 160 200 Figure 17.17: Force-­‐‑deflection response of Belleville spring given by Eq. (17.54). © 2014 CRC Press Belleville Spring Stacks (a) (b) Figure 17.18: Stacking of Belleville springs. (a) in parallel; (b) in series. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Wave Springs (a) (b) (c) Figure 17.19: Examples of common wave spring configurations. (a) Common crest-­‐‑to-­‐‑ crest orientation; (b) crest-­‐‑to-­‐‑crest orientation with shim ends; (c) nested wave springs. Source: Courtesy of Smalley Co. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Multiple Wave Factor Waves per turn, Nw 2.0-­‐‑4.0 4.5-­‐‑6.5 7.0-­‐‑9.5 > 9.5 Multiple wave factor, Kw 3.88 2.9 2.3 2.13 Table 17.4: Multiple wave factor, Kw, used to calculate wave spring stiffness. Source: Courtesy Smalley Co. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Case Study: Progressive Die Ram Blanking punch Pilot Piercing punch Stripper Scrap Die Strip Stop Slug Part Strip (b) Finished washer Scrap First operation (a) Figure 17.20: Illustration of a simple part that is produced by a progressive die. (a) Schematic illustration of the two-­‐‑station die set needed to produce a washer; (b) sequence of operations to produce an aerosol can lid. Source: From Kalpakjian and Schmid [2008]. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Dickerman Feed Cam Gripping unit (sliding) Spring Gripping unit (fixed) Fixed rear guide Figure 17.21: Dickerman Feed Unit. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press Case Study Results 1.4 1.2 500 Safety factor, ns Maximum force, Pmax, lbf 600 400 300 200 100 0 1.0 0.8 0.6 0.4 0.2 0.04 0.08 0.12 0.16 Wire diameter, d, in. 0.20 0 0.04 0.08 0.12 0.16 Wire diameter, d, in. 0.20 Figure 17.22: Performance of the spring in case study. Fundamentals of Machine Elements, 3rd ed. Schmid, Hamrock and Jacobson © 2014 CRC Press