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.

**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) |
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} |

**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 |
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 | Value | ||

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 |

**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 |

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.

- Courtesy of Associated Spring (1987)., Design Handbook
- Budynas.R , Nisbett.K . (2014) . Shigley's Mechanical Engineering Design . 10th edition. McGraw-Hill

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