Compression springs wikipedia

Compression springs wikipedia DEFAULT

Spring (device)

Elastic object that stores mechanical energy

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.
Force (F) vs extension (s).[citation needed]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

A spring is an elastic object that stores mechanical energy. Springs are typically made of spring steel. There are many spring designs. In everyday use, the term often refers to coil springs.

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 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).


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,[1] in door locks.[2] The first spring powered-clocks appeared in that century[2][3][4] 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.


Battery contacts often have a variable spring
A volute spring. Under compression the coils slide over each other, so affording longer travel.
Selection of various arc springsand arc spring systems (systems consisting of inner and outer arc springs).
Tension springs in a folded line reverberation device.
A torsion bar twisted under load

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 remains the same throughout deflection cycle[5]
  • Variable spring – resistance of the coil to load varies during compression[6]
  • 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 [7]

They can also be classified based on their shape:

  • 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.
  • Garter spring – A coiled steel spring that is connected at each end to create a circular shape.

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 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.
    • Hollow tubing springs can be either extension springs or compression springs. Hollow tubing is filled with oil and the means of changing hydrostatic pressure inside the tubing such as a membrane or miniature piston etc. to harden or relax the spring, much like it happens with water pressure inside a garden hose. Alternatively tubing's cross-section is chosen of a shape that it changes its area when tubing is subjected to torsional deformation – change of the cross-section area translates into change of tubing's inside volume and the flow of oil in/out of the spring that can be controlled by valve thereby controlling stiffness. There are many other designs of springs of hollow tubing which can change stiffness with any desired frequency, change stiffness by a multiple or move like a linear actuator in addition to its spring qualities.
  • 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.
  • 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.[8]

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 compressed gas
  • Ideal Spring – a 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.[9]
  • Mainspring – 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.[10]
  • 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.[11] Used in torsion bar vehicle suspension systems.
  • Wave spring – any of many wave shaped springs, washers, and expanders, including linear springs – oall of which are generally made with flat wire or discs that 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.


Hooke's law[edit]

Main article: Hooke's law

As long as 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:

 F=-kx, \


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 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[edit]

Main article: Harmonic oscillator

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

F = m a \quad \Rightarrow \quad -k x = m a. \,
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,

 - k x = m \frac{d^2 x}{dt^2}. \,

This is a second order linear differential equation for the displacement x as a function of time. Rearranging:

\frac{d^2 x}{dt^2} + \frac{k}{m} x = 0, \,

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

 x(t) = A \sin \left(t \sqrt{\frac{k}{m}} \right) + B \cos \left(t \sqrt{\frac{k}{m}} \right). \,

A and B are arbitrary constants that may be found by considering the initial displacement and velocity of the mass. The graph of this function with B = 0 (zero initial position with some positive initial velocity) is displayed in the image on the right.

Energy dynamics[edit]

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:[12]

{\displaystyle E=\left({\frac {1}{2}}\right)kA^{2}}

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 the attached mass m:[12]

{\displaystyle U=\left({\frac {1}{2}}\right)kx^{2}}

The kinetic energyK 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:[12]

{\displaystyle K=\left({\frac {1}{2}}\right)mv^{2}}

Since there is no energy loss in such a system, energy is always conserved and thus:[12]

{\displaystyle E=K+U}

Frequency & period[edit]

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[13]:

{\displaystyle \omega ={\sqrt {\frac {k}{m}}}}[12]

The period T, the amount of time for the spring-mass system to complete one full cycle, of such harmonic motion is given by:[14]

{\displaystyle T={\frac {2\pi }{\omega }}=2\pi {\sqrt {\frac {m}{k}}}}[12]

The frequencyf, the number of oscillations per unit time, of something in simple harmonic motion is found by taking the inverse of the period:[12]

{\displaystyle f={\frac {1}{T}}={\frac {\omega }{2\pi }}={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}[12]


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

 F_{max} = \frac{E d^4 (L-n d)}{16 (1+\nu) (D-d)^3 n} \


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

Zero-length springs[edit]

Simplified LaCoste suspension using a zero-length spring
Spring length Lvs force Fgraph of ordinary (+), zero-length (0) and negative-length (−) springs with the same minimum length L0and 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[edit]


  1. ^Springs How Products Are Made, 14 July 2007.
  2. ^ abWhite, Lynn Jr. (1966). Medieval Technology and Social Change. New York: Oxford Univ. Press. pp. 126–27. ISBN .
  3. ^Usher, Abbot Payson (1988). A History of Mechanical Inventions. Courier Dover. p. 305. ISBN .
  4. ^Dohrn-van Rossum, Gerhard (1998). History of the Hour: Clocks and Modern Temporal Orders. Univ. of Chicago Press. p. 121. ISBN .
  5. ^Constant Springs Piping Technology and Products, (retrieved March 2012)
  6. ^Variable Spring Supports Piping Technology and Products, (retrieved March 2012)
  7. ^"Springs with dynamically variable stiffness and actuation capability". 3 November 2016. Retrieved 20 March 2018 – via
  8. ^"Door Lock Springs". Retrieved 20 March 2018.
  9. ^"Ideal Spring and Simple Harmonic Motion"(PDF). Retrieved 11 January 2016.
  10. ^Samuel, Andrew; Weir, John (1999). Introduction to engineering design: modelling, synthesis and problem solving strategies (2 ed.). Oxford, England: Butterworth. p. 134. ISBN .
  11. ^Goetsch, David L. (2005). Technical Drawing. Cengage Learning. ISBN .
  12. ^ abcdefgh"13.1: The motion of a spring-mass system". Physics LibreTexts. 17 September 2019. Retrieved 19 April 2021.
  13. ^"Harmonic motion". Retrieved 19 April 2021.
  14. ^"simple harmonic motion | Formula, Examples, & Facts". Encyclopedia Britannica. Retrieved 19 April 2021.

