Underdamped spring–mass system
Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations. 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.
The damping of a system can be described as being one of the following:

Overdamped

The system returns (exponentially decays) to equilibrium without oscillating.

Critically damped

The system returns to equilibrium as quickly as possible without oscillating.

Underdamped

The system oscillates (at reduced frequency compared to the undamped case) with the amplitude gradually decreasing to zero.

Undamped

The system oscillates at its natural resonant frequency (ω_{o}).
For example, consider a door that uses a spring to close the door once open. This can lead to any of the above types of damping depending on the strength of the damping. If the door is undamped it will swing back and forth forever at a particular resonant frequency. If it is underdamped it will swing back and forth with decreasing size of the swing until it comes to a stop. If it is critically damped then it will return to closed as quickly as possible without oscillating. Finally, if it is overdamped it will return to closed without oscillating but more slowly depending on how overdamped it is. Different levels of damping are desired for different types of systems.
Contents

Linear damping 1

Example: mass–spring–damper 1.1

System behavior 1.2

Critical damping (ζ = 1) 1.2.1

Overdamping (ζ > 1) 1.2.2

Underdamping (0 ≤ ζ < 1) 1.2.3

Alternative models 2

Errors in popular usage 3

See also 4

References 5

Books 6

External links 7
Linear damping
A particularly mathematically useful type of damping is linear damping. Linear damping occurs when a potentially oscillatory variable is damped by an influence that opposes changes in it, in direct proportion to the instantaneous rate of change, velocity or time derivative, of the variable itself. In engineering applications it is often desirable to linearize nonlinear drag forces. This may be done by finding an equivalent work coefficient in the case of harmonic forcing. In nonharmonic cases, restrictions on the speed may lead to accurate linearization.
In physics and engineering, damping may be mathematically modeled as a force synchronous with the velocity of the object but opposite in direction to it. If such force is also proportional to the velocity, as for a simple mechanical viscous damper (dashpot), the force F may be related to the velocity v by

F = cv \, ,
where c is the damping coefficient, given in units of newtonseconds per meter.
This force may be used as an approximation to the friction caused by drag and may be realized, for instance, using a dashpot. (This device uses the viscous drag of a fluid, such as oil, to provide a resistance that is related linearly to velocity.) Even when friction is related to v^2, if the velocity is restricted to a small range, then this nonlinear effect may be small. In such a situation, a linearized friction coefficient c_{lin} may be determined which produces little error.
When including a restoring force (such as due to a spring) that is proportional to the displacement x and in the opposite direction, and by setting the sum of these two forces equal to the mass of the object times its acceleration creates a secondorder differential equation whose terms can be rearranged into the following form:

\frac{d^2x}{dt^2} + 2\zeta\omega_0\frac{dx}{dt} + \omega_0^2 x = 0,
where ω_{0} is the undamped angular frequency of the oscillator and ζ is a constant called the damping ratio. This equation is valid for many different oscillating systems, but with different formulas for the damping ratio and the undamped angular frequency.
The value of the damping ratio ζ determines the behavior of the system such that ζ = 1 corresponds to being critically damped with larger values being overdamped and smaller values being underdamped. If ζ = 0, the system is undamped.
Example: mass–spring–damper
Mass attached to a spring and damper.
An ideal mass–spring–damper system with mass m, spring constant k, and viscous damper of damping coefficient c is subject to an oscillatory force

F_\mathrm{s} =  k x \,
and a damping force

F_\mathrm{d} =  c v =  c \frac{dx}{dt} =  c \dot{x}.
The values can be in any consistent system of units; for example, in SI units, m in kilograms, k in newtons per meter, and c in newtonseconds per meter or kilograms per second.
Treating the mass as a free body and applying Newton's second law, the total force F_{tot} on the body is

F_\mathrm{tot} = ma = m \frac{d^2x}{dt^2} = m \ddot{x}.
where a is the acceleration of the mass and x is the displacement of the mass relative to a fixed point of reference.
Since F_{tot} = F_{s} + F_{d},

m \ddot{x} = kx + c\dot{x}.
This differential equation may be rearranged into

\ddot{x} + { c \over m} \dot{x} + {k \over m} x = 0.\,
The following parameters are then defined:

