Finite-state stiffness programming under stroke and electrical-load constraints forvery-low-frequency electromagnetic energy harvesters

Abstract

Very-low-frequency vibration energy harvesters are constrained simultaneously by resonance detuning, finite electromechanical coupling, allowable stroke and retuning energy. This study asks how much structural stiffness programming adds once the electrical load is optimized under the same physical limits. Delivered power to a low-inductance electromagnetic load is used throughout, and the displacement limit is enforced directly in the steady-state dynamics; no post-hoc response clipping is used. For an arbitrary fixed stiffness state, an analytical optimum for non-negative external load resistance is derived together with a finite-coupling no-impact feasibility condition. Finite-state stiffness programming is then compared with fixed tuning, load programming, a resource-normalized passive resonator-array proxy and ideal continuous stiffness/load programming. In a high-resolution 0.5–4 Hz moderate benchmark, worst-case constrained coverage is 0.486 for fixed-stiffness load programming, 0.872 for two stiffness states with load programming and 0.934 for three states. In a tighter-stroke regime, load programming alone reaches 0.875 and two stiffness states reach 0.926, showing that the incremental value of stiffness programming is regime-dependent. Physical QZS uncertainty is propagated through the positive and negative stiffness components rather than through their small difference; under the stated 1000-sample uncertainty scenario, the three-state design gives median and fifth-percentile worst-case coverage of 0.867 and 0.681. The retuning study uses an explicit five-cycle neighbour-search algorithm, charges every probe and return, and is repeated over 200 independent 1% measurement-noise realizations. Median time within 90% of the instantaneous finite-state oracle is 88.5%, median time-averaged power ratio is 96.8%, and median net/oracle energy ratio is 95.6%. The results separate the roles of electrical damping and structural retuning and identify regimes in which stiffness programming adds material benefit beyond load programming.