Further reading[edit]

  • Sclater, Neil. (2011). "Spring and screw devices and mechanisms." Mechanisms and Mechanical Devices Sourcebook. 5th ed. New York: McGraw Hill. pp. 279–299. ISBN 9780071704427. Drawings and designs of various spring and screw mechanisms.
  • Parmley, Robert. (2000). "Section 16: Springs." Illustrated Sourcebook of Mechanical Components. New York: McGraw Hill. ISBN 0070486174 Drawings, designs and discussion of various springs and spring mechanisms.
  • Warden, Tim. (2021). “Bundy 2 Alto Saxophone.” This saxophone is known for having the strongest tensioned needle springs in existence.

External links[edit]


Spring (device)

Hooke's lawmodels the properties of springs for small changes in length
Helicalor coilsprings designed for tension
Compression springs store energy when compressed

A spring is a device typically made of metal, usually steel.

The metal can be compressed (squeezed). When the compression force is removed, the spring returns to its original length. The metal is usually spring steel, and it is wound tightly. There is lot of varitions with size and type, example some springs have designed for pulling, not pushing. Gas-springs are often used with vehicles tailgate.

History[change | change source]

Simple non-coiled springs were used throughout human history e.g., the bow. In the Bronze Age more advanced 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 is cast.

Coiled springs appeared early in the 15th century,[1] in door locks.[2] The first spring powered-clocks appeared in that century.[2][3][4] They evolved into the first large watches by the 16th century.

In 1676 British physicist Robert Hooke discovered the principle behind springs' action, that the force it exerts is proportional to its extension, now called Hooke's law.

References[change | change source]

  1. Yakima municipal court
  2. Basic parent functions
  3. Emma stark youtube
  4. Walmart battery pack for iphone

The spring constant describes at Compression springs , Tension springs and Torsion springs the increase in force in relation to the Suspension travel , or with torsion springs in relation to the angle of rotation. It is also called Spring rate , Spring hardness or spring stiffness and defines the hardness of a spring. With the Spring characteristic the course of a spring constant is shown. Is the spring constant linear, ie the Spring force increases evenly with the load on the spring, then the spring characteristic is straight (b.). If, on the other hand, the spring force increases disproportionately or disproportionately with increasing load, then we speak of a progressive (a.) Or degressive (c.) Spring characteristic. The spring constant for compression springs and tension springs is specified in the unit Newton / millimeter and for torsion springs as the spring torque rate in Newton millimeters.


Spring characteristics

The following therefore applies to the spring constant with a straight spring characteristic:

Compression and tension springs

R = \frac<wpml_curved wpml_value='F2-F1'></wpml_curved><wpml_curved wpml_value='s2-s1'></wpml_curved>


R = \frac<wpml_curved wpml_value='M2-M1'></wpml_curved>{\alpha2-\alpha1}


Formulas for calculating the spring constant for compression springs, tension springs and torsion springs:

Formula spring constant compression springs (N / mm) ( Formula collection compression springs )

R=\frac{Gd^<wpml_curved wpml_value='4'></wpml_curved>}{8D^<wpml_curved wpml_value='3'></wpml_curved>n}

Formula spring constant tension springs (N / mm) ( Formula collection extension springs )

R=\frac{Gd^<wpml_curved wpml_value='4'></wpml_curved>}{8D^<wpml_curved wpml_value='3'></wpml_curved>n}=\frac<wpml_curved wpml_value='F-F0'></wpml_curved><wpml_curved wpml_value='s'></wpml_curved>

Formula spring torque rate leg springs (Nmm) ( Torsion springs formula collection )

 R_<wpml_curved wpml_value='M'></wpml_curved>=\frac<wpml_curved wpml_value='M'></wpml_curved>{\alpha}=\frac{d^<wpml_curved wpml_value='4'></wpml_curved>E}<wpml_curved wpml_value='3667Dn'></wpml_curved>

Formula explanation:
α = angle of rotation (°)
d = wire diameter (mm)
D = mean coil diameter (mm)
E = modulus of elasticity (N / mm²) ( E-modulus of different spring steels )
F = spring force (N)
G = sliding and shear modulus (N / mm²) ( G-module made from different spring steels )
F0 = internal preload force
M = torque (Nmm)
n = number of resilient coils (pieces)
R = spring constant (N / mm)
RM = spring torque rate (Nmm)
s = spring deflection (mm)

Tensile test of metal springs - © Von Menner - Wikipedia

The spring constant can also be determined by means of a tensile test. The spring is pulled apart with a force (F) and the spring deflection / Spring work (s2) measured. This gives the spring constant in Newtons / millimeter.

R = \frac<wpml_curved wpml_value='F'></wpml_curved><wpml_curved wpml_value='s'></wpml_curved>

However, this tensile test should be made with different forces in order to obtain an accurate measurement result.


Further information

Tagged on: Calculate spring rate    Calculate the spring constant    Determine the spring rate    Festigkeitsnachweis    Hooke's Law    Spring body    Spring constant    Spring constant compression spring    Spring rate    Spring rate compression spring    Spring rate travel    Spring torque rate    Torsion spring moment

jürgen mugrauerCompression springs, Knowledge, Wire springs

Compression Springs

Coil spring

Mechanical device which stores energy

A selection of conical coil springs

A coil spring is a mechanical device which is typically used to store energy and subsequently release it, to absorb shock, or to maintain a force between contacting surfaces. They are made of an elastic material formed into the shape of a helix which returns to its natural length when unloaded.

Under tension or compression, the material (wire) of a coil spring undergoes torsion. The spring characteristics therefore depend on the shear modulus, not Young's Modulus.

A coil spring may also be used as a torsion spring: in this case the spring as a whole is subjected to torsion about its helical axis. The material of the spring is thereby subjected to a bending moment, either reducing or increasing the helical radius. In this mode, it is the Young's Modulus of the material that determines the spring characteristics.

Metal coil springs are made by winding a wire around a shaped former – a cylinder is used to form cylindrical coil springs.

Illustration of various arc springsand arc spring systems (systems consisting of inner and outer arc springs).

