Nonlinear Structural Systems Research Articles

Feedback Stabilization of Chain-Like MDOF Nonlinear Structural Systems Under Purely Parametric Gaussian White Noises

A feedback control method is proposed to asymptotically stabilize the chain-like multi-degree-of-freedom (MDOF) nonlinear structural system under purely parametric Gaussian white noises. First, the motion equation of a chain-like nonlinear structural system with [Formula: see text] coupled elements and an undetermined control law is derived. It is reduced to a one-dimensional partially averaged Itô stochastic differential equation of the controlled Hamiltonian by applying the stochastic averaging method, which bypasses the challenge of calculating multiple Lyapunov exponents. Next, based on the dynamical programming principle, the dynamical programming equation of the averaged system with an undetermined cost function is established and addressed to obtain the optimal control law. Then, the Lyapunov exponent of the completely averaged Itô equation is derived. For the given requirement of stabilizing the system, the Lyapunov exponent is entirely fixed and used to analyze the stability of the originally chain-like MDOF nonlinear structural system. A controlled chain-like 8-DOF nonlinear structural system is taken as an example to illustrate the application and effectiveness of the proposed procedure and the effect of control forces on stabilizing the system.

An evaluation of shear effect on the behavior of geometrically nonlinear structural system

An evaluation of shear effect on the behavior of geometrically nonlinear structural system

Spectral incremental dynamic methodology for nonlinear structural systems endowed with fractional derivative elements subjected to fully non-stationary stochastic excitation

A novel spectral incremental dynamic analysis methodology for analysing structural response in nonlinear systems with fractional derivative elements is presented, aligning with modern seismic design codes, like Eurocode 8. Drawing inspiration from the concept of fully non-stationary stochastic processes, the vector of the imposed seismic excitations is characterised by time and frequency evolving power spectra stochastically compatible with elastic response spectra of specified damping ratio and ground acceleration. The proposed method efficiently determines the nonlinear system time-dependent probability density functions for the non-stationary system response amplitude by employing potent nonlinear stochastic dynamics concepts, such as stochastic averaging and statistical linearisation. Unlike traditional incremental dynamic analysis curves found in the literature, the herein proposed method introduces a three-dimensional alternative counterpart, that of stochastic engineering demand parameter surfaces, providing with higher-order statistics of the system response. An additional noteworthy aspect involves the derivation of response evolutionary power spectra as function of spectral acceleration, offering a deeper insight into the underlying system dynamics. Besides its capabilities, the method maintains the coveted element of a particularly low associated computational cost, increasing its attractiveness and practicality among diverse applications of engineering interest. Numerical examples comprising the bilinear hysteretic model endowed with fractional derivative elements subject to an Eurocode 8 elastic design spectrum demonstrate the capabilities and reliability of the proposed methodology. Its accuracy is assessed by juxtaposing the derived results with germane Monte Carlo Simulation data.

Machine learning modelling of structural response for different seismic signal characteristics: A parametric analysis

The present study investigates the best seismic parameters for modeling the dynamic response of various non-linear structural systems by comparing different Machine Learning (ML) algorithms. A total of 400 synthetic excitations were generated and analyzed against 23 seismic parameters. These signals were used in a step-by-step numerical analysis to calculate the dynamic responses of 1000 single-degree-of-freedom (SDOF) systems with varying mechanical properties. The data obtained from these responses were processed using 20 ML algorithms, including linear regression, tree, support vector machine (SVM), boosted and bagged trees, and artificial neural network (ANN). Each ML algorithm used a single seismic parameter as input to determine the most predictive parameters for modeling structural responses, defining the high predictive seismic parameters (HPSP) set. To validate the obtained results, the most effective model predictions have been compared with the results of the parametric step-by-step analyses performed for a new group of natural ground motions. The findings demonstrate that with a properly calibrated training phase, considering the specific site hazard and selecting seismic parameters from the HPSP set, the ML model can accurately estimate seismic responses whit a significantly reduced computational effort. This study underscores the potential of integrating ML techniques into the performance-based seismic design approach.

Time-frequency analysis of nonlinear structures under nonstationary excitation by directly solving structural dynamic equation through absolute displacement

The time-frequency evolution of seismic ground motion energy under strong earthquakes, which can be described by the evolutionary power spectral density (EPSD) matrix, will amplify the most unfavorable responses of members in nonlinear structural systems. However, the existing general finite element software fails to reduce the dimension of the EPSD matrix and then decouple time from frequency in frequency domain analysis. This failure greatly limits the simulation and calculation of multi-dimensional nonlinear structures with multiple degrees of freedom by the stochastic vibration theory. In this study, the pseudo excitation method (PEM), which can direct solve structural dynamic equations using absolute displacement, is used to derive the self-spectral and cross-spectral densities of seismic ground motions in one-dimensional and multi-dimensional cases. In MATLAB, the EPSD matrix is transformed into a four-dimensional uniform modulation excitation matrix independent of the time variable. Nonlinear iteration is conducted and the precise integration method (PIM) is employed through a transient analysis module. Furthermore, a simple one-dimensional example is presented to verify the correctness of the proposed algorithm herein. Finally, as an example, we select important parameters of the coherence model at different supports of a half-through three-span concrete-filled steel tubular (CFST) tied-arch bridge in view of real engineering conditions, and calculate the time-varying power spectrum and variance of the responses of arch ribs, arch feet and the bridge deck system. The results show that the theoretical derivation and the algorithm constructed in this paper can simulate the time-delay effect of the nonlinear structural system. This proves the feasibility of general finite element software in solving and analyzing random responses to multi-dimensional and multi-support non-stationary excitation of nonlinear structures.

