Active Geometric Modulation of Nonlinear Energy-Harvesting Branches in a Triple-Hybrid Variable-Length Pendulum Harvester

This study investigates active geometric modulation in a variable-length pendulum energy harvester combining radial electromagnetic, rotational electromagnetic, and piezoelectric transduction. A reciprocal seven-state electromechanical model is formulated and analysed using phase-aligned continuation, transverse Floquet stability, Lyapunov diagnostics, physical load variation, and paired control-on/control-off energy accounting. A minimal threshold-based shift of the radial spring equilibrium serves to reveal the branch-support mechanism rather than to provide a final control strategy. The results show that an established finite-amplitude branch may persist below the local instability boundary of the inactive response, while piezoelectric loading can modify this boundary through electromechanical back-action. All three transduction channels contribute to gross electrical output, and physical load matching increases that output. Nevertheless, the continuous modulation remains energetically unfavourable after actuator, power-conditioning, and auxiliary demands are included and does not robustly retain the branch under the tested nonstationary excitations. The results therefore define requirements for phase-aware, adaptive, latching, or regenerative implementations with substantially lower actuation work.