Damping conditions for multi degrees of freedom

[gopi9]

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.

 

[ShayanJ]

You should specify the configuration exactly and then we can write the equations of motion!

 

[AlephZero]

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 $\omega$).