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Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Beams
Jorge L Suzuki1, Ehsan Kharazmi2, Pegah Varghaei1
1Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824; Department of Computational Mathematics, Science, and Engineering, Michigan State University, East Lansing, MI 48824.
Journal of Computational and Nonlinear Dynamics
|July 14, 2022
Summary
Fractional models reveal how anomalous material changes affect mechanical vibrations. This study analyzes a nonlinear viscoelastic beam, uncovering unique dynamic behaviors dependent on the fractional order.
Area of Science:
- Mechanical Engineering
- Materials Science
- Applied Mathematics
Background:
- Fractional calculus models anomalous materials, showing sensitivity to microstructural changes.
- Understanding the impact of anomalous rheology on nonlinear dynamics is crucial for advanced materials.
Purpose of the Study:
- Investigate the propagation of anomalous rheology in nonlinear mechanical systems.
- Analyze the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam.
Main Methods:
- Hamilton's principle to derive equations of motion for a distributed-order fractional viscoelastic model.
- Spectral decomposition and L1-difference scheme for the linear counterpart.
- Method of multiple scales for solving the nonlinear system.
Main Results:
- Derived a nonlinear time-fractional ordinary differential equation for beam vibration.
- Identified alpha-dependent anomalous dynamic qualities.
- Observed power-law decay rates, amplitude super-sensitivity, and bifurcation in steady-state amplitude.
Conclusions:
- Fractional models accurately capture complex dynamics in anomalous viscoelastic materials.
- The study provides insights into the relationship between material microstructure and system-level behavior.
- Results highlight the importance of fractional order in predicting nonlinear dynamic responses.
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