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Math Help - Simple Harmonic Motion Derivation of frequency

  1. #1
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    Simple Harmonic Motion Derivation of frequency

    Ok i know how to deduce the angular frequency for the basic problem of:
    \frac{d^{2}x}{dt^{2}} = -\frac{kx}{m}
    You simple do:
    x = -\alpha cos(\omega t) + \beta sin(\omega t)
    x ' =...
    x'' = ...

    Hence  x'' = -\omega^{2}x
    So \omega = \sqrt{\frac{k}{m}}

    But how do i do the same for the equation:
    \frac{d^{2}x}{dt^{2}} = -\frac{kx}{m} - \frac{b}{m}\frac{dx}{dt} + \frac{F_{0}}{m}Cos(\alpha t)
    I know it should be:

    \omega = \sqrt{\frac{k}{m}-b^{2}/4m^{2}}

    However, with the damping expression i don't think i'll be able to use the same approach

    any help is much appreciated
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by imagemania View Post
    Ok i know how to deduce the angular frequency for the basic problem of:
    \frac{d^{2}x}{dt^{2}} = -\frac{kx}{m}
    You simple do:
    x = -\alpha cos(\omega t) + \beta sin(\omega t)
    x ' =...
    x'' = ...

    Hence  x'' = -\omega^{2}x
    So \omega = \sqrt{\frac{k}{m}}

    But how do i do the same for the equation:
    \frac{d^{2}x}{dt^{2}} = -\frac{kx}{m} - \frac{b}{m}\frac{dx}{dt} + \frac{F_{0}}{m}Cos(\alpha t)
    I know it should be:

    \omega = \sqrt{\frac{k}{m}-b^{2}/4m^{2}}

    However, with the damping expression i don't think i'll be able to use the same approach

    any help is much appreciated
    The solution at large times is a sinusoid at the driving frequency.

    CB
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