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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|力学代写mechanics代考|Moiré Patterns Formed by Circular, Radial

Line gratings are mainly used in moiré analysis for measurement of rectilinear components of displacement. Circular and radial gratings may also be used for measurement of displacements in polar coordinates. In this section, we will study the moiré patterns formed by superposition of circular, radial, and line gratings by using the general equations of moiré fringes created by two families of curves $S(x$, $y)=k$ and $R(x, y)=l$ developed in Sect. 3.4. Superpositions of circular and line gratings create patterns of extreme beauty.

A circular grating consists of equispaced concentric circles. Consider two circular gratings of pitches $p$ and $p(1+\lambda)$ with their centers at distance $2 c$ apart. Refer both gratings to a system of Cartesian coordinates with the $x$-axis passing through the centers of the circles of the gratings and its origin at the mid-distance between the centers of the gratings. The equations of the two circular gratings are given by
$$\begin{gathered} (x-c)^2+y^2=k^2 p^2 \quad(k=\pm 1, \pm 2, \ldots) \ (x+c)^2+y^2=l^2 p^2(1+\lambda)^2(l=\pm 1, \pm 2, \ldots) \end{gathered}$$
Using Eq. (3.7) we obtain the following equation for the moiré fringes obtained by the superposition of the two gratings
\begin{aligned} &\left{\left[(x+c)^2+y^2\right]+\left(x-c)^2+y^2\right^2-m^2 p^2(1+\lambda)^2\right}^2 \ &=4(1+\lambda)^2\left[(x+c)^2+y^2\right]\left[(x-c)^2+y^2\right] \quad(m=\pm 1, \pm 2, \ldots) \end{aligned}
The moiré pattern formed by two circular gratings of different pitches and at a distance apart for $\lambda>0$ is shown in Fig. 3.9. Using Eq. (3.15) the commutation moiré boundary is obtained as
$$x^2+y^2=c^2$$

## 物理代写|力学代写mechanics代考|Measurement of In-Plane Displacements

So far we have studied the moiré patterns formed by superposition of two line, circular, or radial gratings. Particular attention was paid to line gratings of different pitches at an angle to each other. We established the equations between the pitches of the two gratings and their inclination angle on one hand and the angle of inclination and the distance between the resulting moiré fringes on the other hand.

We will now consider the general case of two line gratings one of which is attached to a two-dimensional specimen (Fig. 3.12). The specimen grating $(S G)$ follows the deformation of the specimen, while the master grating $(R G)$ is superposed to $S G$. The moiré fringes formed from the superposition of the gratings $S G$ and $R G$ are the shortest diagonals of the quadrangles formed by the gratings. Suppose that before deformation of the specimen the lines of the two gratings coincide. After deformation of the specimen, the points at which the lines of order $q-1, q, q+1$ of $S G$ intersect the lines of order $q-1, q, q+1$ of $R G$ do not move in a direction perpendicular to the lines of $R G$, and, therefore, belong to the same moiré fringe of zero order. For the same reason, the point at the intersection of line of order $q-1$ of $\mathrm{SG}$ with the line of order $q$ of $R G$ has moved a distance equal to the pitch $p$ of $R G$ in a direction perpendicular to the lines of $R G$. This point belongs to the moiré fringe of first order. To the same fringe of first order belong the points at the intersection of the line $q$ of $S G$ with the line $q+1$ of $R G$, the line $q+1$ of $\mathrm{SG}$ with the line $q+2$ of $R G$, etc. Similarly, the points at the intersections of the lines $q-2, q-1, q, q+1$ of $S G$ with the lines $q, q+1, q+2, q+3$ of $R G$, respectively, belong to the same fringe of second order. These points have been moved a distance $2 p$. The $n$th fringe is produced by a relative displacement of $n p$.

Thus, the moire fringes are the loci of points with relative displacement in the direction perpendicular to the lines of the reference grating at the deformed state of the specimen equal to an integer number of the pitch of the master grating. Each fringe is characterized by a parameter, $n$, which is called fringe order.

# 力学代考

## 物理代写|力学代写mechanics代考|Moiré Patterns Formed by Circular, Radial

$$(x-c)^2+y^2=k^2 p^2 \quad(k=\pm 1, \pm 2, \ldots)(x+c)^2+y^2=l^2 p^2(1+\lambda)^2(l=\pm 1, \pm 2, \ldots)$$

$\$ \begin{aligned \&=4\left(1+\backslash \backslash\right. ambda) 2 \backslash left \left[(x+c)^{\wedge} 2+y^{\wedge} 2 \backslash\right. right] \backslash left \left[(x c)^{\wedge} 2+y^{\wedge} 2 \backslash\right. right ] \backslash quad (m=\backslash \operatorname{pm} 1, \backslash \operatorname{lpm} 2, \backslash \backslash dots ) lend{对齐} Themoir 伩皀pattern formedbytwocirculargratingsofdifferentpitchesandatadistanceapartfor \ \lambda>0 \isshowninFig. 3.9. UsingEq. (3.15)thecom \mathrm{x}^{\wedge} 2+\mathrm{y}^{\wedge} 2=\mathrm{c}^{\wedge} \mathrm{s} \ \

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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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