Antiresonance

Last updated April 03, 2023

In the physics of coupled oscillators, antiresonance, by analogy with resonance, is a pronounced minimum in the amplitude of an oscillator at a particular frequency, accompanied by a large, abrupt shift in its oscillation phase. Such frequencies are known as the system's antiresonant frequencies, and at these frequencies the oscillation amplitude can drop to almost zero. Antiresonances are caused by destructive interference, for example between an external driving force and interaction with another oscillator.

Antiresonance in electrical engineering

In electrical engineering, antiresonance is the condition for which the reactance vanishes and the impedance of an electrical circuit is very high, approaching infinity.

In an electric circuit consisting of a capacitor and an inductor in parallel, antiresonance occurs when the alternating current line voltage and the resultant current are in phase.^[1] Under these conditions the line current is very small because of the high electrical impedance of the parallel circuit at antiresonance. The branch currents are almost equal in magnitude and opposite in phase.^[2]

Antiresonance in coupled oscillators

The simplest system in which antiresonance arises is a system of coupled harmonic oscillators, for example pendula or RLC circuits.

Consider two harmonic oscillators coupled together with strength $g$ and with one oscillator driven by an oscillating external force $F$ . The situation is described by the coupled ordinary differential equations

{\begin{aligned}{\ddot {x}}_{1}+2\gamma _{1}{\dot {x}}_{1}-2g\omega _{1}x_{2}+\omega _{1}^{2}x_{1}&=2F\cos \omega t\\{\ddot {x}}_{2}+2\gamma _{2}{\dot {x}}_{2}-2g\omega _{2}x_{1}+\omega _{2}^{2}x_{2}&=0\end{aligned}}

where the $ω i$ represent the resonance frequencies of the two oscillators and the $γ i$ their damping rates. Changing variables to the complex parameters:

{\begin{aligned}\alpha _{1}&=\omega _{1}x_{1}+i{\frac {p_{1}}{m_{1}}}\\\alpha _{2}&=\omega _{2}x_{2}+i{\frac {p_{2}}{m_{1}}}\end{aligned}}

allows us to write these as first-order equations:

{\begin{aligned}{\dot {\alpha }}_{1}&=i\omega _{1}\alpha _{1}-\gamma _{1}(\alpha _{1}-\alpha _{1}^{*})-ig{\tfrac {\omega _{1}}{\omega _{2}}}(\alpha _{2}+\alpha _{2}^{*})+iF(e^{i\omega t}+e^{-i\omega t})\\{\dot {\alpha }}_{2}&=i\omega _{2}\alpha _{2}-\gamma _{2}(\alpha _{2}-\alpha _{2}^{*})-ig{\tfrac {\omega _{2}}{\omega _{1}}}(\alpha _{1}+\alpha _{1}^{*})\end{aligned}}

We transform to a frame rotating at the driving frequency

\alpha _{i}\rightarrow \alpha _{i}e^{-i\omega t}

yielding

{\begin{aligned}{\dot {\alpha }}_{1}&=i\Delta _{1}\alpha _{1}-\gamma _{1}(\alpha _{1}-\alpha _{1}^{*}e^{2i\omega t})-ig{\tfrac {\omega _{1}}{\omega _{2}}}(\alpha _{2}+\alpha _{2}^{*}e^{2i\omega t})+iF(1+e^{2i\omega t})\\{\dot {\alpha }}_{2}&=i\Delta _{2}\alpha _{2}-\gamma _{2}(\alpha _{2}-\alpha _{2}^{*}e^{2i\omega t})-ig{\tfrac {\omega _{2}}{\omega _{1}}}(\alpha _{1}+\alpha _{1}^{*}e^{2i\omega t})\end{aligned}}

where we have introduced the detunings $Δ i = ω - ω i$ between the drive and the oscillators' resonance frequencies. Finally, we make a rotating wave approximation, neglecting the fast counter-rotating terms proportional to $e 2 iωt$ , which average to zero over the timescales we are interested in (this approximation assumes that $ω + ω i ≫ ω - ω i$ , which is reasonable for small frequency ranges around the resonances). Thus we obtain:

{\begin{aligned}{\dot {\alpha }}_{1}&=i(\Delta _{1}+i\gamma _{1})\alpha _{1}-ig{\tfrac {\omega _{1}}{\omega _{2}}}\alpha _{2}+iF\\{\dot {\alpha }}_{2}&=i(\Delta _{2}+i\gamma _{2})\alpha _{2}-ig{\tfrac {\omega _{2}}{\omega _{1}}}\alpha _{1}\end{aligned}}

Without damping, driving or coupling, the solutions to these equations are:

\alpha _{i}(t)=\alpha _{i}(0)e^{i\Delta t}

which represent a rotation in the complex $α$ plane with angular frequency $Δ$ .

