# 统计代写|主成分分析代写Principal Component Analysis代考|Comparing Approximate and Exact Solutions for Tapered Elastic Bar Displacement

In the provided solution, you have analyzed the displacement of the end of a tapered elastic bar under tension in three different ways:

(a) Single Element Model:

Assuming the cross-sectional area AAA as 3A04\frac{3A_0}{4} 4 3A 0 ​

​ at the mid-length, the effective spring constant kkk for the element is 3A0E4L\frac{3A_0E}{4L} 4L 3A 0 ​ E ​ . With a nodal force F1=0F_1 = 0F 1 ​ =0 due to the fixed support and F2=PF_2 = PF 2 ​ =P due to the applied tensile load, the displacement U2U_2U 2 ​ at the free end is calculated to be 4PL3A0E\frac{4PL}{3A_0E} 3A 0 ​ E 4PL ​ . (b) Two Equal-Length Elements Model:

For each element, the cross-sectional areas A1A_1A 1 ​ and A2A_2A 2 ​ are respectively 7A08\frac{7A_0}{8} 8 7A 0 ​

​ and 5A08\frac{5A_0}{8} 8 5A 0 ​

​ , leading to their respective spring constants k1k_1k 1 ​ and k2k_2k 2 ​ . The equilibrium equations for the two-element system are solved, considering that there is no external load at the central node. The displacement U3U_3U 3 ​ at the free end is obtained to be 48PL35A0E\frac{48PL}{35A_0E} 35A 0 ​ E 48PL ​ . (c) Exact Solution Using Integration:

By applying equilibrium and considering the varying cross-sectional area A(x)=A0(1−2xL)A(x) = A_0\left(1 – \frac{2x}{L}\right)A(x)=A 0 ​ (1− L 2x ​ ), the exact axial strain εx\varepsilon_xε x ​ is integrated along the length of the bar. The exact displacement UexactU_{exact}U exact ​ at the free end is calculated to be 2PLEA0ln⁡(2)\frac{2PL}{EA_0\ln(2)} EA 0 ​ ln(2) 2PL ​ . Comparing the approximate solutions from (a) and (b) to the exact solution (c):

The single-element solution has a relative error of around 1.333−1.3861.386×10 1.386 1.333−1.386 ​ ×10 The two-element solution has an error of approximately It’s crucial to note that while the displacement converges quickly with the two-element solution, the computed axial stress profile does not match the exact solution precisely. There is a visible discontinuity at the connecting node between the two elements, which would reduce as the number of elements increases and the mesh becomes finer. In practical finite element analysis, it’s essential to verify the convergence of derived quantities such as stresses to ensure accurate predictions.

### MATLAB代写

MATLAB 是一款高性能的技术计算语言，集成了计算、可视化和编程环境于一体，以熟悉的数学符号表达问题和解决方案。MATLAB 的基本数据元素是一个不需要维度的数组，使得能够快速解决带有矩阵和向量公式的多种技术计算问题，相比使用 C 或 Fortran 等标量非交互式语言编写的程序，效率大大提高。MATLAB 名称源自“矩阵实验室”（Matrix Laboratory）。最初开发 MATLAB 的目标是为了提供对 LINPACK 和 EISPACK 项目的矩阵软件的便捷访问，这两个项目代表了当时矩阵计算软件的先进技术。经过长期发展和众多用户的贡献，MATLAB 已成为数学、工程和科学入门及高级课程的标准教学工具，在工业界，MATLAB 是高效研究、开发和分析的理想选择。MATLAB 提供了一系列名为工具箱的特定应用解决方案集，这对广大 MATLAB 用户至关重要，因为它们极大地扩展了 MATLAB 环境，使其能够解决特定类别问题。工具箱包含了针对特定应用领域的 MATLAB 函数（M 文件），涵盖信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等诸多领域。