# 统计代写|主成分分析代写Principal Component Analysis代考|The Principle of Minimum Potential Energy and its Application in Elastic Systems

The principle of minimum potential energy states that for an elastic body in stable equilibrium under external loads and satisfying geometric boundary conditions, the total potential energy is minimized among all possible displacement states. The total potential energy consists of two parts: the strain energy (Ue) stored in the body due to deformation and the potential energy of applied loads (UF), which is the negative of the mechanical work done by these loads.

In the given example, the elastic strain energy is derived from the spring constants and nodal displacements of the system, while the potential energy of applied forces is determined by the work done by these forces. By setting the partial derivatives of the total potential energy with respect to each displacement to zero according to the principle, we obtain a set of algebraic equations equivalent to those derived using Castigliano’s First Theorem. These equations, when represented in matrix form, yield the same stiffness matrix.

Furthermore, the text introduces the global displacement vector {U} and the global stiffness matrix [K], which are key concepts in the finite element method. The strain energy of the system can be equivalently expressed through a matrix triple product involving the transpose of the displacement vector, the stiffness matrix, and the displacement vector itself. This relationship holds true for any elastic system and provides a systematic way to derive the stiffness matrix once the strain energy function is known in terms of nodal displacements.

### MATLAB代写

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