The steady-state solution can be found by setting ${\dot {\alpha }}_{1}={\dot {\alpha }}_{2}=0$ , which gives:

{\begin{aligned}\alpha _{1,ss}&={\frac {-F(\Delta _{2}+i\gamma _{2})}{(\Delta _{1}+i\gamma _{1})(\Delta _{2}+i\gamma _{2})-g^{2}}}\\\alpha _{2,ss}&={\frac {\omega _{2}}{\omega _{1}}}{\dfrac {-Fg}{(\Delta _{1}+i\gamma _{1})(\Delta _{2}+i\gamma _{2})-g^{2}}}\end{aligned}}

Examining these steady state solutions as a function of driving frequency, it is evident that both oscillators display resonances (peaks in amplitude accompanied by positive phase shifts) at the two normal mode frequencies. In addition, the driven oscillator displays a pronounced dip in amplitude between the normal modes which is accompanied by a negative phase shift. This is the antiresonance. Note that there is no antiresonance in the undriven oscillator's spectrum; although its amplitude has a minimum between the normal modes, there is no pronounced dip or negative phase shift.

Interpretation as destructive interference

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 remain almost stationary.

Complicated coupled systems

The frequency response function (FRF) of any linear dynamic system composed of many coupled components will in general display distinctive resonance-antiresonance behavior when driven.^[3]

As a rule of thumb, it can be stated that as the distance between the driven component and the measured component increases, the number of antiresonances in the FRF decreases.^[4] For example, in the two-oscillator situation above, the FRF of the undriven oscillator displayed no antiresonance. Resonances and antiresonances only alternate continuously in the FRF of the driven component itself.

Applications

An important result in the theory of antiresonances is that they can be interpreted as the resonances of the system fixed at the excitation point.^[4] This can be seen in the pendulum animation above: the steady-state antiresonant situation is the same as if the left pendulum were fixed and could not oscillate. An important corollary of this result is that the antiresonances of a system are independent of the properties of the driven oscillator; that is, they do not change if the resonance frequency or damping coefficient of the driven oscillator are altered.

This result makes antiresonances useful in characterizing complex coupled systems which cannot be easily separated into their constituent components. The resonance frequencies of the system depend on the properties of all components and their couplings, and are independent of which is driven. The antiresonances, on the other hand, are dependent upon everything except the component being driven, therefore providing information about how it affects the total system. By driving each component in turn, information about all of the individual subsystems can be obtained, despite the couplings between them. This technique has applications in mechanical engineering, structural analysis,^[5] and the design of integrated quantum circuits.^[6]

In electrical engineering antiresonance is used in wave traps, which are sometimes inserted in series with antennas of radio receivers to block the flow of alternating current at the frequency of an interfering station, while allowing other frequencies to pass.^[7]^[8]

In nanomechanical systems, the sideband spectra of a driven nonlinear mode with its eigenfrequency being modulated at a low frequency (<1 kHz) shows prominent antiresonance line shapes in the power spectra, which can be controlled through the vibration state. The antiresonance frequency can be utilized to characterize the thermal fluctuation and the squeezing parameter of the nonlinear system. ^[9]

Related Research Articles

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:

<span class="mw-page-title-main">Oscillation</span> Repetitive variation of some measure about a central value

Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms.

In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability.

The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:

In physics, the Rabi cycle is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of quantum computing, condensed matter, atomic and molecular physics, and nuclear and particle physics can be conveniently studied in terms of two-level quantum mechanical systems, and exhibit Rabi flopping when coupled to an oscillatory driving field. The effect is important in quantum optics, magnetic resonance and quantum computing, and is named after Isidor Isaac Rabi.

In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. These may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called (causal) Green's functions.

