PX285 - G6 - summary

a vector form of the euler-lagrange equation:

\begin{matrix} (1) & M \ddot{\vec{x}} = - K \vec{x} \end{matrix}

the inertia and the stiffness matrices:

\begin{array}{r} M_{i j} = \frac{\partial^{2} T}{\partial {\dot{x}}_{i} \partial {\dot{x}}_{j}} \\ K_{i j} = \frac{\partial^{2} V}{\partial x_{i} \partial x_{j}} \end{array}

seeking oscillatory solutions, the most general form is:

\vec{x} (t) = A \vec{X} e^{i ω t}

where, $\vec{X} = ({\vec{x}}_{i}, {\vec{x}}_{2}, \dots)$ , where ${\vec{x}}_{n}$ are the positions of individual masses

substituting this into equation $(1)$ , and solving it will reveal of number of modes that is equal to the number of coordinates, $λ$
the general solution is the sum of all the individual solutions (modes), which may not have the same frequency

\vec{x} (t) = \sum_{λ} A^{(λ)} {\vec{X}}^{(λ)} e^{i ω^{(λ)} t}

\begin{matrix} \dot{\vec{x}} = A \vec{X} = i ω A \vec{X} e^{i ω t} \\ \ddot{\vec{x}} = A \vec{X} = - ω^{2} A \vec{X} e^{i ω t} \equiv - ω^{2} x \end{matrix}

\begin{aligned} M (- ω^{2} \vec{x}) & = - K \vec{x}) \\ - ω^{2} M \vec{x} & = - K \vec{x} \\ (K - ω^{2} M) \vec{x} & = \vec{0} \\ A e^{i ω t} (K - ω^{2} M) \vec{X} & = \vec{0} \end{aligned}

$A e^{i ω t}$ is non-zero if $A \neq 0$

\begin{matrix} \underset{⏟}{(K - ω^{2} M)} \vec{X} = \vec{0} \\ B \vec{X} = 0 \end{matrix}

this has solutions when $det B = 0$ , ie: $det (K - ω^{2} M) = 0$
this is an equation, polynomial in $ω^{2}$ , and it is called the secular equation
it can be used to determine possible squared frequencies $ω^{2}$
$ω^{2}$ are the eigenvalues, and $\vec{X}$ are the eigenvectors