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 ̎¡žres;
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)
ŠŽ›’Š•Ȧ™ŽŒ’ęŒŠ’˜n
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