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. 3^rd edition. Massachusetts: Academic Press, 2011.

Gravitational constant

Weight density of spring material

Shear modulus

Mass density of material

Allowable shear stress

Number of inactive coils

Applied Load

Minimum spring deflection

Lower limit of surge wave frequency

Limit on outer diameter of coil

Spring Constant

Shear stress

Wahl stress concentration factor

Frequency of surge waves

Minimum deflection

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