Search results
Results from the WOW.Com Content Network
In the case of the unit step, the overshoot is just the maximum value of the step response minus one. Also see the definition of overshoot in an electronics context . For second-order systems, the percentage overshoot is a function of the damping ratio ζ and is given by [ 3 ]
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:
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.
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]
= 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 tuning application, for instance, is an example of band-pass filtering. The RLC filter is described as a second-order circuit, meaning that any voltage or current in the circuit can be described by a second-order differential equation in circuit analysis. The three circuit elements, R, L and C, can be combined in a number of different ...
Bode's ideal control loop frequency response has the fractional integrator shape and provides the iso-damping property around the gain crossover frequency. This is due to the fact that the phase margin and the maximum overshoot are defined by one parameter only (the fractional power of ), and are independent of open-loop gain.
Different damping ratios produce different SRSs for the same shock waveform. 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.