This paper proposes a novel approach for structural damage identification with a limited number of modal measurements and ultra‐sparse Bayesian regression. An iterative Cross Modal Strain Energy (CMSE) method is proposed for establishing the linear regression model. It can be applied to enlarge the number of available modes, thus alleviating the problem of insufficient mode orders. In addition, the proposed approach can also be used to solve the identification of incomplete measurements by combining with a dynamic condensation process. The condensed system matrices of the damaged structure are updated iteratively by the identified damage severity vector. A major contribution of this study is that the most advanced Bayesian linear regression estimator, called Horseshoe (HS), is first introduced to provide an ultra‐sparse regularization. Owning to the particular choice of a half‐Cauchy prior to the global and local scale hyper‐parameters, using HS can provide a sparser solution than the Bayesian lasso (BL). Therefore, this approach is extremely suitable for structural damage identification with sparse solutions in nature. Other advantages of using HS consist of the easy implementation of Gibbs sampling, effective convergence rate and hyper‐parameter tuning, good stability, and the ability to conduct the indeterminate inverse identification arising from insufficient mode order. The effectiveness and performance of the proposed approach are validated by numerical and experimental studies, considering the effects of measurement noise and a limited number of modal measurements with an insufficient number of mode orders and incomplete measurements.

中文翻译：

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具有有限模态测量和超稀疏贝叶斯回归的结构损伤识别

本文提出了一种用于结构损伤识别的新颖方法，该方法具有数量有限的模态测量和超稀疏贝叶斯回归。提出了一种迭代交叉模态应变能（CMSE）方法来建立线性回归模型。它可以用于扩大可用模式的数量，从而减轻模式顺序不足的问题。此外，所提出的方法还可以用于结合动态冷凝过程来解决不完整测量的识别问题。损坏结构的压缩系统矩阵通过已识别的损坏严重性向量进行迭代更新。这项研究的主要贡献在于，首先引入了最先进的贝叶斯线性回归估计量，即马蹄（HS），以提供超稀疏正则化。由于在全局和局部范围的超参数之前都选择了半隐式设计，因此使用HS可以提供​​比贝叶斯套索（BL）更为稀疏的解决方案。因此，这种方法非常适合自然界中稀疏解决方案的结构损伤识别。使用HS的其他优点包括：易于实施Gibbs采样，有效的收敛速度和超参数调整，良好的稳定性以及由于模式阶数不足而进行不确定的逆识别的能力。考虑到测量噪声和有限数量的模态测量以及不足的模序数量和不完整的测量，通过数值和实验研究验证了该方法的有效性和性能。