belleville spring

Spring (device)

A spring is a flexible elastic object used to store mechanical energy. Springs are usually made out of hardened 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).

The rate 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 has units of force divided by distance, for example lbf/in or N/m. Torsion springs have units of force multiplied by distance 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.


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 is cast.

Coiled springs appeared in the 15th century, in locks. The first spring powered-clocks appeared in that century and evolved into the first large watches by the 16th century. Taqi al-Din built a spring-powered astronomical clock in 1559.

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.


Springs are classified according their properties.

Depending on load they may be classified as:

  • Tension/Extension spring
  • Compression spring
  • Torsional spring

In tension/extension and compression there is axial load. On the other hand in the torsional spring there is torsional force.

Depending on spring material it can be classified as:

  • Wire/Coil spring
  • Flat spring

The most common types of spring are:

  • Cantilever spring - a spring which is fixed only at one end.
  • Coil spring or helical spring - a spring (made by winding a wire around a cylinder) and the conical spring - these are types of torsion spring, because the wire itself is twisted when the spring is compressed or stretched. These are in turn of two types:
    • Compression springs are designed to become shorter when loaded. Their turns are not touching in the unloaded position, and they need no attachment points.
      • A volute spring is a compression spring in the form of a cone, designed so that under compression the coils are not forced against each other, thus permitting longer travel.
    • Tension springs are designed to become longer under load. Their turns are normally touching in the unloaded position, and they have a hook, eye or some other means of attachment at each end.

Other types include:

  • Torsion spring - any spring designed to be twisted rather than compressed or extended. Used in torsion bar vehicle suspension systems.
  • Negator spring - a thin flat metal band that is coiled similar to a tape rule. This type of spring produces a constant force throughout a long displacement.
  • Wave spring - a high stiffness spring


Hooke's law

Most springs (not stretched or compressed beyond the elastic limit) 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:

x is the displacement vector - the distance and direction in which the spring is deformed
F is the resulting force vector - the magnitude and direction of the restoring force the spring exerts
k is the spring constant or force constant of the spring.

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.

There are also linear springs which don't follow Hooke's law: a Negator spring (the spring that a self retracting tape measure uses) provides a constant force.

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

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

- 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.


In classical physics, a spring can be seen as a device that stores 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.

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; and therefore the force — which is the derivative of energy with respect to displacement — will approximate a linear function.

Popular mechanics

Contrary to popular belief, springs do not appreciably "creep" or get "tired" with age alone. Spring steel has a very high resistance to creep under normal loads. For instance, in a car engine, valve springs typically undergo about a quarter billion cycles of compression-decompression over the engine's life time and exhibit no noticeable change in length or loss of strength. But for good measure, springs can be replaced when doing a valve job. The sag observed in some older automobiles suspension is usually due to the springs being occasionally compressed beyond their yield point, causing plastic deformation. This can happen when the vehicle hits a large bump or pothole, especially when heavily loaded. Most vehicles will accumulate a number of such impacts over their working life, leading to a lower ride height and eventual bottoming-out of the suspension. In addition, frequent exposure to road salt accelerates corrosion, leading to premature failure of the springs in the car's suspension. Weakening of a spring is usually an indication that it is close to complete failure.



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