page_title: Mechanical amplifier
section_title: Parametric resonators
hierarchical_section_title: Mechanical amplifier / Resonating amplifiers / Parametric resonators
caption_attribution_description: A girl on a garden swing. Original caption: Bincadeira de criança eu também me acabei nesse balanço...
context_page_description: A mechanical amplifier, or a mechanical amplifying element, is a linkage mechanism that amplifies the magnitude of mechanical quantities such as force, displacement, velocity, acceleration and torque in linear and rotational systems. In some applications, mechanical amplification induced by nature or unintentional oversights in man-made designs can be disastrous. When employed appropriately, it can help to magnify small mechanical signals for practical applications.
  No additional energy can be created from any given mechanical amplifier due to conservation of energy. Claims of using mechanical amplifiers for perpetual motion machines are false, due to either a lack of understanding of the working mechanism or a simple hoax.
context_section_description: Parametric resonance is the physical phenomenon where an external excitation, at a specific frequency and typically orthogonal to the plane of displacement, introduces a periodic modulation in one of the system parameters resulting in a buildup in oscillatory amplitude. It is governed by the Mathieu equation. The following is a damped Mathieu equation.
  where δ is the squared of the natural frequency and ε is the amplitude of the parametric excitation.
  The first order or the principal parametric resonance is achieved when the driving/excitation frequency is twice the natural frequency of a given system. Higher orders of parametric resonance are observed either at or at submultiples of the natural frequency. For direct resonance, the response frequency always matches the excitation frequency. However, regardless of which order of parametric resonance is activated, the response frequency of parametric resonance is always in the vicinity of the natural frequency. Parametric resonance has the ability to exhibit higher mechanical amplification than direct resonance when operating at favourable conditions, but usually has a longer build up/transient state.
  “The parametric resonator provides a very useful instrument that has been developed by a number of researchers, in part because a parametric resonator can serve as a mechanical amplifier, over a narrow band of frequencies.”
caption: Swing is essentially a pendulum, which can be driven into either direct or parametric resonance depending on the nature of excitation and boundary conditions.
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