2018-10-24

Contents

  • Oscillations
  • Waves
  • Application of Fourier Ideas
  • Optics
  • Interference of light

Canonical form of Damped SHM

\ddot{x}+\gamma\dot{x}+{\omega_0} ^2x=0

  • the spontaneous frequency
    \omega_0 = \sqrt {\frac{k}{m}}

  • the damping coefficient
    \gamma = \frac{b}{m}

  • Quality factor: the number of radians of phase elapsed as the amplitude falls to e^{-\frac{1}{2}}
    Q=\frac{\omega_0}{\gamma}

  • the system is critically damped when
    \gamma = 2 \omega_0

Light damping Q>0.5

  • damped oscillation frequency
    \omega_f = \sqrt {{\omega_0} ^2 - \frac{\gamma^2}{4}} = \omega_0 \sqrt {1- \frac{Q^2}{4}}

Heavy damping Q<0.5

Critical damping Q=0.5

  • Notice the corresponding solution forms of three different cases

Driven harmonic oscillation

\ddot{x}+\gamma\dot{x}+{\omega_0} ^2x=\frac{F}{m}

  • The response function (1. obtain the harmonic x solution; 2. divide solution by F). can naturally devise the velocity and acceleration response function
    R(\omega) = \frac{1}{m[({\omega_0}^2 - {\omega}^2) + i\gamma \omega]} = \frac{{(\omega_0}^2 - {\omega}^2) - i\gamma \omega}{m[({\omega_0}^2 - {\omega}^2)^2 + \gamma ^2 \omega ^2]}

  • resonance frequency (differentiate to get)
    \omega_a = \omega_0 \sqrt {1 - \frac{\gamma^2}{2 {\omega_0} ^2}} = \omega_0 \sqrt {1 - \frac{1}{2Q^2}}

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