antiresonance - wikipedia, the free encyclopedia

10/17/2014 Antiresonance - Wikipedia, the free encyclopedia

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Steady-state amplitude and phase of two coupledharmonic oscillators as a function of frequency.

AntiresonanceFrom Wikipedia, the free encyclopedia

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

Antiresonances can occur in all types of coupled oscillator systems, including mechanical, acoustic,electromagnetic and quantum systems. They have important applications in the characterization of complexcoupled systems.

Contents

1 Antiresonance in coupled oscillators2 Interpretation as destructive interference3 Complex coupled systems4 Applications5 Other uses6 See also7 References

Antiresonance in coupled oscillators

The simplest system in which antiresonance arises is asystem of coupled harmonic oscillators, for examplependula or RLC circuits.

Consider two harmonic oscillators coupled togetherwith strength and with one oscillator driven by anoscillating external force . The situation is describedby the coupled ordinary differential equations

where the represent the resonance frequencies of the two oscillators and the their damping rates.Changing variables to the complex parameters , allowsus to write these as first-order equations:



Animation showing time evolution to theantiresonant steady-state of two coupledpendula. The red arrow represents adriving force acting on the left pendulum.

We transform to a frame rotating at the driving frequency , yielding

where we have introduced the detunings between the drive and the oscillators' resonancefrequencies. Finally, we make a rotating wave approximation, neglecting the fast counter-rotating termsproportional to , which average to zero over the timescales we are interested in (this approximationassumes that , which is reasonable for small frequency ranges around the resonances).Thus we obtain:

Without damping, driving or coupling, the solutions to these equations are , whichrepresent a rotation in the complex plane with angular frequency .

The steady-state solution can be found by setting , which gives:

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

Interpretation as destructive interference

The reduced oscillation amplitude at an antiresonance can beregarded as due to destructive interference or cancellation offorces acting on the oscillator.

In the above example, at the antiresonance frequency theexternal driving force acting on oscillator 1 cancels the forceacting via the coupling to oscillator 2, causing oscillator 1 toremain almost stationary.

Complex coupled systems

The frequency response function (FRF) of any linear dynamicsystem composed of many coupled components will in generaldisplay distinctive resonance-antiresonance behavior whendriven.[1]

As a rule of thumb, it can be stated that as the distance between the driven component and the measuredcomponent increases, the number of antiresonances in the FRF decreases.[2] For example, in the two-oscillator situation above, the FRF of the undriven oscillator displayed no antiresonance. Resonances andantiresonances only alternate continuously in the FRF of the driven component itself.

Applications



Example frequency-response function of adynamical system with several degrees of freedom,showing distinct resonance-antiresonance behaviorin both amplitude and phase.

An important result in the theory of antiresonances is that they can be interpreted as the resonances of thesystem fixed at the excitation point.[2] This can be seen in the pendulum animation above: the steady-stateantiresonant situation is the same as if the left pendulum were fixed and could not oscillate. An importantcorollary of this result is that the antiresonances of a system are independent of the properties of the drivenoscillator; i.e. they do not change if the resonance frequency or damping coefficient of the driven oscillatorare altered.

This result makes antiresonances useful in characterizing complex coupled systems which cannot be easilyseparated into their constituent components. Theresonance frequencies of the system depend on theproperties of all components and their couplings, andare independent of which is driven. The antiresonances,on the other hand, are independent of the componentbeing driven, therefore providing information abouthow it affects the total system. By driving eachcomponent in turn, information about all of theindividual subsystems can be obtained, despite thecouplings between them. This technique hasapplications in mechanical engineering, structuralanalysis,[3] and the design of integrated quantumcircuits.[4]

Other uses

In electrical engineering, the word antiresonance mayalso be used to refer to the condition for which theimpedance 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 thealternating current line voltage and the resultant current are in phase.[5] Under these conditions the linecurrent is very small because of the high electrical impedance of the parallel circuit at antiresonance. Thebranch currents are almost equal in magnitude and opposite in phase.[6]

The principle of antiresonance is used in wave traps, which are sometimes inserted in series with antennasof radio receivers to block the flow of alternating current at the frequency of an interfering station, whileallowing other frequencies to pass.[7][8]

See also

ResonanceOscillatorResonance (alternating-current circuits)Tuned mass damper

References

1. ^ Ewins, D. J. (1984). Modal Testing: Theory and Practice. New York: Wiley.

2. ^ a b Wahl, F.; Schmidt, G.; Forrai, L. (1999). "On the significance of antiresonance frequencies in experimentalstructural analysis". Journal of Sound and Vibration 219 (3): 379. Bibcode:1999JSV...219..379W(http://adsabs.harvard.edu/abs/1999JSV...219..379W). doi:10.1006/jsvi.1998.1831(http://dx.doi.org/10.1006%2Fjsvi.1998.1831).

3. ^ Sjvall, P.; Abrahamsson, T. (2008). "Substructure system identification from coupled system test data".Mechanical Systems and Signal Processing 22: 15. Bibcode:2008MSSP...22...15S(http://adsabs.harvard.edu/abs/2008MSSP...22...15S). doi:10.1016/j.ymssp.2007.06.003(http://dx.doi.org/10.1016%2Fj.ymssp.2007.06.003).

4. ^ Sames, C.; Chibani, H.; Hamsen, C.; Altin, P. A.; Wilk, T.; Rempe, G. (2014). "Antiresonance Phase Shift inStrongly Coupled Cavity QED". Physical Review Letters 112: 043601. arXiv:1309.2228



(https://arxiv.org/abs/1309.2228). Bibcode:2014PhRvL.112d3601S(http://adsabs.harvard.edu/abs/2014PhRvL.112d3601S). doi:10.1103/PhysRevLett.112.043601(http://dx.doi.org/10.1103%2FPhysRevLett.112.043601).

5. ^ Kinsler, Lawrence E. et al. - Fundamentals of Acoustics - Wiley, 4 ed, Hardcover, ISBN 0-471-84789-5, 1999,p46

6. ^ Balanis, Constantine A. - Antenna Theory: Analysis and Design - Wiley-Interscience; 3 ed, Hardcover, ISBN0-471-66782-X, 2005, p195

7. ^ Pozar, David M. - Microwave Engineering - Wiley, Hardcover, ISBN 0-471-44878-8, 2004 p2758. ^ Sayre, Cotter W. - Complete Wireless Design - McGraw-Hill Professional; 2 ed, Hardcover, ISBN 0-07-

154452-6, 2008, p4

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