  Degeneracy points of parameter-dependent Hermitian matrices play a
fundamental role in quantum physics, as illustrated by the concept of Berry
phase in quantum dynamics, the Weyl semimetals in condensed-matter physics, and
the robust ground-state degeneracies in topologically ordered quantum systems.
Here, we construct simple ball-and-spring mechanical systems, whose
eigenfrequency degeneracies mimic the behaviour of degeneracy points of
electronic band structures. These classical-mechanical arrangements can be
viewed as de-quantized versions of Weyl Josephson circuits, i.e.,
superconducting nanostructures proposed recently to mimic band structure
effects of Weyl semimetals. In the mechanical setups we study, we identify
degeneracy patterns beyond simple Weyl points, including the chirality flip
effect and a quadratic degeneracy point. Our theoretical work is a step toward
simple and illustrative table-top experiments exploring topological and
differential geometrical aspects of physics.
