科学与工程计算分析的傅里叶级数（基于）多尺度方法：VII。Pasternak地基梁弹性弯曲的傅里叶级数多尺度解
Fourier series (based) multiscale method for computational analysis in science and engineering: VII. Fourier series multiscale solution for elastic bending of beams on Pasternak foundations
摘要：
傅里叶级数多尺度方法是一种简洁有效的多尺度计算分析方法。第七篇文章将弹性地基上梁的结构分析推广到梁横向挠度的四阶线性微分方程的多尺度分析，其中规定了一般的边界条件和宽谱的模型参数。为此，将解函数表示为边界函数和内部函数的线性组合，以保证所得到的级数表达式一致收敛，且在四阶以下是逐项可微的.同时，内函数对应于特解，边界函数对应于满足控制微分方程齐次形式的通解。由于通解恰当地解释了微分方程的含义，因此期望更好地捕捉方程的解的空间特性。以线性无关的齐次微分方程解为边界函数和内函数，在解区间上采用一维半幅Fourier正弦级数，导出了Pasternak地基梁弯曲问题的多尺度傅里叶级数解.然后通过数值算例研究了傅里叶级数多尺度解的收敛特性，并证明了Pasternak地基梁弯曲问题在宽谱模型参数下的多尺度特性。
英文摘要：
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. In the seventh paper, the usual structural analysis of beams on an elastic foundation is extended to a thorough multiscale analysis for a fourth order linear differential equation for transverse deflection of the beam, where general boundary conditions and a wide spectrum of model parameters are prescribed. For this purpose, the solution function is expressed as a linear combination of the boundary function and the internal function, to ensure the series expression obtained uniformly convergent and termwise differentiable up to fourth order. Meanwhile, the internal function corresponds to the particular solution, and the boundary function corresponds to the general solution which satisfies the homogeneous form of the governing differential equation. Since the general solution has appropriately interpreted the meaning of the differential equation, the spatial characteristics of the solution of the equation are expected to be better captured. With the boundary function and the internal function selected specifically as combination of the linearly independent homogeneous solutions of the differential equation, and one-dimensional half-range Fourier sine series over the solution interval, the Fourier series multiscale solution of the bending problem of a beam on the Pasternak foundation is derived. And then the convergence characteristics of the Fourier series multiscale solution are investigated with numerical examples, and the multiscale characteristics of the bending problem of a beam on the Pasternak foundation are demonstrated for a wide spectrum of model parameters.
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