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• Foundations of Data Science 数据科学基础

## 物理代写|力学代写mechanics代考|Sharpening of Moiré Fringes

According to the phenomenon of geometric moiré, the difference of displacements between two points on successive fringes is equal to the pitch of the grating. The density of the grating cannot exceed 40 lines $/ \mathrm{mm}$ or the pitch cannot be smaller than $2.5 \times 10^{-2} \mathrm{~mm}$. For higher grating densities or smaller grating pitches, diffraction effects are introduced. Thus, the sensitivity of the moiré method is limited to the above value of the pitch of the grating. This is a serious drawback of the method compared with other methods of displacement measurement, e.g., electrical resistance strain gages. Attempts have been made to increase the sensitivity of moiré-fringe-sharpening and moiré-fringe-multiplication methods. Moiré-fringesharpening permits a more precise location of the position of the fringe and enhances the accuracy of the moiré method. In this section, we will present a method of moiré-fringe-sharpening.

According to Eq. (3.5) superposition of two parallel line gratings of pitches, $p$ and $p(1+\lambda)$ results in moiré fringes at distances $f \approx p / \lambda_{\text {. }}$ The light intensity is zero at the center of a dark fringe and increases linearly up to the adjacent bright fringe where becomes maximum. It then diminishes to zero at the next dark fringe. Figure $3.18 \mathrm{a}$ shows the triangular light intensity distribution along a line perpendicular to the moiré fringes formed by two intersecting line gratings of $50 \%$ transmittance (the width of the opaque bars of the gratings is equal to the width of the transparent slits).
Minima or zero intensity occur at dark fringes and maxima of $50 \%$ of the intensity of the incident light occurs at bright fringes.

The above triangular intensity distribution can be altered by changing the ratio of the width of the opaque bars to the width of the transparent slits in both gratings. Consider the case where the one grating has wide opaque bars and narrow transparent slits, while the other grating has narrow opaque bars and wide transparent slits. Two gratings are called complementary when the ratio of the width of the opaque bars to the width of the transparent slits of one is reciprocal to the other. Figure $3.18 \mathrm{~b}$ shows the light intensity distribution along a line perpendicular to the moiré fringes formed by two intersecting complementary line gratings. Note that the intensity pattern has a trapezoidal symmetric shape with minima of zero intensity at dark fringes and maxima of intensity $50 \%$ over a width $f(b-c) / p$, where $b$ is the width of the opaque bars, $c$ is the width of the transparent slits, $p$ is the pitch and $f$ is the interfringe spacing. The moiré fringes of Fig. 3.18b are sharper than those of Fig. 3.18a. Sharpening of moiré fringes can also be achieved with non-complimentary gratings. However, the difference of intensity between maxima and minima is reduced.

## 物理代写|力学代写mechanics代考|Moiré of Moiré

According to the moiré effect when two indexed families of curves $S(x, y)=k$ and $R(x, y)=l$ are superposed a third family $M(x, y)=m$ given by $k \pm l=m$ is generated (Eq. (3.7)). The two families of curves corresponding to plus and minus signs are called the additive and subtractive moiré patterns, respectively. The additive moiré pattern is not usually observed, while the subtractive moiré pattern dominates. The moiré phenomenon results in subtracting quantities represented by two families of curves. Moiré fringe patterns can be regarded as indexed families of curves. Thus, superposition of two moiré patterns results in a new moiré pattern called moiré of moiré. The moiré of moiré effect is used quite often in experimental mechanics. We refer two cases.

Consider the moiré fringes of $u$-displacements along the $x$-axis. Each fringe represents the loci of equal values of $u$. Obtain a copy of the moiré pattern of $u-$ displacements and shift it by $\Delta x$ along the $x$-direction. The two superposed families of curves form a new family of fringes which represent the partial derivatives of $u$ with respect to $x(\partial u / \partial x)$. Similarly, we can obtain the derivatives of $u$ with respect to $y(\partial u / \partial y)$ if we shift the moiré pattern of $u$-displacements along the $y$-axis, and take the moiré of the two patterns. In the same way, we obtain the derivatives $\partial v / \partial x$, $\partial v / \partial x$ of the $v$ displacement along the $x$ – and $y$-directions. Thus, the moiré of moiré effect can be used in the differentiation of a function.

As a second case, consider the interference fringe pattern formed by light rays reflected from the front and rear faces of a transparent specimen. Due to the variation of the thickness of the specimen (of the order of the wavelength of light) interference fringes are formed even though the specimen is unloaded. When the specimen is loaded an interference pattern is formed due to the variation of the stress-optical retardation of the light rays reflected from the two faces of the specimen. The interference pattern from the loaded and unloaded specimens presents the variation of the absolute stress-optical retardation of the specimen due to loading and the thickness of the specimen, respectively. When two such patterns are superposed with the specimen unloaded and loaded moiré fringes are obtained which indicate the isochromatics (loci of equal difference of principal stresses) and isopachics (loci of equal sum of the principal stresses) families of curves. This case of interference will be studied in Chap. $8 .$

# 力学代考

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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