Sample Problem : Design of A Compression Spring which works under cyclic loading
An aswound helical compression spring, made of music wire, has a wire size of
0.080
in, an outside coil diameter of 10/16 in, a free length of 4
in, 22 active coils, and both ends
squared and ground. The spring is unpeened. This spring is to be assembled with a
preload of 8 lbf and will operate with a maximum load of 22 lbf during use.
(a) Estimate the factor of safety guarding against fatigue failure using a torsional
Gerber fatigue failure criterion with Zimmerli data.
(b) Repeat part (a) using the Sines torsional fatigue criterion (steady stress component
has no effect), with Zimmerli data.
(c) Repeat using a torsional Goodman failure criterion with Zimmerli data.
(d) Estimate the critical frequency of the spring.
Solution:
Step 1 : Write down input parameters which are
defined in the sample example.
INPUT PROPERTIES SUMMARY 
Parameter 
Symbol 
Value 
Unit 
Wire diameter 
d 
0.080 
inch 
Spring outer diameter 
OD 
0.625 
inch 
Spring free length (height) 
L_{f} 
4 
inch 
Number of active 
N_{a} 
22 
 
End types for compression spring 
Both ends squared and ground 
Maximum cyclic force 
F_{max} 
22 
lbf 
Minimum cyclic force (preload) 
F_{min} 
8 
lbf 
Material 
Music wire (unpeened) 
Density 
ρ 
0.283 
lb/in^{3} 
RESULTS 
Parameter 
Symbol 
Value 
Unit 
Wahl factor 
K_{w} 
1.22 
 
Shear stress amplitude

τ_{a} 
23.14 
ksi 
Midrange shear stress

τ_{m} 
49.58 
Ultimate tensile strength of material 
S_{ut} 
289.35 
Shearing ultimate strength 
S_{su} 
193.86 
Endurance limit ( according to Gerber) 
S_{e} 
38.01 
Endurance limit (according to Goodman) 
S_{e} 
48.79 
Strength amplitude component ( according to Gerber) 
S_{sa} 
32.96 
Strength amplitude component ( according to Sines) 
S_{sa} 
34.95 
Strength amplitude component ( according to Goodman) 
S_{sa} 
31.7 
Factor of safety (Acc. to Gerber)^{+} 
fos_{gerber} 
1.42 
 
Factor of safety (Acc. to Sines)^{+} 
fos_{sines} 
1.51 
Factor of safety (Acc.to Goodman)^{+} 
fos_{goodman} 
1.37 
Material ASTM No. 
A228 
Step 3 : For the calculation of critical working frequency of the spring,
spring rate shall be calculated. Visit
rate based compression spring design calculator and calculate
spring rate (k). Results generated by the calculator are as follows.
RESULTS 
Parameter 
Symbol 
Value 
Unit 
DIMENSIONAL PARAMETERS 
Number of active coils 
N_{a} 
22 
 
Number of total coils 
N_{t} 
24 
Spring index 
C* 
6.81 
Spring rate 
k 
16.89 
lbf/inch 
Wire diameter 
d 
0.08 
inch 
Spring outer diameter 
OD 
0.625 
Spring mean diameter 
D 
0.545 
Spring inner diameter 
ID 
0.465 
Outer diameter at solid length 
OD_{at solid}*** 
0.63 
Spring free length (height) 
L_{f} 
4 
Spring solid height 
L_{s} 
1.92 
Maximum deflection (L_{f} to
L_{s}) 
Δx 
2.08 
Pitch at free length 
p** 
0.17 
SPRING MATERIAL & STRESS RELEATED PARA 
Step 4 : Visit
critical frequency of compression springs and solve the problem. Results
generated by the calculator are as follows.
RESULTS 
Parameter 
Symbol 
Value 
Unit 
Natural frequency of spring 
f^{+} 
174.428 
Hz 
Mass of the active coils 
m 
0.054 
lb 
Spring index 
C* 
6.812 
 
OD 
Spring outer diameter 
0.625 
inch 
D 
Spring mean diameter 
0.545 
ID 
Spring inner diameter 
0.465 
Conclusion:
The natural frequency of the spring is calculated as 174.4 Hz. According to
Ref1, the cyclic loading frequency shall be minimally 13 times smaller than
natural frequency. Based on this information, maximum cyclic loading frequency
is calculated as 174.4/13= 13.4 Hz. If the loading frequency is lower than 13.4
Hz, the effect of resonance is negligible and fos values given in Step2
can be used.
If a spring cannot be designed
to have a natural frequency more than 13 times operating frequency, or if the
spring is to serve as a vibration damping device, it must utilize one of several methods of energy absorption.
Reference: