2D+1 degree of freedom equivalent circuit model for LiNbO₃/metal/LiNbO₃ bimorph bending cantilever

In this paper, we extend the single degree of freedom (1DOF) model to a lumped circuit model derived analytically with three degrees of freedom (2D+1DOFs) for modeling the mechanical-to-electrical conversion by a serial bimorph piezoelectric cantilever. The model is based on a minimal finite element analysis interpolation of the beam bending with two mechanical nodes and four symmetrical electrical nodes. The model satisfies Hamilton's principle, Euler-Bernoulli kinematic assumptions and constitutive piezoelectric equations. The 2D+ 1DOFs are the transverse displacement, the rotation at the tip end, and the voltage output. The DOFs result from the application of three external actions: the vertical tip force, the tip torque and the net electrical charge on the electrodes. The dynamic balance of the bimorph is described by three electromechanical coupled equations that are equivalently rendered by two mutually coupled electronic resonant lumped circuits driven by the mechanical DOFs, and an output circuit involving the charge and the external load. As compared to 1DOF equivalent circuit, this model introduces analytically and parametrically the geometry of the beam and its properties for transient and frequency-domain analysis using computer-aided electronics software. The results of the model are compared to the simulations done by using commercial finite element modeling software and the experimental measurements of a symmetric LiNbO₃/stainless steel/LiNbO₃ serial bimorph beam resonating at 35.7 Hz. Monte Carlo simulations including uncertainty on geometrical parameters show less than 2 % of uncertainty on open circuit voltage and resonance frequency. Our lead-free bimorph shows an electromechanical coupling of k² = 5.2 %.