The Newmark-beta method is a method of numerical integration used to solve certain differential equations. It is widely used in numerical evaluation of the dynamic response of structures and solids such as in finite element analysis to model dynamic systems. The method is named after Nathan M. Newmark, former Professor of Civil Engineering at the University of Illinois at Urbana–Champaign, who developed it in 1959 for use in structural dynamics. The semi-discretized structural equation is a second order ordinary differential equation system,

The Havriliak–Negami relaxation is an empirical modification of the Debye relaxation model in electromagnetism. Unlike the Debye model, the Havriliak–Negami relaxation accounts for the asymmetry and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymers, by adding two exponential parameters to the Debye equation:

The Duffing equation, named after Georg Duffing (1861–1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by

In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).

In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a surface embedded in Euclidean space. It is named after French mathematician Jean Gaston Darboux.

The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

The multivariate stable distribution is a multivariate probability distribution that is a multivariate generalisation of the univariate stable distribution. The multivariate stable distribution defines linear relations between stable distribution marginals. In the same way as for the univariate case, the distribution is defined in terms of its characteristic function.

The Einstein–Hilbert action for general relativity was first formulated purely in terms of the space-time metric. To take the metric and affine connection as independent variables in the action principle was first considered by Palatini. It is called a first order formulation as the variables to vary over involve only up to first derivatives in the action and so doesn't overcomplicate the Euler–Lagrange equations with higher derivative terms. The tetradic Palatini action is another first-order formulation of the Einstein–Hilbert action in terms of a different pair of independent variables, known as frame fields and the spin connection. The use of frame fields and spin connections are essential in the formulation of a generally covariant fermionic action which couples fermions to gravity when added to the tetradic Palatini action.

The Elliott formula describes analytically, or with few adjustable parameters such as the dephasing constant, the light absorption or emission spectra of solids. It was originally derived by Roger James Elliott to describe linear absorption based on properties of a single electron–hole pair. The analysis can be extended to a many-body investigation with full predictive powers when all parameters are computed microscopically using, e.g., the semiconductor Bloch equations or the semiconductor luminescence equations.

Magnetic resonance is a quantum mechanical resonant effect that can appear when a magnetic dipole is exposed to a static magnetic field and perturbed with another, oscillating electromagnetic field. Due to the static field, the dipole can assume a number of discrete energy eigenstates, depending on the value of its angular momentum quantum number. The oscillating field can then make the dipole transit between its energy states with a certain probability and at a certain rate. The overall transition probability will depend on the field's frequency and the rate will depend on its amplitude. When the frequency of that field leads to the maximum possible transition probability between two states, a magnetic resonance has been achieved. In that case, the energy of the photons composing the oscillating field matches the energy difference between said states. If the dipole is tickled with a field oscillating far from resonance, it is unlikely to transition. That is analogous to other resonant effects, such as with the forced harmonic oscillator. The periodic transition between the different states is called Rabi cycle and the rate at which that happens is called Rabi frequency. The Rabi frequency should not be confused with the field's own frequency. Since many atomic nuclei species can behave as a magnetic dipole, this resonance technique is the basis of nuclear magnetic resonance, including nuclear magnetic resonance imaging and nuclear magnetic resonance spectroscopy.

Vasiliev equations are formally consistent gauge invariant nonlinear equations whose linearization over a specific vacuum solution describes free massless higher-spin fields on anti-de Sitter space. The Vasiliev equations are classical equations and no Lagrangian is known that starts from canonical two-derivative Frønsdal Lagrangian and is completed by interactions terms. There is a number of variations of Vasiliev equations that work in three, four and arbitrary number of space-time dimensions. Vasiliev's equations admit supersymmetric extensions with any number of super-symmetries and allow for Yang–Mills gaugings. Vasiliev's equations are background independent, the simplest exact solution being anti-de Sitter space. It is important to note that locality is not properly implemented and the equations give a solution of certain formal deformation procedure, which is difficult to map to field theory language. The higher-spin AdS/CFT correspondence is reviewed in Higher-spin theory article.

Cavity optomechanics is a branch of physics which focuses on the interaction between light and mechanical objects on low-energy scales. It is a cross field of optics, quantum optics, solid-state physics and materials science. The motivation for research on cavity optomechanics comes from fundamental effects of quantum theory and gravity, as well as technological applications.

This article summarizes several identities in exterior calculus.

