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Damping ratio

Classicalmechanics

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Underdamped springmass

system with

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Definition

The effect of varying damping ratio on a second-order system.

The damping ratio is a parameter, usually

denoted by (zeta),[1]

that characterizes the

frequency response of a second order

ordinary differential equation. It is

particularly important in the study of control

theory. It is also important in the harmonic

oscillator.

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 mass m,

damping coefficient c, and spring constant k,

it can be defined as the ratio of the damping

coefficient in the system's differentialequation to the critical damping coefficient:

where the system's equation of motion is

and the corresponding critical damping coefficient is

or

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

Derivation

Using the natural frequency of the simple harmonic oscillator and the definition of the damping ratio

above, we can rewrite this as:

This equation can be solved with the approach.

where Cand s are both complex constants. That approach assumes a solution that is oscillatory and/or decaying

exponentially. Using it in the ODE gives a condition on the frequency of the damped oscillations,

Undamped: Is the case where corresponds to the undamped simple harmonic oscillator, and in that case

the solution looks like , as expected.

Underdamped: Ifs is a complex number, then the solution is a decaying exponential combined with an

oscillatory portion that looks like . This case occurs for , and is referred to as

underdamped.

http://en.wikipedia.org/w/index.php?title=Complex_numberhttp://en.wikipedia.org/w/index.php?title=Simple_harmonic_oscillatorhttp://en.wikipedia.org/w/index.php?title=Natural_frequencyhttp://en.wikipedia.org/w/index.php?title=Damping_coefficienthttp://en.wikipedia.org/w/index.php?title=Harmonic_oscillatorhttp://en.wikipedia.org/w/index.php?title=Harmonic_oscillatorhttp://en.wikipedia.org/w/index.php?title=Control_theoryhttp://en.wikipedia.org/w/index.php?title=Control_theoryhttp://en.wikipedia.org/w/index.php?title=Ordinary_differential_equationhttp://en.wikipedia.org/w/index.php?title=Frequency_responsehttp://en.wikipedia.org/w/index.php?title=File%3A2nd_Order_Damping_Ratios.svg

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Overdamped: Ifs is a real number, then the solution is simply a decaying exponential with no oscillation. This

case occurs for , and is referred to as overdamped.

Critically damped:The case where is the border between the overdamped and underdamped cases, and is

referred to as critically 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

The factors Q, damping ratio , and exponential decay rate are related such that[]

When a second-order system has (that is, when the system is underdamped), it has two complex conjugate

poles that each have a real part of ; that is, the decay rate parameter represents the rate of exponential decay of

the oscillations. A lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for

long times.[2]

For example, a high quality tuning 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

The damping ratio is also related to the logarithmic decrement for underdamped vibrations via the relation

This relation is only meaningful for underdamped systems because the logarithmic decrement is defined as the

natural log of the ratio of any two successive amplitudes, and only underdamped systems exhibit oscillation.

References

http://en.wikipedia.org/w/index.php?title=Logarithmic_decrementhttp://en.wikipedia.org/w/index.php?title=Tuning_forkhttp://en.wikipedia.org/w/index.php?title=Exponential_decayhttp://en.wikipedia.org/w/index.php?title=Real_parthttp://en.wikipedia.org/w/index.php?title=Complex_conjugate

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Article Sources and Contributors 4

Article Sources and ContributorsDamping ratio Source: http://en.wikipedia.org/w/index.php?oldid=549634874 Contributors: ASSami, AndrewDressel, Apoorvajsh, Barkman, BillC, Btyner, Cheeto81, ChrisGualtieri, Craigerv,

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Image Sources, Licenses and ContributorsFile:Damped spring.gif Source: http://en.wikipedia.org/w/index.php?title=File:Damped_spring.gifLicense: Public Domain Contributors: Oleg Alexandrov

File:2nd Order Damping Ratios.svg Source: http://en.wikipedia.org/w/index.php?title=File:2nd_Order_Damping_Ratios.svgLicense: Public Domain Contributors: Inductiveload

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