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Damping conditions for multi degrees of freedom

  1. Dec 26, 2013 #1
    I know that d^2<4mk for underdamped, d^2>4mk for overdamped and d^2=4mk for critically damped. This is true if there is only 1 mass and spring and damper. How to use these equations if I have 2 mass, 3 spring and 3 dampers. That is d,m,k are in 2x2 matrices. Please some one help me with this.
     
  2. jcsd
  3. Dec 26, 2013 #2

    ShayanJ

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    Gold Member

    You should specify the configuration exactly and then we can write the equations of motion!
     
  4. Dec 26, 2013 #3

    AlephZero

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    In the most general case, the damped vibration modes of the system are the eigenvalues and vectors of the quadratic eigenproblem ##(m + \lambda d + \lambda^2k)x = 0##. The solutions will be of the form ##x(t) = X e^{(-\sigma + i \omega) t} ## where ##X## is a complex eigenvector, ##-\sigma + i \omega## is the complex eigenvalue, and the physical solution ##x(t)## is the real part of the right hand side of the equation. ##\sigma## represents the amount of damping, and ##\omega## the damped oscillation frequency (which is 0 for an overdamped system)

    If you get lucky, the eigenvectors will be real, and the same as the eigenvectors of the undamped system ##(m - \omega_u^2 k)x = 0## (but of course the eigenvalues will be different, and the undamped frequency ##\omega_u## is not equal to the damped frequency ##\omega##).
     
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