Abstract
This work establishes the mathematical, physical, and electronic design foundations for the Adaptive Resonant Piezoelectric Energy Harvester (ARPEH), a novel electromechanical platform engineered to reclaim low-frequency, non-stationary ambient mechanical energy and convert it into regulated electrical power assets. We provide first-principles physical derivations governing structural vibration sources across key operational domains: quarter-car suspension dynamics driven by stochastic ISO 8608 road roughness spectra with explicit contour integrals for kinetic dissipation; coupled multi-field piezo-elastic constitutive equations for longitudinal 33-mode stacks mapping continuum fields into discrete lumped electromechanical parameter systems; and aeroelastic wing flutter governed by Euler-Bernoulli beam theory with unsteady Theodorsen aerodynamics. To overcome the historical frequency mismatch penalty quantified as a $10^8$-fold power degradation, we introduce a dual-stage pre-conditioning framework comprising a high-efficiency cycloidal cam-based mechanical frequency multiplier and a non-linear secondary resonator governed by a hardening Duffing potential with proven bandwidth enhancement exceeding a factor of 2.8. The coupled electromechanical system is cast into an explicit non-linear state-space system matrix and solved via adaptive high-order Runge-Kutta numerical integration. Down-circuit electrical extraction employs a specialized Zero-Voltage-Switching (ZVS) Maximum Power Point Tracking (MPPT) resonant converter operating as a high-frequency LC tank circuit with the transducer's internal clamped capacitance, achieving \SI{96.4}{\percent} conversion efficiency.