Duffing-type digitally programmable nonlinear synthetic inductance for piezoelectric structures

Piezoelectric shunt damping techniques using linear circuits (e.g. resistive-inductive) and switching circuits (e.g. synchronized switch on inductor) have been extremely well studied for suppressing resonant vibrations in flexible structures. Both analog circuits and synthetic impedance circuits with digital control have been explored for linear concepts. In a parallel body of work, from the domain of mechanical methods of vibration attenuation, it is also known that leveraging nonlinearities (e.g. stiffness nonlinearity) can enhance the frequency bandwidth and offer amplitude-dependent suppression over a range of frequencies. However, the existing piezoelectric shunt damping techniques have been mostly limited to linear or switching nonlinear circuits, with the exception of a few nonlinear capacitance efforts. This work aims to introduce cubic inductance to emulate Duffing-type hardening nonlinearity in the shunt circuit with precise digital programming and tuning capability. Experiments are performed on a piezoelectric bimorph cantilever under base excitation for concept demonstration and model validation. First, linear frequency response functions of the cantilever are obtained for the short- and open-circuit conditions, and for linear resistive-inductive synthetic shunt damping, to confirm the standard linear behavior and electromechanical model parameters. Then, cubic inductance is introduced to the circuit and nonlinear experiments (up- and down-frequency sweep) are conducted. Cantilever tip to base motion transmissibility frequency response is measured along with piezoelectric voltage to base motion counterpart for a range of base excitation amplitudes. The distortion of the frequency response curves with increased base acceleration levels is observed. The nonlinear cubic coefficient is then varied to alter the manifestation of nonlinear frequency response at a given base excitation level, demonstrating the ease of tuning and triggering the nonlinear behavior on demand by means of the digitally-controlled synthetic impedance shunt. Nonlinear electromechanical model simulations are also validated against the experiments, yielding a very good agreement.