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## 数学代写|有限元代写Finite Element Method代考|Solution Process

Looking at the mesh in Figure 8.6 , one can see that quadrilateral shell elements are used. Therefore, the equations for a linear, quadrilateral shell element must be formulated by ABAQUS. As before, the formulation of the element matrices would require information from the nodal cards and the element connectivity cards. The element type used here is S4, representing four nodal shell elements. There are other types of shell elements available in the ABAQUS element library.

After the nodal and element cards, next to be considered would be the property and material cards. The properties for the shell element used here must be defined, which in this case includes the material used and the thickness of the shell elements. The material cards are similar to those of the case study in Chapter 7 except that here the density of the material must be included, since we are not carrying out a static analysis as in Chapter 7.
The boundary $(\mathrm{BC})$ cards then define the boundary conditions on the model. In this problem, we would like to obtain only the flexural vibration modes of the motor, hence the components of displacements in the plane of the motor are not actually required. As mentioned, this is very much the characteristic of the plate elements. Therefore, DOFs 1, 2 and 6 corresponding to the $x$ and $y$ displacements, and rotation about the $z$ axis, is constrained. The other boundary condition would be the constraining of the displacements of the nodes at the centre of the motor.

Without the need to define any external loadings, the control cards then define the type of analysis ABAQUS would carry out. ABAQUS uses the sub-space iteration scheme by default to evaluate the eigenvalues of the equation of motion. This method is a very effective method of determining a number of lowest eigenvalues and corresponding eigenvectors for a very large system of several thousand DOFs. The procedure is outlined in the case study in Chapter 5. Finally, the output control cards define the necessary output required by the analyst.

## 数学代写|有限元代写Finite Element Method代考|Result and Discussion

Using the input file above, an eigenvalue extraction is carried out in ABAQUS. The output is extracted from the ABAQUS results file showing the first eight natural frequencies and tabulated in Table 8.1. The table also shows results obtained from using triangular elements as well as a finer mesh of quadrilateral elements. It is interesting to note that for certain modes, the eigenvalues and hence the frequencies are repetitive with the previous one. This is due to the symmetry of the circular rotor structure. For example, modes 1 and 2 have the same frequency, and looking at their corresponding mode shapes in Figures 8.7 and 8.8, respectively, one would notice that they are actually of the same shape but bending at a plane $90^{\circ}$ from each other. As such, many consider this as one single mode. Therefore, though eight eigenmodes are extracted, it is effectively equivalent to only five eigenmodes However, to be consistent with the result file from ABAQUS, all the modes extracted will be shown here. Figure 8.9 to 8.14 show the other mode shapes from this analysis. Remember that, since the in-plane displacements are already constrained, these modes are only the flexural modes of the rotor.

Comparing the natural frequencies obtained using 768 triangular elements with those obtained using the quadrilateral elements, one can see that the frequencies are generally higher using the triangular elements. Note that for the same number of nodes, using the quadrilateral elements requires half the number of elements. The results obtained using 384 quadrilateral elements do not differ much from those that use 1280 elements. This again

## 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。