DAMPEN encyclopedia topics

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. 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

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. See Ohm's law.

This force is an (raw) approximation to the friction caused by drag.

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 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.

Example: mass-spring-damper

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_mathrm{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.

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 . 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 $scriptstyle gamma$ is, in general, a complex number.

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

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

Solving for $scriptstyle gamma$ we find:

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

System behavior

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 $scriptstylegamma$ has one real solution, two real solutions, or two complex conjugate solutions.

Critical damping

When ζ = 1, there is a double root $scriptstylegamma$ (defined above), which 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, with only one root $scriptstyle gamma$ , there is in addition to the solution $x = e^{gamma t},$ a solution $x = te^{gamma t},$ :

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, $scriptstylegamma$ there are two different real roots; 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:

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, $scriptstylegamma$ is complex, and the system is under-damped. In this situation, the system will oscillate at the natural damped frequency $scriptstyleomega_mathrm{d}$ , which is a function of the natural frequency and the damping ratio.

In this case, the solution can be generally written as:

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}}(zetaomega_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.

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 (x i 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, 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.

Also requiring complex analysis, but quite more general the fractional model includes both the viscous and hysteretic models but allows also 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 .

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). 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.

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.

Skanking is when a note is isolated by left hand damping of the two strings adjacent to the fully fretted string, producing the desired note (the adjacent strings are scratched). The technique is especially popular among ska, rocksteady and reggae guitarists, who use it with virtually every riddim they play on.

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

Piano

On a piano, damping is controlled by the sustain pedal, with the strings being damped unless the pedal is pressed.

Gamelan

Damping is also important in most percussion instruments in the gamelan, especially the sarons and gendérs. 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. 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.