SUMMARY

Low-frequency applications of piezoelectric shunt circuits have triggered the research on implementing synthetic impedance circuits to avoid the need for large analog circuit components. Synthetic impedance is a type of a voltage-controlled current source used to emulate arbitrary circuit elements by digital signal processing. Most of the existing efforts on synthetic impedance shunts have been limited to standard scenarios of emulating linear circuit components, a typical example being synthetic inductance circuits (emulating a dynamic vibration absorber) to avoid the use of large coils when targeting low frequencies in flexible structures. In this research, we explore nonlinear synthetic impedance-based piezoelectric structures through experiments and modeling, by introducing polynomial nonlinearities in the circuit domain. In the first part of this work, we introduce a hardening Duffing oscillator by means of a cubic synthetic inductance. The inherent capacitance of the piezoelectric elements emulates mass-like behavior, while the linear and cubic inductances form the stiffness component, and resistive elements are added for damping and stability, all in parallel connection. The synthetic Duffing circuit is used as a nonlinear shunt circuit for a linear (stiff) piezoelectric cantilever, and it is shown that the cubic nonlinearity can be programmed by varying the respective gain in the synthetic impedance circuit to change the bifurcation (jump) points in the frequency response. A nonlinear electromechanical model is developed, and numerical simulations are performed via time-domain solution. Alternative solution approaches in the proposed work are the use of the method of harmonic balance as well as discontinuous frequency response estimation by means of machine learning for enhanced computational efficiency. Experiments are guided by numerical simulations throughout the work, and the underlying nonlinear electromechanical models are validated for various excitation amplitudes and nonlinear coefficients. Following the hardening inductor case, the proposed tasks include the softening inductor counterpart (i.e., softening Duffing oscillator), the nonlinear energy sink, as well as the combination of nonlinear circuits with geometrically nonlinear vibrations. The programmable nonlinear synthetic impedance circuits analyzed and experimentally demonstrated here are expected to open a new avenue in the domain of vibration control.