Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. In the sixth paper, exact analysis of the wave propagation in a beam with rectangular cross section is extended to a thorough multiscale analysis for a system of completely coupled second order linear differential equations for modal functions, where general boundary conditions are prescribed. For this purpose, the modal function each is expressed as a linear combination of the corner function, the two boundary functions and the internal function, to ensure the series expressions obtained uniformly convergent and termwise differentiable up to second order. Meanwhile, the sum of the corner function and the internal function corresponds to the particular solution, and the two boundary functions correspond to the general solutions which satisfy the homogeneous form of the equations. Since the general solutions have appropriately interpreted the meaning of the differential equations, the spatial characteristics of the solution of the equations are expected to be better captured in separate directions. With the corner function, the two boundary functions and the internal function selected specifically as polynomials, one-dimensional full-range Fourier series along the x2 (or x1)-direction, and two-dimensional full-range Fourier series, the Fourier series multiscale solution of the wave propagation in a beam with rectangular cross section is derived. For the beam with various boundary conditions, computation and analysis are performed, and the propagation characteristics of elastic waves in the beam are presented. The newly proposed accurate wave model has laid a solid foundation for simultaneous control of coupled waves in the beam and establishment of guided wave NDE techniques.
翻译：将在本系列论文中制定一套简单、高效的多尺度分析方法,即多尺度计算法。在第六份论文中,对带矩形截面的横梁波波传播进行精确分析,以进行彻底的多尺度分析,以建立一个完全同时的第二顺序线性差异方程式系统,用于模式函数,在其中规定一般边界条件。为此,每个模式函数以角函数、两个边界函数和内部函数的线性组合形式表示,以确保以一致和术语方式获得的序列表达,直到第二顺序。与此同时,角函数和内部导向导函数与特定解决方案相对应,而两个边界函数则与满足等式形式的一般解决方案相对应。由于一般解决方案对差异方程式的含义作了适当的解释,因此,每个模型的空间特性预计将以不同方向更好地捕捉到。由于角函数、两个边界函数和内部函数被具体选定为聚度、一米波全波至第二顺序。在 x2 (or x1) 和四面级的滚流模型和四面级序列的滚动模型是四面级模型和四面级的滚流模型,四面级的模型和四面级的滚流的滚流的计算是全流的跨级计算。
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