Nonlinear free vibrations of Timoshenko–Ehrenfest beams using finite element analysis and direct scheme

In this work, nonlinear free vibrations of fully geometrically exact Timoshenko–Ehrenfest beams are investigated. First, the exact strong form of the Timonshenko–Ehrenfest beam, considering the geometrical nonlinearity, is derived, and the required formulations are obtained. Since the strong forms o...

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Bibliographic Details
Published in:Nonlinear dynamics 2024-05, Vol.112 (9), p.7199-7213
Main Authors: Firouzi, Nasser, Lenci, Stefano, Amabili, Marco, Rabczuk, Timon
Format: Article
Language:English
Subjects:
Automotive Engineering
Classical Mechanics
Control
Deformation
Dynamical Systems
Eigenvalues
Eigenvectors
Engineering
Finite element analysis
Finite element method
Fourier transforms
Free vibration
Investigations
Mechanical Engineering
Nonlinearity
Original Paper
Resonant frequencies
Shear strain
Stiffness matrix
Vibration
Vibration analysis
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Summary:In this work, nonlinear free vibrations of fully geometrically exact Timoshenko–Ehrenfest beams are investigated. First, the exact strong form of the Timonshenko–Ehrenfest beam, considering the geometrical nonlinearity, is derived, and the required formulations are obtained. Since the strong forms of governing equations are highly nonlinear, a nonlinear finite element analysis (FEA) is employed to obtain the weak form. The FEA is utilized to compute natural frequencies and mode shapes; the direct scheme is adopted to solve the eigenvalue problem which is obtained by eliminating nonlinear terms. Then, each eigenvector is normalized, and the nonlinear stiffness matrix is derived and the nonlinear free vibration analysis is carried out. A recursive procedure is adopted to proceed until the convergence criterion is satisfied. Finally, the applicability of the proposed formulation is provided with some examples and results are compared with those available in the literature.
ISSN:0924-090X
1573-269X
DOI:10.1007/s11071-024-09403-3