Normal mode

Remarks

The function $x(t) = A \cos(\omega t + \varphi) v$ is a solution if and only if $K v = \omega^2 M v$, so the mode shapes and the squared natural frequencies are the generalized eigenvectors and eigenvalues of the pair $(K, M)$. Each coordinate of a normal mode is a simple harmonic motion or is identically zero. If $K$ is positive definite, there are $n$ natural frequencies $0 < \omega_1 \le \cdots \le \omega_n$, counted with multiplicity, with mode shapes $v_1, \ldots, v_n$ that form a basis of $\R^n$ and satisfy $v_i^T M v_j = 0$ for $i \ne j$.

Example: two unit masses joined to walls and to each other by three springs of stiffness $1$: $M = I_2$ and $K = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$. For $v = (1, 1)$, $K v = v$, so $\omega = 1$; for $v = (1, -1)$, $K v = 3v$, so $\omega = \sqrt 3$.

Non-example: in the same system, $x(t) = \cos(t) \, (1, 0)$ is not a normal mode, since $K (1, 0)^T = (2, -1)^T$ is not a multiple of $(1, 0)^T$, so it is not even a solution.