Posted on Categories:Theoretical mechanics, 物理代写, 理论力学

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## 物理代写|理论力学代写Theoretical Mechanics代考|Rotation on a Fixed Axis

As shown in Fig. 8.3, if we impose a moment on a door, the door will be closed with respect to a fixed axis. This type of motion is also a basic motion, termed as rotation with respect to a fixed axis. In this rotation process, there always exists a fixed axis on the rigid body. This type of motion takes some special features. If we look through the positive direction of $z$-axis in Fig. 8.3, we can find the orbit of an arbitrary point on the rigid body in rotation, which is actually a circle.

To depict the motion of the rigid body, we introduce the angular displacement $\varphi$, whose sign also obeys the right-hand screw law. The angular velocity can be defined as
$$\omega=\lim _{\Delta t \rightarrow 0} \frac{\Delta \varphi}{\Delta t}=\dot{\varphi} .$$
The angular velocity can also be related with the frequency via
$$\omega=2 \pi f .$$
Introducing the concept of rotation velocity $n$ with the unit of $1 / \mathrm{min}$, one has

$$\omega=\frac{2 \pi n}{60}=\frac{\pi n}{30} .$$
Similarly, the angular acceleration can be further defined as
$$\varepsilon=\lim _{\Delta t \rightarrow 0} \frac{\Delta \omega}{\Delta t}=\dot{\omega}=\ddot{\varphi} .$$
The above relation can be analogous to the displacement, velocity, and acceleration defined in the last chapter.

If the rigid body is in rotation with respect to an axis, any point on the rigid body is in a circular motion. As shown in Fig. 8.4, the radius $r$ is the vertical distance from the axis to the arbitrary point, $s$ is the arc length, and $\varphi$ is the corresponding angular displacement. We then have the following geometric relation:
$$s=r \varphi .$$
Taking derivatives on both sides of the above equation, one has
\begin{aligned} &v=\dot{s}=r \dot{\varphi}=\omega r, \ &a_{\tau}=\dot{v}=\ddot{s}=r \ddot{\varphi}=\varepsilon r . \end{aligned}

## 物理代写|理论力学代写Theoretical Mechanics代考|Relative Velocity

We consider two points on the planar figure $A$ and $B$, which have the velocity $v_{A}$ 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 $v_{B A}$, 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 $\boldsymbol{v}_{B A}$ 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.\boldsymbol{v}{B A}\right|{A B} \ &=\left.\boldsymbol{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|理论力学代写Theoretical Mechanics代考|Rotation on a Fixed Axis

$$\omega=\lim {\Delta t \rightarrow 0} \frac{\Delta \varphi}{\Delta t}=\dot{\varphi}$$ 角速度也可以通过以下方式与频率相关 $$\omega=2 \pi f$$ 介绍旋转速度的概苡以 $1 / \mathrm{min}$,一个有 $$\omega=\frac{2 \pi n}{60}=\frac{\pi n}{30}$$ 类似地，角加速度可以进一步定义为 $$\varepsilon=\lim {\Delta t \rightarrow 0} \frac{\Delta \omega}{\Delta t}=\dot{\omega}=\ddot{\varphi} .$$

$$s=r \varphi .$$

$$v=\dot{s}=r \dot{\varphi}=\omega r, \quad a_{\tau}=\dot{v}=\ddot{s}=r \ddot{\varphi}=\varepsilon r .$$

## 物理代写|理论力学代写Theoretical Mechanics代考|Relative Velocity

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

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

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

## 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。