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<p>This work proposes and employs a numerical method based on Galerkin approximations for temporal fractional-order derivatives to analyze the stability of parametrically excited damped structures. More clearly, employing fractional-order constitutive models for damping, the Galerkin approximations for fractional-order derivatives are extended to multi-degree-of-freedom (MDOF) systems, beyond the single degree of freedom (SDOF) systems studied in the literature. The finite-dimensional approximation of the fractional-order derivative is demonstrated to be both computationally efficient and accurate compared to alternative approaches, thereby enabling efficient stability analysis over a wide range of system parameters, particularly for MDOF systems. To establish the efficacy of the proposed method, parametric excitation of an Euler–Bernoulli beam with fractional-order damping models is considered. The system matrices for the Euler–Bernoulli beam are developed following a finite element (FE) discretization using <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(C^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> shape functions for transverse displacement. The numerical approach developed here reduces the fractional-order partial differential equation (PDE) to a finite set of ordinary differential equations (ODEs). Thereafter, stability analysis for beams is carried out using Floquet theory. The results obtained from Floquet theory are validated against direct numerical simulations. Furthermore, instabilities are characterized via bifurcation analysis. Although the present work focuses on beams, the proposed methodology can be extended to a general analysis involving fractional-order derivatives owing to its computational efficiency and general applicability.</p>

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Stability analysis of parametrically excited structures with fractional-order damping

  • 期刊论文

  • 发表日期：

    2026年7月25日

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  • C. G. Aditya,
  • Adireddi Balaji,
  • Sai Sidhardh,
  • C. P. Vyasarayani

摘要

This work proposes and employs a numerical method based on Galerkin approximations for temporal fractional-order derivatives to analyze the stability of parametrically excited damped structures. More clearly, employing fractional-order constitutive models for damping, the Galerkin approximations for fractional-order derivatives are extended to multi-degree-of-freedom (MDOF) systems, beyond the single degree of freedom (SDOF) systems studied in the literature. The finite-dimensional approximation of the fractional-order derivative is demonstrated to be both computationally efficient and accurate compared to alternative approaches, thereby enabling efficient stability analysis over a wide range of system parameters, particularly for MDOF systems. To establish the efficacy of the proposed method, parametric excitation of an Euler–Bernoulli beam with fractional-order damping models is considered. The system matrices for the Euler–Bernoulli beam are developed following a finite element (FE) discretization using \(C^1\) C 1 shape functions for transverse displacement. The numerical approach developed here reduces the fractional-order partial differential equation (PDE) to a finite set of ordinary differential equations (ODEs). Thereafter, stability analysis for beams is carried out using Floquet theory. The results obtained from Floquet theory are validated against direct numerical simulations. Furthermore, instabilities are characterized via bifurcation analysis. Although the present work focuses on beams, the proposed methodology can be extended to a general analysis involving fractional-order derivatives owing to its computational efficiency and general applicability.