Frames, shells, and hybrid structures with early damages, such as early cracks, often behave as extremely weak nonlinear systems, among which the nonlinearity is difficult to be detected, especially if the system response is affected by the noise. To avoid these damages becoming catastrophic failures, developing effective incipient damages detection methods is important. The nonlinear output frequency response functions (NOFRFs) and associated indexes can be considered as one kind of the prospective detection tools, which are usually determined from the established nonlinear autoregressive with exogenous inputs (NARX) model. However, the hyperparameters in the NARX model are difficult to be determined so that the identification accuracy cannot be guaranteed. Therefore, it is important to develop more accurate methods to estimate the NOFRFs and their associated indicators for damage detection. Motivated by the powerful learning abilities of convolutional neural networks (CNN) and long short-term memory (LSTM) networks, a novel method based on NOFRFs and the CNN-LSTM model for detecting the early damages in structures is proposed. By applying the beat excitation, the response of the structure is divided into two components, where the approximately linear component is used to estimate the frequency characteristic of the linear component by the classical linear model and the nonlinear component is used to establish the CNN-LSTM model. By calculating the responses of the two models, the NOFRFs and associated indexes can be accurately estimated, and then the early damage can be detected. Simulation and experimental studies verify the potential and effectiveness of the novel method proposed in this article.

中文翻译：

---

使用非线性输出频率响应函数和 CNN-LSTM 模型检测结构中的早期损坏

具有早期损伤（如早期裂纹）的框架、壳体和混合结构通常表现为极弱的非线性系统，其中非线性难以检测，尤其是在系统响应受噪声影响的情况下。为了避免这些损坏成为灾难性故障，开发有效的早期损坏检测方法很重要。非线性输出频率响应函数（NOFRFs）和相关指标可以被认为是一种前瞻性的检测工具，它们通常是从建立的具有外生输入的非线性自回归（NARX）模型中确定的。然而，NARX模型中的超参数难以确定，无法保证识别精度。所以，开发更准确的方法来估计 NOFRF 及其相关的损伤检测指标非常重要。受卷积神经网络 (CNN) 和长短期记忆 (LSTM) 网络强大学习能力的启发，提出了一种基于 NOFRF 和 CNN-LSTM 模型的用于检测结构早期损坏的新方法。通过应用拍频激励，将结构的响应分为两个分量，其中近似线性分量用于通过经典线性模型估计线性分量的频率特性，非线性分量用于建立CNN-LSTM模型。通过计算两种模型的响应，可以准确估计NOFRFs和相关指标，进而检测早期损伤。