隐变量模型及其在贝叶斯运营模态分析的应用

The Bayesian FFT algorithm is the latest generation of algorithms for operational modal analysis and has received extensive attention for its advantages such as high accuracy, fast calculation speed, and effective uncertainty measurement. However, existing methods need to adopt different optimization algorithms for different situations (sparse modes, dense modes, multi-step tests, etc.), and the programming implementation is extremely complex. Therefore, this paper aims to propose a unified framework for the Bayesian FFT algorithm in different situations and achieve efficient solution of modal parameters. Firstly, regarding the structural modal response as a latent variable, a latent variable model framework for single-step and multi-step Bayesian modal identification tests is established; then, for the proposed latent variable model, the Expectation-Maximization algorithm is used to achieve unified Bayesian inference of modal parameters in various situations, the latent variable is used to decouple the modal parameter optimization process, and the Louis equation is used to indirectly obtain the Hessian matrix of the likelihood function; finally, through two actual engineering test cases and comparison with existing methods, the accuracy and efficiency of the proposed method are verified. The analysis results show that the algorithm proposed in this paper has the same results as existing methods, but its derivation is simple and easy to program, and it has obvious computational advantages especially for dense modal identification problems. This paper establishes a unified latent variable model framework for Bayesian modal identification, simplifies the originally cumbersome and lengthy derivation to a large extent, improves computational efficiency, and also provides the possibility for applying algorithms such as variational Bayes and Gibbs sampling to solve Bayesian modal identification.

贝叶斯FFT 算法是运营模态分析的最新一代算法，以其准确性高、计算速度快、可有效进行不确定性度量等.优点受到广泛关注。然而，现有方法针对不同情况（稀疏模态、密集模态、多步测试等）需采用不同优化算法，且编程.实现极为复杂。为此，本文旨在提出针对不同情况下贝叶斯FFT 算法的统一框架，并实现模态参数的高效求解。.首先，视结构模态响应为隐变量，建立贝叶斯模态识别单步测试和多步测试的隐变量模型框架；然后，针对提出的隐.变量模型运用期望最大化算法实现各种情况下模态参数的统一贝叶斯推断，利用隐变量解耦模态参数优化过程，采.用Louis 等式间接求取似然函数的Hessian 矩阵；最后，通过两个实际工程测试案例，并与现有方法对比，验证所提.方法的准确性和高效性。分析结果表明，本文所提算法与现有方法结果相同，但其推导简单易编程，尤其对于密集.模态识别问题具有明显的计算优势。本文为贝叶斯模态识别建立起统一的隐变量模型框架，在很大程度上简化本.来繁琐且冗长的推导，提高计算效率，同时也为应用变分贝叶斯、吉布斯采样等算法求解贝叶斯模态识别提供了.可能。