# 统计代写|主成分分析代写Principal Component Analysis代考|Mathematical Notes: Developing Mass Matrix for Beam Element Flexural Vibration & Frequency Convergence Analysis

This passage discusses the development of the mass matrix for a beam element undergoing flexural vibration. The process starts with applying Newton’s second law of motion to a differential element of the beam, taking into account the time-dependent loads and the distributed mass along the beam. By simplifying the equations and applying the small-deflection assumption, the governing equation for dynamic beam deflection (Equation 10.72) is derived.

The deflection of the beam is then discretized using interpolation functions, which are now time-dependent. The Galerkin’s method is applied to Equation 10.72 to generate residual equations (Equation 10.74). These equations include terms from the stiffness matrix, the mass matrix, and nodal force vectors.

To obtain the consistent mass matrix for the beam element, the term related to acceleration is integrated over the element length (Equation 10.77), resulting in a four-by-four mass matrix (Equation 10.78) that accounts for the distributed mass of the beam element.

When combined with the stiffness matrix and force vector, the finite element equations of motion (Equation 10.79) are formed. These equations govern the dynamic behavior of the beam element.

An example calculates the natural circular frequencies of a cantilevered beam with a single finite element. By setting up the appropriate boundary conditions, the frequency equation is derived and solved. It is observed that the finite element approximation closely matches the exact solution for the fundamental frequency, while higher-order frequencies require a more refined mesh (more elements) to achieve better accuracy.

The passage highlights that the number of natural frequencies and mode shapes computable depends on the number of degrees of freedom in the finite element model. Increasing the number of elements generally leads to better convergence of computed frequencies to their exact values, with lower frequencies typically converging faster. A practical guideline suggests that for accurately capturing the first P modes of vibration, one should calculate at least 2P modes. This acknowledges the computational challenge and inefficiency in computing all possible frequencies for large finite element models.

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