= Normal modes of three freely ended coupled pendulums
{title2=$\omega^2=g/L+(k/m)\lambda,\quad\lambda=0,1,3$}
For three equal <simple pendulums> in a straight open chain with springs only between neighbours, the spring stiffness matrix in displacement coordinates is $k\begin{pmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{pmatrix}$. Its <eigenvectors> $(1,1,1)$, $(1,0,-1)$ and $(1,-2,1)$ have respective dimensionless <eigenvalues> $0,1,3$. Adding the gravitational restoring term yields the displayed squared <eigenfrequencies>. These are free spring ends: fixed wall springs would change the matrix and mode shapes.
Back to article page
