A **spring** is an elastic object used to store mechanical energy. Springs are usually made out of spring steel. There are a large number of spring designs; in everyday usage the term often refers to coil springs.

## Contents

When a spring is compressed or stretched from its resting position, it exerts an opposing force approximately proportional to its change in length (this approximation breaks down for larger deflections). The *rate* or *spring constant* of a spring is the change in the force it exerts, divided by the change in deflection of the spring. That is, it is the gradient of the force versus deflection curve. An extension or compression spring's rate is expressed in units of force divided by distance, for example lbf/in or N/m. A torsion spring is a spring that works by twisting; when it is twisted about its axis by an angle, it produces a torque proportional to the angle. A torsion spring's rate is in units of torque divided by angle, such as N·m/rad or ft·lbf/degree. The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. The stiffness (or rate) of springs in parallel is additive, as is the compliance of springs in series.

Springs are made from a variety of elastic materials, the most common being spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after fabrication. Some non-ferrous metals are also used including phosphor bronze and titanium for parts requiring corrosion resistance and beryllium copper for springs carrying electrical current (because of its low electrical resistance).

## History

Simple non-coiled springs were used throughout human history, e.g. the bow (and arrow). In the Bronze Age more sophisticated spring devices were used, as shown by the spread of tweezers in many cultures. Ctesibius of Alexandria developed a method for making bronze with spring-like characteristics by producing an alloy of bronze with an increased proportion of tin, and then hardening it by hammering after it was cast.

Coiled springs appeared early in the 15th century, in door locks. The first spring powered-clocks appeared in that century and evolved into the first large watches by the 16th century.

In 1676 British physicist Robert Hooke discovered Hooke's law which states that the force a spring exerts is proportional to its extension.

## Types

Springs can be classified depending on how the load force is applied to them:

**Tension/extension spring**– the spring is designed to operate with a tension load, so the spring stretches as the load is applied to it.

**Compression spring**– is designed to operate with a compression load, so the spring gets shorter as the load is applied to it.

**Torsion spring**– unlike the above types in which the load is an axial force, the load applied to a torsion spring is a torque or twisting force, and the end of the spring rotates through an angle as the load is applied.

**Constant spring**- supported load will remain the same throughout deflection cycle

**Variable spring**- resistance of the coil to load varies during compression

They can also be classified based on their shape:

**Coil spring**– this type is made of a coil or helix of round wire.

**Flat spring**– this type is made of a flat spring steel.

**Machined spring**– this type of spring is manufactured by machining bar stock with a lathe and/or milling operation rather than a coiling operation. Since it is machined, the spring may incorporate features in addition to the elastic element. Machined springs can be made in the typical load cases of compression/extension, torsion, etc.

**Serpentine spring**- a zig-zag of thick wire - often used in modern upholstery/furniture.

The most common types of spring are:

**Cantilever spring**– a spring which is fixed only at one end.

**Coil spring**or

**helical spring**– a spring (made by winding a wire around a cylinder) is of two types:

*Tension*or

*extension springs*are designed to become longer under load. Their turns (loops) are normally touching in the unloaded position, and they have a hook, eye or some other means of attachment at each end.

*Compression springs*are designed to become shorter when loaded. Their turns (loops) are not touching in the unloaded position, and they need no attachment points.

**Volute spring**- a compression coil spring in the form of a cone so that under compression the coils are not forced against each other, thus permitting longer travel.

**Hairspring**or

**balance spring**– a delicate spiral spring used in watches, galvanometers, and places where electricity must be carried to partially rotating devices such as steering wheels without hindering the rotation.

**Leaf spring**– a flat spring used in vehicle suspensions, electrical switches, and bows.

**V-spring**– used in antique firearm mechanisms such as the wheellock, flintlock and percussion cap locks. Also door-lock spring, as used in antique door latch mechanisms.

Other types include :

**Belleville washer**or

**Belleville spring**– a disc shaped spring commonly used to apply tension to a bolt (and also in the initiation mechanism of pressure-activated landmines).

**Constant-force spring**— a tightly rolled ribbon that exerts a nearly constant force as it is unrolled.

**Gas spring**– a volume of gas which is compressed.

**Ideal Spring**– the notional spring used in physics: it has no weight, mass, or damping losses. The force exerted by the spring is proportional to the distance the spring is stretched or compressed from its relaxed position.

**Mainspring**– a spiral ribbon shaped spring used as a power store in clockwork mechanisms: watches, clocks, music boxes, windup toys, and mechanically powered flashlights

**Negator spring**– a thin metal band slightly concave in cross-section. When coiled it adopts a flat cross-section but when unrolled it returns to its former curve, thus producing a constant force throughout the displacement and

*negating*any tendency to re-wind. The most common application is the retracting steel tape rule.

