It has been well explored in the literature that intentionally introduced (designed) nonlinearities offer significant advantages in vibration absorption and isolation problems. In the context of using piezoelectric shunt damping for vibration attenuation in flexible structures, synthetic impedance circuits with digital control enable precise tuning capabilities and eliminate the need for bulky analog circuit components when targeting low frequencies, and they have been studied mostly for linear circuits. In this work, we explore nonlinear synthetic impedance-based Duffing shunts with hardening and softening nonlinearities, connected to a cantilever under base excitation in its geometrically and materially linear regime. We demonstrate a shunt of nonlinear impedance equivalent to a positive or negative cubic inductor connected in parallel to a linear inductor and a linear resistor. This shunt, along with the inherent piezoelectric capacitance, provides digital analog of a hardening- or softening-type nonlinear stiffness, linear stiffness, linear damping, and mass, as a synthetic Duffing circuit. Electromechanical modeling of a piezoelectric cantilevered beam shunted to the Duffing oscillator with hardening or softening inductance is performed. Numerical simulations for single-term and multi-term harmonic balance solutions are performed for periodic response. These approximate analytical predictions are compared against time-domain numerical simulations and are experimentally validated for various mechanical excitation amplitudes and nonlinear coefficients. Overall, a very good agreement is obtained between the model simulations and experimental results, in addition to demonstrating the unique digital programming capabilities of this nonlinear shunt.
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