Unknown Structural Parameters Research Articles

Nonparametric identification of multi-degree-of-freedom nonlinear systems from partially measured responses under uncertain dynamic excitations

With the rapid advent of new materials and novel structures, it becomes difficult, if not impossible, to accurately model and simulate the nonlinear response of complex systems under uncertain dynamic excitations based on close-formed nonlinear functions and parametric identification. In this study, a two-step structural nonlinearity localization and identification approach for multi-degree-of-freedom (MDOF) nonlinear systems under uncertain dynamic excitations is developed by integrating an extended Kalman filter with unknown inputs into the equivalent linearized systems. In the first step, unmeasured responses and excitations are estimated as well as unknown structural parameters and nonlinearity locations are identified by fusing acceleration with displacement time histories at the observed degrees of freedom (DOFs). In the second step, the nonlinear restoring force of the detected nonlinear structural members is identified nonparametrically using three polynomial models, including a power series polynomial model (PSPM), a double Chebyshev polynomial model (DCPM), and a Legendre polynomial model (LPM). Linear multi-story shear frames controlled by nonlinear magnetorheological (MR) dampers are modelled computationally to demonstrate the generality of the proposed methodology. The multi-source uncertainties considered in these representative examples include the location and the type of nonlinearities represented by a Bingham model and a modified Dahl model of the dampers, the location of response measurements, the location and intensity of dynamic excitations, the level of measurement noise, and the initial assignment of structural parameters. The acceleration, velocity, and displacement time histories of structures can be evaluated accurately with a maximum error of 2.62% even with the presence of 8% measurement noise, while the external excitations can be estimated within an error of 1.77%. The location of nonlinear elements can be detected correctly. The structural parameters, the NRFs provided by MR dampers and the corresponding energy dissipation can be identified with a maximum error of 2.08%, 1.19% and 0.39%, respectively, even 8% measurement noise and very rough initial assignment of structure parameters (−70%) are considered. Moreover, the numerical results change little (<0.20%) even the initial assignment of structural parameters varies from 50% to 30% of their original values, no matter which nonparametric model is employed. Results indicate that the presented algorithm can effectively identify unmeasured dynamic responses, structural parameters, unknown excitations, nonlinear locations, and NRF of nonlinear elements in a nonparametric way.

Output-only structural damage identification based on Q-learning hybrid evolutionary algorithm and response reconstruction technique

Various structural damage identification methods have been developed and employed, while the absence of input excitation measurements may pose a huge challenge in their application since the input forces such as seismic load, traffic load, wind load, are not directly measurable. To address this issue, in the present paper, a novel output-only structural damage detection approach based on Q-learning hybrid evolutionary algorithm (QHEA) and response reconstruction technique is presented. On the one hand, the external excitation and structural acceleration responses are reconstructed with the aid of response reconstruction and Tikhonov regularization techniques in time domain. Structural damage identification can be formulated as an optimization-based inverse problem. On the other hand, a new optimization framework QHEA integrating Jaya algorithm, differential evolution, Q-learning algorithm is developed as search tool. The unknown structural parameters and unmeasured input force are iteratively updated by using two different measurement sets until the reconstructed acceleration responses agree well with the measured responses. Numerical examples involving a cantilever beam structure under single excitation and a simply-supported 51-bar truss structure under multiple excitations, as well as an experimental five-floor steel frame structure in the laboratory are carried out to validate the effectiveness of the proposed method. The final results demonstrate that the proposed method can accurately detect damage locations and quantify damage extents without the information of input excitation. In addition, the superiority of QHEA over other heuristic algorithms, the uncertainties of measurement noise and modeling errors on damage identification results are further examined.

