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Bernstein polynomials in analyzing nonlinear forced vibration of curved fractional viscoelastic beam with viscoelastic boundaries

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Abstract

In this paper, an effective numerical Bernstein polynomials operational matrix is exploited to study the nonlinear forced vibration of fractional viscoelastic curved beam with viscoelastic nonlinear boundary conditions, for the first time. The Caputo fractional derivative is employed to incorporate viscoelastic material having nonlinear behavior. Based on Euler–Bernoulli beam theory and von Kármán geometric nonlinearity, the governing equation of viscoelastic curved beam which is a nonlinear integro-partial fractional differential equation is introduced. Based on the collocation method of Bernstein polynomials approximation, the generalized integer-order operational matrices of differentiation and integration as well as fractional-order operational matrix of differentiation are derived. Using Bernstein polynomials operational matrices (BPOM), the nonlinear static problem is discretized; then, it is solved via Newton's method. To compute the linear vibration mode, the linear vibration problem is discretized via BPOM and it is solved as a linear eigenvalue problem. Discretized by the Galerkin approximation, the fractional-order nonlinear integro-partial differential equation is transformed into a nonlinear fractional-order duffing-type equation. The fractional-order duffing equation is discretized using BPOM resulting nonlinear eigenvalue problem, which can be easily solved by pseudo-arc length continuation algorithm. The effects of the Caputo fractional derivative order, foundation parameters, amplitude of initial curvature, viscoelastic parameters and amplitude of excitation force on the nonlinear forced vibration of the viscoelastic beam are investigated through a detailed parametric study. The proposed procedure is supportive in the analysis and design of curved viscoelastic structure with non-classical viscoelastic boundary conditions under the dynamic mechanical loads.

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Authors and Affiliations

  1. Department of Engineering Mathematics, Faculty of Engineering, Zagazig University, Ash Sharqiyah, Egypt

    N. Mohamed & S. A. Mohamed

  2. Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia

    M. A. Eltaher

  3. Mechanical Design and Production Department, Faculty of Engineering, Zagazig University, Ash Sharqiyah, Egypt

    M. A. Eltaher

  4. Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Erasmo Carrera

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  1. N. Mohamed
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Correspondence to M. A. Eltaher.

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Mohamed, N., Eltaher, M.A., Mohamed, S.A. et al. Bernstein polynomials in analyzing nonlinear forced vibration of curved fractional viscoelastic beam with viscoelastic boundaries. Acta Mech (2024). https://doi.org/10.1007/s00707-024-03954-7

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