  This study establishes a non-Bloch band theory for time-modulated discrete
mechanical systems. We consider simple mass-spring chains whose stiffness is
periodically modulated in time. Using the temporal Floquet theory, the system
is characterized by linear algebraic equations in terms of Fourier
coefficients. This allows us to employ a standard linear eigenvalue analysis.
Unlike non-modulated linear systems, the time modulation makes the coefficient
matrix non-Hermitian, which gives rise to, for example, parametric resonance,
non-reciprocal wave transmission, and non-Hermitian skin effects. In
particular, we study finite-length chains consisting of spatially periodic
mass-spring units and show that the standard Bloch band theory is not valid for
estimating their eigenvalue distribution. To remedy this, we propose a
non-Bloch band theory based on a generalized Brillouin zone. The proposed
theory is verified by some numerical experiments.
