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The formula for the Q factor is: =, where M is the mass, k is the spring constant, and D is the damping coefficient, defined by the equation F damping = −Dv, where v is the velocity. [ 24 ] Acoustical systems
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:
Low-Q circuits are therefore damped and lossy and high-Q circuits are underdamped and prone to amplitude extremes if driven at the resonant frequency. [a] Q is related to bandwidth; low-Q circuits are wide-band and high-Q circuits are narrow-band. In fact, it happens that Q is the inverse of fractional bandwidth
= 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 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 ...
Zero damping will produce a maximum response. Very high damping produces a very boring SRS: A horizontal line. The level of damping is demonstrated by the "quality factor", Q which can also be thought of transmissibility in sinusoidal vibration case. Relative damping of 5% results in a Q of 10.
There is no unit designation for transmissibility, although it may sometimes be referred to as the Q factor. The transmissibility is used in calculation of passive hon efficiency. The lesser the transmissibility the better is the damping or the isolation system.
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.