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MATLAB是一个编程和数值计算平台，被数百万工程师和科学家用来分析数据、开发算法和创建模型。

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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
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|matlab代写|Vibration of floating bodies

Consider a solid cylinder of radius $a$ that is partially submerged in a bath of pure water as shown in Figure 2.2.2. Let us find the motion of this cylinder in the vertical direction assuming that it remains in an upright position.

If the displacement of the cylinder from its static equilibrium position is $x$, the weight of water displaced equals $A g \rho_{w} x$, where $\rho_{w}$ is the density of the water and $g$ is the gravitational acceleration. This is the restoring force according to the Archimedes principle. The mass of the cylinder is Ah $\rho$, where $\rho$ is the density of cylinder. From Newton’s second law, the equation of motion is
$$\rho A h x^{\prime \prime}+A g \rho_{w} x=0,$$
or
$$x^{\prime \prime}+\frac{\rho_{w} g}{\rho h} x=0 .$$
From Equation 2.2.20 we see that the cylinder will oscillate about its static equilibrium position $x=0$ with a frequency of
$$\omega=\left(\frac{\rho_{w} g}{\rho h}\right)^{1 / 2} .$$
When both $A$ and $B$ are nonzero, it is often useful to rewrite the homogeneous solution, Equation 2.2.5, as
$$x(t)=C \sin (\omega t+\varphi)$$
to highlight the amplitude and phase of the oscillation. Upon employing the trigonometric angle-sum formula, Equation $2.2 .22$ can be rewritten
$$x(t)=C \sin (\omega t) \cos (\varphi)+C \cos (\omega t) \sin (\varphi)=A \cos (\omega t)+B \sin (\omega t) .$$
From Equation 2.2.23, we see that $A=C \sin (\varphi)$ and $B=C \cos (\varphi)$. Therefore,
$$A^{2}+B^{2}=C^{2} \sin ^{2}(\varphi)+C^{2} \cos ^{2}(\varphi)=C^{2},$$
and $C=\sqrt{A^{2}+B^{2}}$. Similarly, $\tan (\varphi)=A / B$. Because the tangent is positive in both the first and third quadrants and negative in both the second and fourth quadrants, there are two possible choices for $\varphi$. The proper value of $\varphi$ satisfies the equations $A=C \sin (\varphi)$ and $B=C \cos (\varphi)$

## 数学代写|matlab代写|DAMPED HARMONIC MOTION

Free harmonic motion is unrealistic because there are always frictional forces that act to retard motion. In mechanics, the drag is often modeled as a resistance that is proportional to the instantaneous velocity. Adopting this resistance law, it follows from Newton’s second law that the harmonic oscillator is governed by
$$m \frac{d^{2} x}{d t^{2}}=-k x-\beta \frac{d x}{d t},$$
where $\beta$ is a positive damping constant. The negative sign is necessary since this resistance acts in a direction opposite to the motion.

Dividing Equation $2.3 .1$ by the mass $m$, we obtain the differential equation of free damped motion,
$$\frac{d^{2} x}{d t^{2}}+\frac{\beta}{m} \frac{d x}{d t}+\frac{k}{m} x=0,$$ or
$$\frac{d^{2} x}{d t^{2}}+2 \lambda \frac{d x}{d t}+\omega^{2} x=0 .$$
We have written $2 \lambda$ rather than just $\lambda$ because it simplifies future computations. The auxiliary equation is $m^{2}+2 \lambda m+\omega^{2}=0$, which has the roots
$$m_{1}=-\lambda+\sqrt{\lambda^{2}-\omega^{2}}, \quad \text { and } \quad m_{2}=-\lambda-\sqrt{\lambda^{2}-\omega^{2}}$$
From Equation 2.3.4 we see that there are three possible cases which depend on the algebraic sign of $\lambda^{2}-\omega^{2}$. Because all of the solutions contain the damping factor $e^{-\lambda t}$, $\lambda>0, x(t)$ vanishes as $t \rightarrow \infty$.

# matlab代写

## 数学代写|matlab代写|Vibration of floating bodies

$$\rho A h x^{\prime \prime}+A g \rho_{w} x=0,$$

$$x^{\prime \prime}+\frac{\rho_{w} g}{\rho h} x=0 .$$

$$\omega=\left(\frac{\rho_{w} g}{\rho h}\right)^{1 / 2} .$$

$$x(t)=C \sin (\omega t+\varphi)$$

$$x(t)=C \sin (\omega t) \cos (\varphi)+C \cos (\omega t) \sin (\varphi)=A \cos (\omega t)+B \sin (\omega t) .$$

$$A^{2}+B^{2}=C^{2} \sin ^{2}(\varphi)+C^{2} \cos ^{2}(\varphi)=C^{2},$$

## 数学代写|matlab代写|DAMPED HARMONIC MOTION

$$m \frac{d^{2} x}{d t^{2}}=-k x-\beta \frac{d x}{d t},$$

$$\frac{d^{2} x}{d t^{2}}+\frac{\beta}{m} \frac{d x}{d t}+\frac{k}{m} x=0,$$

$$\frac{d^{2} x}{d t^{2}}+2 \lambda \frac{d x}{d t}+\omega^{2} x=0 .$$

$$m_{1}=-\lambda+\sqrt{\lambda^{2}-\omega^{2}}, \quad \text { and } \quad m_{2}=-\lambda-\sqrt{\lambda^{2}-\omega^{2}}$$

## 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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