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 Cycle
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
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
Oil-tempered wire
0.020-0.500
0.50-12
0.186
149
1880
Hard-drawn wirec
0.028-0.500
0.70-12
0.192
136
1750
d
Chromium vanadium
0.032-0.437
0.80-12
0.167
169
2000
Chromium silicone
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
i
fin sh.
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
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
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Spring Equations
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
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
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'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.
Fundamentals of Machine Elements, 3rd ed.
© 2014 CRC Press
Schmid, Hamrock and Jacobson
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Dimensions and Preload
Figure 17.10: Important
dimensions of a helical extension
spring.
Figure 17.11: Preferred range of
preload stress for various spring
indexes.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Torsion Springs
Figure 17.12: Helical torsion spring.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
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).
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Belleville Spring Stacks
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
Figure 17.19: Examples of common wave spring configurations. (a) Common crest-tocrest 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
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
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
Figure 17.21: Dickerman Feed Unit.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press
Case Study Results
Figure 17.22: Performance of the spring in case study.
Fundamentals of Machine Elements, 3rd ed.
Schmid, Hamrock and Jacobson
© 2014 CRC Press