Buckling in compressed elastic strips: an analogue to parametric instabilities

Parametric instabilities are a well studied feature of periodically driven dynamic systems—the canonical example being a pendulum with varying string length. In particular, when parameters of the system fall within a specific domain, any applied perturbation results in the same periodic limit cycle—a lock-in effect. While parametric instabilities arise in initial-value problems, here we realize an analogous effect within a boundary-value problem. We analytically study the compressed 1D elastic beam on the Winkler foundation (a foundation of elastic springs), and see that its ODE has clear parallels to systems which exhibit parametric instabilities. Using simulations and experiments, we also study this system's 3D analogue—a thin elastic strip compressed along its bottom edge. In our static system, the "driven periodic loading" is replicated through periodically modulated stiffness—either in the springs of the Winkler foundation or the height of the elastic strip. We find that the buckling patterns caused by the compression of these structures produces lock-in behavior, where we see pure sine waves of a single wavelength, rather than quasi-periodic combinations of sines. This wavelength lock-in phenomenon could be interesting for enhancing control in patterning techniques based on elastic buckling.