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Abstract Fractional models and their parameters are sensitive to intrinsic microstructural changes in anomalous materials. We investigate how such physics-informed models propagate the evolving anomalous rheology to the nonlinear dynamics of mechanical systems. In particular, we study the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam, under base excitation and free vibration, where the viscoelasticity is described by a distributed-order fractional model. We employ Hamilton's principle to obtain the equation of motion with the choice of specific material distribution functions that recover a fractional Kelvin–Voigt viscoelastic model of order α. Through spectral decomposition in space, the resulting time-fractional partial differential equation reduces to a nonlinear time-fractional ordinary differential equation, where the linear counterpart is numerically integrated through a direct L1-difference scheme. We further develop a semi-analytical scheme to solve the nonlinear system through a method of multiple scales, yielding a cubic algebraic equation in terms of the frequency. Our numerical results suggest a set of α-dependent anomalous dynamic qualities, such as far-from-equilibrium power-law decay rates, amplitude super-sensitivity at free vibration, and bifurcation in steady-state amplitude at primary resonance.
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Analytical Study of Nonlinear Flexural Vibration of a Beam with Geometric, Material and Combined Nonlinearities
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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- Award ID(s):
- 2148646
- PAR ID:
- 10559871
- Publisher / Repository:
- MDPI
- Date Published:
- Journal Name:
- Vibration
- Volume:
- 7
- Issue:
- 2
- ISSN:
- 2571-631X
- Page Range / eLocation ID:
- 464 to 478
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/vibration7020025
