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Damping

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Influence on an oscillating physical system which reduces or prevents its oscillation

This article is about damping in oscillatory systems. For other uses, seeDamping (disambiguation).

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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 (seeviscous drag),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.

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.

Thedamping ratio is adimensionless measure, amongst other measures, that characterises how damped a system is. It is denoted byζ ("zeta") and varies 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.

Oscillation cases

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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]

Damped sine wave

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Not to be confused withDamped wave (radio transmission).

Plot of a damped sinusoidal wave represented as the function $y(t)=e^{-t}\cos(2\pi t)$

{\displaystyle y(t)=e^{-t}\cos(2\pi t)} — Plot of a damped sinusoidal wave represented as the function $y(t)=e^{-t}\cos(2\pi t)$

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 the maximum points of each successive curve are connected, the result resembles an exponential decay function. The general equation for an exponentially damped sinusoid may be represented as: $y(t)=Ae^{-\lambda t}\cos(\omega t-\varphi )$ where:

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

Other important parameters include:

Frequency: $f=\omega /(2\pi )$ , the number of cycles per time unit. It is expressed in inverse time units $t^{-1}$ , orhertz.
Time constant: $\tau =1/\lambda$ , 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 $\ln(2)/\lambda$ which is approximately $0.693/\lambda$ .
Damping ratio: $\zeta$ is a non-dimensional characterization of the decay rate relative to the frequency, approximately $\zeta =\lambda /\omega$ , or exactly $\zeta =\lambda /{\sqrt {\lambda ^{2}+\omega ^{2}}}<1$ .
Q factor: $Q=1/(2\zeta )$ is another non-dimensional characterization of the amount of damping; highQ indicates slow damping relative to the oscillation.

Damping ratio

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Thedamping ratio is a dimensionless parameter, usually denoted byζ (Greek letter zeta),^[7] that characterizes the extent of damping in a second-order ordinarydifferential equation. It is particularly important in the study ofcontrol theory. It is also important in theharmonic oscillator. The greater the damping ratio, the more damped a system is.

Undamped systems have a damping ratio of 0.
Underdamped systems have a value of less than one.
Critically damped systems have a damping ratio of 1.
Overdamped systems have a damping ratio greater than 1.

The damping ratio expresses the level of damping in a system relative to critical damping and can be defined using the damping coefficient:

\zeta ={\frac {c}{c_{c}}}={\frac {\text{actual damping}}{\text{critical damping}}},

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

Taking the simple example of amass-spring-damper model with massm, damping coefficientc, andspring constantk, where $x {\displaystyle x}$ represents thedegree of freedom, the system'sequation of motion is given by:

m{\ddot {x}}+c{\dot {x}}+kx=0

.^[8]

The corresponding critical damping coefficient is: $c_{c}=2{\sqrt {km}}$

and thenatural frequency of the system is: $\omega _{n}={\sqrt {\frac {k}{m}}}$

Using these definitions, the equation of motion can then be expressed as:

{\ddot {x}}+2\zeta \omega _{n}{\dot {x}}+\omega _{n}^{2}x=0.

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

x(t)=Ce^{st},

whereC ands are bothcomplex constants, withs satisfying

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:

Phase portrait of damped oscillator, with increasing damping strength. It starts at undamped, proceeds to underdamped, then critically damped, then overdamped.

Undamped: Is the case where $\zeta =0$ corresponds to the undamped simple harmonic oscillator, and in that case the solution looks like $\exp(i\omega _{n}t)$ , 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 ${\textstyle \exp \left(i\omega _{n}{\sqrt {1-\zeta ^{2}}}t\right)}$ . This case occurs for $\ 0\leq \zeta <1$ , 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 $\zeta >1$ , 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 $\zeta =1$ 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).

Q factor and decay rate

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

\zeta ={\frac {1}{2Q}}={\alpha  \over \omega _{n}}.

When a second-order system has $\zeta <1$ (that is, when the system is underdamped), it has twocomplex conjugate poles that each have areal part of $-\alpha$ ; 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.

Logarithmic decrement

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

\zeta ={\frac {\delta }{\sqrt {\delta ^{2}+\left(2\pi \right)^{2}}}}

where

\delta =\ln {\frac {x_{0}}{x_{1}}}

wherex₀ andx₁ are amplitudes of any two successive peaks.

As shown in the right figure:

\delta =\ln {\frac {x_{1}}{x_{3}}}=\ln {\frac {x_{2}}{x_{4}}}=\ln {\frac {x_{1}-x_{2}}{x_{3}-x_{4}}}

where $x_{1}$ , $x_{3}$ are amplitudes of two successive positive peaks and $x_{2}$ , $x_{4}$ are amplitudes of two successive negative peaks.