Coil springs for vehicles are typically made of hardened steel. A machine called an auto-coiler takes spring wire that has been heated so it can easily be shaped. It is then fed onto a lathe that has a metal rod with the desired coil spring size. The machine takes the wire and guides it onto the spinning rod as well as pushing it across the rod to form multiple coils. The spring is then ejected from the machine and an operator will put it in oil to cool off. The spring is then tempered to lose the brittleness from being cooled. The coil size and strength can be controlled by the lathe rod size and material used. Different alloys are used to get certain characteristics out of the spring, such as stiffness, dampening and strength [1]

Spring rate[edit]

Spring rate is the measurement of how much a coil spring can hold until it compresses 1 inch (2.54 cm). The spring rate is normally specified by the manufacture. If a spring has a rate of 100 then the spring would compress 1 inch with 100 pounds (45 kg) of load.[2]


Types of coil spring are:

  • Tension/extension coil springs, designed to resist stretching. They usually have a hook or eye form at each end for attachment.
  • Compression coil springs, designed to resist being compressed. A typical use for compression coil springs is in carsuspension systems.
    • Volute springs are used as heavy load compression springs. A strip of plate is rolled into the shape of both a helix and a spiral. When compressed, the strip is stiffer edge-on than a wire coil, but the spiral arrangement allows the turns to overlap rather than bottoming out on each other.
    • Arc springs (bow springs) are a special form of coil spring which was originally developed for use in the dual-mass flywheel of internal combustion engine drive trains. The force is applied through the ends of the spring. A torque {\displaystyle M=F\cdot r} can be transmitted around an axis via the force F directed along this helical axis and the lever arm to the system center point r.
  • Torsion springs, designed to resist twisting actions. Often associated to clothes pegs or up-and-over garage doors.


Coil springs have many applications; notable ones include:

Coil springs are commonly used in vehicle suspension. These springs are compression springs and can differ greatly in strength and in size depending on application. A coil spring suspension can be stiff to soft depending on the vehicle it is used on. Coil spring can be either mounted with a shock absorber or mounted separately. Coil springs in trucks allow them to ride smoothly when unloaded and once loaded the spring compresses and becomes stiff. This allows the vehicle to bounce less when loaded. Coil spring suspension is also used high performance cars so that the car can absorb bumps and have low body roll. In off-road vehicles they are used because of their range of travel they allow at the wheel.
  • Coil springs in the engine
Coil springs are used in the engine. These springs are compression springs and play an important role in closing the valves that feed air and let exhaust gasses out of the combustion chamber. The spring is attached to a rocker that is connected to the valve.
  • Coil spring in valvetrain

Tension and extension coil springs of a given material, wire diameter and coil diameter exert the same force when fully loaded; increased number of coils merely (linearly) increases free length and compressed/extended length.

See also[edit]


External links[edit]


Springs wikipedia compression


A spring is a device that changes its shape in response to an external force, returning to its original shape when the force is removed. The energy expended in deforming the spring is stored in it and can be recovered when the spring returns to its original shape. Generally, the amount of the shape change is directly related to the amount of force exerted. If too large a force is applied, however, the spring will permanently deform and never return to its original shape.


There are several types of springs. One of the most common consists of wire wound into a cylindrical or conical shape. An extension spring is a coiled spring whose coils normally touch each other; as a force is applied to stretch the spring, the coils separate. In contrast, a compression spring is a coiled spring with space between successive coils; when a force is applied to shorten the spring, the coils are pushed closer together. A third type of coiled spring, called a torsion spring, is designed so the applied force twists the coil into a tighter spiral. Common examples of torsion springs are found in clipboards and butterfly hair clips.

Still another variation of coiled springs is the watch spring, which is coiled into a flat spiral rather than a cylinder or cone. One end of the spring is at the center of the spiral, and the other is at its outer edge.

Some springs are fashioned without coils. The most common example is the leaf spring, which is shaped like a shallow arch; it is commonly used for automobile suspension systems. Another type is a disc spring, a washer-like device that is shaped like a truncated cone. Open-core cylinders of solid, elastic material can also act as springs. Non-coil springs generally function as compression springs.


Very simple, non-coil springs have been used throughout history. Even a resilient tree branch can be used as a spring. More sophisticated spring devices date to the Bronze Age, when eyebrow tweezers were common in several cultures. During the third century B.C., Greek engineer Ctesibius of Alexandria developed a process for making "springy bronze" by increasing the proportion of tin in the copper alloy, casting the part, and hardening it with hammer blows. He attempted to use a combination of leaf springs to operate a military catapult, but they were not powerful enough. During the second century B.C., Philo of Byzantium, another catapult engineer, built a similar device, apparently with some success. Padlocks were widely used in the ancient Roman empire, and at least one type used bowed metal leaves to keep the devices closed until the leaves were compressed with keys.

The next significant development in the history of springs came in the Middle Ages. A power saw devised by Villard de Honnecourt about 1250 used a water wheel to push the saw blade in one direction, simultaneously bending a pole; as the pole returned to its unbent state, it pulled the saw blade in the opposite direction.

Coiled springs were developed in the early fifteenth century. By replacing the system of weights that commonly powered clocks with a wound spring mechanism, clockmakers

A diagram depicting spring coiling done by a CNC machine.

A diagram depicting spring coiling done by a CNC machine.

were able to fashion reliable, portable timekeeping devices. This advance made precise celestial navigation possible for ocean-going ships.

In the eighteenth century, the Industrial Revolution spurred the development of mass-production techniques for making springs. During the 1780s, British locksmith Joseph Bramah used a spring winding machine in his factory. Apparently an adaptation of a lathe, the machine carried a reel of wire in place of a cutting head. Wire from the reel was wrapped around a rod secured in the lathe. The speed of the lead screw, which carried the reel parallel to the spinning rod, could be adjusted to vary the spacing of the spring's coils.

Common examples of current spring usage range from tiny coils that support keys on cellular phone touchpads to enormous coils that support entire buildings and protect them from earthquake vibration.

Raw Materials

Steel alloys are the most commonly used spring materials. The most popular alloys include high-carbon (such as the music wire used for guitar strings), oil-tempered low-carbon, chrome silicon, chrome vanadium, and stainless steel.

Other metals that are sometimes used to make springs are beryllium copper alloy, phosphor bronze, and titanium. Rubber or urethane may be used for cylindrical, non-coil springs. Ceramic material has been developed for coiled springs in very high-temperature environments. One-directional glass fiber composite materials are being tested for possible use in springs.


