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Normal modes, 2 masses, 2 springs

  1. Jul 20, 2007 #1
    1. The problem statement, all variables and given/known data
    hey, this is my first time using this forum and was wondering if i could have some help with this normal modes question.

    suspended from a ceiling is in order: a spring of spring constant ([tex]k_{2}[/tex]), a mass (m), a spring with spring constant ([tex]k_{1}[/tex]), and another mass (m)

    calulate the normal mode frequencies [tex]\alpha_{1}[/tex], [tex]\alpha_{1}[/tex] sorry can't find the omega symbol.

    attempted solution:
    let the displacement of the higher mass be x and the displacement of the lower mass be y.

    m[tex]\ddot{x}[/tex]=-[tex]k_{2}[/tex]x + [tex]k_{1}[/tex](y-x)

    [tex]\left| \alpha^{2} - (\frac{k_{2}}{m} + \frac{k_{1}}{m}) -----\frac{k_{1}}{m} \right|[/tex]
    [tex]\left| \frac{k_{1}}{m} -------- \alpha^{2} - \frac{k_{1}}{m} \right|[/tex] = 0

    determinant = 0

    so [tex]\alpha^{2}[/tex] = 1/2[ ([tex]\frac{k_{2}}{m}[/tex]+ [tex]\frac{2k_{1}}{m}[/tex]) +- sqrt[([tex]\frac{k_{2}}{m}[/tex]+ [tex]\frac{2k_{1}}{m}[/tex])^2 – 4([tex]\frac{k_{2}*k_{1}}{m^2}[/tex])]]

    after simplifying it doesn't work when I use to try and solve for the amplitude ratios.
    please advise, thanks.
    p.s. is there a simpler equation writer i could download and use here or should i just persevere and try to use this embedded one?
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
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