DAMPING

'Damping' is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations of an oscillatory system.

Contents

Explanation

Example: mass-spring-damper

Differential equation

System behavior

Critical damping

Over-damping

Under-damping

'Solution'

'Other models'

See also

External links

Explanation

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. Thus, for a simple mechanical damper, the force 'F' may be related to the velocity 'v' by
:$$
old{F} = -c old{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 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 viscous damper of damping coeficient ''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 rac{dx}{dt}

Treating the mass as a free body and applying , we have:
:$$
sum 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 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, 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 $gamma$ 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 γ, 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 $gamma$ has one real solution, two real solutions, or
two complex conjugate solutions.

Critical damping

When $zeta=1$, $gamma$ (defined above) 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. An automobile suspension has a damping near critical damping (slightly higher for "hard" suspensions and slightly less for "soft" ones)

Over-damping

When $zeta\; >\; 1$, $gamma$ 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 $gamma$ is complex, and the system is ''under-damped''. In this situation, the system will oscillate at the natural damped frequency $omega\_mathrm\{d\}=omega\_0sqrt\{1-zeta^2\}$, which is a function of the natural frequency and the damping ratio.
----

'Solution'

In the underdamped case,
the solution can be generally written as:
:$$
x (t) = A e^{- zeta omega_0 t} cos( omega_mathrm{d} t + arphi)

where
:$$
omega_mathrm{d} = omega_0 sqrt{1 - zeta^2 }

represents the ''natural damped frequency'' of the system, and ''A'' and φ are determined by the initial conditions of the system (usually the initial position and velocity of the mass).
In the critically damped case, the solution takes the form
:$$
x(t) = (A+Bt)e^{-omega_0 t}

where ''A'' and ''B'' are again determined by the initial conditions.
----

'Other models'

Viscous damping models, although widely - and somehow abusively - used, are not the only damping models. A wide range of models can be found in specialized literature, but one of them should be refered 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 and not the velocity. In such case, the differential equation that describes the movement becomes:
:$m\; ddot\{x\}\; +\; h\; x\; i\; +\; k\; x\; =\; 0$
where ''h'' is the hysteretic damping coeficient 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).
Although requiring complex analysis to solve the equation, this model reproduces the real behaviour of many vibrating structures more closely than the viscous model.

See also

★ Damping ratio

★ Damping factor

★ RLC circuit

★ Oscillator

★ Harmonic oscillator

★ Simple harmonic motion

★ Resonance

★ Audio system measurements

★ Bridging attenuation

★ Matching attenuation

★ Tuned mass damper

★ Vehicle suspension

★ Vibration

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