Various mathematical equations have been developed to describe the properties of springs, based on such factors as wire composition and size, spring coil diameter, the number of coils, and the amount of expected external force. These equations have been incorporated into computer software to simplify the design process.

The Manufacturing Process

The following description focuses on the manufacture of steel-alloy, coiled springs.


  • 1 Cold winding. Wire up to 0.75 in (18 mm) in diameter can be coiled at room temperature using one of two basic techniques. One consists of winding the wire around a shaft called an arbor or mandrel. This may be done on a dedicated spring-winding machine, a lathe, an electric hand drill with the mandrel secured in the chuck, or a winding machine operated by hand cranking. A guiding mechanism, such as the lead screw on a lathe, must be used to align the wire into the desired pitch (distance between successive coils) as it wraps around the mandrel.

    Alternatively, the wire may be coiled without a mandrel. This is generally done with a central navigation computer (CNC) machine.

    Examples of different types of springs.

    Examples of different types of springs.

    The wire is pushed forward over a support block toward a grooved head that deflects the wire, forcing it to bend. The head and support block can be moved relative to each other in as many as five directions to control the diameter and pitch of the spring that is being formed.

    For extension or torsion springs, the ends are bent into the desired loops, hooks, or straight sections after the coiling operation is completed.

  • 2 Hot winding. Thicker wire or bar stock can be coiled into springs if the metal is heated to make it flexible. Standard industrial coiling machines can handle steel bar up to 3 in (75 mm) in diameter, and custom springs have reportedly been made from bars as much as 6 in (150 mm) thick. The steel is coiled around a mandrel while red hot. Then it is immediately removed from the coiling machine and plunged into oil to cool it quickly and harden it. At this stage, the steel is too brittle to function as a spring, and it must subsequently be tempered.


  • 3 Heat treating. Whether the steel has been coiled hot or cold, the process has created stress within the material. To relieve this stress and allow the steel to maintain its characteristic resilience, the spring must be tempered by heat treating it. The spring is heated in an oven, held at the appropriate temperature for a predetermined time, and then allowed to cool slowly. For example, a spring made of music wire is heated to 500°F (260°C) for one hour.


  • 4 Grinding. If the design calls for flat ends on the spring, the ends are ground at this stage of the manufacturing process. The spring is mounted in a jig to ensure the correct orientation during grinding, and it is held against a rotating abrasive wheel until the desired degree of flatness is obtained. When highly automated equipment is used, the spring is held in a sleeve while both ends are ground simultaneously, first by coarse wheels and then by finer wheels. An appropriate fluid (water or an oil-based substance) may be used to cool the spring, lubricate the grinding wheel, and carry away particles during the grinding.
  • 5 Shot peening. This process strengthens the steel to resist metal fatigue and cracking during its lifetime of repeated flexings. The entire surface of the spring is exposed to a barrage of tiny steel balls that hammer it smooth and compress the steel that lies just below the surface.
  • 6 Setting. To permanently fix the desired length and pitch of the spring, it is fully compressed so that all the coils touch each other. Some manufacturers repeat this process several times.
  • 7 Coating. To prevent corrosion, the entire surface of the spring is protected by painting it, dipping it in liquid rubber, or plating it with another metal such as zinc or chromium. One process, called mechanical plating, involves tumbling the spring in a container with metallic powder, water, accelerant chemicals, and tiny glass beads that pound the metallic powder onto the spring surface.

    Alternatively, in electroplating, the spring is immersed in an electrically conductive liquid that will corrode the plating metal but not the spring. A negative electrical charge is applied to the spring. Also immersed in the liquid is a supply of the plating metal, and it is given a positive electrical charge. As the plating metal dissolves in the liquid, it releases positively charged molecules that are attracted to the negatively charged spring, where they bond chemically. Electroplating makes carbon steel springs brittle, so shortly after plating (less than four hours) they must be baked at 325-375°F (160-190°C) for four hours to counteract the embrittlement.

  • 8 Packaging. Desired quantities of springs may simply be bulk packaged in boxes or plastic bags. However, other forms of packaging have been developed to minimize damage or tangling of springs. For example, they may be individually bagged, strung onto wires or rods, enclosed in tubes, or affixed to sticky paper.

Quality Control

Various testing devices are used to check completed springs for compliance with specifications. The testing devices measure such properties as the hardness of the metal and the amount of the spring's deformation under a known force. Springs that do not meet the specifications are discarded. Statistical analysis of the test results can help manufacturers identify production problems and improve processes so fewer defective springs are produced.

Approximately one-third of defective springs result from production problems. The other two-thirds are caused by deficiencies in the wire used to form the springs. In 1998, researchers reported the development of a wire coilability test (called FRACMAT) that could screen out inadequate wire prior to manufacturing springs.

Computer-operated coiling machines improve quality in two ways. First, they control the diameter and pitch of the spring more precisely than manual operations can. Second, through the use of piezoelectric materials, whose size varies with electrical input, CNC coiling heads can precisely adjust in real time to measurements of spring characteristics. As a result, these intelligent machines produce fewer springs that must be rejected for not meeting specifications.

The Future

Demands of the rapidly growing computer and cellular phone industries are pushing spring manufacturers to develop reliable, cost-effective techniques for making very small springs. Springs that support keys on touchpads and keyboards are important, but there are less apparent applications as well. For instance, a manufacturer of test equipment used in semiconductor production has developed a microspring contact technology. Thousands of tiny springs, only 40 mils (0.040 in or 1 mm) high, are bonded to individual contact points of a semiconductor wafer. When this wafer is pressed against a test instrument, the springs compress, establishing highly reliable electrical connections.

Medical devices also use very small springs. A coiled spring has been developed for use in the insertion end of a catheter or an endoscope. Made of wire 0.0012 in (30 micrometers or 0.030 mm) in diameter, the spring is 0.0036 in (0.092 mm) thick—about the same as a human hair. The Japanese company that developed this spring is attempting to make it even smaller.

The ultimate miniaturization accomplished so far was accomplished in 1997 by an Austrian chemist named Bernard Krautler. He built a molecular spring by stringing 12 carbon atoms together and attaching a vitamin B12 molecule to each end of the chain by means of a cobalt atom. In the relaxed state the chain has a zigzag shape; when it is wetted with water, however, it kinks tightly together. Adding cyclodextrin causes the chain to return to its relaxed state. No practical application of this spring has yet been found, but research continues.

