Damping

Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include viscous drag in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and bikes.
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 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 or attenuate. 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 by "ζ" (zeta), that can vary from undamped (), underdamped () through critically damped () to overdamped ().
The behaviour of oscillating systems is often of interest in a diverse range of disciplines that include control engineering, chemical 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.
Depending on the amount of damping present, a system exhibits different oscillatory behaviors.
Damped sine wave.
A damped sine wave or damped sinusoid is a sinusoidal function whose amplitude approaches zero as time increases, corresponding to the underdamped case of damped second-order systems, or underdamped second-order differential equations.
Damped sine waves are commonly seen in science and engineering, wherever a harmonic oscillator is losing energy 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, and that usually assumed, is the form found in linear systems, which is exponential damping, in which the outer envelope of the successive peaks is an exponential decay curve. The general equation for an exponentially damped sinusoid may be represented as:
where:
Other important parameters include:
Damping ratio definition.
The "damping ratio" is a parameter, usually denoted by "ζ" (zeta), 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:
where the system's equation of motion is
and the corresponding critical damping coefficient is
or
where
The damping ratio is dimensionless, being the ratio of two coefficients of identical units.
Derivation.
Using the natural frequency of a harmonic oscillator formula_22 and the definition of the damping ratio above, we can rewrite this as:
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.
where "C" and "s" are both complex constants, with "s" satisfying
Two such solutions, for the two values of "s" satisfying the equation, can be combined to make the general real solutions, with oscillatory and decaying properties in several regimes:
"Q" factor and decay rate.
The "Q" factor, damping ratio "ζ", and exponential decay rate α are related such that
When a second-order system has formula_33 (that is, when the system is underdamped), it has two complex conjugate poles that each have a real part of formula_34; that is, the decay rate parameter formula_35 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. 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.
For underdamped vibrations, the damping ratio is also related to the logarithmic decrement formula_36. The damping ratio can be found for any two peaks, even if they are not adjacent. For adjacent peaks:
where "x"0 and "x"1 are amplitudes of any two successive peaks.
As shown in the right figure:
where formula_40, formula_41 are amplitudes of two successive positive peaks and formula_42, formula_43 are amplitudes of two successive negative peaks.
Percentage overshoot.
In control theory, overshoot refers to an output exceeding its final, steady-state value. For a step input, the percentage overshoot (PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, the "overshoot" is just the maximum value of the step response minus one.
The percentage overshoot (PO) is related to damping ratio ("ζ") by:
Conversely, the damping ratio ("ζ") that yields a given percentage overshoot is given by: