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This article explores the nonlinear vibration of beams with different types of nonlinearities. The beam vibration was modeled using Hamilton’s principle, and the equation of motion was solved using method of multiple time scales. Three models were developed assuming (a) geometric nonlinearity, (b) material nonlinearity and (c) combined geometric and material nonlinearity. The material nonlinearity also included both third and fourth nonlinear elasticity terms. The frequency response equation of these models were further evaluated quantitatively and qualitatively. The models capture the hardening effect, i.e., increase in resonant frequency as a function of forcing amplitude for geometric nonlinearity, and the softening effect, i.e., decrease in resonant frequency for material nonlinearity. The model is applied on the first three bending modes of the cantilever beam. The effect of the fourth-order material nonlinearity was smaller compared to the third-order term in the first mode, whereas it is significantly larger in second and third mode. The combined nonlinearity models shows a discontinuous frequency shift, which was resolved by utilizing a set of transition assumptions. This results in a smooth transition between the material and geometric zones in amplitude. These parametric models allow us to fine tune the nonlinear response of the system by changing the physical properties such as geometry, linear and nonlinear elastic properties.
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This content will become publicly available on May 2, 2026
Steady-state nonlinear dynamics of a flexible beam with large deformation under oscillatory flow
This work investigates the steady-state nonlinear dynamics of a large-deformation flexible beam model under oscillatory flow. A flexible beam dynamics model combined with hydrodynamic loading is employed using large deformation beam theory. The equations of motion discretised using the high-order finite element method (FEM) are solved in the time domain using the efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method. The arc-length continuation method and Hsu’s method trace stable and unstable solutions. The numerical results are in accordance with the physical experimental results and reveal multiple resonance phenomena. Low-order resonances exhibit hardening due to geometric nonlinearity, while higher-order resonances transition from softening to hardening influenced by inertia and geometric nonlinearity. A strong coupling between tensile and bending deformation is observed. The axial deformation is dominated by inertia, while bending resonance is influenced by an interplay between inertia, structure stiffness, and fluid drag. Finally, the effects of two dimensionless parameters, Keulegan and Carpenter number (KC) and Cauchy number (Ca), on the response of the flexible beam are discussed.
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- Award ID(s):
- 2327916
- PAR ID:
- 10644692
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of fluids and structures
- Volume:
- 136
- ISSN:
- 0889-9746
- Page Range / eLocation ID:
- 104327
- Subject(s) / Keyword(s):
- Nonlinear dynamics Fluid-solid coupling Large deformation beams
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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