Resolving Sensor Misalignment Errors

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:

$\displaystyle F=-kx$,

where

$\displaystyle x$ is the displacement vector – the distance from its equilibrium length.

$\displaystyle F$ is the resulting force vector – the magnitude and direction of the restoring force the spring exerts

$\displaystyle k$ 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.

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

$\displaystyle F=ma\quad \Rightarrow \quad -kx=ma.\,$

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,

$\displaystyle -kx=m\frac d^2xdt^2.\,$

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

$\displaystyle \frac d^2xdt^2+\frac kmx=0,\,$

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

$\displaystyle x(t)=A\sin \left(t\sqrt \frac km\right)+B\cos \left(t\sqrt \frac km\right).\,$

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

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:^[14]

$\displaystyle E=\left(\frac 12\right)kA^2$

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 its displacement x:^[14]

$\displaystyle U=\left(\frac 12\right)kx^2$

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:^[14]

$\displaystyle K=\left(\frac 12\right)mv^2$

Since there is no energy loss in such a system, energy is always conserved and thus:^[14]

$\displaystyle E=K+U$

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^[15]:

$\displaystyle \omega =\sqrt \frac km$^[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:^[16]

$\displaystyle T=\frac 2\pi \omega =2\pi \sqrt \frac mk$^[14]

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:^[14]

$\displaystyle f=\frac 1T=\frac \omega 2\pi =\frac 12\pi \sqrt \frac km$^[14]

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.

The force of a fully compressed spring is:

$\displaystyle F_max=\frac Ed^4(L-nd)16(1+\nu )(D-d)^3n\$

where

E – Young's modulus

d – spring wire diameter

L – free length of spring

n – number of active windings

$\displaystyle \nu$ – Poisson ratio

D – spring outer diameter.

Zero-length spring is a term for a specially designed coil spring that would exert zero force if it had zero length. That is, in a line graph of the spring's force versus its length, the line passes through the origin. A real coil spring will not contract to zero length because at some point the coils touch each other. "Length" here is defined as the distance between the axes of the pivots at each end of the spring, regardless of any inelastic portion in-between.

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, the manufacture of springs is typically not accurate enough to produce springs with tension consistent enough for applications that use zero length springs, so they 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.

• Shock absorber
• Slinky, helical spring toy
• Volute spring

• 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.

• Paredes, Manuel (2013). "How to design springs". insa de toulouse. Retrieved 13 November 2013.
• Wright, Douglas. "Introduction to Springs". Notes on Design and Analysis of Machine Elements. Department of Mechanical & Material Engineering, University of Western Australia. Retrieved 3 February 2008.
• Silberstein, Dave (2002). "How to make springs". Bazillion. Archived from the original on 18 September 2013. Retrieved 3 February 2008.
• Springs with Dynamically Variable Stiffness (patent)
• Smart Springs and their Combinations (patent)