# 统计代写|主成分分析代写Principal Component Analysis代考|”Orthogonality of Principal Modes in Multi-Degree-of-Freedom Systems: Decoupling Equations of Motion”

The passage you’ve provided explains the orthogonality of the principal modes of vibration in a multiple degrees-of-freedom system. This property is crucial because it allows for the decoupling of the equations of motion, making it easier to analyze the individual contributions of each mode to the overall response.

When the system is subject to free vibration (no external forces), the motion can be described by Equation 10.99. Upon substituting the modal solutions, which are sinusoidal functions with natural frequencies ωi\omega_iω i ​ , into this equation, we find that for different modes, their modal amplitude vectors multiplied by the mass matrix and the stiffness matrix are orthogonal when acted upon by different natural frequencies. This leads to the condition in Equation 10.105, which states that for distinct modes i≠ji \neq ji  =j, the inner product of the modal amplitude vectors with the mass matrix is zero.

By defining the modal matrix [A][A][A] whose columns are the modal amplitude vectors, and computing the matrix triple product [S]=[A]T[M][A][S] = [A]^T [M] [A][S]=[A] T [M][A], we see that [S][S][S] is diagonal due to the orthogonality property. Normalizing the modal amplitude vectors ensures that the diagonal terms of [S][S][S] are unity, thereby creating an orthonormal basis.

The text then introduces a change of variables to transform the original nodal displacements q{q}q into generalized displacements p{p}p using the normalized modal matrix. This transformation simplifies the equations of motion such that they separate into independent single-degree-of-freedom equations, each characterized by one of the natural frequencies ωi2\omega_i^2ω i 2 ​ .

In the specific example, the modal matrix [A][A][A] is normalized, and it is verified that [A]T[M][A]=[I][A]^T [M] [A] = [I][A] T [M][A]=[I] (the identity matrix) and [A]T[K][A][A]^T [K] [A][A] T [K][A] is a diagonal matrix with entries being the squares of the natural frequencies.

The process of normalization involves scaling each modal amplitude vector so that its dot product with the mass matrix equals 1. After normalization, the stiffness matrix’s action on the modal amplitudes, when represented in the modal space, results in another diagonal matrix with entries ωi2\omega_i^2ω i 2 ​ .

In conclusion, the orthogonality and normalization of the modal amplitude vectors enable the transformation of the coupled equations of motion into uncoupled equations, which greatly simplifies the analysis of the dynamic behavior of the system. In the provided example, the author demonstrates how to calculate and normalize the modal matrix for a specific system, verifying that the transformed matrices satisfy the expected properties.

### MATLAB代写

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