\omega_0 = \sqrt{ k \over m }

\zeta = { c \over 2 \sqrt{m k} }.
The first parameter, ω_{0}, is called the (undamped) natural frequency of the system. The second parameter, ζ, is called the damping ratio. The natural frequency represents an angular frequency, expressed in radians per second. The damping ratio is a dimensionless quantity.
The differential equation now becomes

\ddot{x} + 2 \zeta \omega_0 \dot{x} + \omega_0^2 x = 0.\,
Continuing, we can solve the equation by assuming a solution x such that:

x = e^{\gamma t}\,
where the parameter \gamma (gamma) is, in general, a complex number.
Substituting this assumed solution back into the differential equation gives

\gamma^2 + 2 \zeta \omega_0 \gamma + \omega_0^2 = 0 \, ,
which is the characteristic equation.
Solving the characteristic equation will give two roots
\gamma_{\pm}=\zeta \omega_0 \pm \omega_0 \sqrt{\zeta ^2  1}.
The solution to the differential equation is thus^{[1]}

x(t) = Ae^{\gamma_+ t} + Be^{\gamma_ t} \, ,
where A and B are determined by the initial conditions of the system:

A = x(0)+\frac{\gamma_+x(0)\dot{x}(0)}{\gamma_\gamma_+}

B = \frac{\gamma_+x(0)\dot{x}(0)}{\gamma_\gamma_+}.
System behavior
Time dependence of the system behavior on the value of the damping ratio ζ, for undamped (blue), underdamped (green), critically damped (red), and overdamped (cyan) cases, for zerovelocity initial condition.
The behavior of the system depends on the relative values of the two fundamental parameters, the natural frequency ω_{0} and the damping ratio ζ. In particular, the qualitative behavior of the system depends crucially on whether the quadratic equation for γ has one real solution, two real solutions, or two complex conjugate solutions.
Critical damping (ζ = 1)
When ζ = 1, there is a double root γ (defined above), which is real. The system is said to be critically damped. A critically damped system converges to zero as fast as possible without oscillating (although overshoot can occur). An example of critical damping is the door closer seen on many hinged doors in public buildings. The recoil mechanisms in most guns are also critically damped so that they return to their original position, after the recoil due to firing, in the least possible time.
In this case, with only one root γ, there is in addition to the solution x(t) = e^{γt} a solution x(t) = te^{γt}:^{[4]}

x(t) = (A+Bt)\,e^{\omega_0 t} \,
where A and B are determined by the initial conditions of the system (usually the initial position and velocity of the mass):

A = x(0) \,

B = \dot{x}(0)+\omega_0x(0) \,
Overdamping (ζ > 1)
When ζ > 1, the system is overdamped and there are two different real roots. An overdamped doorcloser takes longer to close than a critically damped door does.
The solution to the motion equation is:^{[5]}

x(t) = Ae^{\gamma_+ t} + Be^{\gamma_ t}
where A and B are determined by the initial conditions of the system:

A = x(0)+\frac{\gamma_+x(0)\dot{x}(0)}{\gamma_\gamma_+}

B = \frac{\gamma_+x(0)\dot{x}(0)}{\gamma_\gamma_+}.
Underdamping (0 ≤ ζ < 1)
Finally, when 0 < ζ < 1, γ is complex, and the system is underdamped. In this situation, the system will oscillate at the natural damped frequency ω_{d}, which is a function of the natural frequency and the damping ratio. To continue the analogy, an underdamped door closer would close quickly, but would hit the door frame with significant velocity, or would oscillate in the case of a swinging door.
In this case, the solution can be generally written as:^{[6]}

x (t) = e^{ \zeta \omega_0 t} (A \cos\,(\omega_\mathrm{d}\,t) + B \sin\,(\omega_\mathrm{d}\,t ))\,
where

\omega_\mathrm{d} = \omega_0 \sqrt{1  \zeta^2 }\,
represents the damped frequency or ringing frequency of the system,^{[7]} and A and B are again determined by the initial conditions of the system:

A = x(0)\,

B = \frac{1}{\omega_\mathrm{d}}(\zeta\omega_0x(0)+\dot{x}(0)).\,
This "damped frequency" is not to be confused with the damped resonant frequency or peak frequency ω_{peak}.^{[8]} This is the frequency at which a moderately underdamped (ζ < 1/√2) simple 2ndorder harmonic oscillator has its maximum gain (or peak transmissibility) when driven by a sinusoidal input. The frequency at which this peak occurs is given by:

