Damping ratio

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Underdamped spring–mass system with ζ<1

In engineering, the damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static 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. The damping ratio is a measure of describing how rapidly the oscillations decay from one bounce to the next.

The behaviour of oscillating systems is often of interest in a diverse range of disciplines that include control engineering, mechanical engineering, structural engineering and electrical 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 an electric motor, but a normalised, or non-dimensionalised approach can be convenient in describing common aspects of behavior.

Oscillation cases[edit]


Definition[edit]

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

The damping ratio is a parameter, usually denoted by ζ (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 differential equation to the critical damping coefficient:

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

where the system's equation of motion is

 m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0

and the corresponding critical damping coefficient is

 c_c = 2\sqrt{km}

or

 c_c=2m\omega_n

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


Formulae[edit]

zeta=sqrt (1-( (log (mp) )*( log (mp) )) / (pi*pi) )

Where ,


Derivation[edit]

Using the natural frequency of the simple harmonic oscillator \omega_n = \sqrt{k/m} and the definition of the damping ratio above, we can rewrite this as:

 \frac{d^2x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2 x = 0.

This equation can be solved with the approach.

 x(t)=C e^{s t},\,

where C and 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,

 s = -\omega_n (\zeta \pm \sqrt{\zeta^2-1}).

Q factor and decay rate[edit]

Main article: Q factor

The factors Q, damping ratio ζ, and exponential decay rate α are related such that[2]

 \zeta = \frac{1}{2 Q} = { \alpha \over \omega_0 }.

When a second-order system has \zeta < 1 (that is, when the system is underdamped), it has two complex conjugate poles that each have a real part of \alpha; that is, the decay rate parameter \alpha 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.[3] 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[edit]

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

 \zeta = \frac{\delta}{\sqrt{(2\pi)^2+\delta^2}} \qquad \text{where} \qquad \delta \triangleq \ln\frac{x_1}{x_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.

See also[edit]

References[edit]

  1. ^ Alciatore, David G. (2007). Introduction to Mechatronics and Measurement Systems (3rd ed.). McGraw Hill. ISBN 978-0-07-296305-2. 
  2. ^ William McC. Siebert. Circuits, Signals, and Systems. MIT Press. 
  3. ^ 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.