**Progressive rate coil springs**– A coil spring with a variable rate, usually achieved by having unequal pitch so that as the spring is compressed one or more coils rests against its neighbour.

**Rubber band**– a tension spring where energy is stored by stretching the material.

**Spring washer**– used to apply a constant tensile force along the axis of a fastener.

**Torsion spring**– any spring designed to be twisted rather than compressed or extended. Used in torsion bar vehicle suspension systems.

**Wave spring**– the name applies to a multitude of wave shaped springs, washers and expanders, including linear springs, all of which are generally made with flat wire or discs which are

*marcelled*according to industrial terms, usually by die-stamping, into a wavy regular pattern resulting in curvilinear lobes. Round wire wave springs exist as well. Types include : wave washer, single turn wave spring, multi-turn wave spring, linear wave spring, marcel expander, interlaced wave spring and nested wave spring amongst others.

## Hooke's law

As long as they are not stretched or compressed beyond their elastic limit, most springs obey Hooke's law, which states that the force with which the spring pushes back is linearly proportional to the distance from its equilibrium length:

where

*x*is the displacement vector – the distance and direction the spring is deformed from its equilibrium length.

*F*is the resulting force vector – the magnitude and direction of the restoring force the spring exerts

*k*is the

**rate**,

**spring constant**or

**force constant**of the spring, a constant that depends on the spring's material and construction. The negative sign indicates that the force the spring exerts is in the opposite direction from its displacement

Coil springs and other common springs typically obey Hooke's law. There are useful springs that don't: springs based on beam bending can for example produce forces that vary nonlinearly with displacement.

If made with constant pitch (wire thickness), conical springs will have a variable rate. However, a conical spring can be made to have a constant rate by creating the spring with a variable pitch. A larger pitch in the larger-diameter coils and a smaller pitch in the smaller-diameter coils will force the spring to collapse or extend all the coils at the same rate when deformed.

## Simple harmonic motion

Since force is equal to mass, *m*, times acceleration, *a*, the force equation for a spring obeying Hooke's law looks like:

The mass of the spring is small in comparison to the mass of the attached mass and is ignored. Since acceleration is simply the second derivative of x with respect to time,

This is a second order linear differential equation for the displacement

the solution of which is the sum of a sine and cosine:

## Theory

In classical physics, a spring can be seen as a device that stores potential energy, specifically elastic potential energy, by straining the bonds between the atoms of an elastic material.

Hooke's law of elasticity states that the extension of an elastic rod (its distended length minus its relaxed length) is linearly proportional to its tension, the force used to stretch it. Similarly, the contraction (negative extension) is proportional to the compression (negative tension).

This law actually holds only approximately, and only when the deformation (extension or contraction) is small compared to the rod's overall length. For deformations beyond the elastic limit, atomic bonds get broken or rearranged, and a spring may snap, buckle, or permanently deform. Many materials have no clearly defined elastic limit, and Hooke's law can not be meaningfully applied to these materials. Moreover, for the superelastic materials, the linear relationship between force and displacement is appropriate only in the low-strain region.

Hooke's law is a mathematical consequence of the fact that the potential energy of the rod is a minimum when it has its relaxed length. Any smooth function of one variable approximates a quadratic function when examined near enough to its minimum point as can be seen by examining the Taylor series. Therefore, the force—which is the derivative of energy with respect to displacement—will approximate a linear function.

Force of fully compressed spring

where

E – Young's modulus d – spring wire diameter L – free length of spring n – number of active windings## Zero-length springs

"Zero-length spring" is a term for a specially designed coil spring that would exert zero force if it had zero length; if there were no constraint due to the finite wire diameter of such a helical spring, it would have zero length in the unstretched condition. That is, in a line graph of the spring's force versus its length, the line passes through the origin. Obviously a coil spring cannot contract to zero length because at some point the coils will touch each other and the spring will not be able to shorten any more. Zero length springs are made by manufacturing a coil spring with built-in tension (A twist is introduced into the wire as it is coiled, in the manufacturing process. This works because a coiled spring "unwinds" as it is stretched.), so if it *could* contract further, the equilibrium point of the spring, the point at which its restoring force is zero, occurs at a length of zero. In practice, zero length springs are made by combining a "negative length" spring, made with even more tension so its equilibrium point would be at a "negative" length, with a piece of inelastic material of the proper length so the zero force point would occur at zero length.

A zero length spring can be attached to a mass on a hinged boom in such a way that the force on the mass is almost exactly balanced by the vertical component of the force from the spring, whatever the position of the boom. This creates a horizontal "pendulum" with very long oscillation period. Long-period pendulums enable seismometers to sense the slowest waves from earthquakes. The LaCoste suspension with zero-length springs is also used in gravimeters because it is very sensitive to changes in gravity. Springs for closing doors are often made to have roughly zero length so that they will exert force even when the door is almost closed, so it will close firmly.