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Damping
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EXPLANATION


In Physics and Engineering , damping is Mathematically Modelled as a Force with magnitude proportional to that of the Velocity of the object but opposite in direction to it. Thus, for a simple mechanical damper, the force F is related to the velocity '''v''' by

:\bold{F} = -B \bold{v}

:where ''B'' is the ''damper constant''.

This relationship is perfectly analogous to Electrical Resistance . See Ohm's Law .

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 resonant 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 damper constant ''B'' (in newton- Second s per meter) can be described with the following formulae:

:F_\mathrm{s} \ \ = \ \ - k x
:F_\mathrm{d} \ \ = \ \ - B v \ \ = \ \ - B \dot{x} \ \ = \ \ - B rac{dx}{dt}

Treating the mass as a , we have:

:\Sigma\ F \ \ = \ \ ma\ \ = \ \ m \ddot{x} \ \ = \ \ m rac{d^2x}{dt^2}

where ''a'' is the Acceleration (in meters per second2) of the mass and x is the Displacement (in meters) of the mass relative to a fixed point of reference.


Differential equation

The equations of motion combine to form a second-order Differential Equation for displacement ''x'' as a function of time ''t'' (in Seconds ):

:m \ddot{x} + B \dot{x} + k x = 0

Rearranging, we have

:\ddot{x} + { B \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
:\sigma = { B \over 2m }

The first parameter, \omega_0 , is called the (undamped) Natural Frequency of the system.
The second, \sigma , is called the '' Damping Factor ''. Both parameters represent Angular Frequencies and have for units of measure Radians per second.

The differential equation now becomes:

:\ddot{x} + 2 \sigma \dot{x} + \omega_0^2 x = 0

Continuing, we can solve the equation by assuming
:\ x = e^{\gamma t} \

:where \ \gamma \ is, in general, a Complex Number .

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

:\gamma^2 + 2 \sigma \gamma + \omega_0^2 = 0

Solving for \ \gamma \ , we find:

:\gamma = - \sigma \pm \sqrt{\alpha^2 - \omega_0^2}


System behavior

The behavior of the system depends on the relative values of the two fundamental parameters, the natural frequency ω0 and the damping factor σ. (Also known as the damping ratio, given the symbol ζ, as defined as the systems ''closeness'' to critical damping )


Critical damping

When \sigma^2 - \omega_0^2 = 0, \gamma is real and the system is ''critically damped''. An example of critical damping is the door-closer seen on many hinged doors in public buildings.


Over-damping

When \sigma^2 - \omega_0^2 > 0, \gamma is still real, but now the system is said to be ''over-damped''. An overdamped door-closer will take longer to close the door than a critically damped door closer.


Under-damping

Finally, when \sigma^2 - \omega_0^2 < 0 , \ \gamma \ is Complex , and the system is ''under-damped''. In this situation, the system will oscillate at the damped frequency, which is a function of the natural frequency and the damping factor.

The solution can be generally written as:

: x (t) \ = \ A e^{- \sigma t} cos( \omega_\mathrm{d} t + \phi)

where

: \omega_\mathrm{d} = \sqrt{\omega_0^2 - \sigma^2 }

represents the ''damped frequency'' of the system, and ''A'' and \phi are determined by the initial conditions of the system (usually the initial position and velocity of the mass).


DAMPING FACTOR

This term refers to the amount of Damping in an oscillatory system and is usually used to describe loudspeaker damping by an amplifier. In the case of loudspeaker systems, damping factor also describes the ability of the amplifier to control undesirable movement of the speaker cone near the resonant frequency of the speaker system


Audio loudspeaker example


A speaker diaphragm has mass, and the surround has stiffness. Together these form a Resonant system and the cone may resonate in response to short audio Pulse s.

A high damping factor indicates that an amplifier will have greater control over the movement of the speaker cone, particularly in the bass region where the resonant frequency of the speaker system will lie. This damping gives a "tight bass" sound from the sound system.


Effect of cable resistance


The damping factor is affected to a small extent by the resistance of the speaker cables. The higher the resistance of the speaker cables, the lower the damping factor. A large damping factor(greater than about 10) is no advantage. Thus provided that the return path of the cables measures less than about 0.8 Ω , thicker or better cables will make no perceptible difference. The difference in damping with a factor of 10 is the difference between 8 Ω and 8.8 Ω, which is unlikely to give more than a fraction of a dB difference in sound pressure level at the low frequency resonance of the speaker.

For ) is generally smaller than 0.1 Ω (ohms), and can be seen from the point of view of the loudspeaker as a near short-circuit. This will very rapidly absorb any unwanted currents induced by the mechanical resonance of the speaker's voice coil, acting as a very effective 'brake' on the speaker (just as a short circuit across the terminals of a generator will make it very hard to turn), thus keeping it under control.

This is called voltage Bridging . Z_\mathrm{load} >> Z_\mathrm{source}.

The loudspeaker's Load Impedance (input impedance) of Z_\mathrm{load} is usually around 4 to 8 Ω although other impedance speakers are available.

Solving for Z_\mathrm{source}:
:
Z_\mathrm{source} = rac{Z_\mathrm{load}}{DF}



Amplifier output impedance


The output impedance of an amplifier can therefore be calculated from the damping factor and the loudspeaker impedance. Note that modern amplifiers, employing relatively high levels of Negative Feedback , generally exhibit extremely low output impedances — one of the many consequences of using feedback. Thus "damping factor" figures in themselves do not say very much about the quality of a system. Given the controversy that has surrounded the topic of feedback for many years, some may see a high damping factor as a mark of poor quality becuase it implies a high level of NFB in the amplifier.


Zero electrical damping factor

There is also a good case to be made for a loudspeaker system with zero electrical damping. In the case of a speaker in a small sealed box, there is no way to reduce the system resonance below a few hundred Hz (even if the speaker itself has a very low free air resonance) because of the low compliance of the air in the box. However, because of the high system resonance, other losses being equal, the acoustic cone damping will be high. All that is needed is then to electrically drive the speaker from an amplifier with a high o/p Z (ie a current source) for a flat response all the way down to dc.

One advantage of the system is that whether it is being opearted above or below the system resonance, the cone excursion is current controlled and is therefore linear (independent of surround nonlinearities). The only slight problem with this scheme is that the efficiency may not be too high.


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