NationMaster - Encyclopedia: 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. Oscillation is the variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ...

1 Explanation
2 Example: mass-spring-damper
- 2.1 Differential equation
- 2.2 System behavior
3 Solution
4 See also
5 External links

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 Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... Engineering is the design, analysis, and/or construction of works for practical purposes. ... A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system. ... In physics, force is an influence that may cause a body to accelerate. ... This article or section does not cite its references or sources. ...

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

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

This relationship is perfectly analogous to electrical resistance. See Ohm's law. Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... poo A voltage source, V, drives an electric current, I , through resistor, R, the three quantities obeying Ohms law: V = IR Ohms law states that, in an electrical circuit, the current passing through a conductor from one terminal point on the conductor to another terminal point on the...

This force is an approximation to the friction caused by drag. Friction is the force that opposes the relative motion or tendency toward such motion of two surfaces in contact. ... An object falling through a gas or liquid experiences a force in direction opposite to its motion. ...

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. This article or section does not adequately cite its references or sources. ... The violin is a bowed string instrument with four strings tuned in perfect fifths. ... A vibration in a string is a wave. ...

Example: mass-spring-damper

A mass attached to a spring and a 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 damper constant c (in Newton-seconds per meter) can be described with the following formula: Image File history File links Mass-Spring-Damper. ... Image File history File links Mass-Spring-Damper. ... The international prototype, made of platinum-iridium, which is kept at the BIPM under conditions specified by the 1st CGPM in 1889. ... The newton (symbol: N) is the SI unit of force. ... The metre, or meter (symbol: m) is the SI base unit of length. ... The newton (symbol: N) is the SI unit of force. ...

$F_mathrm{s} = - k x$

$F_mathrm{d} = - c v = - c dot{x} = - c frac{dx}{dt}$

Treating the mass as a free body and applying Newton's second law, we have: Free body is the generic term used by physicists to describe some thing—be it a bowling ball, a spaceship, pendulum, a television, or anything else—which can be seen as moving as a single mass. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

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

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. Acceleration is the time rate of change of velocity, and at any point on a velocity-time graph, it is given by the slope of the tangent to that point basicly. ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ...

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): A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... Look up second in Wiktionary, the free dictionary. ...

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

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. This article is about resonance in physics. ... The damping ratio is a parameter Î¶ that characterizes the frequency response of a second order ordinary differential equation. ... It has been suggested that this article or section be merged into Angular velocity. ... In mathematics and physics, the radian is a unit of angle measure. ... In the physical sciences, a dimensionless number (or more precisely, a number with the dimensions of 1) is a quantity which describes a certain physical system and which is a pure number without any physical units; it does not change if one alters ones system of units of measurement...

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

$x = e^{gamma t}$

where the parameter $gamma$ is, in general, a complex number. The factual accuracy of this article is disputed. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

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 $γ$ has one real solution, two real solutions, or two complex conjugate solutions. Image File history File links Critial_damping. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

Critical damping

When $ζ = 1$ , $gamma$ (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

Over-damping

When $ζ > 1$ , $gamma$ 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. Image File history File links Overdamping. ...

Under-damping

Finally, when $ζ < 1,$ $gamma$ is complex, and the system is under-damped. In this situation, the system will oscillate at the damped frequency $omega_mathrm{d}=omega_0sqrt{1-zeta^2}$ , which is a function of the natural frequency and the damping ratio. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

Solution

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

$x (t) = A e^{- zeta omega_0 t} cos( omega_mathrm{d} t + varphi)$

where

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

represents the 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.

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

Categories: Electronics terms | Control theory | Ordinary differential equations | Dynamics

Results from FactBites:

damping: Definition and Much More from Answers.com (1218 words)
	Viscous damping is caused by such energy losses as occur in liquid lubrication between moving parts or in a liquid forced through a small opening by a piston, as in automobile shock absorbers.
	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.
	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.

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	The main effect of damping in a loudspeaker is to reduce the SPL produced by the loudspeaker's diaphragm moving because of its own inertia after the signal stops.
	Electrical damping is determined by the resistance load on the loudspeaker, which is the sum of the loudspeaker cable resistance, and the amplifier output impedance.
	Electrical damping or Damping Factor (DF) is calculated by dividing the loudspeaker's voice coil resistance by the sum of the loudspeaker cable resistance and amplifier output impedance.

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