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Characteristic vibration frequencies of a system

  1. Feb 27, 2012 #1

    fluidistic

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    1. The problem statement, all variables and given/known data
    I'm trying to solve an example in Boas's book (Mathematical methods in the physical sience page 423) but by using Goldstein's (Classical mechanics) method.
    The mechanical system consists of 3 springs (with constants k) and 2 masses (with mass m) this way: |----m----m----|, where | are walls.


    2. Relevant equations and approach
    Find kinetic and potential energy. Then write both of them in terms of matrices. Then the characteristic frequencies are calculated from [itex]|V-\omega ^2 T |=0[/itex].


    3. The attempt at a solution
    Like Boas found, if [itex]x_1[/itex] and [itex]x_2[/itex] are the displacement of the masses from their equilibrium position, then [itex]T=\frac{m}{2}(\dot x ^2 + \dot y^2 )[/itex]. And [itex]T=k(x^2+y^2-xy)[/itex].
    In order to find the matrices T and V, I write [itex]T=\begin{pmatrix} \dot x_1 & \dot x_2 \end{pmatrix} \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} \dot x_1 \\ \dot x_2 \end{pmatrix}[/itex]. This gives me [itex]A=D=\frac{m}{2}[/itex] and [itex]C=D=0[/itex].
    I do a similar algebra for V, namely [itex]V=\begin{pmatrix} x_1 & x_2 \end{pmatrix} \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}[/itex]. This gives me [itex]A=D=k[/itex] while [itex]C+B=-1[/itex]. Here I am not 100% sure but I think that since I'm using normal coordinates (still not 100% sure I'm really doing it), C must equal B. Assuming this I get [itex]C=B=-1/2[/itex].
    I calculated [itex]|V-\omega ^2 T |=0[/itex], this gave me [tex]\left ( k-\frac{m\omega ^2}{2}-\frac{1}{2} \right )\left ( k-\frac{m\omega ^2}{2}+\frac{1}{2} \right )=0[/tex].
    So that either [itex]\omega = \pm \sqrt {\frac{2k-1}{m}}[/itex] or [itex]\omega = \pm \sqrt {\frac{2k+1}{m}}[/itex]. Therefore I get 4 possible frequencies for the characteristic frequencies. I know I should only get 2 though, so something is already wrong. Looking at Boas, he/she found [itex]\omega _1 =\sqrt {\frac{k}{m}}[/itex] and [itex]\omega _2 =\sqrt {\frac{3k}{m}}[/itex].
    I'm desperate in knowing where is/are my error(s).
     
    Last edited: Feb 27, 2012
  2. jcsd
  3. Feb 27, 2012 #2

    fzero

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    You should find that [itex]B=C=k/2[/itex]. Also [itex]\omega<0[/itex] doesn't lead to a new solution (your basis functions are appropriately odd or even).
     
  4. Feb 27, 2012 #3

    fluidistic

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    Thanks for pointing this out. I realize I forgot a factor "k".
    So now I find the solutions provided by the book and thank you for your comment about a negative omega, I didn't know this.
     
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