\omega_{peak} = \omega_0\sqrt{1  2\zeta^2}.
For an underdamped system, the value of ζ can be found by examining the logarithm of the ratio of succeeding amplitudes of a system. This is called the logarithmic decrement.
Alternative models
Viscous damping models, although widely used, are not the only damping models. A wide range of models can be found in specialized literature. One is the socalled "hysteretic damping model" or "structural damping model".
When a metal beam is vibrating, the internal damping can be better described by a force proportional to the displacement but in phase with the velocity. In such case, the differential equation that describes the free movement of a singledegreeoffreedom system becomes:

m \ddot{x} + h x i + k x = 0
where h is the hysteretic damping coefficient and i denotes the imaginary unit; the presence of i is required to synchronize the damping force to the velocity (xi being in phase with the velocity). This equation is more often written as:

m \ddot{x} + k ( 1 + i \eta ) x = 0
where η is the hysteretic damping ratio, that is, a measure of the fraction of energy lost in each cycle of the vibration.

\eta = Ed / ( \pi k X^2 )
where Ed is the energy lost and X is the amplitude of the cycle
Although requiring complex analysis to solve the equation, this model reproduces the real behavior of many vibrating structures more closely than the viscous model.
A more general model that also requires complex analysis, the fractional model not only includes both the viscous and hysteretic models, but also allows for intermediate cases (useful for some polymers):

m \ddot{x} + A \frac{d^r x}{dt^r} i + k x = 0
where r is any number, usually between 0 (for hysteretic) and 1 (for viscous), and A is a general damping (h for hysteretic and c for viscous) coefficient.
Nonlinear damping
Nonlinear passive damping offers important advantages compared to purely linear designs.^{[9]} Nonlinear damping using an odd function, for example cubic damping, allows the user to damp the resonance without increasing the energy in the frequency response tails and hence overcomes several limitations of a purely linear design.
Errors in popular usage
It has become common in popular English, especially in science fiction,^{[10]}^{[11]} to substitute the word dampening when the concept of damping is intended. Defined as to make damp or to stifle,^{[12]} dampening can be correctly used to describe depressing the intensity of an emotion, but should not be used to describe the reduction in amplitude of a force, a harmonic oscillation, or similar physical processes or phenomena. For such phenomena, "damping" is the correct word.^{[13]}^{[14]}^{[15]}
See also
References

^ MathWorldA Wolfram Web Resource

^ Katsuhiko Ogata (2005). System Dynamics (4th ed.). University of Minnesota. p. 617.

^

^ Weisstein, Eric W. "Damped Simple Harmonic MotionCritical Damping." From MathWorldA Wolfram Web Resource. [3]

^ Weisstein, Eric W., "Damped Simple Harmonic MotionOverdamping.", MathWorld.

^ Weisstein, Eric W. "Damped Simple Harmonic MotionUnderdamping." From MathWorldA Wolfram Web Resource. [4]

^ Lincoln D. Jones (2003). Electrical Engineering License Review (8th ed.). Dearborn Trade Publishing. p. 6‑15.

^ Millard F. Beatty (2006). Principles of engineering mechanics. Birkhäuser. p. 167.

^ Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and SpatioTemporal Domains". Wiley, 2013

^ The Star Trek Transcripts  Metamorphosis

^ The Movie Transcripts  Star Trek Generations

^ dampening  Collins English Dictionary  Complete & Unabridged 10th Edition

^ Lawrence E. Kinsler (1982). Fundamentals of Acoustics (3, illustrated ed.). Wiley. p. 7.

^ J. P. Den Hartog (1985). Mechanical Vibrations. Courier Dover. p. 7.

^ Leonard Meirovitch (2002). Fundamentals of Vibrations. McGrawHill Higher Education. pp. 25–27.
Books
Komkov, Vadim (1972) Optimal control theory for the damping of vibrations of simple elastic systems. Lecture Notes in Mathematics, Vol. 253. SpringerVerlag, BerlinNew York.
External links

Calculation of the matching attenuation,the damping factor, and the damping of bridging

Damping Matlab scripts
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.