Dual state-parameter estimation of continuous structural systems with physics-informed parallel neural networks

Vibration-based structural identification is a common approach to infer structural integrity by, for example, estimating the state of the structural system and the associated model parameters that can then be used to predict performance. To this end, finite element (FE) model updating approaches have been widely employed. However, in the context of continuous systems, FE model updating can be computationally intensive and prone to truncation and numerical quadrature errors in the variational forms. Alternatively, purely data-based identification approaches, for example, artificial neural networks (NNs), have been suggested, yet such methods have been shown to be susceptible to overfitting. To overcome these challenges, this paper develops and demonstrates a physics-informed parallel neural networks (PIPNNs) framework for the purpose of solving dual state-parameter estimation problems of continuous structural systems that can be represented by a system of partial differential equations (PDEs). Specifically, the governing PDEs and associated continuity and equilibrium conditions of the system are embedded as soft constraints into the loss function of the NNs. The PIPNNs learn to approximate the PDEs and the unknown structural parameters by minimizing the physics-informed loss function. In the context of continuous systems, the PIPNNs are used to estimate the unknown structural parameters and then be used to estimate the full state of the system, for example, displacement along a multi-span bridge or a floor system. For general applicability, the supports and boundary conditions that produce discontinuities must be considered in the computational domain. To accomplish this, the computational domain is divided into subdomains, and each subdomain is represented as an individual governing PDE adjoining adjacent subdomains by means of compatibility and equilibrium conditions and solved synchronously by the PIPNNs. As such, each NN provides a unique representation in each subdomain while also satisfying additional constraints of continuity and differentiability at the interface between two subdomains. The framework is verified and validated against an alternative independent model, and its accuracy is assessed through the application to several illustrative examples.

Advances in dynamic load identification based on data-driven techniques

Dynamic loads on engineering structures are often difficult to measure directly. Therefore, indirect identification methods based on dynamic responses are commonly used. However, this approach is often influenced by ill-conditioned matrices, noise interference, unknown structural and/or material parameters, and difficulty of constructing transfer functions when the traditional physics-based model is used. To address these issues, significant strides have been made in data-driven identification of dynamic loads, which are model-free and independent of structural characteristics. This paper tries to present a comprehensive review of dynamic load identification methods based on data-driven techniques, covering two aspects: load localization and load reconstruction. Features of the widely used data-driven techniques such as the geometric method, reference database method, machine learning methods including SVM-based methods and ANN-based methods and deep learning methods are discussed in detail. Additionally, this paper offers insight into the challenges and prospects of the data-driven techniques for dynamic load identification. The review aims to provide valuable insights for identifying dynamic loads in complex structures based on data-driven techniques and suggests future research directions.

A Critical Introduction to the Usual Robust Control Framework in Macroeconomics

By applying robust control the decision maker wants to make good decisions when his model is only a good approximation of the true one. Such decisions are said to be robust to model misspecification. In this paper it is shown that the application of the usual robust control framework in discrete time problems is associated with some interesting, if not unexpected, results. Results that have far reaching consequences when robust control is applied sequentially, say every year in fiscal policy or every quarter (month) in monetary policy. This is true when unstructured uncertainty à la Hansen and Sargent is used, both in the case of a “probabilistically sophisticated” and a non- “probabilistically sophisticated” decision maker, or when uncertainty is related to unknown structural parameters of the model.

Metamodel-assisted hybrid optimization strategy for model updating using vibration response data

In this study, an effective and novel method, termed Metamodel Assisted Hybrid of Particle Swarm Optimization with Genetic Algorithm (MA-HPSOGA), is developed to identify unknown structural dynamic parameters. The method first constructs four popular metamodels to substitute the computationally expensive numerical analysis based on the Latin hypercube sampling method and probabilistic finite element analysis, and their accuracy is assessed by R-squared. Subsequently, a suitable and low-cost metamodel is selected in combination with a hybrid optimization strategy by incorporating Genetic Algorithm (GA) into Particle Swarm Optimization (PSO). Two examples with measured vibration response data and different levels of complexity are used to verify the effectiveness and practicality of the presented method. The results showed that polynomial chaos expansion assisted HPSOGA has the highest computational efficiency and accuracy in the four coupled methods. Besides, compared to the conventional iteration-based dynamic parameter identification methods, the presented method shows an overwhelming advantage in terms of computational efficiency. Furthermore, the performance of HPSOGA is compared with its sub-algorithms, showing that the hybrid strategy offers faster convergence and stronger robustness. Our findings reveal that the MA-HPSOGA may be used as a promising method for achieving high-efficiency model updating in large-scale complex structures.

