Spring (device)

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Helical coil springs designed for tension Springs 009.jpg
Helical coil springs designed for tension
A heavy-duty coil spring designed for compression and tension Ressort de compression.jpg
A heavy-duty coil spring designed for compression and tension
The English longbow - a simple but very powerful spring made of yew, measuring 2 m (6 ft 7 in) long, with a 470 N (105 lbf) draw weight, with each limb functionally a cantilever spring. Englishlongbow.jpg
The English longbow – a simple but very powerful spring made of yew, measuring 2 m (6 ft 7 in) long, with a 470 N (105 lbf) draw weight, with each limb functionally a cantilever spring.
Force (F) vs extension (s).
Spring characteristics: (1) progressive, (2) linear, (3) degressive, (4) almost constant, (5) progressive with knee Federkennlinie.svg
Force (F) vs extension (s). Spring characteristics: (1) progressive, (2) linear, (3) degressive, (4) almost constant, (5) progressive with knee
A machined spring incorporates several features into one piece of bar stock Machined Spring.jpg
A machined spring incorporates several features into one piece of bar stock
Military booby trap firing device from USSR (normally connected to a tripwire) showing spring-loaded firing pin Russian - MUV pull fuze.jpg
Military booby trap firing device from USSR (normally connected to a tripwire) showing spring-loaded firing pin

A spring is a device consisting of an elastic but largely rigid material (typically metal) bent or molded into a form (especially a coil) that can return into shape after being compressed or extended. [1] Springs can store energy when compressed. In everyday use the term often refers to coil springs, but there are many different spring designs. Modern springs are typically manufactured from spring steel. An example of a non-metallic spring is the bow, made traditionally of flexible yew wood, which when drawn stores energy to propel an arrow.


When a conventional spring, without stiffness variability features, 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 or N/m or lbf/in. 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 manufacture. Some non-ferrous metals are also used, including phosphor bronze and titanium for parts requiring corrosion resistance, and low-resistance beryllium copper for springs carrying electrical current.


Simple non-coiled springs have been 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 springs out of an alloy of bronze with an increased proportion of tin, hardened by hammering after it was cast.

Coiled springs appeared early in the 15th century, [2] in door locks. [3] The first spring powered-clocks appeared in that century [3] [4] [5] and evolved into the first large watches by the 16th century.

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


A spiral torsion spring, or hairspring, in an alarm clock. Alarm Clock Balance Wheel.jpg
A spiral torsion spring, or hairspring, in an alarm clock.
Battery contacts often have a variable spring Sanyo MR-110 Battery Contacts (36717564412).jpg
Battery contacts often have a variable spring
A volute spring. Under compression the coils slide over each other, so affording longer travel. Volute spring1.jpg
A volute spring. Under compression the coils slide over each other, so affording longer travel.
Vertical volute springs of Stuart tank Volutespring.jpg
Vertical volute springs of Stuart tank
Selection of various arc springs and arc spring systems (systems consisting of inner and outer arc springs). Bogenfedern und Bogenfedersysteme.jpg
Selection of various arc springs and arc spring systems (systems consisting of inner and outer arc springs).
Tension springs in a folded line reverberation device. Reverb-3.jpg
Tension springs in a folded line reverberation device.
A torsion bar twisted under load Torsion-Bar with-load.jpg
A torsion bar twisted under load
Leaf spring on a truck Leafs1.jpg
Leaf spring on a truck


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
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 remains the same throughout deflection cycle [6]
Variable spring
Resistance of the coil to load varies during compression [7]
Variable stiffness spring
Resistance of the coil to load can be dynamically varied for example by the control system, some types of these springs also vary their length thereby providing actuation capability as well [8]

They can also be classified based on their shape:

Flat spring
Made of a flat spring steel.
Machined spring
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.
Garter spring
A coiled steel spring that is connected at each end to create a circular shape.

