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Resonance is produced depending on the boundary conditions imposed on a wave. Resonance can be produced in a string under tension with symmetrical boundary conditions (i.e., has a node at each end). A node is defined as a fixed point where the string does not move. The symmetrical boundary conditions result in some frequencies resonating and producing standing waves, while other frequencies interfere destructively. Sound waves can resonate in a hollow tube, and the frequencies of the sound...
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If a driven oscillator needs to resonate at a specific frequency, then very light damping is required. An example of light damping includes playing piano strings and many other musical instruments. Conversely, to achieve small-amplitude oscillations as in a car's suspension system, heavy damping is required. Heavy damping reduces the amplitude, but the tradeoff is that the system responds at more frequencies. Speed bumps and gravel roads prove that even a car's suspension system is not...
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The parallel RLC circuit is an arrangement where the resistor (R), inductor (L), and capacitor (C) are all connected to the same nodes and, as a result, share the same voltage across them. The parallel RLC circuit is analyzed in terms of admittance (Y), which reflects the ease with which current can flow. The admittance is given by:
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Series resonance occurs in a circuit containing inductive (L), capacitive (C), and resistive (R) elements connected sequentially. At the resonance frequency, the inductive and capacitive reactances are equal in magnitude but opposite in sign, effectively canceling each other. This causes the circuit's impedance is minimal, primarily determined by the resistance R. The resonant frequency of an RLC circuit is defined as:
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Comment on "Slow passage through resonance".

Claire Bourquard1, Nicolas Noiray1

  • 1CAPS Laboratory, Department of Mechanical and Process Engineering, ETH Zürich, Switzerland.

Physical Review. E
|November 28, 2019
PubMed
Summary

A study on damped harmonic oscillators with frequency chirps found resonance occurs when the instantaneous frequency equals the natural frequency, not midway as previously stated. This clarifies critical frequency ramp rates for amplitude and frequency measurements.

Area of Science:

  • Physics
  • Mechanical Engineering
  • Nonlinear Dynamics

Background:

  • A previous study by Park et al. proposed a condition for resonance onset in linearly damped harmonic oscillators with linear frequency chirps.
  • The prior work suggested resonance occurs when the forcing frequency is midway between the initial and natural frequencies.

Purpose of the Study:

  • To correct the resonance condition for a linearly damped harmonic oscillator under a linear frequency chirp.
  • To clarify the relationship between instantaneous frequency and resonance in driven harmonic oscillators.
  • To differentiate critical frequency ramp rates for various resonance measurements.

Main Methods:

  • Numerical analysis of oscillator response to a frequency-chirped forcing function.
  • Analytical derivation of the resonance condition.

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  • Comparison of theoretical predictions with simulation results.

    • Main Results:

    • The resonance condition stated by Park et al. was found to be inaccurate.
    • Resonance in this system occurs when the instantaneous forcing frequency approaches the natural frequency (ωn).
    • The instantaneous frequency ramps twice as fast as previously reported, leading to an earlier resonance onset.

    Conclusions:

    • The critical frequency for resonance onset is the natural frequency, not the average of initial and natural frequencies.
    • Distinctions exist between critical ramp rates for measuring resonance amplitude versus resonance frequency and damping.
    • This work provides a more accurate understanding of resonance phenomena in time-varying frequency systems.