# 统计代写|主成分分析代写Principal Component Analysis代考|”Distributed Mass Effects on Longitudinal Vibration: Transition from Lumped-Mass to Finite Element Analysis”

The passage above describes the transition from analyzing lumped-mass systems to considering the effects of distributed mass in structural vibrations, specifically focusing on the longitudinal vibration of a bar element. In a lumped-mass system, each mass is directly associated with a node, and the mass matrix is diagonal. However, when dealing with solid structures, mass is distributed throughout the geometry, necessitating consideration of distributed mass properties.

In the case of a longitudinally vibrating bar element, the dynamic behavior is governed by the one-dimensional wave equation (Equation 10.52):

E∂2u∂x2=ρ∂2u∂t2E \frac{tial^2 u}{tial x^2} = \rho \frac{tial^2 u}{tial t^2}E ∂x 2

∂ 2 u ​ =ρ ∂t 2

∂ 2 u ​

Discretizing the displacement function u(x,t)u(x,t)u(x,t) in terms of nodal displacements and interpolation functions, the Galerkin’s method is applied to derive the residual equations (Equation 10.54), which lead to the formulation of both the stiffness matrix ([k][k][k]) and the consistent mass matrix ([m][m][m]). The consistent mass matrix accounts for the distributed mass of the element based on its density and cross-sectional area.

The final dynamic finite element equations for the bar element combine the stiffness and consistent mass matrices (Equation 10.60):

[m]u¨+[k]u=f[m]{\ddot{u}} + [k]{u} = {f}[m] u ¨ +[k]u=f

The text goes on to illustrate how to calculate the natural frequencies of the bar element. For free vibration, the nodal forces are set to zero, leading to a frequency equation (Equation 10.61) that resembles the eigenvalue problem seen earlier for simpler systems. The solution to this frequency equation provides the natural frequencies.

An example demonstrates the determination of natural circular frequencies for a solid circular shaft fixed at one end, modeled using two finite elements. After assembling the global stiffness and consistent mass matrices, applying the boundary condition (fixed at one end), and assuming a sinusoidal response, the frequency equation is derived and solved to give two natural frequencies.

Lastly, the text compares the results from using a consistent mass matrix to those from a lumped mass matrix approach, highlighting that the former provides upper bounds for natural frequencies and can offer more accuracy, while the latter is computationally simpler but might not guarantee bounds. Despite this, lumped mass matrices are frequently employed in practice, especially for bar and beam elements, due to their computational efficiency and acceptable prediction accuracy.

### MATLAB代写

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