A broad class of mechanical and physical systems is governed by linear differential equations with coefficients that vary periodically in time or space. Such periodic modulation gives rise to spectral lock-in regions that govern the system’s response. These lock-in regimes arise, for example, as parametric instabilities in dynamical systems and as band gaps in waves propagating through periodic media. Despite their ubiquity, these spectral mechanisms remain largely unexplored and unexploited in mechanical engineering. In the first part of this talk, I will introduce the concept of spectral lock-in at the heart of the stability analysis of Floquet systems. I will show how it can be exploited to enhance parametric instabilities and to enrich buckling patterns in elastic structures subjected to periodically varying compressive stresses. In the second part, I will present a class of systems with periodic coefficients that exhibit discrete spectra directly analogous to those of the stationary Schrödinger equation. I will demonstrate how these ideas can be leveraged to design new quantum-inspired functionalities in engineering systems, including the dynamical stabilization of mechanical oscillators and the control of wave propagation in periodic media.