# Compression Spring Example

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 of Compression Spring Design Problem:

Step 1 : Write down input parameters.

 INPUT PROPERTIES SUMMARY Parameter Value 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 Value 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 Value 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

Step 4 : Visit critical frequency of compression springs  and solve the problem. Results generated by the calculator are as follows.

 RESULTS Parameter Value 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.