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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Antiresonance ： ウィキペディア英語版 Antiresonance In the physics of coupled oscillators, antiresonance, by analogy with resonance, is a pronounced minimum in the amplitude of one oscillator at a particular frequency, accompanied by a large 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. 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 complex coupled systems. ==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 :$$ \begin \ddot_1 + 2\gamma_1 \dot_1 - 2g \omega_1 x_2 + \omega_1^2 x_1 &=& 2F\cos\omega t \\ \ddot_2 + 2\gamma_2 \dot_2 - 2g \omega_2 x_1 + \omega_2^2 x_2 &=& 0 \end where the $\backslash omega\_i$ represent the resonance frequencies of the two oscillators and the $\backslash gamma\_i$ their damping rates. Changing variables to the complex parameters $\backslash alpha\_1\; =\; \backslash omega\_1\; x\_1\; +\; ip\_1/m\_1$, $\backslash alpha\_2\; =\; \backslash omega\_2\; x\_2\; +\; ip\_2/m\_1$ allows us to write these as first-order equations: :$$ \begin \dot_1 &=& i\omega_1 \alpha_1 - \gamma_1(\alpha_1 - \alpha_1^ *) - ig\tfrac(\alpha_2 + \alpha_2^ *) + iF(e^+e^) \\ \dot_2 &=& i\omega_2 \alpha_2 - \gamma_2(\alpha_2 - \alpha_2^ *) - ig\tfrac(\alpha_1 + \alpha_1^ *) \end We transform to a frame rotating at the driving frequency $\backslash alpha\_i\; \backslash rightarrow\; \backslash alpha\_i\; e^$, yielding :$$ \begin \dot_1 &=& i\Delta_1 \alpha_1 - \gamma_1(\alpha_1 - \alpha_1^ * e^) - ig\tfrac(\alpha_2 + \alpha_2^ * e^) + iF(1+e^) \\ \dot_2 &=& i\Delta_2 \alpha_2 - \gamma_2(\alpha_2 - \alpha_2^ * e^) - ig\tfrac(\alpha_1 + \alpha_1^ * e^) \end where we have introduced the detunings $\backslash Delta\_i\; =\; \backslash omega\; -\; \backslash omega\_i$ between the drive and the oscillators' resonance frequencies. Finally, we make a rotating wave approximation, neglecting the fast counter-rotating terms proportional to $e^$, which average to zero over the timescales we are interested in (this approximation assumes that $\backslash omega\; +\; \backslash omega\_i\; \backslash gg\; \backslash omega\; -\; \backslash omega\_i$, which is reasonable for small frequency ranges around the resonances). Thus we obtain: :$$ \begin \dot_1 &=& i (\Delta_1 + i\gamma_1) \alpha_1 - ig\tfrac\alpha_2 + iF \\ \dot_2 &=& i (\Delta_2 + i\gamma_2) \alpha_2 - ig\tfrac\alpha_1 \end Without damping, driving or coupling, the solutions to these equations are $\backslash alpha\_i(t)\; =\; \backslash alpha\_i(0)\; e^$, which represent a rotation in the complex $\backslash alpha$ plane with angular frequency $\backslash Delta$. The steady-state solution can be found by setting $\backslash dot\_1\; =\; \backslash dot\_2\; =\; 0$, which gives: :$$ \begin \alpha_ &=& \dfrac \\ \alpha_ &=& \dfrac\dfrac \end 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Antiresonance」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.