Details

The formula for the spring constant of a helical spring with constant pitch and constant winding diameter adapted for steel wire is: N/cm, where is the coil diameter, is the wire diameter, and is the number of active windings.

An active winding is a winding that has not been collapsed (has height larger than the wire diameter). In a cylindrical spring, under increasing load, all windings collapse simultaneously, but in shaped springs this is not the case. Conical springs have their windings collapse in sequence, one at a time, from large to small diameters. For hyperboloidal springs, the larger windings collapse first in groups of two (except in the case of an odd number of windings when the smallest one will collapse last by itself).

A shaped spring is taken as a stack of cylindrical springs with varying outer diameters. The effective spring constant is then that of a set of springs in series and the result of the formula: . For the springs in this Demonstration, this is: where represents the radius of the spring winding in the function of its position along its central axis.

Snapshots 1, 2, and 3: These are the cylindrical, conical, and barrel-shaped versions of the spring in the thumbnail. All have the same free height, number of windings, wire diameter, and maximum coil diameter. Only their coil shapes are different. Look at the differences in the spring constant, load needed to get approximately the same deflection, weight, and length of wire used.

Snapshots 4, 5, and 6: The same three springs as Snapshots 1, 2, and 3 but now fully compressed to their "solid height". Again, there is a big difference between the cylindrical and shaped versions.

Snapshots 7, 8, and 9: These show how in the cylindrical spring all windings deform equally and collapse simultaneously and how the windings of the shaped springs deform differently and collapse sequentially.