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Hooke's law

Principle of physics that states that the force (F) needed to extend or compress a spring by some distance X scales linearly with respect to that distance

Hooke's law: the force is proportional to the extension
Bourdon tubesare based on Hooke's law. The force created by gas pressureinside the coiled metal tube above unwinds it by an amount proportional to the pressure.
The balance wheelat the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period.

Hooke's law is a law of physics that states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, Fs = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), 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 Latinanagram.[1][2] He published the solution of his anagram in 1678[3] as: ut tensio, sic vis ("as the extension, so the force" or "the extension is proportional to the force"). Hooke states in the 1678 work that he was aware of the law since 1660.

Hooke's equation holds (to some extent) in many other situations where an elastic body is deformed, such as wind blowing on a tall building, and a musician plucking a string of a guitar. An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean.

Hooke's law is only a first-order linear approximation to the real response of springs and other elastic bodies to applied forces. It must eventually fail once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before those elastic limits are reached.

On the other hand, Hooke's law is an accurate approximation for most solid bodies, as long as the forces and deformations are small enough. For this reason, Hooke's law is extensively used in all branches of science and engineering, and is the foundation of many disciplines such as seismology, molecular mechanics and acoustics. It is also the fundamental principle behind the spring scale, the manometer, the galvanometer, and the balance wheel of the mechanical clock.

The modern theory of elasticity generalizes Hooke's law to say that the strain (deformation) of an elastic object or material is proportional to the stress applied to it. However, since general stresses and strains may have multiple independent components, the "proportionality factor" may no longer be just a single real number, but rather a linear map (a tensor) that can be represented by a matrix of real numbers.

In this general form, Hooke's law makes it possible to deduce the relation between strain and stress for complex objects in terms of intrinsic properties of the materials it is made of. For example, one can deduce that a homogeneous rod with uniform cross section will behave like a simple spring when stretched, with a stiffness k directly proportional to its cross-section area and inversely proportional to its length.

Formal definition[edit]

For linear springs[edit]

Consider a simple helical spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is Fs. Suppose that the spring has reached a state of equilibrium, where its length is not changing anymore. Let x be the amount by which the free end of the spring was displaced from its "relaxed" position (when it is not being stretched). Hooke's law states that

{\displaystyle F_{s}=kx}

or, equivalently,

{\displaystyle x={\frac {F_{s}}{k}}}

where k is a positive real number, characteristic of the spring. Moreover, the same formula holds when the spring is compressed, with Fs and x both negative in that case. According to this formula, the graph of the applied force Fs as a function of the displacement x will be a straight line passing through the origin, whose slope is k.

Hooke's law for a spring is sometimes, but rarely, stated under the convention that Fs is the restoring force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes

{\displaystyle F_{s}=-kx}

since the direction of the restoring force is opposite to that of the displacement.

General "scalar" springs[edit]

Hooke's spring law usually applies to any elastic object, of arbitrary complexity, as long as both the deformation and the stress can be expressed by a single number that can be both positive and negative.

For example, when a block of rubber attached to two parallel plates is deformed by shearing, rather than stretching or compression, the shearing force Fs and the sideways displacement of the plates x obey Hooke's law (for small enough deformations).

Hooke's law also applies when a straight steel bar or concrete beam (like the one used in buildings), supported at both ends, is bent by a weight F placed at some intermediate point. The displacement x in this case is the deviation of the beam, measured in the transversal direction, relative to its unloaded shape.

The law also applies when a stretched steel wire is twisted by pulling on a lever attached to one end. In this case the stress Fs can be taken as the force applied to the lever, and x as the distance traveled by it along its circular path. Or, equivalently, one can let Fs be the torque applied by the lever to the end of the wire, and x be the angle by which that end turns. In either case Fs is proportional to x (although the constant k is different in each case.)

Vector formulation[edit]

In the case of a helical spring that is stretched or compressed along its axis, the applied (or restoring) force and the resulting elongation or compression have the same direction (which is the direction of said axis). Therefore, if Fs and x are defined as vectors, Hooke's equation still holds and says that the force vector is the elongation vector multiplied by a fixed scalar.

General tensor form[edit]

Some elastic bodies will deform in one direction when subjected to a force with a different direction. One example is a horizontal wood beam with non-square rectangular cross section that is bent by a transverse load that is neither vertical nor horizontal. In such cases, the magnitude of the displacement x will be proportional to the magnitude of the force Fs, as long as the direction of the latter remains the same (and its value is not too large); so the scalar version of Hooke's law Fs = −kx will hold. However, the force and displacement vectors will not be scalar multiples of each other, since they have different directions. Moreover, the ratio k between their magnitudes will depend on the direction of the vector Fs.

Yet, in such cases there is often a fixed linear relation between the force and deformation vectors, as long as they are small enough. Namely, there is a functionκ from vectors to vectors, such that F = κ(X), and κ(αX1 + βX2) = ακ(X1) + βκ(X2) for any real numbers α, β and any displacement vectors X1, X2. Such a function is called a (second-order) tensor.

With respect to an arbitrary Cartesian coordinate system, the force and displacement vectors can be represented by 3 × 1 matrices of real numbers. Then the tensor κ connecting them can be represented by a 3 × 3 matrix κ of real coefficients, that, when multiplied by the displacement vector, gives the force vector:

{\displaystyle \mathbf {F} \,=\,{\begin{bmatrix}F_{1}\\F_{2}\\F_{3}\end{bmatrix}}\,=\,{\begin{bmatrix}\kappa _{11}&\kappa _{12}&\kappa _{13}\\\kappa _{21}&\kappa _{22}&\kappa _{23}\\\kappa _{31}&\kappa _{32}&\kappa _{33}\end{bmatrix}}{\begin{bmatrix}X_{1}\\X_{2}\\X_{3}\end{bmatrix}}\,=\,{\boldsymbol {\kappa }}\mathbf {X} }

That is,

{\displaystyle F_{i}=\kappa _{i1}X_{1}+\kappa _{i2}X_{2}+\kappa _{i3}X_{3}}

for i = 1, 2, 3. Therefore, Hooke's law F = κX can be said to hold also when X and F are vectors with variable directions, except that the stiffness of the object is a tensor κ, rather than a single real number k.

