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The general equation for an exponentially damped sinusoid may be represented as: = () ... is the natural frequency of the system. The damping ratio is ...
This is called the damped resonance frequency or the damped natural frequency. It is the frequency the circuit will naturally oscillate at if not driven by an external source. The resonance frequency, ω 0 , which is the frequency at which the circuit will resonate when driven by an external oscillation, may often be referred to as the undamped ...
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.
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.
If a frictional force proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can: Oscillate with a frequency lower than in the undamped case, and an amplitude decreasing with time (underdamped oscillator).
Natural frequency, measured in terms of eigenfrequency, is the rate at which an oscillatory system tends to oscillate in the absence of disturbance. A foundational example pertains to simple harmonic oscillators, such as an idealized spring with no energy loss wherein the system exhibits constant-amplitude oscillations with a constant frequency.
Similar to the Fourier transform, Prony's method extracts valuable information from a uniformly sampled signal and builds a series of damped complex exponentials or damped sinusoids. This allows the estimation of frequency, amplitude, phase and damping components of a signal.
Resonance occurs in a damped driven oscillator when ω = ω 0, that is, when the driving frequency is equal to the natural frequency of the system. When this occurs, the denominator of the amplitude is minimized, which maximizes the amplitude of the oscillations.