# 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.

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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)

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

Spring Materials Table 17.1: Typical properties of common spring materials. Source: Adapted from Relvas [1996].

Spring Material Properties
Table 17.2: Coefficients used in Eq. (17.2) for selected spring materials.

Helical Coil 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.

Wire Stresses and Correction
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. Figure 17.4: Comparison of the Wahl and Bergstraesser curvature correction factors used for helical springs. The transverse shear factor is also shown.

Compression Spring Ends
Figure 17.5: Four end types commonly used in compression springs. (a) Plain; (b) plain and ground; (c) squared; (d) squared and ground.

Deflection 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.

Spring Equations Table 17.3: Useful formulas for compression springs with four end conditions.

Deflection – Graphical Representation
Figure 17.7: Graphical representation of deflection, force and length for four spring positions.

Spring Buckling Figure 17.8: Critical buckling conditions for parallel and nonparallel ends of compression springs. Source: Engineering Guide to Spring Design, Barnes Group, Inc., [1987].

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. 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. 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. 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. 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.

Design Procedure 17.1 (concluded)
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. At the free height, the spring has no restraining force, and therefore a spring should have at least some preload. 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. 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's 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 better design.

Extension Spring Ends 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.

Figure 17.10: Important dimensions of a helical extension spring. Figure 17.11: Preferred range of preload stress for various spring indexes.

Torsion Springs Figure 17.12: Helical torsion spring.

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.

Gas Springs Figure 17.15: Gas springs. (a) A collection of gas springs. Note that the springs are available with a wide variety of end attachments and strut lengths. Source: Courtesy of Newport Engineering Associates, Inc. (b) Schematic illustration of a typical gas spring.

Belleville Springs Figure 17.16: Typical Belleville spring. (a) Isometric view of Belleville spring; (b) cross section, with key dimensions identified. Figure 17.17: Force-deflection response of Belleville spring given by Eq. (17.54).

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

Wave Springs 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.

Multiple Wave Factor Table 17.4: Multiple wave factor, Kw, used to calculate wave spring stiffness. Source: Courtesy Smalley Co.

Case Study: Progressive Die
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].

Dickerman Feed Figure 17.21: Dickerman Feed Unit.

Case Study Results Figure 17.22: Performance of the spring in case study.

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