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I Finding eigenvalues for 4 DOF system

  1. Feb 15, 2017 #1
    Hey all,

    I've derived a fourth order dynamic system as represented by the following:
    Capture.jpg
    I need to determine the eigenvalues for this system to check whether they're purely real with no imaginary components.

    How should I go about doing this? I have done eigenvalue problems in the past, but not to this extent. Would I just determine the eigenvalues for each matrix then do the addition of them?

    Note: all the variables (excluding theta's and x's) have constant values.

    Any help would be appreciated, thanks!
     
  2. jcsd
  3. Feb 18, 2017 #2

    jambaugh

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    Here's how I would approach it, first introduce auxiliary variables and reduce the system to first order. For example replace [itex] \ddot{\theta}_2 \to \dot{\omega}_2[/itex] and [itex]\dot{\theta}_2 \to \omega[/itex] with the additional equation [itex]\dot{\theta}_2 = \omega_2[/itex]. You'll end up with an 8x8 system but it will be first order, of the form:

    [tex] \dot{\mathbf{u}} = A \mathbf{u} + \mathbf{b}[/tex]
    With [itex] \mathbf{u} = ( \theta_2,\theta_3, x_2, x_5, \omega_2,\omega_3, v_2,v_5)^T[/itex].

    You can then focus on finding or qualifying the eigen-values of the matrix [itex]A[/itex].
     
  4. Feb 22, 2017 #3
    Thank you for your help Jambaugh!
     
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