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物理代写|振动力学代写Vibration Mechanics代考|ENGIN4301 Regularity

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物理代写|振动力学代写Vibration Mechanics代考|Regularity

The term “regularity” usually means the repeatability of identical or similar shapes in geometry. For instance, the mast of satellite NuSTAR for detecting black-holes in universe was composed of many identical and deployable cells as shown in Fig. 2.2. These identical cells enabled the mast to be stowed into a very small volume before the launch of space mission and then deployed with a telescope at one end in orbit, exhibiting the beautiful regularity.

During the elementary course of vibrations, one has touched upon many regular patterns. For example, the periodic vibration in Fig. $2.3$ appears almost everywhere, exhibiting the regular time history in Fig. 2.3a and the beautiful phase portrait in Fig. 2.3b. The other example is the quasi-periodic vibration shown in Fig. 2.4a. This vibration consists of two harmonic vibrations with very close frequencies and exhibits a regular beat phenomenon in the time history.

In the theory of linear vibrations, many formulae show regularity. For example, if the mode shape vectors in Eq. (2.2.4) are normalized with respect to the modal mass coefficients first and then used to define a mode shape matrix as follows
$$\overline{\boldsymbol{\Phi}} \equiv\left[\overline{\boldsymbol{\varphi}}_1 \overline{\boldsymbol{\varphi}}_2 \cdots \overline{\boldsymbol{\varphi}}_n\right] \in \mathbb{R}^{n \times n}, \quad \overline{\boldsymbol{\varphi}}_r \equiv \frac{1}{\sqrt{M_r}} \boldsymbol{\varphi}_r, \quad r=1,2, \ldots, n$$
it is easy to express Eq. (2.2.5) as a regular form as follows
$$\overline{\boldsymbol{\Phi}}^{\mathrm{T}} \boldsymbol{M} \overline{\boldsymbol{\Phi}}=\boldsymbol{I}_n \in \mathbb{R}^{n \times n}, \quad \overline{\boldsymbol{\Phi}}^{\mathrm{T}} \boldsymbol{K} \overline{\boldsymbol{\Phi}}=\boldsymbol{\Omega}^2 \equiv \underset{1 \leq r \leq n}{\operatorname{diag}}\left[\omega_r^2\right] \in \mathbb{R}^{n \times n}$$

物理代写|振动力学代写Vibration Mechanics代考|Singularity

Singularity here refers to an extreme case of strange phenomenon. In the history of mechanics, the earliest singularity, which received much attention, might be either the static buckling of a rod under compression or the resonance of a suspended bridge. These singular phenomena did not exhibit any evidence in advance, but occurred abruptly. Later on, engineers encountered many similar phenomena, such as the flutter of an aircraft wing and the gallop of a steel bridge of long span.

Most singular phenomena brought about disasters to humankind. However, once the physical mechanism behind a singular phenomenon is clear, the utilizations of the singular phenomenon enable people to enjoy the beauty of singularity.

Example 2.2.4 Consider an SDoF system subject to a harmonic excitation of basement and define the displacement transmissibility $T_d$ for the system as the ratio of displacement amplitude of the system to the displacement amplitude of the basement. Figure $2.9$ shows the transmissibility $T_d$ with an increase of dimensionless excitation frequency $\lambda \equiv \omega / \omega_n$ for several damping ratio $\zeta$ of the system.

In history, people observed the resonance of such a system when the excitation frequency $\omega$ approached the natural frequency $\omega_n$ of the system, namely, a peak at $\lambda \approx 1$ in Fig. 2.9. As most structures had a tiny damping ratio, say, $\zeta \in(0.001,0.05)$, the resonant peak was very high and gave rise to disasters. When people got to know the rule of displacement transmissibility of the above system, they could not only avoid or reduce the resonance, but also utilize the lower displacement transmissibility in a frequency range of $\lambda>\sqrt{2}$ in order to design various vibration isolation systems.
In the studies of vibration problems, some singularities come from the assumptions for modeling a real system. For instance, the un-damped model of a real system exhibits infinitely large resonance, and an idealized system with exact symmetry has repeated natural frequencies. The following example demonstrates such a singularity from a simplified system with symmetry.

物理代写|振动力学代写振动力学代考|规律性

$$\overline{\boldsymbol{\Phi}} \equiv\left[\overline{\boldsymbol{\varphi}}_1 \overline{\boldsymbol{\varphi}}_2 \cdots \overline{\boldsymbol{\varphi}}_n\right] \in \mathbb{R}^{n \times n}, \quad \overline{\boldsymbol{\varphi}}_r \equiv \frac{1}{\sqrt{M_r}} \boldsymbol{\varphi}_r, \quad r=1,2, \ldots, n$$
，则很容易将Eq.(2.2.5)表示为正则形式，如下所示
$$\overline{\boldsymbol{\Phi}}^{\mathrm{T}} \boldsymbol{M} \overline{\boldsymbol{\Phi}}=\boldsymbol{I}_n \in \mathbb{R}^{n \times n}, \quad \overline{\boldsymbol{\Phi}}^{\mathrm{T}} \boldsymbol{K} \overline{\boldsymbol{\Phi}}=\boldsymbol{\Omega}^2 \equiv \underset{1 \leq r \leq n}{\operatorname{diag}}\left[\omega_r^2\right] \in \mathbb{R}^{n \times n}$$

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。