# 统计代写|主成分分析代写Principal Component Analysis代考|”Mode Superposition Method in Finite Element Analysis: A Study on Harmonic Forcing Functions”

The mode superposition method leverages the orthogonality of the principal modes of vibration to simplify the analysis of a finite element model subjected to harmonic forcing functions. When the system is exposed to a harmonic force F(t)=F0sin⁡(ωft)F(t) = F_0 \sin(\omega_f t)F(t)=F 0 ​ sin(ω f ​ t), the method breaks down the complex system dynamics into P uncoupled ordinary differential equations, each associated with a single generalized coordinate pip_ip i ​ .

Given a system with P degrees of freedom, the steady-state response of each generalized coordinate is derived as a sum of particular solutions corresponding to each harmonic force component present in the forcing function. According to Equation 10.122, the generalized displacement pi(t)p_i(t)p i ​ (t) is expressed as a weighted sum of harmonic oscillations with frequencies matching those of the forcing terms, where the weights are determined by the modal participation factors (elements of the normalized modal matrix).

In the context of Example 10.3 with a three degrees-of-freedom system and a downward force applied only to mass 2, the generalized forces are computed through the transformation involving the transpose of the normalized modal matrix. Each generalized coordinate has its own ODE, which can be solved independently to yield the time-dependent generalized displacements.

Once these generalized displacements are obtained, the actual nodal displacements are recovered by transforming them back to the physical coordinate system using the inverse of the normalized modal matrix, as per Equation 10.112.

The provided calculations show that the displacement responses of masses 1, 2, and 3 are sinusoidal functions of time with amplitudes proportional to F0F_0F 0 ​ and inversely proportional to the difference between the square of the forcing frequency ωf\omega_fω f ​ and the square of the natural frequencies ωi2\omega_i^2ω i 2 ​ .

Several key points are highlighted:

Each mass undergoes a sinusoidal oscillation at the frequency of the forcing function. The magnitudes of the displacements reflect the influence of the system’s natural modes, as evident from the involvement of the modal participation factors and the natural frequencies. These solutions describe only the forced motion of the masses; any free vibrations would be superimposed on this forced response. Damping effects are not considered in this simplified model. Although the mode superposition method appears intricate when done manually, it is well-suited to computational implementation and can be efficiently executed using digital computers. Further discussion about computer-based applications of the method is reserved for subsequent materials.

### 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。