Asymptotic Wave Solutions for Euler-Bernoulli and Timoshenko Beam by Ray Method and Stationary Phase Method

Numercial integration;Flexural waves;Asymptotic solutions;Ray method;Stationary phase method;Wave propagation;Beam theory

Liang, Aihua ; Taofang Zeng, Committee Member,Mohammad N. Noori, Committee Member,Fuh-Gwo Yuan, Committee Chair,Liang, Aihua ; Taofang Zeng ; Committee Member ; Mohammad N. Noori ; Committee Member ; Fuh-Gwo Yuan ; Committee Chair

University:North Carolina State University

关键词: Numercial integration;    Flexural waves;    Asymptotic solutions;    Ray method;    Stationary phase method;    Wave propagation;    Beam theory;   

Others  :  https://repository.lib.ncsu.edu/bitstream/handle/1840.16/2604/etd.pdf?sequence=1&isAllowed=y

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【 摘 要 】

The behavior of flexural (bending, transverse) waves in beam structures is of basic importance in the stress wave theory and has been studied for many years. Many methods have been attempted to understand the flexural waves in beams based on Euler-Bernoulli and Timoshenko beam theories. Accurate numerical evaluations for transient waves in structures are usually difficult due to the complexity of the governing equations.The present research was aimed at developing asymptotic wave solutions that can effectively describe one-dimensional flexural waves based on Euler-Bernoulli beam theory and Timoshenko beam theory.Two methods were introduced to obtain the asymptotic wave solutions for Timoshenko and Euler-Bernoulli beams. One is stationary phase method, which derives the asymptotic solution directly from the wave solution that can be expressed in terms of the Fourier integral. From the features obtained from the stationary phase method, the ray method is introduced to seek the asymptotic wave solutions from the governing equations of wave motion. The asymptotic solutions obtained by the two methods, as expected, are consistent, and the attributes of the wave motion are very clear to comprehend in the asymptotic solutions. The amplitude function of the wave is composed of an arbitrary function, which can be determined by initial and/or boundary conditions, and a time decay rate proportional to t -1/2. The wave motion depends on one parameter t along the ray x/t=c8.The numerical calculations were then applied to initial-value problems, boundary-value problems. The results show that the asymptotic solutions predict the transient wave behavior very closely for the long time response in beam structures.