A Predictive, Invertible Branched-Beam Model for Monolithic Dual-Resonance Slitted Piezoelectric Energy Harvesters

A piezoelectric cantilever with a longitudinal slit splits one bending resonance into two closely spaced peaks. This is attractive for two-frequency sources. However, existing models fit, rather than predict, the resonances. We develop a predictive branched-beam model where the harvester consists of a full-width root splitting into two prongs with independent tip masses. The two resonances are the roots of a closed-form characteristic equation without fitted parameters, the anti-resonance is the transmission zero of the same branched model, and the formulation reduces to the classical cantilever in the no-slit limit. The second (anti-symmetric) mode is weakly tunable and is located near 45 Hz; the inversion leaves the first resonance free and pins the second effectively. It is predictive and inverted to size the tip masses. Three-dimensional finite-element analysis, which tracks 27.2 → 22.3 Hz (against a finite-element 28.9 → 23.5 Hz as the mass doubles) within about 6%, and the prong kinematics, confirm the tip-mass trend. The anti-resonance is called the charge cancellation effect and can be controlled by wiring the electrodes. For a two-line source (25/45 Hz), the design, tuned to the two lines, yields up to about 1.9× the power of a size-matched single-peak beam with two matched lines (1.6× when the second dominates, ≈1× for broadband), with the second peak being intrinsically smaller. A Monte-Carlo study shows a power coefficient of variation of about 72% with manufacturing and damping scatter, the two layer thicknesses accounting for about 83% of the variance, motivating post-fabrication tip-mass trimming. The model is rigorously validated against three-dimensional electromechanical finite-element analysis; experimental validation on a physical prototype is identified as the essential next step. This work transforms the slitted harvester into a designable monolithic dual-frequency device.