Chapter 17: Springs. Fundamentals of Machine Elements, 3 rd ed. Schmid, Hamrock and Jacobson CRC Press

Transcription

1 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)

2 Stress Cycle Stress U U Strain Figure 17.1: Stress- strain curve for one complete cycle.

3 Spring Materials Elastic Shear Density, Maximum modulus, modulus,, service E, G, kg/m 3 temperature, n GPa (Mpsi) GPa (MPsi) (lbm/in 3 ) C ( F) Principal characteristics High- carbon steels Music wire (ASTM A228) 207 (30.0) 79.3 (11.5) 7840 (0.283) 120 (248) High strength; excellent fatigue life Hard drawn (ASTM A227) 207 (30.0) 79.3 (11.5) 7840 (0.283) 120 (248) General purpose use; poor fatigue life Stainless steels Martensitic (AISI 410, 420) 200 (29.0) 75.8 (11.0) 7750 (0.280) 250 (482) Unsatisfactory for subzero applications Austenitic (AISI 301, 302) 193 (28.0) 68.9 (9.99) 7840 (0.283) 315 (600) Good strength at moderate temperatures; low stress relaxation Copper- based alloys Spring brass (ASTM B134) 110 (15.9) 41.4 (6.00) 8520 (0.308) 90 (194) Low cost; high conductivity; poor mechanical properties Phosphor bronze (ASTM B159) 103 (14.9) 43.4 (6.29) 8860 (0.320) 90 (194) Ability to withstand repeated res; popular alloy Beryllium copper (ASTM B197) 131 (19.0) 44.8 (6.50) 8220 (0.297) 200 (392) High yield and fatigue strength; hardenable Nickel- based alloys Inconel (31.0) 75.8 (11.0) 8500 (0.307) 315 (600) Good strength; high corrosion resistance Inconel X- 750 Ni- Span C 214 (31.0) 186 (27.0) 75.8 (11.0) 66.2 (9.60) 8250 (0.298) 8140 (0.294) 600 (1110) 90 (194) 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].

4 Spring Material Properties Size range Exponent, Constant, A p Material in. mm m ksi MPa Music wire a Oil-tempered wire b Hard-drawn wire c Chromium vanadium d Chromium silicon e stainless steel Phosphor-bronze f a Surface is smooth and free from defects and has a bright, lustrous b Surface hasaslight heat-treating scale that must be removed before plating. Surface is smooth and bright with no visible marks. d Aircraft-quality tempered wire; can also be obtained annealed. 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.

5 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.

6 Wire Stresses and Correction 1.5 d d 1.4 d (a) Spring axis d (b) Spring axis Spring factor K d K b K w (c) D/2 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. (d) D/2 Spring Index, C Figure 17.4: Comparison of the Wahl and Bergstraesser curvature correction factors used for helical springs. The transverse shear factor is also shown.

7 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.

8 Deflection (P = 0) P r P o P s l f li g a l o l s (a) (b) (c) (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.

9 Spring Equations Term Plain Type of spring end Plain and ground Squared or closed Squared and ground Number of end coils, N e Total number of active coils, N a N t N t 1 N t 2 N t 2 Free length, l f pn a + d p(n a + 1) pn a + 3 d pn a + 2 d Solid length, l s d(n t + 1) dn t d(n t + 1) dn t Pitch at free length, p (l f d)/n a l f / (N a + 1) ( l f 3d)/N a (l f 2d)/N a Table 17.3: Useful formulas for compression springs with four end conditions.

10 Deflection Graphical Representation l f l i l o Length, l P s 0 l s Spring force, P P o P i 0 0 δ i δ o δ s Deflection, δ Figure 17.7: Graphical representation of deflection, force and length for four spring positions.

11 Spring Buckling Ratio of deflection to free length, /l f 0.80 Stable 0.60 Unstable 0.40 Stable 0.20 Unstable Parallel ends Nonparallel ends Ratio of free length to mean coil diameter, l f /D Figure 17.8: Critical buckling conditions for parallel and nonparallel ends of compression springs. Source: Engineering Guide to Spring Design, Barnes Group, Inc., [1987].

12 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.

13 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.

14 P P Extension Spring d r 3 A d Ends r 1 r 2 r 4 B (a) P (b) P d r 3r1 A d r 2 r 4 (c) (d) B 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.

15 Dimensions and Preload d o l f g a d i l l l b Preload stress, MPa Preferred range Preload stress, ksi l h Spring index Figure 17.10: Important dimensions of a helical extension spring. Figure 17.11: Preferred range of preload stress for various spring indexes.

16 Torsion Springs P a P d D Figure 17.12: Helical torsion spring.

17 Leaf Spring Rear shock absorber Spring shackle Spring eye Brake drum Leaf spring Figure 17.13: Illustration of a leaf spring used in an automotive application. Figure 17.14: Leaf spring. (a) Triangular plate, cantilever spring; (b) equivalent multiple- leaf spring.

18 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.

19 Belleville Springs h D i D o t Percent force to flat Height-tothickness ratio, (a) (b) Figure 17.16: Typical Belleville spring. (a) Isometric view of Belleville spring; (b) cross section, with key dimensions identified Percent deflection to flat Figure 17.17: Force- deflection response of Belleville spring given by Eq. (17.54).

20 Belleville Spring Stacks (a) (b) Figure 17.18: Stacking of Belleville springs. (a) in parallel; (b) in series.

21 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.

22 Multiple Wave Factor Waves per turn, N w Multiple wave factor, K w > Table 17.4: Multiple wave factor, K w, used to calculate wave spring stiffness. Source: Courtesy Smalley Co.

23 Case Study: Progressive Die Ram Blanking punch Pilot Scrap Die Stop Piercing punch Stripper Strip Slug Part Strip Finished washer Scrap First operation (a) (b) 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].

24 Dickerman Feed Cam Gripping unit (sliding) Spring Gripping unit (fixed) Fixed rear guide Figure 17.21: Dickerman Feed Unit.

25 Case Study Results Maximum force, P max, lbf Safety factor, n s Wire diameter, d, in. Wire diameter, d, in. Figure 17.22: Performance of the spring in case study.