Adaptive backstepping control design for ATMD systems in nonlinear structures with nonlinear disturbance and parametric uncertainties

Active tuned mass damper (ATMD) devices are recommended in various structures to reduce vibrations. The performance of the ATMD system is closely tied to the control design utilized, according to research in the literature. The designed control input will take into account as much system dynamics as possible, making it possible to use it as a generalized controller in all similar systems. Nonlinear behavior is the natural behavior of engineering structures, and in order to obtain a more realistic and high-performance ATMD system, it can be considered as an appropriate approach to design the controller by considering the structural nonlinearities. In addition, unexpected external influences and parameter uncertainties must be taken into account during control design to ensure that control systems perform successfully in similar systems in all conditions. Therefore, in this study, the nonlinear model of the multi-story building was reconstructed by adding nonlinear disturbance to represent unknown external effects. Band-limited white noise is used as a disturbance function and Gaussian white noise is added to measured states in this study. To obtain a robust controller, unknown structural parameters are compensated for by using adaptive compensation terms. With the controller design supported by the Lyapunov-based stability analysis, the stability of the vibrating structure featuring an ATMD is theoretically guaranteed while achieving the main control goal. Performance analyses of the designed controllers are carried out with simulation studies. The efficiency of the developed Lyapunov-based controller in dampening the unwanted vibrations that occurred on the building is seen in simulation results.

Output-only structural identification with random decrement technique

Most of existing time domain methods fail to identify the unknown structural parameter in the ambient condition, because the force location is unknown and the number of unknown forces is usually larger than the number of sensors. A novel output-only structural identification method based on the random decrement (RD) technique and covariance matrix adaptation evolutionary strategy (CMA-ES) is presented. The RD functions of a structure under ambient excitation are extracted and treated as free vibration responses for structural parameter identification. The unknown structural parameters are identified by minimizing the discrepancy between the measured and calculated RD functions with the CMA-ES. Unlike traditional output-only methods, which require the estimation of the input excitation in the process, the proposed method does not require the identification of the input forces, thereby significantly reducing the computational cost. Numerical studies on two- and three-dimensional trusses and experimental studies on a steel frame are presented to demonstrate the accuracy and efficiency of the proposed approach. The effects of measurement noise and measurement duration on the identified results are examined. Results show that the proposed method can obtain satisfactory results even with 20% measurement noise and structural damage less than 10% can be detected. Comparison studies with the genetic algorithm (GA) and particle swarm optimization (PSO) are presented to verify the computational efficiency of the proposed method. Results also show that the proposed CMA-ES has a faster convergence speed than the GA and PSO method. The results demonstrate that the proposed technique can be used to detect damage severity and location in structures under ambient conditions.

Finite element model updating and damage identification using semi-rigidly connected frame element and optimization procedure: An experimental validation

Owing to ever-increasing complexity of engineering structures, developing a methodology for the early detection of defects has become crucial to ensure their long-term safety and reliability with the least amount of expense. There are always discrepancies between experimental and numerical modal data because of unknown structural parameters and uncertainties. The finite element (FE) model updating techniques attempt to minimize the differences by adjusting the unknown parameters of the FE model. Therefore, the FE model updating methods are essential for developing a baseline model and accurate damage identifications in subsequent steps. This paper employs the semi-rigidly connected frame element (S-RCFE) instead of the standard Euler-Bernoulli beam element for assembling the FE model of the experimental beam and establishing a high-fidelity numerical model. The S-RCFE with extra design parameters, including the end fixity factor of all connections, enables us to achieve a reasonable agreement between experimental and numerical models through the optimization-based procedure. In FE model updating step, two objective functions based on modified total modal assurance criterion (MTMAC) and changes in natural frequencies are used to minimize by three optimization algorithms, viz, grey wolf optimizer (GWO), gradient-based optimization (GBO), and an improved version of GWO (IGWO). The influence of the S-RCFE and standard Euler-Bernoulli beam on the model updating accuracy is also examined, and the efficiency of S-RCFE is evaluated. The statistical results reveal that GWO-MTMAC and IGWO-MTMAC can be successfully implemented for FE model updating with almost the same performance. However, IGWO provides the most reliable results with a relatively extensive computation time for damage identification in all scenarios. In some damage scenarios, the GWO and GBO perform comparably with very similar running time. Data used in this article can be found at https://github.com/Samir-Khatir/Data-modal-analysis-of-healthy-and-cracked-beam.git

Model reduction technique for Bayesian model updating of structural parameters using simulated modal data

