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

The moiré methods studied in the previous sections of this chapter concerned with inplane displacements $u, v$ and strains $\varepsilon_x, \varepsilon_y, \gamma_{x y}$. It was assumed that the out-of-plane displacements are small and they do not affect the in-plane displacements. In this section, we will present a method for measuring out-of-plane displacements independently from in-plane displacements. Determination of out-of-plane displacements is important in-plane stress problems because the out-of-plane strain is proportional to the sum of the two in-plane principal (or normal) stresses.

For the determination of out-of-plane displacements, we will present the shadow moiré method. The method uses the superposition of a grating and its shadow on the surface of the specimen. The surface of the specimen is coated with a matte finish and a master grating is placed in front of the surface (Fig. 3.16). The shadow of the master grating on the surface of the specimen constitutes the specimen grating. When a collimated beam of light is directed at an angle to the master grating interference of the master grating and its shadow on the surface takes place, and moiré fringes are formed.

Let $i$ be the angle at which the collimated beam of light impinges on the specimen and $\rho$ be the angle of viewing the obtained fringe pattern. Both angles are measured with respect to the normal to the master grating. Consider two adjacent fringe locations created from two points on the surface of the specimen at distances $w_1$ and $w_2$ from the master grating. The displacement of the “specimen grating” (the shadow of the master grating on the surface of the body) at direction perpendicular to the lines of the master grating, is
$$\delta=(C D)=\left(w_2-w_1\right)(\tan i+\tan \rho)$$
For two points at successive moiré fringes, their relative displacement is equal to the pitch $p$ of the master grating. Thus, we have
$$\left(w_2-w_l\right)(\tan i+\tan \rho)-p$$

## 物理代写|力学代写mechanics代考|Measurement of Out-Of-Plane Slopes

Measurement of out-of-plane slopes of bending plates is important for the determination of stresses. The plate curvatures are obtained by simple differentiation of the slopes and stresses are linearly related to the curvatures. The plate curvatures $\rho_x$ and $\rho_y$ are related to the out-of-plane slopes by the following equations
$$\frac{1}{\rho_x}=-\frac{\partial^2 w}{\partial x^2}, \frac{1}{\rho_y}=-\frac{\partial^2 w}{\partial y^2}$$
and the normal stresses $\sigma_x$ and $\sigma_y$ are related to the curvatures by
$$\sigma_x=\frac{E z}{1-v^2}\left(\frac{1}{\rho_x}+v \frac{1}{\rho_y}\right), \sigma_y=\frac{E z}{1-v^2}\left(\frac{1}{\rho_y}+v \frac{1}{\rho_x}\right)$$
where $E$ is the modulus of elasticity, $v$ is the Poisson’s ratio, and $z$ is the coordinate variable along the thickness of the plate.

The out-of-plane displacement analysis presented in the previous section can be used for the determination of the stresses by twice differentiating the displacements. A direct determination of slopes allows the calculation of curvatures by performing one differentiation. A moiré method that gives directly the slopes $\partial w / \partial x$ and $\partial w / \partial y$ were developed by Ligtenberg.

The method is based on measurement of the displacement of the image of a master grating placed far from the plate after it is reflected from the surface of the plate. The reflected image interferes with the master grating yielding fringes which represent the partial slopes of the plate. The specimen surface is made reflective instead of matte as in the previous case of measurement of out-of-plane displacements. The reflected image of the grating on the specimen does not depend on the angle of the incident light, and therefore, collimated light is not needed.

Figure 3.17 presents the optical arrangement of the Ligtenberg method. A coarse master grating is placed at a large distance $d$ from the plate under study. The plate is viewed from a camera placed at a hole in the center of the grating. Consider a point $P$ of the plate which is reflected at point $Q$ on the screen. When the plate is deformed point $P$ moves to point $P^{\prime}$ which is reflected to point $R$ on the screen.

# 力学代考

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

$$\delta=(C D)=\left(w_2-w_1\right)(\tan i+\tan \rho)$$

$$\left(w_2-w_l\right)(\tan i+\tan \rho)-p$$

## 物理代写|力学代写mechanics代考|Measurement of Out-Of-Plane Slopes

$$\frac{1}{\rho_x}=-\frac{\partial^2 w}{\partial x^2}, \frac{1}{\rho_y}=-\frac{\partial^2 w}{\partial y^2}$$

$$\sigma_x=\frac{E z}{1-v^2}\left(\frac{1}{\rho_x}+v \frac{1}{\rho_y}\right), \sigma_y=\frac{E z}{1-v^2}\left(\frac{1}{\rho_y}+v \frac{1}{\rho_x}\right)$$

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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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