# 统计代写|主成分分析代写Principal Component Analysis代考|Mathematical Notes: Deriving Mass Matrices and Equations of Motion in Finite Element Analysis Using Energy-Based Approach

This text describes the transition from deriving mass matrices for simple systems like springs and bars to more complex structures using an energy-based approach rooted in Lagrangian mechanics. Starting with the simple harmonic oscillator, the conservation of mechanical energy principle is applied to derive its equation of motion, which aligns with Newton’s second law.

For a general three-dimensional body, the kinetic energy of a differential mass is calculated and then integrated over the entire volume to find the total kinetic energy of the body. When the displacement field is discretized using finite element interpolation functions, the element kinetic energy is transformed into a matrix form involving nodal velocities and the mass matrix.

In the example provided, a two-dimensional rectangular element is considered. The mass matrix is formulated using interpolation functions in natural coordinates and integrating over the element volume. The resulting mass matrix is a symmetric and consistent 8×8 matrix, though only a 4×4 submatrix contains non-zero terms due to the nature of the interpolation functions and the geometry of the element.

The text then moves onto developing the global equations of motion for a finite element model subjected to dynamic loading. By considering the total mechanical energy and its rate of change, the equations of motion are derived, leading to a system of ordinary differential equations that couples displacement and velocity through the mass and stiffness matrices.

These coupled equations are essential for analyzing the natural frequencies and mode shapes of the structure, which are eigenvalues and eigenvectors of the frequency equation derived from the system’s homogeneous equations. The free-vibration response of the system is a superposition of these natural vibration modes, where the amplitude and phase angle of each mode are determined by the initial conditions.

Modal analysis plays a crucial role in understanding the response of a structural system to harmonic (sinusoidal) forcing functions, which is often implemented in finite element software packages. The solution of the free-vibration problem provides insights into the natural frequencies and mode shapes, which are foundational to characterizing the dynamic behavior of the structure.

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