Damping

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

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

[edit] 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.

[edit] Example: mass-spring-damper

A mass attached to a spring and a damper. (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 B (in newton-seconds per meter) can be described with the following formula:

$F_\mathrm{s} \ \ = \ \ - k x$

$F_\mathrm{d} \ \ = \ \ - B v \ \ = \ \ - B \dot{x} \ \ = \ \ - B \frac{dx}{dt}$

Treating the mass as a free body and applying Newton's second law, we have:

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

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.

[edit] 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

$\zeta = { B \over 2 \sqrt{k m} }.$

The first parameter, ω₀, is called the (undamped) natural frequency of the system. The second, ζ, is called the damping factor. The natural frequency represents an angular frequency and has for units of measure radians per second. The damping factor 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

$\ 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:

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

[edit] 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 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.

[edit] 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.

[edit] 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.

[edit] 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 $\omega_\mathrm{d}=\omega_0\sqrt{1-\zeta^2}$ , which is a function of the natural frequency and the damping factor.

[edit] 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

$x(t) \ = \ (A+Bt)e^{-\zeta\omega_0 t}$