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The Application of the Modified Lindstedt–Poincaré Method to Solve the Nonlinear Vibration Problem of Exponentially Graded Laminated Plates on Elastic Foundations

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  • Mahmure Avey

    (Department of Mathematical Engineering, Graduate School of Istanbul Technical University, Maslak 34469, Istanbul, Turkey
    Analytical Information Resources Center of UNEC, Azerbaijan State Economics University, Baku AZ1001, Azerbaijan
    Application and Research Center, Istanbul Ticaret University, Beyoglu 34445, Istanbul, Turkey)

  • Francesco Tornabene

    (Department of Innovation Engineering, University of Salento, I-73100 Lecce, Italy)

  • Nigar Mahar Aslanova

    (Department of Mathematics, Azerbaijan University of Architecture and Construction, Baku AZ1073, Azerbaijan)

  • Abdullah H. Sofiyev

    (Department of Mathematics, Istanbul Ticaret University, Beyoglu 34445, Istanbul, Turkey
    Scientific Research Department, Azerbaijan University of Architecture and Construction, Baku AZ1073, Azerbaijan
    Scientific Research Centers for Composition Materials of UNEC, Azerbaijan State Economic University, Baku AZ1001, Azerbaijan)

Abstract

The solution of the nonlinear (NL) vibration problem of the interaction of laminated plates made of exponentially graded orthotropic layers (EGOLs) with elastic foundations within the Kirchhoff–Love theory (KLT) is developed using the modified Lindstedt–Poincaré method for the first time. Young’s modulus and the material density of the orthotropic layers of laminated plates are assumed to vary exponentially in the direction of thickness, and Poisson’s ratio is assumed to be constant. The governing equations are derived as equations of motion and compatibility using the stress–strain relationship within the framework of KLT and von Karman-type nonlinear theory. NL partial differential equations are reduced to NL ordinary differential equations by the Galerkin method and solved by using the modified Lindstedt–Poincaré method to obtain unique amplitude-dependent expressions for the NL frequency. The proposed solution is validated by comparing the results for laminated plates consisting of exponentially graded orthotropic layers with the results for laminated homogeneous orthotropic plates. Finally, a series of examples are presented to illustrate numerical results on the nonlinear frequency of rectangular plates composed of homogeneous and exponentially graded layers. The effects of the exponential change in the material gradient in the layers, the arrangement and number of the layers, the elastic foundations, the plate aspect ratio and the nonlinearity of the frequency are investigated.

Suggested Citation

  • Mahmure Avey & Francesco Tornabene & Nigar Mahar Aslanova & Abdullah H. Sofiyev, 2024. "The Application of the Modified Lindstedt–Poincaré Method to Solve the Nonlinear Vibration Problem of Exponentially Graded Laminated Plates on Elastic Foundations," Mathematics, MDPI, vol. 12(5), pages 1-22, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:5:p:749-:d:1349797
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    References listed on IDEAS

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    1. Fatima Zohra Zaoui & Djamel Ouinas & Belkacem Achour & Mabrouk Touahmia & Mustapha Boukendakdji & Enamur R. Latifee & Ahmed A. Alawi Al-Naghi & Jaime Aurelio Viña Olay, 2022. "Mathematical Approach for Mechanical Behaviour Analysis of FGM Plates on Elastic Foundation," Mathematics, MDPI, vol. 10(24), pages 1-29, December.
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