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Tay, Mareels and Moore (1998) defined settling time as "the time required for the response curve to reach and stay within a range of certain percentage (usually 5% or 2%) of the final value." [ 2 ] Mathematical detail
where is the relaxation time of the particle (the time constant in the exponential decay of the particle velocity due to drag), is the fluid velocity of the flow well away from the obstacle, and is the characteristic dimension of the obstacle (typically its diameter) or a characteristic length scale in the flow (like boundary layer thickness). [1]
[5] In air, the same theory can be used to explain why small water droplets (or ice crystals) can remain suspended in air (as clouds) until they grow to a critical size and start falling as rain (or snow and hail). [6] Similar use of the equation can be made in the settling of fine particles in water or other fluids. [citation needed]
This is the formula for length contraction. As there existed a proper time for time dilation, there exists a proper length for length contraction, which in this case is ℓ. The proper length of an object is the length of the object in the frame in which the object is at rest.
As an example of this formula, if Δ = 1/e 4 = 1.8 %, the settling time condition is t S = 8 τ 2. In general, control of overshoot sets the time constant ratio, and settling time t S sets τ 2 . [ 5 ] [ 6 ] [ 7 ]
First order LTI systems are characterized by the differential equation + = where τ represents the exponential decay constant and V is a function of time t = (). The right-hand side is the forcing function f(t) describing an external driving function of time, which can be regarded as the system input, to which V(t) is the response, or system output.
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For applications in control theory, according to Levine (1996, p. 158), rise time is defined as "the time required for the response to rise from x% to y% of its final value", with 0% to 100% rise time common for underdamped second order systems, 5% to 95% for critically damped and 10% to 90% for overdamped ones. [6]