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Optimal design of conical springs

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Abstract

Mechanical systems often use springs to store energy though their axial length must sometimes be significantly reduced. This leads to the use of conical springs as they are able to telescope. Designers of mechanical systems can call on a large number of tools to assist them though most of these are merely validation tools requiring concomitant trial and error strategies. Optimization strategies can be applied to provide synthesis assistance tools for which the designer simply specifies his requirement. Thus the tool directly indicates the spring best suited to standards and requirements. Recent advances in the study of constant pitch conical springs have provided analytical expressions of their behavior even in the non-linear phase. Considering this, we have used optimization strategies to provide a synthesis tool for conical spring design. An example of application is presented. The tool introduced here is thus a synthesis assistance tool that can be of considerable interest for designers who require a conical spring in their design.

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Abbreviations

A :

least compressed state of the spring

B :

most compressed state of the spring

D 1 :

small mean diameter (mm)

D 2 :

large mean diameter (mm)

D f :

diameter limiting free coils and coils at solid (mm)

D iL :

large inner diameter (mm)

D iS :

small inner diameter (mm)

D oL :

large outer diameter (mm)

D oS :

small outer diameter (mm)

d :

wire diameter (mm)

F 1 :

minimum operating load (N)

F 2 :

maximum operating load (N)

F c :

solid load (N)

F T :

transition operating load (N)

f e :

natural frequency of surge waves (Hz)

G :

wire torsion modulus (N/mm2)

k :

stress correction factor

L 0 :

free length (mm)

L 1 :

maximum operating length (mm)

L 2 :

minimum operating length (mm)

L a :

free length of active coils (mm)

L c :

solid length (mm)

L K :

buckling length (mm)

L n :

minimal operating length to keep within the operating range (mm)

L s :

solid length of active coils (mm)

L T :

transition length (mm)

M :

spring mass (g)

r m :

radial pitch of the spring (mm)

n a :

number of active coils

n i :

number of inactive coils for the ends related to L 0

n f :

number of coils free to deflect during a compression phase

n T :

total number of coils

R :

spring rate of the linear range (N/mm)

R m :

ultimate tensile strength of the wire (N/mm2)

S h :

spring travel (mm)

w 1 :

spring index related to D 1

w 2 :

spring index related to D 2

α 2 :

percentage of R m at L 2

α c :

percentage of R m at L c

α F :

fatigue life factor

γ :

maximum helix angle (°)

ν :

end fixation factor

S:

from the specifications sheet

M:

from manufacturer’s constraints or standards

U:

upper limit

L:

lower limit

References

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Acknowledgments

We would like to thank Schneider Electric S.A. for their financial and technical support.

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Correspondence to Manuel Paredes.

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Paredes, M., Rodriguez, E. Optimal design of conical springs. Engineering with Computers 25, 147–154 (2009). https://doi.org/10.1007/s00366-008-0112-3

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  • DOI: https://doi.org/10.1007/s00366-008-0112-3

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