Inphysical systems,damping is the loss ofenergy of anoscillating system bydissipation.^[1][2] Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation.^[3] Examples of damping includeviscous damping in a fluid (seeviscousdrag),surface friction,radiation,^[1]resistance inelectronic oscillators, and absorption and scattering of light inoptical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur inbiological systems andbikes^[4] (ex.Suspension (mechanics)). Damping is not to be confused withfriction, which is a type of dissipative force acting on a system. Friction can cause or be a factor of damping.

Thedamping ratio is adimensionless measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position ofstatic equilibrium. A mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system tends to return to its equilibrium position, but overshoots it. Sometimes losses (e.g. frictional) damp the system and can cause the oscillations to gradually decay in amplitude towards zero orattenuate. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next.

The damping ratio is a system parameter, denoted byζ ("zeta"), that can vary fromundamped (ζ = 0),underdamped (ζ < 1) throughcritically damped (ζ = 1) tooverdamped (ζ > 1).

The behaviour of oscillating systems is often of interest in a diverse range of disciplines that includecontrol engineering,chemical engineering,mechanical engineering,structural engineering, andelectrical engineering. The physical quantity that is oscillating varies greatly, and could be the swaying of a tall building in the wind, or the speed of anelectric motor, but a normalised, or non-dimensionalised approach can be convenient in describing common aspects of behavior.

Depending on the amount of damping present, a system exhibits different oscillatory behaviors and speeds.

• Where the spring–mass system is completely lossless, the mass would oscillate indefinitely, with each bounce of equal height to the last. This hypothetical case is calledundamped.
• If the system contained high losses, for example if the spring–mass experiment were conducted in aviscous fluid, the mass could slowly return to its rest position without ever overshooting. This case is calledoverdamped.
• Commonly, the mass tends to overshoot its starting position, and then return, overshooting again. With each overshoot, some energy in the system is dissipated, and the oscillations die towards zero. This case is calledunderdamped.
• Between the overdamped and underdamped cases, there exists a certain level of damping at which the system will just fail to overshoot and will not make a single oscillation. This case is calledcritical damping. The key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time.^[5]

Adamped sine wave ordamped sinusoid is asinusoidal function whose amplitude approaches zero as time increases. It corresponds to theunderdamped case of damped second-order systems, or underdamped second-order differential equations.^[6]Damped sine waves are commonly seen inscience andengineering, wherever aharmonic oscillator is losingenergy faster than it is being supplied.A true sine wave starting at time = 0 begins at the origin (amplitude = 0). A cosine wave begins at its maximum value due to its phase difference from the sine wave. A given sinusoidal waveform may be of intermediate phase, having both sine and cosine components. The term "damped sine wave" describes all such damped waveforms, whatever their initial phase.

The most common form of damping, which is usually assumed, is the form found in linear systems. This form is exponential damping, in which the outer envelope of the successive peaks is an exponential decay curve. That is, when you connect the maximum point of each successive curve, the result resembles an exponential decay function. The general equation for an exponentially damped sinusoid may be represented as:$${\displaystyle y(t)=A{e}^{-\lambda t}\cos (\omega t-\phi )}$$ where:

• ${\displaystyle y(t)}$  is the instantaneous amplitude at timet;
• ${\displaystyle A}$  is the initial amplitude of the envelope;
• ${\displaystyle \lambda }$  is the decay rate, in the reciprocal of the time units of the independent variablet;
• ${\displaystyle \phi }$  is the phase angle att = 0;
• ${\displaystyle \omega }$  is theangular frequency.

Other important parameters include:

• Frequency:${\displaystyle f=\omega /(2\pi )}$ , the number of cycles per time unit. It is expressed in inverse time units${\displaystyle t^{-1}}$ , orhertz.
• Time constant:${\displaystyle \tau =1/\lambda }$ , the time for the amplitude to decrease by the factor ofe.
• Half-life is the time it takes for the exponential amplitude envelope to decrease by a factor of 2. It is equal to${\displaystyle \ln (2)/\lambda }$  which is approximately${\displaystyle 0.693/\lambda }$ .
• Damping ratio:${\displaystyle \zeta }$  is a non-dimensional characterization of the decay rate relative to the frequency, approximately${\displaystyle \zeta =\lambda /\omega }$ , or exactly .
• Q factor:${\displaystyle Q=1/(2\zeta )}$  is another non-dimensional characterization of the amount of damping; highQ indicates slow damping relative to the oscillation.

Thedamping ratio is a parameter, usually denoted byζ (Greek letter zeta),^[7] that characterizes thefrequency response of asecond-order ordinary differential equation. It is particularly important in the study ofcontrol theory. It is also important in theharmonic oscillator. In general, systems with higher damping ratios (one or greater) will demonstrate more of a damping effect. Underdamped systems have a value of less than one. Critically damped systems have a damping ratio of exactly 1, or at least very close to it.

The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. For a damped harmonic oscillator with massm, damping coefficientc, andspring constantk, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient:

${\displaystyle \zeta =\frac{c}{c_c}=\frac{\text{actual damping}}{\text{critical damping}},}$ 

where the system's equation of motion is

${\displaystyle m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=0}$ .^[8]

and the corresponding critical damping coefficient is

${\displaystyle c_c=2\sqrt{km}}$ 

or

${\displaystyle c_c=2m\sqrt{\frac{k}{m}}=2m{\omega }_n}$ 

where

${\displaystyle {\omega }_n=\sqrt{\frac{k}{m}}}$  is thenatural frequency of the system.

