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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 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). The behaviour of oscillating systems is often of interest in a diverse range of disciplines that include control engineering , chemical engineering , mechanical ...
Damped oscillation is a typical transient response, where the output value oscillates until finally reaching a steady-state value. In electrical engineering and mechanical engineering, a transient response is the response of a system to a change from an equilibrium or a steady state. The transient response is not necessarily tied to abrupt ...
The underdamped response is a decaying oscillation at frequency ω d. The oscillation decays at a rate determined by the attenuation α. The exponential in α describes the envelope of the oscillation. B 1 and B 2 (or B 3 and the phase shift φ in the second form) are arbitrary constants determined by boundary conditions. The frequency ω d is ...
The equation describes the motion of a damped oscillator with a more complex potential than in simple harmonic motion (which corresponds to the case = =); in physical terms, it models, for example, an elastic pendulum whose spring's stiffness does not exactly obey Hooke's law.
model damped unforced oscillations of a weight on a spring. The displacement will then be of the form () = / (). The constant T (= /) is called the relaxation time of the system and the constant μ is the quasi-frequency.
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 ...
Near the origin = =, the system is unstable, and far from the origin, the system is damped. The Van der Pol oscillator does not have an exact, analytic solution. [13] However, such a solution does exist for the limit cycle if f(x) in the Lienard equation is a constant piece-wise function.