We introduce the augmented Tikhonov regularization method motivated by Bayesian principle to improve the load identification accuracy in seriously ill-posed problems. Firstly, the Green kernel function of a structural dynamic response is established; then, the unknown external loads are identified. In order to reduce the identification error, the augmented Tikhonov regularization method is combined with the Green kernel function. It should be also noted that we propose a novel algorithm to determine the initial values of the regularization parameters. The initial value is selected by finding a local minimum value of the slope of the residual norm. To verify the effectiveness and the accuracy of the proposed method, three experiments are performed, and then the proposed algorithm is used to reproduce the experimental results numerically. Numerical comparisons with the standard Tikhonov regularization method show the advantages of the proposed method. Furthermore, the presented results show clear advantages when dealing with ill-posedness of the problem.

Evaluating dynamic loads in real time is crucial for health monitoring, fault diagnosis and fatigue analysis in aerospace, automotive and earthquake engineering among other vibration related applications. Developing such algorithms can be vital for several safety and performance functionalities. Therefore, over the past few years the identification of dynamic loads has attracted a lot of attention; however, little literature on the online identification can be found. In this paper, we propose an online-identification method of structural dynamic loads so that the dynamic load is evaluated in real time and while the system response is still being measured. This is achieved by significantly improving the identification efficiency while retaining a high accuracy. The proposed method which is based on Kalman filter, is introduced in detail for a finite as well as an infinite number of degrees of freedom. Starting from an initial guess of the state vector we evaluate the error covariance, which then helps to identify the value of the excitation force using a weighted least square method and minimizing the covariance unbiased estimation. This is repeated at certain time intervals i.e., time steps where the state vector is updated in real time as acceleration measurements are updated. The feasibility of the method is validated using numerical simulations and an experimental verification where a detailed LabVIEW (National Instruments Ltd.) implementation is provided.

The distributed dynamic load is difficult to obtain due to the complexity of loads in practical engineering, such as the aerodynamic loads of aircraft and the distributed dynamic loads of sea-crossing bridges. Thus, distributed dynamic load identification is important to deal with these difficulties, which is generally an ill-posed problem considering the inversion of the infinite dynamic loads. The traditional Tikhonov regularization technique is limited on the optimal regularization parameters selection. Consequently, in this paper, we develop a novel distributed dynamic load identification algorithm in combination with the orthogonal polynomials and the Bayesian framework. Thus, the orthogonal polynomial coefficients in the load identification model are regarded as the prior probability distribution of unknown variables in the Bayesian inference. Simultaneously, the posterior probability distribution of the orthogonal polynomial coefficients is derived based on the Bayesian formula and the likelihood function. The regularization parameters and the standard deviation of the response error are also treated as random variables to obtain the corresponding prior distribution in the multi-level Bayesian model. Moreover, the maximum posterior estimate is applied aiming at determining the regularization parameters, as well as the orthogonal polynomial coefficients to reconstruct the distributed dynamic loads. Compared with the Tikhonov regularization, a series of numerical simulations are studied to verify the effectiveness and high accuracy, as well as the noise resistance, and the results illustrate that this approach is effective to reconstruct the distributed dynamic loads.

As an essential component of dynamic loads, traditional time-domain identification methods exhibit notably insufficient accuracy when dealing with distributed dynamic load identification under unknown initial conditions. This paper explores a novel and effective methodology, utilizing the Bayesian framework and orthogonal polynomials fitting, to reconstruct the distributed dynamic loads of thin plate structures over any arbitrary time period under unknown initial conditions. The forced vibration under the orthogonal basis function loads and the free decay vibration after the removal of basis function loads are used to characterize the forced vibration induced by the identified distributed dynamic load and the decay vibration caused by unknown initial conditions, respectively. By integrating structural dynamic responses within a multi-layer Bayesian framework, the time history and spatial distribution of the load over any arbitrary time period are identified. The innovation of this methodology is that the contribution of the initial conditions to the response is independently characterized by the free decay response caused by the removal of the basis function loads, which effectively resolves the issue of insufficient identification accuracy in existing traditional time-domain methods due to unknown initial conditions. Consequently, the accuracy and reliability of the distributed dynamic load identification is significantly enhanced, which provides a new solution for distributed dynamic load identification under unknown initial conditions. Additionally, simulation cases involving various load conditions and noise levels are discussed under unknown initial conditions over arbitrary time periods. The results demonstrate that the proposed method achieves favorable identification accuracy and robustness under unknown initial conditions.