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## 物理代写|理论力学作业代写Theoretical Mechanics代考|Relative Velocity

We consider two points on the planar figure $A$ and $B$, which have the velocity $\boldsymbol{v}{A}$ and $v{B}$, respectively, as schematized in Fig. 9.1. We normally name point $A$ as the base point, as it is a reference point. There is a relative velocity $\boldsymbol{v}{B A}$, with the meaning that $A$ is the base point. As a result, we know that $v{B A}$ is not equal to $v_{A B}$. According to the velocity superposition, one has
$$v_{B}=v_{A}+v_{B A},$$
where the direction of the relative velocity $v_{B A}$ is perpendicular to the line $A B$, as point $B$ rotates with respect to point $A$ in a circular motion. From the above formula, we can solve the velocity of any point on the planar figure if the base point is given. Therefore, this method of velocity composition is called “Method of base point”.
As a consequence, if we decompose the above velocities in the direction of line $A B$, then one has

\begin{aligned} \left.\boldsymbol{v}{B}\right|{A B} &=\left.\boldsymbol{v}{A}\right|{A B}+\left.v_{B A}\right|{A B} \ &=\left.v{A}\right|{A B} . \end{aligned} This means the projections of the velocities of the two points are equal along their connection line. In fact, this is the second method to study the velocity composition, which is termed as “Method of velocity projection”. As shown in Fig. 9.2, if we have known the angles between the velocities and line $A B$, we then have the relation $$v{A} \cos \alpha=v_{B} \cos \beta .$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Instantaneous Center

Next, we will introduce a simple and convenient method, i.e., the instantaneous center method. It is true that there is always one special point on the planar figure with zero velocity. Herein we omit the process of proof, and we only remember this conclusion. This point is called “instantaneous center of velocity”, or “instantaneous center”. If $P$ is the instantaneous center, then $v_{P}=0$. We also adopt the method of base point, where we select the instantaneous center as the base point. Consequently, we have
\begin{aligned} v_{A} &=v_{P}+v_{A P} \ &=v_{A P} \end{aligned}

From the above discussions, we know that the velocity direction of point $A$ is normal to the line $A P$. It should be noticed that $P$ is only a point with an instantaneous zero velocity, and we do not know what the state in the next time will be. At this instantaneous time, point $P$ is similar to a circle center, and point $A$ is in a circular motion only at this time. If the angular velocity of the rigid body is $\omega$, the velocity of point $A$ is
$$v_{A}=v_{A P}=\omega \cdot A P$$
In contrast, how to find the instantaneous center if the velocity is given? Let us look at a planar figure with two velocities of two points, as shown in Fig. 9.3.

# 理论力学代写

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Relative Velocity

$$v_{B}=v_{A}+v_{B A}$$

$$\boldsymbol{v} B|A B=\boldsymbol{v} A| A B+v_{B A}|A B \quad=v A| A B .$$

$$v A \cos \alpha=v_{B} \cos \beta$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Instantaneous Center

$$v_{A}=v_{P}+v_{A P} \quad=v_{A P}$$

$$v_{A}=v_{A P}=\omega \cdot A P$$

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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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