An attempt has been made to study the effectiveness of model reduction technique for Bayesian approach of model updating with incomplete modal data sets. The inverse problems in system identification require the solution of a family of plausible values of model parameters based on available data. Specifically, an iterative model reduction algorithm is proposed based on a non-linear optimization method to solve the transformation parameter such that no prior choices of response parameters are required. The modal ordinates synthesized at the unmeasured degrees of freedom (DOF) from the reduced order model are used for a better estimate of likelihood functions. The reduced-order model is subsequently implemented for updating of unknown structural parameters. The present study also synthesizes the mode shape ordinates at unmeasured DOF from the reduced order model. The efficiency of the proposed model reduction algorithm is further studied by adding noises of varying percentages to the measured modal data sets. The proposed methodology is illustrated numerically to update the stiffness parameters of an eight-story shear building model considering simulated datasets contaminated by Gaussian error as evidence. The capability of the proposed model reduction algorithm coupled with Markov Chain Monte Carlo (MCMC) algorithm is compared with the case where only MCMC algorithm is used to investigate their effectiveness in updating model parameters. The numerical study focuses on the effect of reduced number of measurements for various measurement configurations in estimating the variation of errors in determining the modal data. Subsequently, its effects in reducing the uncertainty of model updating parameters are investigated. The effectiveness of the proposed model reduction algorithm is tested for number of modes equal to the number of master DOFs and gradually decrease of mode numbers from the number of master DOFs.

Extended Kalman filter algorithm for non-roughness and moving damage identification

It is a promising method to identify structural damage using bridge dynamic response under moving vehicle excitation, but the lack of accurate information about road roughness and vehicle parameters will lead to the failure of this method. The paper proposed a step-by-step EKF damage identification method, which transforms the inversion problem of unknown structural parameters under unknown loads (vehicle and road roughness) into two separate inversion problems: moving contact force identification and damage parameters identification. Firstly, the VBI model is converted into bridge vibration model under a moving contact force, and the moving contact force covering the information of road roughness and vehicle parameters can be calculated by EKF iteration. Secondly, the moving contact force identified in the first step is loaded on the bridge as a known condition, and the bridge damage problem is also solved by the EKF method. Numerical analyses of a simply-supported bridge under the moving vehicle are conducted to investigate the accuracy and efficiency of the proposed method. Effects of the vehicle speed, the damage cases, the measurement noise, and the roughness levels on the accuracy of the identification results are investigated. The results demonstrate the proposed algorithm is efficient and robust, and the algorithm can be developed into an effective tool for structural health monitoring of bridges.

Reverse Design of a Novel Coupling Strut for Vibration Attenuation in the Helicopter Cabin

Helicopter gearbox support strut is one of the main research objects in the field of vibration and noise control in helicopter cabins. Aiming to further widen the vibration attenuation range of traditional Bragg periodic struts, a novel type of Local resonance (LR)/Bragg coupling periodic strut with graded parameters as well as the reverse design method is proposed. Combined with the spectral element method (SEM) and the transfer matrix method (TMM), the analytical expression of the transform relationship of longitudinal vibrations through the coupling strut is yielded. The impacts of different parameters on the boundaries of bandgaps are explored according to the results of simulation analysis. On this basis, the gradient of parameters is determined, and then all unknown structural parameters can also be determined. Compared with the traditional Bragg periodic struts and the LR/Bragg coupling periodic strut with non-graded parameters, the presented strut has an obvious advantage of widening the low-frequency bandgaps below 500 Hz.

Prediction of single salt rejection in PES/CMS based membranes

This study explains the modeling of synthesized membranes using the Donnan Steric Pore model (DSPM) based on the Extended Nernst Planck Equation (ENP). Conventionally, structural parameters required to predict the performance of the membranes were determined through tedious experimentation, which in this study are found using a new MATLAB technique. A MATLAB program is used to determine the unknown structural parameters such as effective charge density (Xd), effective pore radius (rp), and effective membrane thickness to porosity ratio (Δx/Ak) by using the single solute rejection and permeation data. It was found that the model predicted the rejection of studied membranes accurately, with the E5C1 membrane exceeding the others (E5, E5C5) for rejection of single and divalent salt's aqueous solutions. The rejection of 100 ppm aqueous solution of NaCl for E5C1 was around 60%, whereas, for an aqueous solution of 100 ppm, CaCl2 rejection reached up to 80% at 10 bar feed pressure. The trend of salt rejection for all three membranes was found to be in the following order: E5C1 > E5C5 > E5, confirming that their structural parameters-controlled ion transport in these membranes. The structural parameters, such as effective pore radius, effective membrane thickness to porosity ratio, and effective charge density for the best performing membrane, i.e., E5C1, were determined to be 0.5 nm, 16 μm, and −6.04 mol/m3,respectively. Finally, it can be asserted that this method can be used to predict the real performance of membranes by significantly reducing the number of experiments previously required for the predictive modeling of nanofiltration-type membranes.

