Damping

A damped spring-mass system. This article is about the harmonic oscillator in classical mechanics

Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system. Oscillation is the repetitive variation typically in Time, of some measure about a central value (often a point of Equilibrium) or between two or more different states

1 Definition
2 Example: mass-spring-damper
- 2.1 Differential equation
- 2.2 System behavior
3 Alternative models
4 In Music
5 See also
6 References
7 External links

Definition

In physics and engineering, damping may be mathematically modelled as a force synchronous with the velocity of the object but opposite in direction to it. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Note The term model has a different meaning in Model theory, a branch of Mathematical logic. In Physics, a force is whatever can cause an object with Mass to Accelerate. In Physics, velocity is defined as the rate of change of Position. Thus, for a simple mechanical damper, the force F may be related to the velocity v by

$\bold{F} = -c \bold{v}$

where c is the viscous damping coefficient, given in units of newton-seconds per meter.

This relationship is perfectly analogous to electrical resistance. Electrical resistance is a ratio of the degree to which an object opposes an Electric current through it measured in Ohms Its reciprocal quantity is See Ohm's law. Ohm's law applies to Electrical circuits it states that the current through a conductor between two points is directly proportional to the

This force is an (raw) approximation to the friction caused by drag. Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a

In playing stringed instruments such as guitar or violin, damping is the quieting or abrupt silencing of the strings after they have been sounded, by pressing with the edge of the palm, or other parts of the hand such as the fingers on one or more strings near the bridge of the instrument. The guitar is a Musical instrument with ancient roots that is used in a wide variety of musical styles The violin is a bowed String instrument with four strings usually tuned in Perfect fifths It is the smallest and highest-pitched member The strings themselves can be modelled as a continuum of infinitesimally small mass-spring-damper systems where the damping constant is much smaller than the resonance frequency, creating damped oscillations (see below). See also Vibrating string. A Vibration in a string is a Wave. Usually a vibrating string produces a Sound whose Frequency in most cases is constant

Example: mass-spring-damper

A mass attached to a spring and damper. The damping coefficient, usually c, is represented by B in this case. The F in the diagram denotes an external force, which this example does not include.

An ideal mass-spring-damper system with mass m (in kilograms), spring constant k (in newtons per meter) and viscous damper of damping coefficient c (in newton-seconds per meter) can be described with the following formula:

F s = - k x

$F_\mathrm{d} = - c v = - c \dot{x} = - c \frac{dx}{dt}$

Treating the mass as a free body and applying Newton's second law, we have:

$\sum F = ma = m \ddot{x} = m \frac{d^2x}{dt^2}$

where a is the acceleration (in meters per second squared) of the mass and x is the displacement (in meters) of the mass relative to a fixed point of reference. The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International Newton-second (symbol Ns, N s or N·s) is the derived SI unit of Impulse and Momentum. Free body is the generic term used by physicists and engineers to describe some thing&mdashbe it a Bowling ball, a Spacecraft, Pendulum Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the In Physics, displacement is the vector that specifies the position of a point or a particle in reference to a previous position or to the origin of the chosen

Differential equation

The above equations combine to form the equation of motion, a second-order differential equation for displacement x as a function of time t (in seconds):

$m \ddot{x} + c \dot{x} + k x = 0.\,$

Rearranging, we have

$\ddot{x} + { c \over m} \dot{x} + {k \over m} x = 0.\,$

Next, to simplify the equation, we define the following parameters:

$\omega_0 = \sqrt{ k \over m }$

and

$\zeta = { c \over 2 \sqrt{k m} }.$

The first parameter, ω₀, is called the (undamped) natural frequency of the system . A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units In Physics, resonance is the tendency of a system to Oscillate at maximum Amplitude at certain frequencies, known as the system's The second parameter, ζ, is called the damping ratio. In Mechanical engineering, the Vibration behaviour of a group of components (a "system" is often of interest The natural frequency represents an angular frequency, expressed in radians per second. Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 The damping ratio is a dimensionless quantity. In Dimensional analysis, a dimensionless quantity (or more precisely a quantity with the dimensions of 1) is a Quantity without any Physical units

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 $\scriptstyle \gamma$ is, in general, a complex number. In Mathematics, Statistics, and the mathematical Sciences a parameter ( G auxiliary measure) is a quantity that defines certain characteristics Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Substituting this assumed solution back into the differential equation, we obtain

