Posted on Categories:Vibration Mechanics, 振动力学, 物理代写

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## 物理代写|振动力学代写Vibration Mechanics代考|Closely Distributed Natural Modes of a Symmetric Structure

There are numerous symmetric structures in engineering. For example, airplanes and ground vehicles must be symmetric about a plane parallel to their moving direction. That is, they have mirror-symmetry. The other example of symmetry covers the bladed disc of an aero-engine in Fig. 1.8a and the radio-telescopic antenna of satellite Spektru-R in Fig. 1.8b, where both structures are exactly identical after they are rotated through an angle of $2 \pi / n$ about a central axis and exhibit the cyclic symmetry or a compound term cyclosymmetry. The designs of those symmetric structures not only show the beauty in appearance, but also meet the requirements of their functions.
It is well known that the system symmetry enables one to simplify the dynamic computation of the system. For a structure with mirror symmetry, it is easy to divide an arbitrary dynamic load to the structure into two parts, namely, a symmetric part and an anti-symmetric part. They excite the symmetric vibration and anti-symmetric vibration, respectively. Hence, one can take a half of the structure and compute respectively the symmetric and anti-symmetric vibrations, and then sum up the two vibrations to get the vibration due to an arbitrary dynamic load. For a structure with cyclosymmetry, the past decades since the 1970 s have seen several approaches to simplify the computation of natural vibrations of the structure. The approaches enable one to compute the natural vibrations of the structure from those of one of $n$ sectors. The bladed disc of an aero-engine or a turbo-engine usually has so many blades that $n>>10$ holds true in practice. These approaches, therefore, can save a great amount of computational cost.

In recent years, engineers have not worried so much about the computational cost with an integration of those approaches based on structure symmetry and fast development of computational techniques. Yet, they have encountered some new problems from structure symmetry as follows.

## 物理代写|振动力学代写Vibration Mechanics代考|Problems of Concern

(1) Problem 5A: Poor repeatability of modal tests of a symmetric structure
The engineering consultations that the author attended dealt with several kinds of symmetric structures. The problems often proposed by engineers were about the natural modes closely distributed in frequency domain and the poor repeatability in modal tests. For instance, the modal test of a bladed disc of a turbo-engine usually led to many sets of natural vibrations with very close frequencies but different mode shapes. One mode shape looked like the rotation of the other mode shape about the central axis by an angle. However, these mode shapes exhibited very poor repeatability in further modal test. As well known, the poor repeatability of industrial products in tests before use is a tough issue. This is particularly a big issue for any space structures in ground vibration tests before launch. Then, what is the reason of the above problem?
(2) Problem 5B: Natural vibrations of a cyclosymmetric structure with the dynamic deformation at the central axis

In the previous researches on the natural vibrations of cyclosymmetric structures, most publications dealt with annular structures shown in Fig. 1.8a, where the central axis of the structure is either clamped or hollowed, and no dynamic deformation needs to be studied. In practice, however, a cyclosymmetric structure may undergo the dynamic deformation at the central axis. A typical example is the radio-telescopic antenna of satellite Spektr-R in Fig. 1.8b. Then, it is reasonable to check the difference between the two kinds of cyclosymmetric systems, and whether the research on the natural vibrations of cyclosymmetric structures of the first kind can be extended to the second kind.

Among the early studies on the natural vibrations of cyclosymmetric structures of the first kind, D. L. Thomas in the UK analyzed the standing waves in the structures and pointed out the existence of standing waves along the circumferential direction. Many people thought that the standing waves in a cyclosymmetric structure of the second kind would be no longer those along the circumferential direction since the central DoFs of the cyclosymmetric structure of the second kind were not fixed. Therefore, it is difficult to extend the vibration analysis of cyclosymmetric structures of the first kind to those of the second kind.

## 物理代写|振动力学代写振动力学代考|关注的问题

(1)问题5A:对称结构模态试验的可重复性差

(2)问题5B:具有中心轴动变形的环对称结构的自然振动

## 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。

Posted on Categories:Vibration Mechanics, 振动力学, 物理代写

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## 物理代写|振动力学代写Vibration Mechanics代考|Problems of Concern

(1) Problem 2A: Abnormal effects of damping coefficients
In the hydro-elastic mounting, the oil passes through a set of orifices and produces damping. Hence, it is possible to control damping effect by adjusting the number of orifices and the radii of orifices. Yet, both computations and experiments showed that the resonance suppression seemed almost unchanged when adjusting these parameters. Therefore, engineers felt strange why the hydro-damping did not produce expected results and contradicted to their intuition of mechanics.
(2) Problem 2B: How to determine the degree of freedom of a system In classical dynamics, the degree of freedom, or the DoF for short, of a system is the minimal number of independent generalized coordinates of the system. Then, the degree of freedom of a lumped parameter system is the number of lumped inertial components, such as a lumped mass. The vibration system in Fig. 1.4b has a single lumped mass and seems to be a system of single degree of freedom, or $S D o F$ for short. Yet, it is not possible to use only the vertical displacement $u_1$ of the lumped mass to describe all motions of the system. That is, it is necessary to use the other two vertical displacements $u_2$ and $u_3$ in Fig. 1.4b to describe the motions of two connection points of dashpots and springs. As such, the system seems to have three degrees of freedom, or 3-DoFs for short. Engineers wondered how many degrees of freedom the system possesses.

