# Damping ratio

damping redirects here. For the technique for musical instruments, seeDamping (music).

\displaystyle \vec F=m\vec a

Inertial/Non-inertial reference frame

Mechanics of planar particle motion

Newtons law of universal gravitation

Angular acceleration/displacement/frequency/velocity

Underdampedspringmass systemwith 1

Inengineering, thedamping ratiois adimensionlessmeasure describing howoscillations in a systemdecay 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 is trying 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 of describing how rapidly the oscillations decay from one bounce to the next.

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

The behaviour of oscillating systems is often of interest in a diverse range of disciplines that includecontrol engineeringchemical Engineeringmechanical engineeringstructural 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.

Where the springmass system is completely lossless, the mass would oscillate indefinitely, with each bounce of equal height to the last. This hypothetical case is called

If the system contained high losses, for example if the springmass experiment were conducted in aviscousfluid, the mass could slowly return to its rest position without ever overshooting. This case is called

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 called

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 called

. The key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time.

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

Thedamping ratiois a parameter, usually denoted by (zeta),[1]that characterizes thefrequency responseof asecond order ordinary differential equation. It is particularly important in the study ofcontrol theory. It is also important in theharmonic 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 massm, damping coefficientc, and spring constantk, it can be defined as the ratio of the damping coefficient in the systems differential equation to the critical damping coefficient:

where the systems equation of motion is

and the corresponding critical damping coefficient is

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

This equation can be solved with the approach.

whereCandsare bothcomplexconstants. 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,

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

corresponds to the undamped simple harmonic oscillator, and in that case the solution looks like

\displaystyle \exp(i\omega _nt)

is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like

\displaystyle \exp \left(i\omega _n\sqrt 1-\zeta ^2t\right)

is a real number, then the solution is simply a decaying exponential with no oscillation. This case occurs for

is the border between the overdamped and underdamped cases, and is referred to as

. 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 ratio , and exponential decay rate are related such that[2]

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.

Introduction to Mechatronics and Measurement Systems

(3rd ed.). McGraw Hill.ISBN978-0-07-296305-2.

Process control engineering: a textbook for chemical, mechanical and electrical engineers

. CRC Press. p.96.ISBN978-2-88124-628-9.

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