Moving force identification (MFI) from dynamic responses of bridges is a typical inverse problem with ill-posedness. Under the efforts of researchers, some regularization methods have been presented to solve the ill-posed problem, but there still lacks an effective index to reveal the ill-posedness of the vehicle–bridge dynamic system such that it can be utilized as a guidance for the regularization parameter selection. In this paper, an ill-posedness indicator (IPI) defined as the ratio of the Fourier coefficient to the singular value is adopted to reveal the ill-posedness in the MFI problem. Simulation results show that the larger the IPI value is, the more obvious the ill-posedness of the vehicle–bridge system equation, namely, the intrinsic factor of ill-posedness in MFI is attributed to very large IPI value. The maximum IPI value increases with the increasing noise level, which leads directly to the ill-posedness of the vehicle–bridge system equation. In addition, a relative percentage error (RPE) is used to select the optimal regularization parameters, while evaluating the ill-posedness existing in the MFI. Using the proposed IPI value, the influence of ill-posedness on identified results is evaluated in this study, which can assist qualitatively and quantitatively in selecting optimal regularization parameters and proper regularization methods.

中文翻译：

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正则化方法的运动力辨识与参数选择的不适定性确定

桥梁动态响应的移动力识别（MFI）是一个典型的不适定反问题。在研究人员的努力下，已经提出了一些正则化方法来解决不适定问题，但仍然缺乏一个有效的指标来揭示车桥动力学系统的不适定性，从而可以作为指导正则化参数选择。在本文中，采用定义为傅里叶系数与奇异值之比的不适定指标（IPI）来揭示MFI问题中的不适定性。仿真结果表明，IPI值越大，车桥系统方程的病态性越明显，即MFI中的病态性的内在因素归因于非常大的IPI值。最大 IPI 值随着噪声水平的增加而增加，这直接导致了车桥系统方程的不适定性。此外，使用相对百分比误差（RPE）来选择最佳正则化参数，同时评估 MFI 中存在的不适定性。本研究使用提出的 IPI 值评估不适定性对识别结果的影响，这有助于定性和定量地选择最佳正则化参数和适当的正则化方法。