Deformation of Noncylindrical Extension Springs

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This Demonstration models the deformation of noncylindrical springs under an applied tension load. An extension spring is designed to get longer when a load is applied to it. The spring coils can be cylindrical, with constant outer diameter of the windings, or noncylindrical, with variable outer diameter. Popular shapes for noncylindrical springs are hyperboloidal (hourglass/barrel) springs or conical springs. When a load is applied, a cylindrical spring has constant deformation of the windings over its entire length. Noncylindrical springs have variable deformation of the windings, depending on their diameter. Larger diameter windings deform more than smaller ones.

Contributed by: Erik Mahieu (August 2011)
Open content licensed under CC BY-NC-SA



The spring constant of a helical spring with constant pitch and winding diameter is taken from WolframAlpha["spring constant",…] and adapted for steel wire: N/cm; is the coil diameter, is the wire diameter, and is the number of windings.

A shaped spring is taken as a stack of cylindrical springs with varying outer diameters. The effective spring constant of a set of springs in series is then given by .

The extension for each partial winding is taken from Hooke's law: .

Snapshot 1: all windings of a cylindrical spring deform equally

Snapshots 2–5: shaped springs deform more in the larger diameter windings and less in those with smaller diameter

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