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.
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 = -B \bold]
- where B is the damper constant.
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-seconds per meter) can be described with the following formulae:
- [F_\mathrm \ \ = \ \ - k x]
- [F_\mathrm \ \ = \ \ - B v \ \ = \ \ - B \dot \ \ = \ \ - B \frac]
- [\Sigma\ F \ \ = \ \ ma\ \ = \ \ m \ddot \ \ = \ \ m \frac]
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 + B \dot + k x = 0]
- [\ddot + \dot + x = 0]
- [\omega_0 = \sqrt]
- [\zeta = ]
The differential equation now becomes:
- [\ddot + 2 \zeta \dot + \omega_0^2 x = 0]
- [\ x = e^ \ ]
Substituting this assumed solution back into the differential equation, we obtain:
- [\gamma^2 + 2 \zeta \gamma + \omega_0^2 = 0]
- [\gamma = - \zeta \pm \sqrt]
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. Zeta, whose dimensions are those of frequency, can therefore be seen to be a quantity that that affects the damped resonant frequency of the system. When zeta is equal to the undamped frequency, the damped frequency is zero, and the system is critically damped. The ratio [ \sigma = \zeta/\omega_0 ] is known as the damping ratio, and describes the system's closeness to critical damping.
Critical damping
When [\zeta^2 - \omega_0^2 = 0] (or, equivalently, σ = 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 [\zeta^2 - \omega_0^2 > 0] (or σ > 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 [ \zeta^2 - \omega_0^2 < 0 ] (or σ < 1), [\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^ cos( \omega_\mathrm t + \phi)]
- [\omega_\mathrm = \sqrt]
See also
- Damping factor
- RLC circuit
- Oscillator
- Harmonic oscillator
- Simple harmonic motion
- Resonance
- Impedance
- Input impedance
- Output impedance
- Voltage divider
- Audio system measurements
- Bridging attenuation
- Matching attenuation
External links
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