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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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Effects of High Frequency Horizontal Base Excitation on a Bistable System
Abstract High frequency excitation (HFE) is known to induce various nontrivial effects, such as system stiffening, biasing, and the smoothing of discontinuities in dynamical systems. These effects become increasingly pertinent in multi-stable systems, where the system’s bias towards a certain equilibrium state can depend heavily on the combination of forcing parameters, leading to stability in some scenarios and instability in others. In this initial investigation, our objective is to pinpoint the specific parameter ranges in which the bistable system demonstrates typical HFE effects, both through numerical simulations and experimental observations. To accomplish this, we utilize the method of multiple scales to analyze the interplay among different time scales. The equation of slow dynamics reveals how the excitation parameters lead to a change in stability of equilibrium points. Additionally, we delineate the parameter ranges where stabilizing previously unstable equilibrium configurations is achievable. We demonstrate the typical positional biasing effect of high-frequency excitation that leads to a shift in the equilibrium points as the excitation parameter is varied. This kind of excitation can enable the active shaping of potential wells. Finally, we qualitatively validate our numerical findings through experimental testing using a simplistic model made with LEGOs.
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- Award ID(s):
- 2145803
- PAR ID:
- 10571597
- Publisher / Repository:
- American Society of Mechanical Engineers
- Date Published:
- ISBN:
- 978-0-7918-8843-8
- Format(s):
- Medium: X
- Location:
- Washington, DC, USA
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1115/DETC2024-143874
