Programmable Modal Signatures in Mass-Spring Systems through Stiffness Asymmetry

Symmetry breaking in a passive mass-spring system with unequal spring constants reshapes the normal mode landscape. For systems with three or more masses (N ≥ 3), we demonstrate that adjusting the unequal spring constants {k_i} divides the stiffness-ratio simplex into regions. Each region is identified by a binary modal signature that encodes nodal parity and edge/localization characteristics. Using eigenvalue analysis with parameter continuation, we track how modal patterns change as the ratios {k_i} are varied. We identify modes that interchange the positions of mass oscillations, resulting in a partitioning of the parameter space into regions, each associated with a unique binary signature. Notably, small amounts of disorder and viscous damping preserve these signatures, suggesting a topology-like protection mechanism in a purely linear system. We also present a constructive synthesis rule: sort the stiffnesses to achieve a desired signature. This approach is validated through prototypes with 3 to 5 masses and molecular dynamics simulations, showcasing controlled switching and resilience to imperfections. These findings connect symmetry breaking to a topological classification without relying on periodicity or nonlinearity, enabling the development of compact, programmable mechanical filters, energy funneling mechanisms, and vibration isolation systems.