Percentage overshoot

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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:

\mathrm {PO} =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:

\zeta ={\frac {-\ln \left({\frac {\rm {PO}}{100}}\right)}{\sqrt {\pi ^{2}+\ln ^{2}\left({\frac {\rm {PO}}{100}}\right)}}}

Examples and applications

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Viscous drag

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Main articles:Viscous damping andDashpot

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]

Damping in electrical systems

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

Magnetic damping

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Main article:Magnetic damping

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 damping

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Main article:Magnetorheological damper

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]

Material damping

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Materials have varying degrees of internal damping properties due to microstructural mechanisms within them. This property is sometimes known asdamping capacity. In metals, this arises due to movements of dislocations, as demonstrated nicely in this video:^[19] Metals, as well as ceramics and glass, are known for having very light material damping. By contrast, polymers have a much higher material damping that arises from the energy loss required to contiually break and reform theVan der Waals forces between polymer chains. The cross-linking inthermoset plastics causes less movement of the polymer chains and so the damping is less.

Material damping is best characterized by the loss factor $\eta$ , given by the equation below for the case of very light damping, such as in metals or ceramics:

\eta =2\zeta {\frac {\omega }{\omega _{n}}}

This is because many microstructural processes that contribute to material damping are not well modelled by viscous damping, and so the damping ratio varies with frequency. Adding the frequency ratio as a factor typically makes the loss factor constant over a wide frequency range.

References

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^^a ^bEscudier, Marcel; Atkins, Tony (2019)."A Dictionary of Mechanical Engineering".Oxford Reference.doi:10.1093/acref/9780198832102.001.0001.ISBN 978-0-19-883210-2.
^Steidel (1971).An Introduction to Mechanical Vibrations. John Wiley & Sons. p. 37.damped, which is the term used in the study of vibration to denote a dissipation of energy
^Crandall, S. H. (January 1970). "The role of damping in vibration theory".Journal of Sound and Vibration.11 (1):3–18, IN1.Bibcode:1970JSV....11....3C.doi:10.1016/s0022-460x(70)80105-5.
^J. P. Meijaard; J. M. Papadopoulos; A. Ruina & A. L. Schwab (2007). "Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review".Proceedings of the Royal Society A.463 (2084):1955–1982.Bibcode:2007RSPSA.463.1955M.doi:10.1098/rspa.2007.1857.S2CID 18309860.lean and steer perturbations die away in a seemingly damped fashion. However, the system has no true damping and conserves energy. The energy in the lean and steer oscillations is transferred to the forward speed rather than being dissipated.
^Urone, Paul Peter; Hinrichs, Roger (2016)."16.7 Damped Harmonic Motion".College Physics. OpenStax – via University of Central Florida.
^Douglas C. Giancoli (2000). [Physics for Scientists and Engineers with Modern Physics (3rd Edition)]. Prentice Hall. p. 387ISBN 0-13-021517-1
^Alciatore, David G. (2007).Introduction to Mechatronics and Measurement (3rd ed.). McGraw Hill.ISBN 978-0-07-296305-2.
^Rahman, J.; Mushtaq, M.; Ali, A.; Anjam, Y.N; Nazir, S. (2014)."Modelling damped mass spring system in MATHLAB Simulink".Journal of Faculty of Engineering & Technology.2.
^William McC. Siebert.Circuits, Signals, and Systems. MIT Press.
^Ming Rao and Haiming Qiu (1993).Process control engineering: a textbook for chemical, mechanical and electrical engineers. CRC Press. p. 96.ISBN 978-2-88124-628-9.
^"Dynamics and Vibrations: Notes: Free Damped Vibrations".
^"Damping Evaluation". 19 October 2015.
^Kuo, Benjamin C & Golnaraghi M F (2003).Automatic control systems (Eighth ed.). NY: Wiley. p. §7.3 p. 236–237.ISBN 0-471-13476-7.
^^a ^b"damping | Definition, Types, & Examples".Encyclopedia Britannica. Retrieved2021-06-09.
^Gupta, B. R. (2001).Principles of Electrical, Electronics and Instrumentation Engineering. S. chand Limited. p. 338.ISBN 9788121901031.
^"Eddy Currents and Magnetic Damping | Physics".courses.lumenlearning.com. Retrieved2021-06-09.
^LEE, DUG-YOUNG; WERELEY, NORMAN M. (June 2000)."Quasi-Steady Herschel-Bulkley Analysis of Electro- and Magneto-Rheological Flow Mode Dampers".Electro-Rheological Fluids and Magneto-Rheological Suspensions. WORLD SCIENTIFIC:579–586.doi:10.1142/9789812793607_0066.ISBN 978-981-02-4258-9.
^Savaresi, Sergio M.; Poussot-Vassal, Charles; Spelta, Cristiano; Sename, Oliver; Dugard, Luc (2010-01-01), Savaresi, Sergio M.; Poussot-Vassal, Charles; Spelta, Cristiano; Sename, Oliver (eds.),"CHAPTER 2 - Semi-Active Suspension Technologies and Models",Semi-Active Suspension Control Design for Vehicles, Boston: Butterworth-Heinemann, pp. 15–39,doi:10.1016/b978-0-08-096678-6.00002-x,ISBN 978-0-08-096678-6, retrieved2023-07-15
^"Bubble Raft Model - part 1".YouTube. 24 October 2016.

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