Identification of centrifugal pump flow based on feature recognition

The flow rate of water pumps is a crucial parameter for diagnosis, optimisation and renovation of transmission and distribution systems. To solve the problems of limited measurement conditions, inadequate consideration of operational wear and tear, and large deviations in the fitted flow curves when obtaining flow rates by direct measurement or characteristic curve fitting, this paper proposes a method for identification of centrifugal pump flow rates based on feature recognition. The method combines the unknown structural parameters of a centrifugal pump with three simple and easy-to-measure operating data: pump head, frequency and power, to obtain the pump characteristics and construct a model for identification of centrifugal pump flow. At the same time, the influence of measurement errors and wear and tear on the model’s accuracy is investigated. The results show that the maximum relative error of the pump operating flow obtained by the method is 2.27%, a slight deviation from the actual measured flow. The maximum relative error of flow rate is 2.7% when a maximum measurement error of ± 0.1%, 0.5% and ± (3%+5) for the required operational data, respectively, indicating that the method has a high resistance to interference. The method considers the different degrees of operating wear and tear of the pumps, saves on equipment input and enables fast and accurate identification of the flow rate of the pumps under different operating conditions.

Efficient Inverse Method for Structural Identification Considering Modeling and Response Uncertainties

The inverse problem analysis method provides an effective way for the structural parameter identification. However, uncertainties wildly exist in the practical engineering inverse problems. Due to the coupling of multi-source uncertainties in the measured responses and the modeling parameters, the traditional inverse method under the deterministic framework faces the challenges in solving mechanism and computing cost. In this paper, an uncertain inverse method based on convex model and dimension reduction decomposition is proposed to realize the interval identification of unknown structural parameters according to the uncertain measured responses and modeling parameters. Firstly, the polygonal convex set model is established to quantify the epistemic uncertainties of modeling parameters. Afterwards, a space collocation method based on dimension reduction decomposition is proposed to transform the inverse problem considering multi-source uncertainties into a few interval inverse problems considering response uncertainty. The transformed interval inverse problem involves the two-layer solving process including interval propagation and optimization updating. In order to solve the interval inverse problems considering response uncertainty, an efficient interval inverse method based on the high dimensional model representation and affine algorithm is further developed. Through the coupling of the above two strategies, the proposed uncertain inverse method avoids the time-consuming multi-layer nested calculation procedure, and then effectively realizes the uncertainty identification of unknown structural parameters. Finally, two engineering examples are provided to verify the effectiveness of the proposed uncertain inverse method.

Kalman Filter for Linear Systems With Unknown Structural Parameters

This brief considers Kalman filter for linear systems with unknown structural parameters. We design a Bayesian parameter identification algorithm based on maximum likelihood (ML) criterion and expectation maximization (EM). Under the identification of structural parameter, we perform the Kalman filter to estimate the states. The Kalman filter and the parameter identification algorithm are interactive to estimate the states and the structural parameters. More specifically, first, we use EM algorithm together with Kalman smoother to estimate the structural parameters. Then, on the basis of results provided by the first step, we employ the classical Kalman filter to predict and update the states, which formulates the final proposed Kalman filter. Performance analysis and simulation results verify the presented filter.

