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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. [ABSTRACT FROM AUTHOR]

The studies on inverse problems exist extensively in aerospace, mechanical, identification, detection, scanning imaging and other fields. Its ill-posed characteristics often lead to large oscillations in the solution of the inverse problem. In this study, the truncated generalized singular value decomposition (TGSVD) method is introduced to identify two kinds of moving forces, single and multi-axial forces. The truncating point is the most influential regularization parameter of TGSVD, which is initially selected by two classic regularization parameter selection criteria, namely, the L-curve criterion and the generalized cross-validation (GCV) criterion. Due to numerical non-uniqueness and noise disturbance in moving force identification (MFI), numerical simulation results show that neither of the two criteria can effectively help select the optimal truncating point of TGSVD. Hence, a relative percentage error (RPE) criterion is proposed for selecting the truncating point of TGSVD. Comparative studies show that the RPE criterion can be used to select the optimal truncating point of TGSVD more accurately against the GCV criterion and L-curve criterion. Moreover, the RPE criterion can be used to reflect the connections between certain properties and the ill-posedness problem existing in MFI, which should be adopted priority for the optimal truncating point selection of TGSVD. [ABSTRACT FROM AUTHOR]

This article introduces and evaluates the piecewise polynomial truncated singular value decomposition algorithm toward an effective use for moving force identification. Suffering from numerical non-uniqueness and noise disturbance, the moving force identification is known to be associated with ill-posedness. An important method for solving this problem is the truncated singular value decomposition algorithm, but the truncated small singular values removed by truncated singular value decomposition may contain some useful information. The piecewise polynomial truncated singular value decomposition algorithm extracts the useful responses from truncated small singular values and superposes it into the solution of truncated singular value decomposition, which can be useful in moving force identification. In this article, a comprehensive numerical simulation is set up to evaluate piecewise polynomial truncated singular value decomposition, and compare this technique against truncated singular value decomposition and singular value decomposition. Numerically simulated data are processed to validate the novel method, which show that regularization matrix L and truncating point k are the two most important governing factors affecting identification accuracy and ill-posedness immunity of piecewise polynomial truncated singular value decomposition. [ABSTRACT FROM AUTHOR]

• A preconditioned least square QR-factorization approach is developed for moving force identification. • The regularization matrix L is introduced to improve the ill-posed problems. • The number of iterations j is introduced to avoid noise disturbance and ensure the robustness. • Preconditioned least square QR-factorization approach is validated through numerical simulation. This paper develops a novel method for moving force identification (MFI) called preconditioned least square QR-factorization (PLSQR) method. The algorithm seeks to reduce the impact of identification errors caused by unknown noise. The biaxial moving forces travel on a simply supported bridge at three different speeds is used to generate numerical simulations to assess the effectiveness and applicability of the algorithm. Results indicate that the method is more robust towards ill-posed problem and has higher identification precision than the conventional time domain method (TDM). In addition, the robustness and ill-posed immunity of PLSQR are directly affected by two kinds of regularization parameters, namely, number of iterations j and regularization matrix L. Compared with the standard form of least square QR-factorization (LSQR), i.e., the regularization matrix L being the identity matrix I n , the PLSQR with the optimal number of iterations j and regularization matrix L has many advantages on MFI and it is more suitable for field trials due to better adaptability with type of sensors and number of sensors. [ABSTRACT FROM AUTHOR]

This paper develops a modified preconditioned conjugate gradient (M-PCG) method for moving force identification (MFI) by improving the conjugate gradient (CG) and preconditioned conjugate gradient (PCG) methods with a modified Gram-Schmidt algorithm. The method aims to obtain more accurate and more efficient identification results from the responses of bridge deck caused by vehicles passing by, which are known to be sensitive to ill-posed problems that exist in the inverse problem. A simply supported beam model with biaxial time-varying forces is used to generate numerical simulations with various analysis scenarios to assess the effectiveness of the method. Evaluation results show that regularization matrix L and number of iterations j are very important influence factors to identification accuracy and noise immunity of M-PCG. Compared with the conventional counterpart SVD embedded in the time domain method (TDM) and the standard form of CG, the M-PCG with proper regularization matrix has many advantages such as better adaptability and more robust to ill-posed problems. More importantly, it is shown that the average optimal numbers of iterations of M-PCG can be reduced by more than 70% compared with PCG and this apparently makes M-PCG a preferred choice for field MFI applications. [ABSTRACT FROM AUTHOR]

