Optimizing the Design of a Helical Spring
Introduction
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.
restart:local γ:withUnitsSimple:
Parameters
Gravitational constant
g≔386 inch s−2:
Weight density of spring material
γ≔0.285 lbf inch−3:
Shear modulus
G≔1.15⋅107 lbf inch−2:
Mass density of material
ρ≔γg
7890.583227kgm3
Allowable shear stress
τ__a≔80000 lbf inch−2:
Number of inactive coils
Q≔2:
Applied Load
P≔10 lbf:
Minimum spring deflection
Δ≔0.5 inch:
Lower limit of surge wave frequency
ω__0≔100 Hz:
Limit on outer diameter of coil
D__0≔1.5inch:
Engineering Relationships
Spring Constant
K≔d4⋅G8⋅D3⋅N
1.44×106d4D3Nlbfin2
Shear stress
τ≔8⋅k⋅P⋅Dπ⋅d3
τ≔80kDπd3lbf
Wahl stress concentration factor
k≔4 D−d4⋅D−d+0.615⋅dD
k≔4D−d4D−4d+0.615dD
Frequency of surge waves
ω≔d2⋅Pi⋅N⋅D2⋅G2⋅ρ
356.746dND2ms
Constraints
Minimum deflection
cons1≔PK≥Δ
500.00×10-3≤176.70×10-9D3Nd4m
The outer diameter of the spring should be smaller than or equal to D0.
cons2≔D+d≤D__0
D+d≤38.10×10-3m
Avoid resonance by making the frequency of surge waves along a spring greater than a minimum defined value.
cons3≔ω≥ω__0
100.00≤356.75dND2m
The shear stress cannot exceed the allowable shear stress.
cons4≔τ≤τ__a
80.004.00D−d4.00D−4.00d+615.00×10-3dDDπd3≤124.00×1061m2
Collect all the constraints
cons≔cons1,cons2,cons3,cons4:
Objective function
Mass of spring
mass≔14⋅N+Q⋅π2⋅D⋅d2⋅ρ
77876.93497N4+12Dd2kgm3
Optimization
bounds≔N=2..15,d=0.05inch..2inch,D=0.25inch..D__0:
Hence the optimized design variables are
Digits≔20: results≔Optimization:-Minimizemass,cons,bounds,iterationlimit=105:
The optimized spring has a weight of
results1
8.92×10-3lb
and dimensions of
results2
D=356.88×10-3in,N=11.29,d=51.70×10-3in
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