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.

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. Thus, for a simple mechanical damper, the force F may be related to the velocity v by



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

Enlarge picture
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 viscous damper of damping coeficient c (in Newton-seconds per meter) can be described with the following formula:





Treating the mass as a free body and applying , we have:



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):



Rearranging, we have



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



and



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



Continuing, we can solve the equation by assuming a solution x such that:



where the parameter is, in general, a complex number.

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



Solving for γ, we find:

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

Critical damping

When ζ = 1, (defined above) is real, the system is said to be 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)

In this case, the solution simplifies to[1]:



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



Over-damping

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

The solution to the motion equation is[2]:



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



Under-damping

Finally, when 0 ≤ ζ < 1, is complex, and the system is under-damped. In this situation, the system will oscillate at the natural damped frequency , which is a function of the natural frequency and the damping ratio.

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



where



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



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:



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). This equation is more often written as:



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.

See also:  and

See also

References

1. ^ Weisstein, Eric W. "Damped Simple Harmonic Motion--Critical Damping." From MathWorld--A Wolfram Web Resource. [1]
2. ^ Weisstein, Eric W. "Damped Simple Harmonic Motion--Overdamping." From MathWorld--A Wolfram Web Resource. [2]
3. ^ Weisstein, Eric W. "Damped Simple Harmonic Motion--Underdamping." From MathWorld--A Wolfram Web Resource. [3]

External links



''For other uses, see oscillator (disambiguation)
Oscillation is the variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states.
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Electrical resistance is a measure of the degree to which an object opposes an electric current through it. The SI unit of electrical resistance is the ohm. Its reciprocal quantity is electrical conductance measured in siemens.
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Friction is the force of two surfaces in contact. It is not a fundamental force, as it is derived from electromagnetic forces between atoms. When contacting surfaces move relative to each other, the friction between the two objects converts kinetic energy into thermal energy, or
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drag (sometimes called resistance) is the force that resists the movement of a solid object through a fluid (a liquid or gas). Drag is made up of friction forces, which act in a direction parallel to the object's surface (primarily along its sides, as friction forces at the
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The violin is a bowed string instrument with four strings tuned in perfect fifths. It is the smallest and highest-pitched member of the violin family of string instruments, which also includes the viola and
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vibrating string produces a sound whose frequency in most cases is constant. Therefore, since frequency characterizes the pitch, the sound produced is a constant note. Vibrating strings are the basis of any string instrument like guitar, cello, or piano.
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resonance is the tendency of a system to oscillate at maximum amplitude at a certain frequency. This frequency is known as the system's resonance frequency. When damping is small, the resonance frequency is approximately equal to the natural frequency of the system, which
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The damping ratio is a parameter, usually denoted by ζ (zeta), that characterizes the frequency response of a second order ordinary differential equation. It is particularly important in the study of control theory. It is also important in the harmonic oscillator.
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angular frequency ω (also referred to by the terms angular speed, radial frequency, and radian frequency) is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity.
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