References

↑ Kinsler, Lawrence E.; et al. (1999). Fundamentals of Acoustics (4th hardcover ed.). Wiley. p. 46. ISBN 0-471-84789-5.
↑ Balanis, Constantine A. (2005). Antenna Theory: Analysis and Design (3rd hardcover ed.). Wiley Interscience. p. 195. ISBN 0-471-66782-X.
↑ Ewins, D. J. (1984). Modal Testing: Theory and Practice. New York: Wiley.
1 2 Wahl, F.; Schmidt, G.; Forrai, L. (1999). "On the significance of antiresonance frequencies in experimental structural analysis". Journal of Sound and Vibration. 219 (3): 379. Bibcode:1999JSV...219..379W. doi:10.1006/jsvi.1998.1831.
↑ Sjövall, P.; Abrahamsson, T. (2008). "Substructure system identification from coupled system test data". Mechanical Systems and Signal Processing. 22: 15. Bibcode:2008MSSP...22...15S. doi:10.1016/j.ymssp.2007.06.003.
↑ Sames, C.; Chibani, H.; Hamsen, C.; Altin, P. A.; Wilk, T.; Rempe, G. (2014). "Antiresonance phase shift in strongly coupled cavity QED". Physical Review Letters. 112: 043601. arXiv: 1309.2228 . Bibcode:2014PhRvL.112d3601S. doi:10.1103/PhysRevLett.112.043601. PMID 24580448.
↑ Pozar, David M. (2004). Microwave Engineering (hardcover ed.). Wiley. p. 275. ISBN 0-471-44878-8.
↑ Sayre, Cotter W. (2008). Complete Wireless Design (2nd hardcover ed.). McGraw-Hill Professional. p. 4. ISBN 0-07-154452-6.
↑ Yang, Fan; Fu, Mengqi; Bosnjak, Bojan; Blick, Robert H.; Jiang, Yuxuan; Scheer, Elke (26 October 2021). "Mechanically Modulated Sideband and Squeezing Effects of Membrane Resonators". Physical Review Letters. 127 (18): 184301. arXiv: 2107.10355 . doi:10.1103/PhysRevLett.127.184301.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Kinsler, Lawrence E.; et al. (1999). Fundamentals of Acoustics (4th hardcover ed.). Wiley. p. 46. ISBN 0-471-84789-5.

[2] Balanis, Constantine A. (2005). Antenna Theory: Analysis and Design (3rd hardcover ed.). Wiley Interscience. p. 195. ISBN 0-471-66782-X.

[3] Ewins, D. J. (1984). Modal Testing: Theory and Practice. New York: Wiley.

[Wahl1999-4] 1 2 Wahl, F.; Schmidt, G.; Forrai, L. (1999). "On the significance of antiresonance frequencies in experimental structural analysis". Journal of Sound and Vibration. 219 (3): 379. Bibcode:1999JSV...219..379W. doi:10.1006/jsvi.1998.1831.

[5] Sjövall, P.; Abrahamsson, T. (2008). "Substructure system identification from coupled system test data". Mechanical Systems and Signal Processing. 22: 15. Bibcode:2008MSSP...22...15S. doi:10.1016/j.ymssp.2007.06.003.

[6] Sames, C.; Chibani, H.; Hamsen, C.; Altin, P. A.; Wilk, T.; Rempe, G. (2014). "Antiresonance phase shift in strongly coupled cavity QED". Physical Review Letters. 112: 043601. arXiv: 1309.2228 . Bibcode:2014PhRvL.112d3601S. doi:10.1103/PhysRevLett.112.043601. PMID 24580448.

[7] Pozar, David M. (2004). Microwave Engineering (hardcover ed.). Wiley. p. 275. ISBN 0-471-44878-8.

[8] Sayre, Cotter W. (2008). Complete Wireless Design (2nd hardcover ed.). McGraw-Hill Professional. p. 4. ISBN 0-07-154452-6.

[9] Yang, Fan; Fu, Mengqi; Bosnjak, Bojan; Blick, Robert H.; Jiang, Yuxuan; Scheer, Elke (26 October 2021). "Mechanically Modulated Sideband and Squeezing Effects of Membrane Resonators". Physical Review Letters. 127 (18): 184301. arXiv: 2107.10355 . doi:10.1103/PhysRevLett.127.184301.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]