# Eigen-Modes for a homogenous, non-conservative mechanical system

$$M\,\ddot{y}+D\,\dot{y}+K\,y=0\\ y=\begin{bmatrix} y_1&y_2&\cdots&y_f \end{bmatrix}^\top\\ M=M^\top>0,\quad D=D^\top,\quad K=K^\top\geq0$$

Equations show a homogeneous non-conservative mechanical system for which I am trying to draw a sketch of eigen-modes. Could anyone please help?

• This is standard theory. Without some more constraints on D in particular, you can't say much except that the eigenvalues and vectors of the quadratic eigenproblem are all complex, the real part of an eigenvalue measures the amount of damping (or exponential growth, i.e. "negative damping") in that mode, and the imaginary part measures the frequency of the mode. In the general case, there is nothing to "draw" except a complex plane with a random set of dots on it! – alephzero Aug 9 at 18:17
• @alephzero For a general $D$ you can also look at the system matrix of the first order differential equation with the state vector $[y^\top \quad \dot{y}^\top]^\top$. If modes are under critically damped, then you will have pairs of complex conjugate eigenvalues and eigenvectors. The real and imaginary parts of these conjugate eigenvectors still have a physical meaning and can both be drawn separately. – fibonatic Aug 9 at 18:56
• Much thanks for your answer but could you please direct to me the image so that I can have more clarity ? – MechNovice Aug 10 at 7:31
• I am not sure what you mean, do you mean the image in your original question? – fibonatic Aug 10 at 9:04
• Thanks again for your answer. I mean the sketch of the eigen modes w.r.t given set of equations. – MechNovice Aug 10 at 9:18