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The following implementation in the Python programming language works for complex arguments and typically gives 13 correct decimal places. Note that omitting the smallest coefficients (in pursuit of speed, for example) gives totally inaccurate results; the coefficients must be recomputed from scratch for an expansion with fewer terms.
A familiar example of parametric oscillation is "pumping" on a playground swing. [ 4 ] [ 5 ] [ 6 ] A person on a moving swing can increase the amplitude of the swing's oscillations without any external drive force (pushes) being applied, by changing the moment of inertia of the swing by rocking back and forth ("pumping") or alternately standing ...
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 ...
For the case of flutter analysis, lift history for the motion obtained from the Wagner analysis (Herbert A. Wagner) with varying frequency of oscillation shows that magnitude of lift decreases and a phase lag develops between the aircraft motion and the unsteady aerodynamic forces. Reduced frequency can be used to explain the amplitude ...
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.
A mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system tends to return to its equilibrium position, but overshoots it. Sometimes losses (e.g. frictional) damp the system and can cause the oscillations to gradually decay in amplitude towards zero or attenuate. The damping ratio ...
The reduced oscillation amplitude at an antiresonance can be regarded as due to destructive interference or cancellation of forces acting on the oscillator. In the above example, at the antiresonance frequency the external driving force F acting on oscillator 1 cancels the force acting via the coupling to oscillator 2, causing oscillator 1 to ...
One particle: N particles: One dimension ^ = ^ + = + ^ = = ^ + (,,) = = + (,,) where the position of particle n is x n. = + = = +. (,) = /.There is a further restriction — the solution must not grow at infinity, so that it has either a finite L 2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): [1] ‖ ‖ = | |.