Optimizing the Design of a Helical Spring
The design optimization of helical springs is of considerable engineering interest, and demands strong solvers. While the number of constraints is small, the coil and wire diameters are raised to higher powers; this makes the optimization difficult for gradient-based solvers working in standard floating-point precision; a larger number of working digits is needed.
Maple lets you increase the number of digits used in calculations; hence numerically difficult problems like this can be solved.
This application minimizes the mass of a helical spring. The constraints include the minimum deflection, the minimum surge wave frequency, the maximum stress, and a loading condition.
The design variables are the diameter of the wire d, the outside diameter of the spring D, and the number of coils N.
Reference: Arora, Jasbir S. Introduction to Optimum Design. 3rd edition. Massachusetts: Academic Press, 2011.
g≔386 inch s−2:
Weight density of spring material
γ≔0.285 lbf inch−3:
G≔1.15⋅107 lbf inch−2:
Mass density of material
Allowable shear stress
τ__a≔80000 lbf inch−2:
Number of inactive coils
Minimum spring deflection
Lower limit of surge wave frequency
Limit on outer diameter of coil
Wahl stress concentration factor
Frequency of surge waves
The outer diameter of the spring should be smaller than or equal to D0.
Avoid resonance by making the frequency of surge waves along a spring greater than a minimum defined value.
The shear stress cannot exceed the allowable shear stress.
Collect all the constraints
Mass of spring
Hence the optimized design variables are
The optimized spring has a weight of
and dimensions of
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