2018-10-26

  • resonance frequency for acceleration response (R(\omega){\omega}^2)

\omega_{acc} = \omega_0 ( {1 - \frac{1}{2Q^2}} ) ^ {-\frac{1}{2}}

  • for completeness, listing the resonance frequency for the three quantities in question

\omega_{A} = \omega_0 ( {1 - \frac{1}{2Q^2}} ) ^ {\frac{1}{2}}

\omega_{velo} = \omega_0

\omega_{acc} = \omega_0 ( {1 - \frac{1}{2Q^2}} ) ^ {-\frac{1}{2}}

  • notice \omega_a \cdot \omega_{acc} = {\omega_0}^2 by comparing with the resonance frequency for displacement as discussed last time

Mean power: \frac{1}{2} |F_0||v_0|\cos ({\phi_F}-{\phi_v})

  • \mathcal{Re}\{\bf A\} \mathcal{Re}\{\bf B\} = \frac{1}{2} \mathcal{Re}\{ {\bf A} {\bf B^{\dagger}} \} is true when \bf A and \bf B have the same time dependence (e.g., both propagating forward/backward in time) in their complex power representation
  • \langle P_{dissipated} \rangle = \frac{1}{2} b{|v_0|}^2, where b=\gamma m

Power resonance and bandwidth

  • Consider the half power points, whose difference gives the bandwidth
    \Delta \omega = \omega_{+} - \omega_{-} = \gamma

Q = \frac{\omega_0}{\gamma} = \frac{\omega_0}{\Delta \omega}

  • this provides a second definition of the quality factor

Canonical form of the LCR circuit equation

\omega_0 = \frac{1}{\sqrt{LC}}

\gamma = \frac{R}{L}

Q = \frac{\omega_0}{\gamma} = \frac{1}{R}\sqrt \frac{L}{C}

\ddot q + \gamma \dot q + {\omega_0}^2 q = \frac {V} {L}

Impedance: Force divided by the velocity

Z = \frac{F}{v}

  • encapsulating both magnitude and phase shift information
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