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<p>This work proposes an analytical solution for a Duffing oscillator augmented with a non-smooth quadratic nonlinearity, a configuration that naturally arises when modeling the resonant behavior of elastic structures including piezoelectric components because of their ferroelastic nonlinear behavior. Such a formulation is particularly relevant for the identification of piezoelectric material properties. Two perturbation techniques—the single-harmonic Harmonic Balance Method (HBM) and the Method of Multiple Scales (MMS)—are employed to derive analytical expressions for the backbone curve, the frequency response and the stability of periodic solutions. These results are benchmarked against a high-fidelity numerical reference obtained via numerical continuation using a 150-harmonic HBM combined with the Asymptotic Numerical Method (ANM) using the software MANLAB. The comparison highlights the remarkable accuracy and broad validity of the HBM-based analytical solution, while also revealing the more restricted applicability of the MMS approach.</p>

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Free and forced response of a Duffing oscillator with a non-smooth quadratic term: analytical perturbative solutions

  • 期刊论文

  • 发表日期：

    2026年6月24日

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  • Charlélie Bertrand,
  • Hugo Fayolle,
  • Christophe Giraud-Audine,
  • Olivier Thomas

摘要

This work proposes an analytical solution for a Duffing oscillator augmented with a non-smooth quadratic nonlinearity, a configuration that naturally arises when modeling the resonant behavior of elastic structures including piezoelectric components because of their ferroelastic nonlinear behavior. Such a formulation is particularly relevant for the identification of piezoelectric material properties. Two perturbation techniques—the single-harmonic Harmonic Balance Method (HBM) and the Method of Multiple Scales (MMS)—are employed to derive analytical expressions for the backbone curve, the frequency response and the stability of periodic solutions. These results are benchmarked against a high-fidelity numerical reference obtained via numerical continuation using a 150-harmonic HBM combined with the Asymptotic Numerical Method (ANM) using the software MANLAB. The comparison highlights the remarkable accuracy and broad validity of the HBM-based analytical solution, while also revealing the more restricted applicability of the MMS approach.