Hooke's law for continuous media[edit]

Main article: linear elasticity

(a) Schematic of a polymer nanospring. The coil radius, R, pitch, P, length of the spring, L, and the number of turns, N, are 2.5 μm, 2.0 μm, 13 μm, and 4, respectively. Electron micrographs of the nanospring, before loading (b-e), stretched (f), compressed (g), bent (h), and recovered (i). All scale bars are 2 μm. The spring followed a linear response against applied force, demonstrating the validity of Hooke's law at the nanoscale.[4]

The stresses and strains of the material inside a continuous elastic material (such as a block of rubber, the wall of a boiler, or a steel bar) are connected by a linear relationship that is mathematically similar to Hooke's spring law, and is often referred to by that name.

However, the strain state in a solid medium around some point cannot be described by a single vector. The same parcel of material, no matter how small, can be compressed, stretched, and sheared at the same time, along different directions. Likewise, the stresses in that parcel can be at once pushing, pulling, and shearing.

In order to capture this complexity, the relevant state of the medium around a point must be represented by two-second-order tensors, the strain tensorε (in lieu of the displacement X) and the stress tensorσ (replacing the restoring force F). The analogue of Hooke's spring law for continuous media is then

{\displaystyle {\boldsymbol {\sigma }}=\mathbf {c} {\boldsymbol {\varepsilon }},}

where c is a fourth-order tensor (that is, a linear map between second-order tensors) usually called the stiffness tensor or elasticity tensor. One may also write it as

{\displaystyle {\boldsymbol {\varepsilon }}=\mathbf {s} {\boldsymbol {\sigma }},}

where the tensor s, called the compliance tensor, represents the inverse of said linear map.

In a Cartesian coordinate system, the stress and strain tensors can be represented by 3 × 3 matrices

{\displaystyle {\boldsymbol {\varepsilon }}\,=\,{\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\end{bmatrix}}\,;\qquad {\boldsymbol {\sigma }}\,=\,{\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}}

Being a linear mapping between the nine numbers σij and the nine numbers εkl, the stiffness tensor c is represented by a matrix of 3 × 3 × 3 × 3 = 81 real numbers cijkl. Hooke's law then says that

{\displaystyle \sigma _{ij}=\sum _{k=1}^{3}\sum _{l=1}^{3}c_{ijkl}\varepsilon _{kl}}

where i,j = 1,2,3.

All three tensors generally vary from point to point inside the medium, and may vary with time as well. The strain tensor ε merely specifies the displacement of the medium particles in the neighborhood of the point, while the stress tensor σ specifies the forces that neighboring parcels of the medium are exerting on each other. Therefore, they are independent of the composition and physical state of the material. The stiffness tensor c, on the other hand, is a property of the material, and often depends on physical state variables such as temperature, pressure, and microstructure.

Due to the inherent symmetries of σ, ε, and c, only 21 elastic coefficients of the latter are independent.[5] This number can be further reduced by the symmetry of the material: 9 for an orthorhombic crystal, 5 for an hexagonal structure, and 3 for a cubic symmetry.[6] For isotropic media (which have the same physical properties in any direction), c can be reduced to only two independent numbers, the bulk modulusK and the shear modulusG, that quantify the material's resistance to changes in volume and to shearing deformations, respectively.

Analogous laws[edit]

Since Hooke's law is a simple proportionality between two quantities, its formulas and consequences are mathematically similar to those of many other physical laws, such as those describing the motion of fluids, or the polarization of a dielectric by an electric field.

In particular, the tensor equation σ = relating elastic stresses to strains is entirely similar to the equation τ = με̇ relating the viscous stress tensorτ and the strain rate tensorε̇ in flows of viscous fluids; although the former pertains to static stresses (related to amount of deformation) while the latter pertains to dynamical stresses (related to the rate of deformation).

Units of measurement[edit]

In SI units, displacements are measured in meters (m), and forces in newtons (N or kg·m/s2). Therefore, the spring constant k, and each element of the tensor κ, is measured in newtons per meter (N/m), or kilograms per second squared (kg/s2).

For continuous media, each element of the stress tensor σ is a force divided by an area; it is therefore measured in units of pressure, namely pascals (Pa, or N/m2, or kg/(m·s2). The elements of the strain tensor ε are dimensionless (displacements divided by distances). Therefore, the entries of cijkl are also expressed in units of pressure.

General application to elastic materials[edit]

Stress–strain curvefor low-carbon steel, showing the relationship between the stress(force per unit area) and strain(resulting compression/stretching, known as deformation). Hooke's law is only valid for the portion of the curve between the origin and the yield point (2).

Objects that quickly regain their original shape after being deformed by a force, with the molecules or atoms of their material returning to the initial state of stable equilibrium, often obey Hooke's law.

Hooke's law only holds for some materials under certain loading conditions. Steel exhibits linear-elastic behavior in most engineering applications; Hooke's law is valid for it throughout its elastic range (i.e., for stresses below the yield strength). For some other materials, such as aluminium, Hooke's law is only valid for a portion of the elastic range. For these materials a proportional limit stress is defined, below which the errors associated with the linear approximation are negligible.

Rubber is generally regarded as a "non-Hookean" material because its elasticity is stress dependent and sensitive to temperature and loading rate.

Generalizations of Hooke's law for the case of large deformations is provided by models of neo-Hookean solids and Mooney–Rivlin solids.

Derived formulae[edit]

Tensional stress of a uniform bar[edit]

A rod of any elastic material may be viewed as a linear spring. The rod has length L and cross-sectional area A. Its tensile stressσ is linearly proportional to its fractional extension or strain ε by the modulus of elasticityE:

{\displaystyle \sigma =E\varepsilon }.

