Spiral Spring Design

Spiral springs, often called clock springs, consist of a flat strip wound into a spiral. Applied torque causes the strip to bend and produces angular deflection, allowing the spring to store and release mechanical energy.

This page provides engineering reference information for flat spiral springs, including strip geometry, torque, angular deflection, spring rate and bending stress. The equations are intended for preliminary design of springs operating within the elastic range.

Not sure whether a spiral spring is suitable for your application?
See the Spring Selection and Design Guide for guidance on selecting between spiral, torsion, compression, extension, disc, leaf and other spring types.

Spiral Spring Principles
Nomenclature
Spiral Spring Geometry
Spring Rate
Spiral Spring Formulae
Design Considerations

Spiral Spring Principles

A spiral spring consists of a strip wound into a substantially flat spiral. One end of the strip is normally attached to a central arbor or shaft and the other to an external member or housing. Relative rotation between the ends bends the strip and produces a resisting torque.

A familiar example is the traditional clock spring. Similar spring arrangements are used where rotational energy storage, return movement or controlled angular displacement is required.

Terminology: A conventional flat spiral or clock spring should not automatically be described as a constant-torque or constant-force spring. Its torque normally changes with angular deflection. Constant-force and constant-torque springs are related strip-spring forms but operate according to different principles.

Spiral Spring Nomenclature

The following notation is used on this page. SI units are shown for consistency. In practical calculations, millimetres and N/mm² are often more convenient provided a consistent system of units is used throughout.

Symbol	Description	SI Unit
D	Outside diameter of spring	m
d	Inside diameter of spring	m
b	Width of spring strip	m
t	Thickness of spring strip	m
n	Number of turns	dimensionless
L	Effective length of spring strip	m
E	Young's modulus	Pa (N/m²)
I	Second moment of area of the spring-strip cross-section	m⁴
F	Applied force producing the spring torque	N
M	Applied moment or torque	N m
θ	Angular deflection	rad
k	Rotational spring rate, M/θ	N m/rad
y	Distance from the neutral axis to the outer surface of the strip; for a rectangular strip, y = t/2	m
σ	Maximum bending stress in the spring strip	Pa (N/m²)

For a rectangular strip section:

I = \frac{b t^{3}}{12}

Spiral Spring Geometry

The principal geometry of a flat spiral spring is shown below. The spring is formed from a strip of width b and thickness t, wound between an inner diameter d and an outside diameter D.

Flat spiral clock spring showing inside diameter, outside diameter and spring strip geometry

The torque may be produced by a tangential force acting at an appropriate radius. Where force F acts tangentially at the outside diameter D, the corresponding torque is:

M = \frac{F D}{2}

More generally, torque is force multiplied by the perpendicular distance between the force line of action and the axis of rotation.

Spiral Spring Rate

The rotational spring rate k is defined here as the torque required per unit angular deflection:

k = \frac{M}{θ}

For an approximately linear elastic characteristic, a larger spring rate means that a greater torque is required to produce a given angular deflection.

Spiral Spring Formulae

Length of Spring Strip

The effective strip length is related to the spring geometry and number of turns:

L = \frac{π n (D + d)}{2}

Spring Rate

For an ideal flat spiral spring operating in elastic bending, the rotational spring rate depends on Young's modulus, strip section and effective strip length:

k = \frac{E b t^{3}}{12 L}

Spring Surface Stress

The maximum elastic bending stress occurs at the outer surface of the strip and is related to applied torque and strip section:

σ = \frac{6 M}{b t^{2}}