# 统计代写|主成分分析代写Principal Component Analysis代考|Derivation of Global Stiffness Matrix for Connected Spring Elements

The derivation of the global stiffness matrix for a system of connected spring elements begins with establishing the equilibrium conditions for each individual element. In the example given, there are two linear spring elements with different spring constants, k1 and k2, connected at node 2. Each spring’s equilibrium is expressed using its respective element stiffness matrix and nodal displacements:

For the first spring (Equation 2.8a):

k1 & -k1 -k1 & k1 \end{bmatrix} \begin{bmatrix} u(1)_1 u(1)_2 \end{bmatrix} = \begin{bmatrix} f(1)_1 f(1)_2 \end{bmatrix} \] For the second spring (Equation 2.8b): $\begin{bmatrix} k2 & -k2 -k2 & k2 \end{bmatrix} \begin{bmatrix} u(2)_1 u(2)_2 \end{bmatrix} = \begin{bmatrix} f(2)_2 f(2)_3 \end{bmatrix}$ These equations are then adjusted to reflect the global nodal displacements U1, U2, and U3 through displacement compatibility conditions (Equation 2.9): $u(1)_1 = U1, \quad u(1)_2 = U2, \quad u(2)_1 = U2, \quad u(2)_2 = U3$ Substituting these relations into the individual element equilibrium equations leads to: $\begin{bmatrix} k1 & -k1 -k1 & k1 \end{bmatrix} \begin{bmatrix} U1 U2 \end{bmatrix} = \begin{bmatrix} f(1)_1 f(1)_2 \end{bmatrix}$ $\begin{bmatrix} k2 & -k2 -k2 & k2 \end{bmatrix} \begin{bmatrix} U2 U3 \end{bmatrix} = \begin{bmatrix} f(2)_2 f(2)_3 \end{bmatrix}$ To combine these into a single system of equations, the matrices are expanded to a 3×3 format while accounting for the non-connected nodes. Adding the expanded equations results in a 3×3 system stiffness matrix [K]: $\begin{bmatrix} k1 & -k1 & 0 -k1 & k1+k2 & -k2 0 & -k2 & k2 \end{bmatrix} \begin{bmatrix} U1 U2 U3 \end{bmatrix} = \begin{bmatrix} F1 F2 F3 \end{bmatrix}$ This system matrix is derived from the equilibrium conditions of each node (Equation 2.14), ensuring that the forces acting on each node balance out. The final system stiffness matrix [K] captures the collective behavior of the interconnected spring elements and maintains properties typical of linear systems: symmetry and singularity due to the absence of constraints against rigid body motion. The process of assembling the global stiffness matrix is achieved by aligning the element stiffness contributions with their corresponding global nodes and combining them appropriately, a methodology that extends to more complex structures with multiple elements. The superposition principle is central to this assembly process, allowing us to build up the system-level behavior from the individual elemental responses.

### 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。