$\gamma^2 + 2 \zeta \omega_0 \gamma + \omega_0^2 = 0.\,$

Solving for γ, we find:

$\gamma = \omega_0( - \zeta \pm \sqrt{\zeta^2 - 1}).$

System behavior

Dependence of the system behavior on the value of the damping ratio

ζ

The behavior of the system depends on the relative values of the two fundamental parameters, the natural frequency ω₀ and the damping ratio ζ. In particular, the qualitative behavior of the system depends crucially on whether the quadratic equation for $\scriptstyle\gamma$ has one real solution, two real solutions, or two complex conjugate solutions. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree.

Critical damping

When ζ = 1, $\scriptstyle\gamma$ (defined above) is real, the system is said to be critically damped. A critically damped system converges to zero faster than any other without oscillating. 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, the solution simplifies to^[1]:

$x(t) = (A+Bt)\,e^{-\omega_0 t} \,$

where A and B are determined by the initial conditions of the system (usually the initial position and velocity of the mass):

$A = x(0) \,$

$B = \dot{x}(0)+\omega_0x(0) \,$

Over-damping

When ζ > 1, $\scriptstyle\gamma$ is still real, but now the system is said to be over-damped. An over-damped door-closer will take longer to close than a critically damped door would.

The solution to the motion equation is^[2]:

$x(t) = Ae^{\gamma_+ t} + Be^{\gamma_- t}$

where A and B are determined by the initial conditions of the system:

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

$B = -\frac{\gamma_+x(0)-\dot{x}(0)}{\gamma_--\gamma_+}.$

Under-damping

Finally, when 0 ≤ ζ < 1, $\scriptstyle\gamma$ is complex, and the system is under-damped. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In this situation, the system will oscillate at the natural damped frequency $\scriptstyle\omega_\mathrm{d}$ , which is a function of the natural frequency and the damping ratio.

In this case, the solution can be generally written as^[3]:

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

where

$\omega_\mathrm{d} = \omega_0 \sqrt{1 - \zeta^2 }\,$

represents the natural damped frequency of the system, and A and B are again determined by the initial conditions of the system:

$A = x(0)\,$

$B = \frac{1}{\omega_\mathrm{d}}(\zeta\omega_0x(0)+\dot{x}(0)).\,$

For an under-damped 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. Logarithmic decrement, δ is used to find the Damping ratio of an underdamped system in the time domain

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, but one of them should be referred here: the so called "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 single-degree-of-freedom 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). A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation 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, the fraction of energy lost in each cycle of the vibration.

Although requiring complex analysis to solve the equation, this model reproduces the real behaviour of many vibrating structures more closely than the viscous model. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex

See also: Friction and Drag

In Music

Guitar

On guitar, damping (also referred to as choking) is a technique where, shortly after playing the strings, the sound is reduced by pressing the right hand palm against the strings, right hand damping (including Palm muting), or relaxing the left hand fingers' pressure on the strings, left hand damping (or Left-hand muting). Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a The guitar is a Musical instrument with ancient roots that is used in a wide variety of musical styles A string is the vibrating element that is the source of vibration in String instruments such as the Guitar, Harp, Piano, and members The palm mute is a playing technique for the Guitar or bass. This technique is known as Pizzicato by Classical guitar players (see Classical Left-hand muting is a performance technique for stringed instruments where the vibration of a string is muffled by the left hand Scratching is where the strings are played while damped, ie, the strings are damped before playing. The term presumably refers to the clunky sound produced. In funk music this is often done over a sixteenth note pattern with occasional sixteenths undamped. Funk is an American musical style that originated in the mid- to late-1960s when African American musicians blended Soul music, Soul In Music, a sixteenth note (American or "German" terminology or semiquaver (also occasionally demiquaver, British or "classical"

Floating is the technique where a chord is sustained past a sixteenth note rather than that note being scratched, the term referring to the manner in which the right hand "floats" over the strings rather than continuing to scratch. This article describes musical chords in traditional Western styles

