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The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient:
= is called the "damping ratio". Step response of a damped harmonic oscillator; curves are plotted for three values of μ = ω 1 = ω 0 √ 1 − ζ 2. Time is in units of the decay time τ = 1/(ζω 0). The value of the damping ratio ζ critically determines the behavior of the system. A damped harmonic oscillator can be:
The logarithmic decrement can be obtained e.g. as ln(x 1 /x 3).Logarithmic decrement, , is used to find the damping ratio of an underdamped system in the time domain.. The method of logarithmic decrement becomes less and less precise as the damping ratio increases past about 0.5; it does not apply at all for a damping ratio greater than 1.0 because the system is overdamped.
The Q factor is a parameter that describes the resonance behavior of an underdamped harmonic oscillator (resonator). Sinusoidally driven resonators having higher Q factors resonate with greater amplitudes (at the resonant frequency) but have a smaller range of frequencies around that frequency for which they resonate; the range of frequencies for which the oscillator resonates is called the ...
A useful parameter is the damping factor, ... plot for the voltages across the elements of an RLC series circuit. Natural frequency ω 0 = 1 rad/s, damping ratio ...
Here damping ratio is always less than one. Critically damped A critically damped response is the response that reaches the steady-state value the fastest without being underdamped. It is related to critical points in the sense that it straddles the boundary of underdamped and overdamped responses. Here, the damping ratio is always equal to one.
is the undamped natural frequency and is the damping ratio. The homogeneous equation for the mass spring system is: The homogeneous equation for the mass spring system is: x ¨ + 2 ζ ω n x ˙ + ω n 2 x = 0 {\displaystyle {\ddot {x}}+2\zeta \omega _{n}{\dot {x}}+\omega _{n}^{2}x=0}
Therefore, the damped and undamped description are often dropped when stating the natural frequency (e.g. with 0.1 damping ratio, the damped natural frequency is only 1% less than the undamped). The plots to the side present how 0.1 and 0.3 damping ratios effect how the system “rings” down over time.