A Perforated-Root Piezoelectric Cantilever Harvester with a Bow-Tie Cellular Substrate: Distributed-Parameter Modelling, Finite-Element Analysis, and Fatigue-Constrained Design

Cellular substrates placed beneath the piezoceramic are increasingly used to raise the output of vibration energy harvesters, yet the enhancement is commonly attributed to auxeticity, and the durability penalty of perforation is seldom quantified. This paper presents a perforated-root piezoelectric cantilever in which a doubly periodic array of bow-tie (double-arrowhead) through-holes occupies the high-curvature root beneath the electrode while the distal span remains solid. Using a cell of positive effective Poisson’s ratio, we show that power gain is a substrate-compliance and strain-relocation effect set by the position of the perforation relative to the electrode rather than by a negative Poisson’s ratio. A segmented distributed-parameter model in which the perforated root is homogenised and enters the beam through its longitudinal effective modulus E1*, with the piezoelectric coupling reduced to its plane-stress value, is derived and tested against three independent finite-element measurements on the explicit hole geometry in ANSYS Parametric Design Language (APDL). Two agree closely: a direct axial-tension test returns E1*/Es=0.270 against 0.268 predicted (0.7%), and the substrate-only fundamental is 21.7 Hz against 21.5 Hz predicted (1.0%). The third does not: with the piezoceramic present, the reduction predicts 47.7 Hz against 36.4 Hz, so the absolute frequency of the complete device is not yet established, and only the trends are relied upon here. At equal overall dimensions and piezoceramic, the baseline cell raises the peak power by about 14% over the solid beam, from 38.6 to 44.0 μW, and lowers the resonance from 57.6 to 50.7 Hz; both effects grow with hole size, reaching 23% and 46.5 Hz at the largest cell examined. A fatigue-constrained formulation, in which the net-section ligament stress bounds the usable fill factor, caps the fill at f≈0.59 under a 1 g excitation and yields a fill factor–load design map for durable operation.