# 统计代写|主成分分析代写Principal Component Analysis代考|”Extension of Mass Matrix Calculation for Two-Dimensional Truss Structures: Axial and Transverse Displacements”

This passage describes the extension of the mass matrix calculation for bar elements in two-dimensional truss structures considering both axial and transverse displacements. In the original bar-element-consistent mass matrix, derived for axial vibrations only, the kinetic energy was calculated along the axis of the element. However, when considering two-dimensional motion, both axial and transverse kinetic energies contribute to the total energy of the element.

In the context provided:

Derivation of Element Mass Matrix: The kinetic energy TTT for a small volume of the bar is expanded to account for both axial uuu and transverse vvv displacements. These displacements are interpolated using the same shape functions N1(x)N_1(x)N 1 ​ (x) and N2(x)N_2(x)N 2 ​ (x) as for the axial displacement. The element kinetic energy is thus represented by a matrix equation that includes contributions from both axial and transverse directions.

Transformation to Global Coordinates: The local mass matrix [m(e_2)][m(e\_2)][m(e_2)] for the two-dimensional element is transformed to the global coordinate system using the rotation matrix [R][R][R], which accounts for the element’s orientation relative to the global axes. However, it turns out that due to mass being an intrinsic scalar quantity, independent of the coordinate system, the mass matrix remains the same in both the local and global frames for a bar element in a truss structure, regardless of its orientation.

Application to Example: An example of a truss structure with eight bar elements is given. The mass matrix for each element is calculated based on its length, cross-sectional area, and material density. The global mass matrix is assembled using the element-to-global displacement relationships and the direct assembly procedure.

Modal Analysis: The objective is to perform modal analysis on the truss structure with no external loads. The global stiffness matrix is already available from previous calculations. The mass matrix is now appropriately constructed to reflect the two-dimensional motion of the bar elements.

Results: The global mass matrix is simplified by applying constraints (fixed boundary conditions), resulting in a reduced mass matrix for the active degrees of freedom. The stiffness matrix for these active DOFs is also extracted. With these matrices, the characteristic determinant is evaluated to find the natural frequencies and modal shapes of the structure.

Discussion: High-frequency natural modes are observed, which are typical for lightweight structures with good stiffness-to-mass ratios. The mode shapes (vectors normalized to the mass matrix) are presented in tabular format and visualized graphically to show the geometric patterns of the structure’s natural vibrations.

To summarize, the inclusion of transverse displacement in the mass matrix calculation for two-dimensional truss structures doesn’t affect the actual mass matrix itself but is crucial for accurately capturing the kinetic energy distribution and thereby determining the correct dynamic behavior of the structure.

### MATLAB代写

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