Common types

The most common types of spring are:

Cantilever spring
A flat spring fixed only at one end like a cantilever, while the free-hanging end takes the load.
Coil spring
Also known as a helical spring. A spring (made by winding a wire around a cylinder) is of two types:
Arc spring
A pre-curved or arc-shaped helical compression spring, which is able to transmit a torque around an axis.
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.
Balance spring
Also known as a hairspring. 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.
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. [9]

Other types

Other types include:

Belleville washer
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 compressed gas.
Ideal spring
An idealised perfect spring with no weight, mass, damping losses, or limits, a concept used in physics. The force an ideal spring would exert is exactly proportional to its extension or compression. [10]
A spiral ribbon-shaped spring used as a power store of 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. [11]
Progressive rate coil springs
A coil spring with a variable rate, usually achieved by having unequal distance between turns 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. [12] Used in torsion bar vehicle suspension systems.
Wave spring
various types of spring made compact by using waves to give a spring effect.


Hooke's law

An ideal spring acts in accordance with Hooke's law, which states that the force with which the spring pushes back is linearly proportional to the distance from its equilibrium length:


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

Most real springs approximately follow Hooke's law if not stretched or compressed beyond their elastic limit.

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 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 forces 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 displacement, x, as a function of time. The amount of time that passes between peaks is called the period. Periodampwave.svg
The displacement, x, as a function of time. The amount of time that passes between peaks is called the period.

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 as a function of time. Rearranging:

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

and are arbitrary constants that may be found by considering the initial displacement and velocity of the mass. The graph of this function with (zero initial position with some positive initial velocity) is displayed in the image on the right.

Energy dynamics

In simple harmonic motion of a spring-mass system, energy will fluctuate between kinetic energy and potential energy, but the total energy of the system remains the same. A spring that obeys Hooke's Law with spring constant k will have a total system energy E of: [13]

Here, A is the amplitude of the wave-like motion that is produced by the oscillating behavior of the spring.

The potential energy U of such a system can be determined through the spring constant k and its displacement x: [13]

The kinetic energy K of an object in simple harmonic motion can be found using the mass of the attached object m and the velocity at which the object oscillates v: [13]

Since there is no energy loss in such a system, energy is always conserved and thus: [13]

Frequency & period

The angular frequency ω of an object in simple harmonic motion, given in radians per second, is found using the spring constant k and the mass of the oscillating object m [14] :


The period T, the amount of time for the spring-mass system to complete one full cycle, of such harmonic motion is given by: [15]


The frequency f, the number of oscillations per unit time, of something in simple harmonic motion is found by taking the inverse of the period: [13]



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 – approximates a linear function.

Force of fully compressed spring


E – Young's modulus
d – spring wire diameter
L – free length of spring
n – number of active windings
Poisson ratio
D – spring outer diameter

Zero-length springs

Simplified LaCoste suspension using a zero-length spring LaCoste suspension seismometer principle.svg
Simplified LaCoste suspension using a zero-length spring
Spring length L vs force F graph of ordinary (+), zero-length (0) and negative-length (-) springs with the same minimum length L0 and spring constant Zero length spring graph.svg
Spring length L vs force F graph of ordinary (+), zero-length (0) and negative-length (−) springs with the same minimum length L0 and spring constant

"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 touch each other and the spring can't 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 during manufacture; this works because a coiled spring "unwinds" as it stretches), 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 exert force even when the door is almost closed, so they can hold it closed firmly.


See also

Related Research Articles

<span class="mw-page-title-main">Force</span> Influence that can cause motion of an object

In physics, a force is an influence that causes the motion of an object with mass to change its velocity, i.e., to accelerate. It can be a push or a pull, always with magnitude and direction, making it a vector quantity. It is measured in the SI unit of newton (N) and represented by the symbol F.

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:

<span class="mw-page-title-main">Oscillation</span> Repetitive variation of some measure about a central value

Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms.

<span class="mw-page-title-main">Potential energy</span> Energy held by an object because of its position relative to other objects

In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. The term potential energy was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to the ancient Greek philosopher Aristotle's concept of potentiality.

In mechanics and physics, simple harmonic motion is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the static equilibrium position and a restoring force on the moving object that is directly proportional to the magnitude of the object's displacement and acts towards the object's equilibrium position. It results in an oscillation, described by a sinusoid which continues indefinitely, if uninhibited by friction or any other dissipation of energy.

