Fractional Vibrations with Applications to Euler-Bernoulli Beams

The book examines vibration phenomena with an emphasis on fractional vibrations using the functional form of linear vibrations with frequency-dependent mass, damping, or stiffness, covering the theoretical analysis potentially applicable to structures and, in particular, ship hulls. Covering the six classes of fractional vibrators and seven classes of fractionally damped Euler-Bernoulli beams that play a major role in hull vibrations, this book presents analytical formulas of all results with concise expressions and elementary functions that set it apart from other recondite studies. The results show that equivalent mass or damping can be negative and depends on fractional orders. Other key highlights of the book include a concise mathematical explanation of the Rayleigh damping assumption, a novel description of the nonlinearity of fractional vibrations, and a new concept of fractional motion, offering exciting additions to the field of fractional vibrations. This title will be a must-read for students, mathematicians, physicists, and engineers interested in vibration phenomena and novel vibration performances, especially fractional vibrations.

Table of Contents:
Part I: Fundamentals 1. Harmonic Vibrations 2. Vibrations Excited by Periodic Forces 3. Fourier Transform and Spectra 4. Responses Excited by Deterministically Aperiodic Forces 5. Vibrations with Multiple Degrees-of-Freedom 6. Vibrations of Distributed Systems and Euler-Bernoulli Beam Part II: Fractional Vibrations 7. Six Classes of Fractional Vibrations 8. Fractional Vibrations of Class I 9. Fractional Vibrations of Class II 10. Class III Fractional Vibrations 11. Fractional Vibrations of Class IV 12. Class V Fractional Vibrations 13. Fractional Vibrations of Class VI 14. Explanation of Rayleigh Damping Assumption based on Fractional Vibrations 15. Mass 16. Vibrators with Variable-Order Fractional Forces Part III: Fractional Euler-Bernoulli Beams 17. Free Response to Longitudinal Vibrations of Uniform Circular Beam with Fractional Coordinates 18. Free Response to Euler-Bernoulli Beam with Fractional Coordinates 19. Forced Response to Euler-Bernoulli Beam with Fractional Coordinates 20. Seven Classes of Fractionally Damped Euler-Bernoulli Beams 21. Forced Response to Damped Euler-Bernoulli Beam with Fractional Inertia Force (Class 1) 22. Forced Response to Damped Euler-Bernoulli Beam with Fractional External Damping Force (Class 2) 23. Forced Response to Damped Euler-Bernoulli Beam with Fractional Internal Damping Force (Class 3) 24. Forced Response to Damped Euler-Bernoulli Beam with Fractional External and Internal Damping Forces (Class 4) 25. Forced Response to Damped Euler-Bernoulli Beam with Fractional Inertia and External Damping Forces (Class 5) 26. Forced Response to Damped Euler-Bernoulli Beam with fractional Inertia and Internal Damping Forces (Class 6) 27. Forced Response to Multi-Fractional Damped Euler-Bernoulli Beam (Class 7) 28. Notes on Fractional Vibrations Part IV: Some Techniques in Vibrations 29. Sampling, Aliasing, Anti-Aliasing Filtering and Time Signal Leakage 30. A Method for Requiring Block Size for Spectrum Measurement of Ocean Surface Waves 31. Time-Frequency Distributions of Encountered Waves using Hilbert-Huang Transform 32. An Optimal Controller of an Irregular Wave Maker 33. On von Kármán Spectrum from a View of Fractal

About the Author :
Ming Li is a professor at Ocean College, Zhejiang University, China, and an emeritus professor at East China Normal University. He has been a contributor for many years to the fields of mathematics, statistics, mechanics, and computer science. His publications with CRC Press also include Multi-Fractal Traffic and Anomaly Detection in Computer Communications and Fractal Teletraffic Modeling and Delay Bounds in Computer Communications.