Sample Problem : Design of A Compression Spring which works under cyclic loading

An as-wound 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) Lf 4 inch
Number of active Na 22 ---
End types for compression spring Both ends squared and ground
Maximum cyclic force Fmax 22 lbf
Minimum cyclic force (preload) Fmin 8 lbf
Material Music wire (unpeened)
Density ρ 0.283 lb/in3

Step 2 : Visit compression spring design for fatigue loading and solve the problem. Results generated by the calculator are as follows.


 RESULTS
Parameter Symbol Value Unit
Wahl factor Kw 1.22 ---
Shear stress amplitude
τa 23.14 ksi
Midrange shear stress
τm 49.58
Ultimate tensile strength of material Sut 289.35
Shearing ultimate strength Ssu 193.86
Endurance limit ( according to Gerber) Se 38.01
Endurance limit (according to Goodman) Se 48.79
Strength amplitude component ( according to Gerber) Ssa 32.96
Strength amplitude component ( according to Sines) Ssa 34.95
Strength amplitude component ( according to Goodman) Ssa 31.7
Factor of safety (Acc. to Gerber)+ fosgerber 1.42 ---
Factor of safety (Acc. to Sines)+ fossines 1.51
Factor of safety (Acc.to Goodman)+ fosgoodman 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 Na 22 ---
Number of total coils Nt 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 ODat solid*** 0.63
Spring free length (height) Lf 4
Spring solid height Ls 1.92
Maximum deflection (Lf to Ls) 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 Ref-1, 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 Step-2 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: