Mixed-axis trajectory of a prestressed spring pair ($X/ϵ=\cos (\sqrt{2}{\omega }_{y}t),Y/ϵ=\cos ({\omega }_{y}t)$)

Two equal masses linked in a line by three identical springs, each stretched to twice its natural length, have longitudinal stiffness matrix $k\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$ and transverse matrix half as large. Equal displacements of both masses select the symmetric modes. Equal longitudinal and transverse release amplitudes produce the title trajectory. It is not closed after one transverse period because the two mode frequencies have irrational ratio $\sqrt{2}$. Higher-order geometric terms are neglected in this small-displacement solution.