= Normal modes of an anchored complete spring graph
{title2=$K=k[(n+1)I-\mathbf1\mathbf1^T]$}
Connect every pair among $n$ moving scalar coordinates and one fixed coordinate by identical <springs> of stiffness $k$. The coupling pattern is the <complete graph> $K_{n+1}$ with one vertex fixed. Its <potential energy> is $k\sum_{i<j}(X_i-X_j)^2/2$. The moving-coordinate <stiffness matrix> is the principal submatrix displayed in the header. For equal <masses> $m$, the uniform <normal mode> has squared <angular frequency> $k/m$, while every sum-zero vector has squared angular frequency $(n+1)k/m$. There are $n-1$ such degenerate modes. This follows directly from the <Laplacian spectrum of a complete graph> and its rank-one matrix form. Constant gravitational forces shift the <equilibrium> without changing the stiffness.
Back to article page