## 物理代写|振动力学代写Vibration Mechanics代考|Basic Ideas of Study

In fact, the two problems come from the same origin. In elementary textbooks, most vibration systems do not have a spring and a dashpot in serial shown in Fig. 1.4b, where the two connection points do not have any lumped mass so that two degrees of freedom of the system degenerate. The contribution of the dashpot to the system damping ratio of the system, thus, changes.

To understand the above issue, consider a dynamic serial system shown in Fig. 1.5, where a dashpot and a spring in serial constitute Maxwell’s fluidic component ${ }^{4,5}$. Apart from the displacement $u_1(t)$ of the mass block, one needs to use displacement $u_2(t)$ to describe the motion of the connection point between the dashpot and the spring. Hence, the system is not an SDoF system. Assume that the system has 2-DoFs, but the lumped mass at the connection point vanishes. Thus, the dynamic equations of the system satisfy
$$\left{\begin{array}{l} m \frac{\mathrm{d}^2 u_1(t)}{\mathrm{d} t^2}+c\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}-\frac{\mathrm{d} u_2(t)}{\mathrm{d} t}\right]=0 \ c\left[\frac{\mathrm{d} u_2(t)}{\mathrm{d} t}-\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]+k u_2(t)=0 \end{array}\right.$$
Eliminating displacement $u_2$ from Eq. (1.2.1) leads to an ordinary differential equation in terms of velocity $\mathrm{d} u_1(t) / \mathrm{d} t$, namely,
$$\frac{1}{k} \frac{\mathrm{d}^2}{\mathrm{~d} t^2}\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]+\frac{1}{c} \frac{\mathrm{d}}{\mathrm{d} t}\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]+\frac{1}{m}\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]=0$$

## 物理代写|振动力学代写振动力学代考|研究的基本思想

$$\left{\begin{array}{l} m \frac{\mathrm{d}^2 u_1(t)}{\mathrm{d} t^2}+c\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}-\frac{\mathrm{d} u_2(t)}{\mathrm{d} t}\right]=0 \ c\left[\frac{\mathrm{d} u_2(t)}{\mathrm{d} t}-\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]+k u_2(t)=0 \end{array}\right.$$

$$\frac{1}{k} \frac{\mathrm{d}^2}{\mathrm{~d} t^2}\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]+\frac{1}{c} \frac{\mathrm{d}}{\mathrm{d} t}\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]+\frac{1}{m}\left[\frac{\mathrm{d} u_1(t)}{\mathrm{d} t}\right]=0$$

## 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。

Posted on Categories:Vibration Mechanics, 振动力学, 物理代写

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## 物理代写|振动力学代写Vibration Mechanics代考|Start of Research-Oriented Study

Vibration refers to the oscillatory motion of a body or a mechanical/structural system about its equilibrium position. It was one of the earliest physical phenomena that ancient scholars observed and studied, and is one of the most important dynamic problems that engineers today need to solve for designing, manufacturing/constructing and operating a great variety of industrial products. As a consequence, vibration has been a kind of popular engineering problems.

The industrial products over the past decades have seen a change from the static design and dynamic validation to the dynamic design. Hence, engineers should lay a solid foundation of vibration mechanics so as to properly solve various dynamic problems of industrial products from the design stage to the operational stage, and make further innovations in future.

In engineering education, vibration mechanics has served as a core for several courses, such as mechanical vibrations and structural dynamics, for different educational programs. The textbooks of those courses look standard and conventional. In order to conduct a kind of research-oriented studies, this chapter assists readers to have a new start of learning vibration mechanics, that is, the research-oriented study starting from open problems.

## 物理代写|振动力学代写Vibration Mechanics代考|Needs for Research-Oriented Study

Recent years have witnessed the rapid development of both computational mechanics and associated software. As such, the dynamic modeling, computation, analysis and design of industrial products have greatly relied on the software of computational mechanics. The above development, thus, has shortened the courses of mechanics in the engineering education of most countries. As a consequence, the courses of mechanical vibrations and structural dynamics have been simpler and simpler, with a great reduction in fundamental theory and experimental training. The above trend has produced a negative influence on the quality of engineering education.

For example, the engineers of new generation usually do not establish the simple models of mechanics for their industrial products so as to analyze the influence of model parameters on the dynamic behaviors of the products, but rely on the software of computational mechanics and conduct a huge amount of numerical simulations. They often present many simulation results to the chief-engineer or the third party in evaluations, but are not able to well explain the simulation results from the physical mechanisms. Some recent failures in aerospace engineering, for instance, have indicated that the engineers of new generation had not gained an insight into the dynamic problems because of the weakness in fundamental concepts and theoretical foundations.

The author has studied the dynamics and control of aerospace structures over the past four decades, and participated in many consultative discussions with aerospace engineers. The above experience has enabled the author to recognize the importance of both fundamental concepts and theoretical foundations in engineering education. Many senior professors and experienced engineers feel that the current courses of mechanical vibrations and structural dynamics are not able to ensure the high quality of engineering education.

In the recent discussions about teaching reforms in China, there are two kinds of appeals from the professors teaching those courses. One is to increase the teaching periods for undergraduate students and return to the conventional model of theoretical and experimental trainings. The other is to increase the related course periods for graduate students so as to solve the problems. These two suggestions, however, may not be feasible in practice.

## 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。