Spring (mathematics)

Definition

A spring wrapped around the z-axis can be defined parametrically by:

x ( u , v ) = ( R + r cos ⁡ v ) cos ⁡ u , y ( u , v ) = ( R + r cos ⁡ v ) sin ⁡ u , z ( u , v ) = r sin ⁡ v + P ⋅ u π ,

where

u ∈ [ 0 ,   2 n π )   ( n ∈ R ) , v ∈ [ 0 ,   2 π ) , R is the distance from the center of the tube to the center of the helix, r is the radius of the tube, P is the speed of the movement along the z axis (in a right-handed Cartesian coordinate system, positive values create right-handed springs, whereas negative values create left-handed springs), n is the number of rounds in circle.

The implicit function in Cartesian coordinates for a spring wrapped around the z-axis, with n = 1 is

( R − x 2 + y 2 ) 2 + ( z + P arctan ⁡ ( x / y ) π ) 2 = r 2 .

The interior volume of the spiral is given by

V = 2 π 2 n R r 2 = ( π r 2 ) ( 2 π n R ) .

Other definitions

Note that the previous definition uses a vertical circular cross section. This is not entirely accurate as the tube becomes increasingly distorted as the Torsion increases (ratio of the speed P and the incline of the tube).

An alternative would be to have a circular cross section in the plane perpendicular to the helix curve. This would be closer to the shape of a physical spring. The mathematics would be much more complicated.

The torus can be viewed as a special case of the spring obtained when the helix degenerates to a circle.