<span class="mw-page-title-main">Young's modulus</span> Mechanical property that measures stiffness of a solid material

Young's modulus, the Young modulus, or the modulus of elasticity in tension or compression, is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied lengthwise. It quantifies the relationship between tensile/compressive stress and axial strain in the linear elastic region of a material and is determined using the formula:

<span class="mw-page-title-main">Hooke's law</span> Physical law: force needed to deform a spring scales linearly with distance

In physics, Hooke's law is an empirical law which states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance—that is, Fs = kx, where k is a constant factor characteristic of the spring, and x is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis. Hooke states in the 1678 work that he was aware of the law since 1660.

<span class="mw-page-title-main">Coil spring</span> Mechanical device which stores energy

The most common type of spring is the coil spring, which is made out of a long piece of metal that is wound around itself. Coil springs were in use in Roman times, evidence of this can be found in bronze Fibulae — the clasps worn by Roman soldiers among others. These are quite commonly found in Roman archeological digs.

<span class="mw-page-title-main">Stiffness</span> Resistance to deformation in response to force

Stiffness is the extent to which an object resists deformation in response to an applied force.

<span class="mw-page-title-main">Torsion spring</span> Type of spring

A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. When it is twisted, it exerts a torque in the opposite direction, proportional to the amount (angle) it is twisted. There are various types:

<span class="mw-page-title-main">Buckling</span> Sudden change in shape of a structural component under load

In structural engineering, buckling is the sudden change in shape (deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have buckled. Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress of a column.

<span class="mw-page-title-main">Fall factor</span> Mathematical ratio relevant to climbing safety

In lead climbing using a dynamic rope, the fall factor (f) is the ratio of the height (h) a climber falls before the climber's rope begins to stretch and the rope length (L) available to absorb the energy of the fall,

<span class="mw-page-title-main">Bending</span>

In applied mechanics, bending characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.

Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, stretched or generally deformed in any manner. Elasticity theory primarily develops formalisms for the mechanics of solid bodies and materials. The elastic potential energy equation is used in calculations of positions of mechanical equilibrium. The energy is potential as it will be converted into other forms of energy, such as kinetic energy and sound energy, when the object is allowed to return to its original shape (reformation) by its elasticity.

<span class="mw-page-title-main">Tension (physics)</span> Pulling force transmitted axially – Opposite of compression

In physics, tension is described as the pulling force transmitted axially by the means of a string, a rope, chain, or similar object, or by each end of a rod, truss member, or similar three-dimensional object; tension might also be described as the action-reaction pair of forces acting at each end of said elements. Tension could be the opposite of compression.

<span class="mw-page-title-main">Elastic pendulum</span>

In physics and mathematics, in the area of dynamical systems, an elastic pendulum is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The system exhibits chaotic behaviour and is sensitive to initial conditions. The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations.

In a real spring–mass system, the spring has a non-negligible mass . Since not all of the spring's length moves at the same velocity as the suspended mass , its kinetic energy is not equal to . As such, cannot be simply added to to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to to correctly predict the behavior of the system.

<span class="mw-page-title-main">Structural engineering theory</span>

Structural engineering depends upon a detailed knowledge of loads, physics and materials to understand and predict how structures support and resist self-weight and imposed loads. To apply the knowledge successfully structural engineers will need a detailed knowledge of mathematics and of relevant empirical and theoretical design codes. They will also need to know about the corrosion resistance of the materials and structures, especially when those structures are exposed to the external environment.

Carbon nanotube springs are springs made of carbon nanotubes (CNTs). They are an alternate form of high density, lightweight, reversible energy storage based on the elastic deformations of CNTs. Many previous studies on the mechanical properties of CNTs have revealed that they possess high stiffness, strength and flexibility. The Young's modulus of CNTs is 1 TPa and they have the ability to sustain reversible tensile strains of 6% and the mechanical springs based on these structures are likely to surpass the current energy storage capabilities of existing steel springs and provide a viable alternative to electrochemical batteries. The obtainable energy density is predicted to be highest under tensile loading, with an energy density in the springs themselves about 2500 times greater than the energy density that can be reached in steel springs, and 10 times greater than the energy density of lithium-ion batteries.


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Further reading