Skanking is where a note is isolated by left hand damping of the two strings adjacent to the fully fretted string producing the desired note, ie the adjacent strings are scratched. A fret is a raised portion on the neck of a Stringed instrument, that extends generally across the full width of the neck See also: Bang/Skank/Cheka. Damping is any effect either deliberately engendered or inherent to a system that tends to reduce the amplitude of Oscillations of an oscillatory system Cheka (real name David Lozada) is a Reggaeton artist from Guayama Puerto Rico. The technique is extremely popular among Reggae, Ska, and Rocksteady guitarists, who uses it with virtually every riddim they play on. Reggae is a Music genre first developed in Jamaica in the late 1960s Ska ( pronounced /ska/ or in Jamaican Patois /skja/ is a Music genre that originated in Jamaica in the late 1950s and which was the precursor Rocksteady is a Music genre that was most popular in Jamaica, starting around 1966 and its Reggae successor was established around 1968 A riddim is an Instrumental version of a song which applies to Jamaican music (mostly dancehall and reggae or other forms of Caribbean music. It is a classical element of this style of music.

Damping is possible on other string instruments by halting the vibration of the strings using the left hand, similar to on a guitar. ^[4]

Piano

On a piano, damping is controlled by the sustain pedal, with the strings being damped unless the pedal is pressed. The piano is a Musical instrument played by means of a keyboard that produces sound by striking steel strings with Felt covered hammers A sustain or sustaining pedal (also damper pedal or loud pedal) is the most commonly used pedal in a modern Piano.

Gamelan

Damping is also important in most percussion instruments in the gamelan, especially the sarons and gendérs. A gamelan is a musical ensemble of Indonesia typically featuring a variety of instruments such as metallophones xylophones drums and gongs bamboo flutes bowed and The saron is a musical instrument of Indonesia, which is used in the Gamelan. A gendér is a type of Metallophone used in Balinese and Javanese Gamelan music On instruments that are played with a single mallet, the left hand is used to damp the previously hit note when a new note is played. A mallet is a type of hammer with a head made of softer materials than the Steel normally used in hammerheads so as to avoid damaging a delicate surface In Music, the term note has two primary meanings 1 a sign used in Musical notation to represent the relative duration and pitch of a Sound; On the gendér, which is played with mallets in both hands, the keys must be damped by the same hand, and it requires practice to master the technique.

References

^ Weisstein, Eric W. In Mechanical engineering, the Vibration behaviour of a group of components (a "system" is often of interest The term damping factor can also refer to the amount of Damping in any oscillatory system or in numerical algorithms In Audio system An RLC circuit (also known as a Resonant circuit tuned circuit or LCR circuit is an Electrical circuit consisting of a Resistor (R an Oscillation is the repetitive variation typically in Time, of some measure about a central value (often a point of Equilibrium) or between two or more different states This article is about the harmonic oscillator in classical mechanics Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. In Physics, resonance is the tendency of a system to Oscillate at maximum Amplitude at certain frequencies, known as the system's Practical The impulse excitation technique is a Nondestructive test method that uses Natural frequency, dimensions and mass of a test-piece to determine Young's Audio system measurements are made for several purposes Designers take measurements so that they can specify the performance of a piece of equipment A tuned mass damper, also known as an active mass damper ( AMD) or harmonic absorber, is a device mounted in structures to prevent discomfort damage or Suspension is the term given to the system of springs, Shock absorbers and linkages that connects a Vehicle to its Wheels Suspension Vibration refers to mechanical Oscillations about an equilibrium point. See also Earthquake engineering In Earthquake engineering, vibration control is a set of technical means aimed to mitigate Seismic Impacts Jet damping or Thrust damping is the effect of rocket exhaust removing energy from the transverse angular motion of a rocket "Damped Simple Harmonic Motion--Critical Damping. " From MathWorld--A Wolfram Web Resource. [1]
^ Weisstein, Eric W. "Damped Simple Harmonic Motion--Overdamping. " From MathWorld--A Wolfram Web Resource. [2]
^ Weisstein, Eric W. "Damped Simple Harmonic Motion--Underdamping. " From MathWorld--A Wolfram Web Resource. [3]
^ Bolton, Ross (2001). Funk Guitar: The Essential Guide. ISBN 0-634-01168-5.

External links

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

A-level Physics experiment on the subject of Damped Harmonic Motion with solution curve graphs

Dictionary

-verb

Present participle of damp.

-noun

The reduction in the magnitude of oscillations by the dissipation of energy
The stabilization of a physical system by reducing oscillation

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