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# 网课代修|Matlab代写|SIMPLE HARMONIC MOTION

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## 网课代修|Matlab代写|SIMPLE HARMONIC MOTION

Second-order, linear, ordinary differential equations often arise in mechanical or electrical problems. The purpose of this section is to illustrate how the techniques that we just derived may be applied to these problems.

We begin by considering the mass-spring system illustrated in Figure $2.2 .1$ where a mass $m$ is attached to a flexible spring suspended from a rigid support. If there were no spring, then the mass would simply fall downward due to the gravitational force $m g$. Because there is no motion, the gravitational force must be balanced by an upward force due to the presence of the spring. This upward force is usually assumed to obey Hooke’s law, which states that the restoring force is opposite to the direction of elongation and proportional to the amount of elongation. Mathematically the equilibrium condition can be expressed $m g=k s$.

Consider now what happens when we disturb this equilibrium. This may occur in one of two ways: We could move the mass either upward or downward and then release it. Another method would be to impart an initial velocity to the mass. In either case, the motion of the mass/spring system would be governed by Newton’s second law, which states that the acceleration of the mass equals the imbalance of the forces. If we denote the downward displacement of the mass from its equilibrium position by positive $x$, then
$$m \frac{d^{2} x}{d t^{2}}=-k(s+x)+m g=-k x,$$
since $k s=m g$. After dividing Equation $2.2 .1$ by the mass, we obtain the second-order differential equation
$$\frac{d^{2} x}{d t^{2}}+\frac{k}{m} x=0,$$
or
$$\frac{d^{2} x}{d t^{2}}+\omega^{2} x=0,$$
where $\omega^{2}=k / m$ and $\omega$ is the circular frequency. Equation $2.2 .3$ describes simple harmonic motion or free undamped motion. The two initial conditions associated with this differential equation are
$$x(0)=\alpha, \quad x^{\prime}(0)=\beta$$

## 网课代修|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 $A h \rho$, 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
$$x^{\prime \prime}+\frac{\rho_{w} g}{\rho h} x=0 .$$

d2Xd吨2+ķ米X=0,

d2Xd吨2+ω2X=0,

X(0)=一个,X′(0)=b

## 网课代修|MATLAB代写|VIBRATION OF FLOATING BODIES

ρ一个HX′′+一个Gρ在X=0,

X′′+ρ在GρHX=0.

## 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。