The modulus of elasticity may often be considered constant. In turn,

\varepsilon = \frac{\Delta L}{L}

(that is, the fractional change in length), and since

{\displaystyle \sigma ={\frac {F}{A}}\,,}

it follows that:

{\displaystyle \varepsilon ={\frac {\sigma }{E}}={\frac {F}{AE}}\,.}

The change in length may be expressed as

{\displaystyle \Delta L=\varepsilon L={\frac {FL}{AE}}\,.}

Spring energy[edit]

The potential energy Uel(x) stored in a spring is given by

{\displaystyle U_{\mathrm {el} }(x)={\tfrac {1}{2}}kx^{2}}

which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over displacement. Since the external force has the same general direction as the displacement, the potential energy of a spring is always non-negative.

This potential Uel can be visualized as a parabola on the Ux-plane such that Uel(x) = 1/2kx2. As the spring is stretched in the positive x-direction, the potential energy increases parabolically (the same thing happens as the spring is compressed). Since the change in potential energy changes at a constant rate:

{\displaystyle {\frac {d^{2}U_{\mathrm {el} }}{dx^{2}}}=k\,.}

Note that the change in the change in U is constant even when the displacement and acceleration are zero.

Relaxed force constants (generalized compliance constants)[edit]

Relaxed force constants (the inverse of generalized compliance constants) are uniquely defined for molecular systems, in contradistinction to the usual "rigid" force constants, and thus their use allows meaningful correlations to be made between force fields calculated for reactants, transition states, and products of a chemical reaction. Just as the potential energy can be written as a quadratic form in the internal coordinates, so it can also be written in terms of generalized forces. The resulting coefficients are termed compliance constants. A direct method exists for calculating the compliance constant for any internal coordinate of a molecule, without the need to do the normal mode analysis.[7] The suitability of relaxed force constants (inverse compliance constants) as covalent bond strength descriptors was demonstrated as early as 1980. Recently, the suitability as non-covalent bond strength descriptors was demonstrated too.[8]

Harmonic oscillator[edit]

See also: Harmonic oscillator

A mass suspended by a spring is the classical example of a harmonic oscillator

A mass m attached to the end of a spring is a classic example of a harmonic oscillator. By pulling slightly on the mass and then releasing it, the system will be set in sinusoidal oscillating motion about the equilibrium position. To the extent that the spring obeys Hooke's law, and that one can neglect friction and the mass of the spring, the amplitude of the oscillation will remain constant; and its frequencyf will be independent of its amplitude, determined only by the mass and the stiffness of the spring:

{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}

This phenomenon made possible the construction of accurate mechanical clocks and watches that could be carried on ships and people's pockets.

Rotation in gravity-free space[edit]

If the mass m were attached to a spring with force constant k and rotating in free space, the spring tension (Ft) would supply the required centripetal force (Fc):

{\displaystyle F_{\mathrm {t} }=kx\,;\qquad F_{\mathrm {c} }=m\omega ^{2}r}

Since Ft = Fc and x = r, then:

k = m \omega^2

Given that ω = 2πf, this leads to the same frequency equation as above:

{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}

Linear elasticity theory for continuous media[edit]

Note: the Einstein summation convention of summing on repeated indices is used below.

Isotropic materials[edit]

For an analogous development for viscous fluids, see Viscosity.

Isotropic materials are characterized by properties which are independent of direction in space. Physical equations involving isotropic materials must therefore be independent of the coordinate system chosen to represent them. The strain tensor is a symmetric tensor. Since the trace of any tensor is independent of any coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor.[9] Thus in index notation:

{\displaystyle \varepsilon _{ij}=\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)}

where δij is the Kronecker delta. In direct tensor notation:

{\displaystyle {\boldsymbol {\varepsilon }}=\operatorname {vol} ({\boldsymbol {\varepsilon }})+\operatorname {dev} ({\boldsymbol {\varepsilon }})\,;\qquad \operatorname {vol} ({\boldsymbol {\varepsilon }})={\tfrac {1}{3}}\operatorname {tr} ({\boldsymbol {\varepsilon }})~\mathbf {I} \,;\qquad \operatorname {dev} ({\boldsymbol {\varepsilon }})={\boldsymbol {\varepsilon }}-\operatorname {vol} ({\boldsymbol {\varepsilon }})}

where I is the second-order identity tensor.

The first term on the right is the constant tensor, also known as the volumetric strain tensor, and the second term is the traceless symmetric tensor, also known as the deviatoric strain tensor or shear tensor.

The most general form of Hooke's law for isotropic materials may now be written as a linear combination of these two tensors:

{\displaystyle \sigma _{ij}=3K\left({\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)+2G\left(\varepsilon _{ij}-{\tfrac {1}{3}}\varepsilon _{kk}\delta _{ij}\right)\,;\qquad {\boldsymbol {\sigma }}=3K\operatorname {vol} ({\boldsymbol {\varepsilon }})+2G\operatorname {dev} ({\boldsymbol {\varepsilon }})}

where K is the bulk modulus and G is the shear modulus.

Using the relationships between the elastic moduli, these equations may also be expressed in various other ways. A common form of Hooke's law for isotropic materials, expressed in direct tensor notation, is [10]

{\displaystyle {\boldsymbol {\sigma }}=\lambda \operatorname {tr} ({\boldsymbol {\varepsilon }})\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}={\mathsf {c}}:{\boldsymbol {\varepsilon }}\,;\qquad {\mathsf {c}}=\lambda \mathbf {I} \otimes \mathbf {I} +2\mu {\mathsf {I}}}

where λ = K − 2/3G = c1111 − 2c1212 and μ = G = c1212 are the Lamé constants, I is the second-rank identity tensor, and I is the symmetric part of the fourth-rank identity tensor. In index notation:

{\displaystyle \sigma _{ij}=\lambda \varepsilon _{kk}~\delta _{ij}+2\mu \varepsilon _{ij}=c_{ijkl}\varepsilon _{kl}\,;\qquad c_{ijkl}=\lambda \delta _{ij}\delta _{kl}+\mu \left(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}\right)}

The inverse relationship is[11]

{\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2\mu }}{\boldsymbol {\sigma }}-{\frac {\lambda }{2\mu (3\lambda +2\mu )}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} ={\frac {1}{2G}}{\boldsymbol {\sigma }}+\left({\frac {1}{9K}}-{\frac {1}{6G}}\right)\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} }

