Optimizing the Design of a Helical Spring - Maple Programming Help

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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

g386 inch s2:

Weight density of spring material

γ0.285 lbf  inch3:

Shear modulus

G1.15107 lbf inch2:

Mass density of material

ργg

7890.583227kgm3

(2.1)

Allowable shear stress

τ__a80000 lbf inch2:

Number of inactive coils

Q2:

Applied Load

P10 lbf:

Minimum spring deflection

Δ0.5 inch:

Lower limit of surge wave frequency

ω__0100 Hz:

Limit on outer diameter of coil

D__01.5inch:

Engineering Relationships

Spring Constant

Kd4G8D3N

1.437500000106d4D3Nlbfin2

(3.1)

Shear stress

τ8kPDπd3

τ80kDπd3lbf

(3.2)

Wahl stress concentration factor

k4 Dd4Dd+0.615dD

k4Dd4D4d+0.615dD

(3.3)

Frequency of surge waves

ωd2PiND2G2ρ

356.7459020dND2ms

(3.4)

Constraints

Minimum deflection

cons1PKΔ

0.51.76695652210-7D3Nd4m

(4.1)

The outer diameter of the spring should be smaller than or equal to D0.

cons2D+dD__0

D+d0.03810000000m

(4.2)

Avoid resonance by making the frequency of surge waves along a spring greater than a minimum defined value.

cons3ωω__0

100356.7459020dND2m

(4.3)

The shear stress cannot exceed the allowable shear stress.

cons4ττ__a

804Dd4D4d+0.615dDDπd32000000000000161291m2

(4.4)

Collect all  the constraints

conscons1,cons2,cons3,cons4:

Objective function

Mass of spring

mass14N+Qπ2Dd2ρ

77876.93497N4+12Dd2kgm3

(5.1)

Optimization

boundsN=2..15,d=0.05inch..2inch,D=0.25inch..D__0:

 

Hence the optimized design variables are

Digits20: resultsOptimization:-Minimizemass,cons,bounds,iterationlimit=105:


The optimized spring has a weight of

results1

0.0089173025695656307766lb

(6.1)

and dimensions of

results2

D=0.35687610134569512407in,N=11.293324278719230408,d=0.051695064873242398171in

(6.2)