This paper proposes a new methodology for moving force identification (MFI) from the responses of bridge deck. Based on the existing time domain method (TDM), the MFI problem eventually becomes solving the linear algebraic equation in the form Ax = b . The vector b is usually contaminated by an unknown error e generating from measurement error, which often called the vector e as ‘‘noise’’. With the ill-posed problems that exist in the inverse problem, the identification force would be sensitive to the noise e . The proposed truncated generalized singular value decomposition method (TGSVD) aims at obtaining an acceptable solution and making the noise to be less sensitive to perturbations with the ill-posed problems. The illustrated results show that the TGSVD has many advantages such as higher precision, better adaptability and noise immunity compared with TDM. In addition, choosing a proper regularization matrix L and a truncation parameter k are very useful to improve the identification accuracy and to solve ill-posed problems when it is used to identify the moving force on bridge. [ABSTRACT FROM AUTHOR]

Four new methods have been developed to overcome the ill-posed problems inherently existing in moving force identification (MFI) in previous studies. This paper is an extension of the work to evaluate the overall performance of these presented methods by numerical simulations and experiment verifications in laboratory. A simply-supported bridge and two types of moving forces are adopted to evaluate the identification accuracy and ill-posed immunity of these new approaches. Bending moment and acceleration responses are measured when the time-varying forces moving across the bridge deck at constant speed. Numerical simulations of both uniaxial and biaxial forces include 12 cases, which are used to compare the identification accuracy and ill-posed immunity of these methods in detail. Finally, a hinge supported steel beam model and a vehicle model were designed and fabricated in laboratory. Then a series of experimental studies on MFI with these four methods are performed in laboratory. Both numerical and experimental results show that these four approaches can accurately identify moving forces with strong robustness and ill-posed immunity. Moreover, the truncated generalized singular value decomposition (TGSVD) method has higher identification accuracy than the piecewise polynomial truncated singular value decomposition (PP-TSVD) method, and the modified preconditioned conjugate gradient (M-PCG) method has higher identification efficiency than the preconditioned least square QR-factorization (PLSQR) method. To summarize, if the first goal in MFI is to improve the identification accuracy, the TGSVD method is recommended due to its high identification accuracy and stability in different cases. If the first goal in MFI is to improve the identification efficiency, the M-PCG method is recommended due to its high identification efficiency and easy to determine the optimal number of iterations. • Comparative studies have been carried out to evaluate the proposed four methods. • A guideline on selecting the best method for practical MFI has been provided. • Studies show that the PP-TSVD method is the worst among these four methods. • The TGSVD method is recommended for improving the identification accuracy. • The M-PCG method is recommended for improving the identification efficiency. [ABSTRACT FROM AUTHOR]

• A preconditioned range restricted GMRES algorithm is developed for MFI. • It has significant advantages in both of the identification efficiency and accuracy. • The accuracy of the new method is very high in highly inaccurate measurement cases. • The new method is verified by numerical simulations and experimental studies. Moving force identification (MFI) is a widely concerned inverse problem in structural dynamics and well-known as intrinsically existing ill-posedness. With the help of Arnoldi process and Krylov subspace method, the generalized minimal residual (GMRES) method can be improved to a range restricted generalized minimal residual (RRGMRES) method. Furthermore, by introducing the smoothing-norm preconditioning, a preconditioned range restricted generalized minimal residual (PRRGMRES) method is proposed to provide a stable solution to the ill-posed dynamic force identification problem. Simulations show that the novel method has significant improvement when compared to the classic time domain method and the RRGMRES method. In addition, to show the effectiveness and advantages of the proposed method, the PRRGMRES method is also compared with a newly-proposed regularization method named the preconditioned least square QR-factorization (PLSQR) method. Simulation results show that the PRRGMRES method has much better robustness and higher computational efficiency than the PLSQR method especially in dealing with highly inaccurate measurement cases. Finally, the accuracy and efficiency of the PRRGMRES method is verified by experimental studies. The PRRGMRES method has good performance in both overcoming ill-posed problems and improving computational efficiency, which should be of the highest priority in adoption for MFI. [ABSTRACT FROM AUTHOR]