The damping ratio is dimensionless, being the ratio of two coefficients of identical units.

Using the natural frequency of aharmonic oscillator${\omega }_n=\sqrt{k/m}$  and the definition of the damping ratio above, we can rewrite this as:

${\displaystyle \frac{d^{2}x}{d{t}^2}+2\zeta {\omega }_n\frac{dx}{dt}+{\omega }_n^{2}x=0.}$ 

This equation is more general than just the mass–spring system, and also applies to electrical circuits and to other domains. It can be solved with the approach

${\displaystyle x(t)=C{e}^{st},}$ 

whereC ands are bothcomplex constants, withs satisfying

${\displaystyle s=-{\omega }_n\left(\zeta \pm i\sqrt{1-{\zeta }^2}\right).}$ 

Two such solutions, for the two values ofs satisfying the equation, can be combined to make the general real solutions, with oscillatory and decaying properties in several regimes:

Undamped

Is the case where${\displaystyle \zeta =0}$  corresponds to the undamped simple harmonic oscillator, and in that case the solution looks like${\displaystyle \exp (i{\omega }_{n}t)}$ , as expected. This case is extremely rare in the natural world with the closest examples being cases where friction was purposefully reduced to minimal values.

Underdamped

Ifs is a pair of complex values, then each complex solution term is a decaying exponential combined with an oscillatory portion that looks like$\exp \left(i{\omega }_n\sqrt{1-{\zeta }^2}t\right)$ . This case occurs for , and is referred to asunderdamped (e.g., bungee cable).

Overdamped

Ifs is a pair of real values, then the solution is simply a sum of two decaying exponentials with no oscillation. This case occurs for${\displaystyle \zeta >1}$ , and is referred to asoverdamped. Situations where overdamping is practical tend to have tragic outcomes if overshooting occurs, usually electrical rather than mechanical. For example, landing a plane in autopilot: if the system overshoots and releases landing gear too late, the outcome would be a disaster.

Critically damped

The case where${\displaystyle \zeta =1}$  is the border between the overdamped and underdamped cases, and is referred to ascritically damped. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism).

TheQ factor, damping ratioζ, and exponential decay rate α are related such that^[9]

${\displaystyle \zeta =\frac{1}{2Q}=\frac{\alpha }{{\omega }_n}.}$ 

When a second-order system has  (that is, when the system is underdamped), it has twocomplex conjugate poles that each have areal part of${\displaystyle -\alpha }$ ; that is, the decay rate parameter represents the rate ofexponential decay of the oscillations. A lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times.^[10] For example, a high qualitytuning fork, which has a very low damping ratio, has an oscillation that lasts a long time, decaying very slowly after being struck by a hammer.

For underdamped vibrations, the damping ratio is also related to thelogarithmic decrement${\displaystyle \delta }$ . The damping ratio can be found for any two peaks, even if they are not adjacent.^[11] For adjacent peaks:^[12]

${\displaystyle \zeta =\frac{\delta }{\sqrt{{\delta }^2+{\left(2\pi \right)}^2}}}$  where${\displaystyle \delta =\ln \frac{x_0}{x_1}}$ 

wherex₀ andx₁ are amplitudes of any two successive peaks.

As shown in the right figure:

${\displaystyle \delta =\ln \frac{x_1}{x_3}=\ln \frac{x_2}{x_4}=\ln \frac{x_1-x_2}{x_3-x_4}}$ 

where${\displaystyle x_1}$ ,${\displaystyle x_3}$  are amplitudes of two successive positive peaks and${\displaystyle x_2}$ ,${\displaystyle x_4}$  are amplitudes of two successive negative peaks.

Incontrol theory,overshoot refers to an output exceeding its final, steady-state value.^[13] For astep input, thepercentage overshoot (PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, theovershoot is just the maximum value of the step response minus one.

The percentage overshoot (PO) is related to damping ratio (ζ) by:

${\displaystyle \mathrm{P}\mathrm{O}=100\exp \left(-\frac{\zeta \pi }{\sqrt{1-{\zeta }^2}}\right)}$ 

Conversely, the damping ratio (ζ) that yields a given percentage overshoot is given by:

${\displaystyle \zeta =\frac{-\ln \left(\frac{\mathrm{P}\mathrm{O}}{100}\right)}{\sqrt{{\pi }^2+{\ln }^2\left(\frac{\mathrm{P}\mathrm{O}}{100}\right)}}}$ 

When an object is falling through the air, the only force opposing its freefall is air resistance. An object falling through water or oil would slow down at a greater rate, until eventually reaching a steady-state velocity as the drag force comes into equilibrium with the force from gravity. This is the concept ofviscous drag, which for example is applied in automatic doors or anti-slam doors.^[14]

Electrical systems that operate withalternating current (AC) use resistors to damp LC resonant circuits.^[14]

Kinetic energy that causes oscillations is dissipated as heat by electriceddy currents which are induced by passing through a magnet's poles, either by a coil or aluminum plate. Eddy currents are a key component ofelectromagnetic induction where they set up amagnetic flux directly opposing the oscillating movement, creating a resistive force.^[15] In other words, the resistance caused by magnetic forces slows a system down. An example of this concept being applied is thebrakes on roller coasters.^[16]

Magnetorheological Dampers (MR Dampers) useMagnetorheological fluid, which changes viscosity when subjected to a magnetic field. In this case, Magnetorheological damping may be considered an interdisciplinary form of damping with both viscous and magnetic damping mechanisms.^[17][18]

• "Damping".Encyclopædia Britannica.
• OpenStax, College. "Physics".Lumen.