Analytical design of 2-DOF piezoelectric cantilevers for vibration energy harvesting

•Two-degree-of-freedom devices are needed for sources with two dominant frequencies.•A model is presented to design piezoelectric harvesters over two resonant frequencies.•The model is experimentally validated with a strongly coupled device.•The proof mass dimensions allow the ratio between frequencies to be adjusted.•The electrodes optimization maximizes the electromechanical coupling coefficients. The design of two-degree-of-freedom vibration energy harvesters is necessary to enable the vibration energy harvesting for signals with two dominant vibration frequencies. This paper presents an analytical method for the design of piezoelectric cantilevers over two resonant frequencies. A two-degree of freedom (2-DOF) analytical model is proposed and exploited to provide general design guidelines: it enables the sizing of the proof mass to tailor the two resonant frequencies and the optimization of the electrodes’ dimensions to maximize the electromechanical coupling coefficients k2 of the harvester at the resonances. Such high electromechanical coupling coefficients allow reaching the maximum power limit at the output of the vibration energy harvesters and, at high values of k2, to tune the resonant frequency using a dedicated electrical circuit (e.g., resistive tuning or non-linear circuit). A prototype of PZT-based cantilever has been designed and tested to validate the model. The prototype features two close resonant frequencies of 35.3 Hz and 53.2 Hz. When the electrodes are properly connected, the electromechanical coupling coefficient k2 reaches 10.4 % for the first mode and 8.7 % for the second mode, allowing for respective frequency bandwidths of 3 Hz (8.4 %) and 3.5 Hz (6.5 %) with resistive frequency tuning at 0.5 m/s2 acceleration. The maximum harvested power is 262 µW and 149 µW, respectively, for excitation frequencies matching the first and second resonances.