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The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next. The damping ratio is a system parameter, denoted by ζ ("zeta"), that can vary from undamped (ζ = 0), underdamped (ζ < 1) through critically damped (ζ = 1) to overdamped (ζ > 1).
= 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 uppercase zeta is not used, because it is normally identical to Latin Z. The lower case letter can be used to represent: The Riemann zeta function in mathematics; The damping ratio of an oscillating system in engineering and physics; The rotational quantity of angular jerk in physics; The effective nuclear charge on an electron in quantum ...
Natural frequency ω 0 = 1 rad/s, damping ratio ζ = 0.4. Sinusoidal steady state is represented by letting s = jω , where j is the imaginary unit . Taking the magnitude of the above equation with this substitution:
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 ...
Lines of constant damping ratio can be drawn radially from the origin and lines of constant natural frequency can be drawn as arccosine whose center points coincide with the origin. By selecting a point along the root locus that coincides with a desired damping ratio and natural frequency, a gain K can be calculated and implemented in the ...
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}
The value that the damping coefficient must reach for critical damping in the mass-spring-damper model is: =. To characterize the amount of damping in a system a ratio called the damping ratio (also known as damping factor and % critical damping) is used. This damping ratio is just a ratio of the actual damping over the amount of damping ...