Therefore, the compliance tensor in the relation εs : σ is

{\displaystyle {\mathsf {s}}=-{\frac {\lambda }{2\mu (3\lambda +2\mu )}}\mathbf {I} \otimes \mathbf {I} +{\frac {1}{2\mu }}{\mathsf {I}}=\left({\frac {1}{9K}}-{\frac {1}{6G}}\right)\mathbf {I} \otimes \mathbf {I} +{\frac {1}{2G}}{\mathsf {I}}}

In terms of Young's modulus and Poisson's ratio, Hooke's law for isotropic materials can then be expressed as

{\displaystyle \varepsilon _{ij}={\frac {1}{E}}{\big (}\sigma _{ij}-\nu (\sigma _{kk}\delta _{ij}-\sigma _{ij}){\big )}\,;\qquad {\boldsymbol {\varepsilon }}={\frac {1}{E}}{\big (}{\boldsymbol {\sigma }}-\nu (\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} -{\boldsymbol {\sigma }}){\big )}={\frac {1+\nu }{E}}{\boldsymbol {\sigma }}-{\frac {\nu }{E}}\operatorname {tr} ({\boldsymbol {\sigma }})\mathbf {I} }

This is the form in which the strain is expressed in terms of the stress tensor in engineering. The expression in expanded form is

{\displaystyle {\begin{aligned}\varepsilon _{11}&={\frac {1}{E}}{\big (}\sigma _{11}-\nu (\sigma _{22}+\sigma _{33}){\big )}\\\varepsilon _{22}&={\frac {1}{E}}{\big (}\sigma _{22}-\nu (\sigma _{11}+\sigma _{33}){\big )}\\\varepsilon _{33}&={\frac {1}{E}}{\big (}\sigma _{33}-\nu (\sigma _{11}+\sigma _{22}){\big )}\\\varepsilon _{12}&={\frac {1}{2G}}\sigma _{12}\,;\qquad \varepsilon _{13}={\frac {1}{2G}}\sigma _{13}\,;\qquad \varepsilon _{23}={\frac {1}{2G}}\sigma _{23}\end{aligned}}}

where E is Young's modulus and ν is Poisson's ratio. (See 3-D elasticity).

Derivation of Hooke's law in three dimensions
The three-dimensional form of Hooke's law can be derived using Poisson's ratio and the one-dimensional form of Hooke's law as follows.

Consider the strain and stress relation as a superposition of two effects: stretching in direction of the load (1) and shrinking (caused by the load) in perpendicular directions (2 and 3),

{\displaystyle {\begin{aligned}\varepsilon _{1}'&={\frac {1}{E}}\sigma _{1}\,,\\\varepsilon _{2}'&=-{\frac {\nu }{E}}\sigma _{1}\,,\\\varepsilon _{3}'&=-{\frac {\nu }{E}}\sigma _{1}\,,\end{aligned}}}

where ν is Poisson's ratio and E is Young's modulus.

We get similar equations to the loads in directions 2 and 3,

{\displaystyle {\begin{aligned}\varepsilon _{1}''&=-{\frac {\nu }{E}}\sigma _{2}\,,\\\varepsilon _{2}''&={\frac {1}{E}}\sigma _{2}\,,\\\varepsilon _{3}''&=-{\frac {\nu }{E}}\sigma _{2}\,,\end{aligned}}}


{\displaystyle {\begin{aligned}\varepsilon _{1}'''&=-{\frac {\nu }{E}}\sigma _{3}\,,\\\varepsilon _{2}'''&=-{\frac {\nu }{E}}\sigma _{3}\,,\\\varepsilon _{3}'''&={\frac {1}{E}}\sigma _{3}\,.\end{aligned}}}

Summing the three cases together (εi = εi′ + εi″ + εi‴) we get

{\displaystyle {\begin{aligned}\varepsilon _{1}&={\frac {1}{E}}{\big (}\sigma _{1}-\nu (\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{2}&={\frac {1}{E}}{\big (}\sigma _{2}-\nu (\sigma _{1}+\sigma _{3}){\big )}\,,\\\varepsilon _{3}&={\frac {1}{E}}{\big (}\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}){\big )}\,,\end{aligned}}}

or by adding and subtracting one νσ

{\displaystyle {\begin{aligned}\varepsilon _{1}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{1}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{2}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{2}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\\\varepsilon _{3}&={\frac {1}{E}}{\big (}(1+\nu )\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}\,,\end{aligned}}}

and further we get by solving σ1

{\displaystyle \sigma _{1}={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {\nu }{1+\nu }}(\sigma _{1}+\sigma _{2}+\sigma _{3})\,.}

Calculating the sum

{\displaystyle {\begin{aligned}\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3}&={\frac {1}{E}}{\big (}(1+\nu )(\sigma _{1}+\sigma _{2}+\sigma _{3})-3\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}){\big )}={\frac {1-2\nu }{E}}(\sigma _{1}+\sigma _{2}+\sigma _{3})\\\sigma _{1}+\sigma _{2}+\sigma _{3}&={\frac {E}{1-2\nu }}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\end{aligned}}}

and substituting it to the equation solved for σ1 gives

{\displaystyle {\begin{aligned}\sigma _{1}&={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {E\nu }{(1+\nu )(1-2\nu )}}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\\&=2\mu \varepsilon _{1}+\lambda (\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\,,\end{aligned}}}

where μ and λ are the Lamé parameters.

Similar treatment of directions 2 and 3 gives the Hooke's law in three dimensions.

In matrix form, Hooke's law for isotropic materials can be written as

{\displaystyle {\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\end{bmatrix}}\,=\,{\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\\gamma _{23}\\\gamma _{13}\\\gamma _{12}\end{bmatrix}}\,=\,{\frac {1}{E}}{\begin{bmatrix}1&-\nu &-\nu &0&0&0\\-\nu &1&-\nu &0&0&0\\-\nu &-\nu &1&0&0&0\\0&0&0&2+2\nu &0&0\\0&0&0&0&2+2\nu &0\\0&0&0&0&0&2+2\nu \end{bmatrix}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\end{bmatrix}}}

where γij = 2εij is the engineering shear strain. The inverse relation may be written as


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