Energetic frontier theory for smart piezoelectric structures with coupled vibration attenuation and energy harvesting

Abstract

Piezoelectric energy harvesters and shunted smart structures are often analyzed separately as power generators and vibration absorbers, even though both functions arise from the same electromechanical coupling. This paper asks a joint design question: for a prescribed residual-vibration level, what is the largest useful electrical power that a passive piezoelectric or equivalent electromechanical attachment can deliver? We formulate the problem as an attenuation-constrained power boundary. A scalarized support functional gives a universal upper bound on the budget frontier and, when its extended-real form is proper, concave, and upper semicontinuous, an exact dual representation. Closed-form harmonic and white-noise SDOF benchmarks then show that the forcing model changes the frontier geometry: resonant harmonic forcing produces an interior knee, whereas the ideal white-noise boundary is monotone. The construction extends to separable modal allocation and, under explicit continuity and uniform-stability assumptions, to covariance-based linear stochastic models. A reduced-order verification is performed on a literature-calibrated PZT-5A/brass/PZT-5A bimorph cantilever. A 20-element beam finite-element model is compared with a first-mode model identified before the frontier sweep and with conventional resistive load choices. The reduced model captures the principal computed frontier features. An illustrative 20 V peak-terminal-voltage admissibility screen shows how excluding high-voltage linear operating points changes the accessible branch; it is not a converter or clamp simulation. A final restricted nonlinear extension establishes stationary-frontier existence under compactness and continuity hypotheses and derives a fixed-frequency Duffing deformation. The resulting framework connects attenuation requirements, useful power, modal participation, and electrical admissibility in a single design map while keeping the assumptions behind each level of the model explicit.