Data-driven approach for post-earthquake condition and reliability assessment with approximate Bayesian computation

Estimating the reliability of civil structures after earthquake events is a challenging problem because the uncertainties of random variables are usually unknown. This paper presents a data-driven approach for post-earthquake reliability assessments of civil structures. The measured vibration data are used to update the probability density functions of random variables. Two approximate Bayesian computation techniques are introduced to generate the posterior probability density functions of structural parameters. One is the likelihood-free Bayesian inference method, in which the weighted sum of squared errors term in the likelihood function is approximated by a Gaussian process model and the posterior probability density functions is generated using the Metropolis–Hastings algorithm. The second method is the variational inference approach, in which the posterior probability density function is approximated by a Gaussian mixture model. Both the proposed techniques can provide an approximation of the posterior probability density functions of the unknown parameters. Furthermore, the updated probability density functions are applied for reliability assessments. Numerical studies on a steel frame structure and a reinforced concrete structure are conducted to verify the accuracy and efficiency of the proposed techniques. Results show that the posterior probability density functions of unknown structural parameters can be updated and the reliability of the structure and components can be estimated.

Bayesian model updating of civil structures with likelihood-free inference approach and response reconstruction technique

Bayesian inference methods typically require a considerable amount of computation time in the calculation of forward models. This limitation restricts the application of Bayesian inference methods for the parameter identification of complex engineering problems. We propose a novel likelihood-free Bayesian inference method for structural parameter identification. An adaptive Gaussian surrogate model (GSM) was integrated with the transitional Markov chain Monte Carlo (MCMC) method for Bayesian inference. The log-likelihood function was approximated with GSM and the transitional MCMC method was used to generate the posterior distribution samples. A response reconstruction technique was combined with the likelihood-free Bayesian inference method for the parameter identification. Both numerical studies and experimental studies were conducted to verify the accuracy and efficiency of the proposed method. The results showed that the proposed method could be used to estimate the posterior probabilities of unknown structural parameters. Additionally, the proposed method was more efficient than the delayed rejection adaptive Metropolis and Gibbs sampling methods.

Probabilistic model updating via variational Bayesian inference and adaptive Gaussian process modeling

The estimation of the posterior probability distribution of unknown parameters remains a challenging issue for model updating with uncertainties. Most current studies are based on stochastic simulation techniques. This paper proposes a novel variational Bayesian inference approach to estimate posterior probability distributions by using the vibration responses of civil engineering structures. An adaptive Gaussian process modeling technique is used to represent the “expensive-to-evaluate” likelihood function, and the unknown posterior probability distribution is represented using a Gaussian mixture model. The evidence lower bound (ELBO) and its gradients can be computed analytically using the built Gaussian process and mixture models. The unknown parameters in the Gaussian mixture model can be identified by maximizing the value of ELBO. The stochastic gradient descent method is applied to perform the optimization. Numerical studies on an eight-story shear-type building and a simply supported beam are conducted to verify the accuracy and efficiency of using the proposed approach for probabilistic model updating and damage identification. Experimental studies on a laboratory steel frame structure are also conducted to validate the proposed approach. Results demonstrate that the posterior probability distributions of the unknown structural parameters can be successfully identified, and reliable probabilistic model updating and damage identification can be achieved.

State-Input System Identification of Tall Buildings under Unknown Seismic Excitations Based on Modal Kalman Filter with Unknown Input

The seismic safety evaluation of tall building structures is necessary, but the multidegree-of-freedom feature of tall buildings increases the difficulty of joint identification of structural states, structural parameters, and unknown seismic excitations. Therefore, the state-input-system identification of tall buildings under unknown seismic excitations is the focus of this paper. To avoid the complex work of establishing a structural motion equation in absolute coordinate system, the simple structural motion equation in relative coordinates is directly adopted, but the measured absolute accelerations are used to establish the observation equation. In this way, the establishment of approximate assumptions is avoided, and direct feedback is not reflected in the observation equation. Also, the modal expansion technique is adopted to reduce the dimension of the motion equations and the size of the structural state to be identified. Different from the previous methods based on modal Kalman filtering, the unknown seismic excitation is treated as an unknown input instead of unknown modal forces in this paper, so the dimension of unknown forces in the identification process is not increased. When structural parameters of tall buildings are known, the generalized modal Kalman filtering with unknown input (GMKF-UI) proposed by the authors can simultaneously identify structural states and unknown seismic excitations by observing partial absolute acceleration responses. When extended to the case of unknown structural parameters, a generalized modal extended Kalman filtering with unknown input (GMEKF-UI) is proposed in this paper to simultaneously identify structural states, the unknown seismic inputs, and tall building systems using only partial absolute acceleration responses. Two numerical examples are used to verify the effectiveness of the proposed method.