Unconditionally Stable Central Difference Dissipative Algorithm for Multi-Directional Real-Time Hybrid Simulations of Large Nonlinear Structural Systems

ABSTRACT The central difference is a popular algorithm used to integrate the equations of motion, yet suffers from two drawbacks: (1) it is only conditionally stable and requires a small-time step to maintain numerical stability; (2) it is non-dissipative, and high-frequency spurious oscillations may appear and compromise the accuracy of the solution. These drawbacks are detrimental to applying the algorithm to the real-time hybrid simulation of large, complex nonlinear structural systems. In this paper, the conventional central difference algorithm is modified to overcome these drawbacks, and the modified algorithm is applied to the real-time hybrid simulation of complex structural systems.

Enhancing the Mobility of Small-Scale Robots Via Nonlinear Structural Springs Exhibiting Negative Stiffness

Abstract Compared to traditional robotic systems, small-scale robots, ranging from several millimeters to micrometres in size, are capable of reaching narrower and vulnerable regions with minimal damage. However, conventional small-scale robots’ limited maneuverability and controlability hinder their ability to effectively navigate in the intricate environments, such as the gastrointestinal tract. Self-propelled capsule robots driven by vibrations and impacts emerge as a promising solution, holding the potentials to enhance diagnostic accuracy, enable targeted drug delivery, and alleviate patient discomfort during gastrointestinal endoscopic procedures. This paper builds upon our previous work on self-propelled capsule robots, exploring the potential of nonlinear connecting springs to enhance its propulsion capabilities. Leveraging a mathematical model for self-propelling robots with a von Mises truss spring, which is verified using a finite element model, we investigate the effects of negative stiffness and snap-back within the nonlinear structural spring on the robots’ propelling speed. Our analysis reveals that the negative stiffness of the von Mises truss can significantly reduce the sensitivity of the propelling speed to excitation frequency. As a result, the capsule robot exhibits a remarkably wider operational band where it maintains a high average propelling speed, surpassing its linear counterpart. This work sheds light on the potential for developing customized nonlinear structural systems for diverse scenarios in small-scale robot applications, opening up new possibilities for enhanced functionality and maneuverability in various biomedical applications.

Enhancing seismic resilience of nonlinear structures through optimally designed additional mass dampers

Reducing dynamic responses in structures subjected to external excitations like earthquakes and strong winds is crucial for ensuring safety and serviceability. Conventional tuned mass dampers (CTMDs) have limitations in mitigating nonlinear vibrations efficiently. This research proposes the use of additional mass dissipation (AMD) devices to control nonlinear dynamic responses more effectively than CTMDs. The AMD concept involves installing an auxiliary mass–damper system atop the primary structure. An analytical approach is developed to determine the optimal design parameters of AMDs for linear and nonlinear single-degree-of-freedom systems using H2 and H∞ optimization techniques. Transfer function and harmonic balance methods are employed to evaluate the dynamic response reduction capabilities. Compared to optimally designed CTMDs, the proposed H2 and H∞ optimized AMDs demonstrate remarkably superior vibration mitigation achieving up to 77 % and 73.71 % greater reductions in dynamic responses, respectively. The Newmark-beta numerical validation confirms AMDs outperforming CTMDs by 16.06 % in minimizing structural vibrations. The novel AMD concept, coupled with the derived optimal design formulations, offers a highly effective vibration control strategy for both linear and nonlinear structural systems under harmonic and random excitations. This analytical solution paves the way for implementing AMDs in practice to enhance structural safety and longevity against external dynamic loads.

A novel time-domain approach for identifying nonlinear structural dynamical system with explicit model based on observer/Kalman filter identification method

In this paper, a novel time-domain method for identifying nonlinear structural dynamical system with explicit model is proposed. By exploiting the observer/Kalman filter identification (OKID) and nonlinear feedback interpretation, the nonlinear parameters of the system are estimated accurately. The effectiveness of the proposed nonlinear OKID (NOKID) method is demonstrated through three numerical examples, namely a five degree-of-freedom system with one nonlinear spring, a three degree-of-freedom system with three nonlinear springs and a nonlinear rectangular membrane structure with all edges clamped. Comparing with the time-domain nonlinear subspace identification (TNSI) method, the proposed NOKID method can identify the nonlinear parameters without system order determination, and the identification results obtained by the proposed method are real numbers and have no dependency on frequency.

Instantaneous frequency identification for nonstationary signals of time-varying structures using enhanced synchroextracting wavelet transform and dynamic optimization

Civil engineering structures under ambient excitations are essentially time-varying and nonlinear structural systems and the resultant response signals exhibit non-stationarity. To reveal the time-varying characteristics of non-stationary signals, time frequency analysis methods with high resolutions are required. Although synchroextracting short time Fourier transform (SESTFT) is able to generate more energy concentrated time frequency representations and allow signal reconstruction, its major disadvantage is the window function is fixed. To address this issue, an enhanced synchroextracting wavelet transform (ESEWT) method is proposed to refine frequency bands by combing synchroextracting and continuous wavelet transform. After that, dynamic optimization (DO) is used to extract instantaneous frequency (IF) curves within the refined frequency bands. Two numerical examples and an experimental study case are investigated to illustrate the effectiveness and accuracy of the proposed method. The results demonstrate that the proposed ESEWT is capable of extracting frequency bands more accurately and its combination with DO identifies IFs of non-stationary signals better than current time frequency analysis methods such as SESTFT.

An Iterative Augmented Unscented Kalman Particle Filter for Simultaneous State-Parameter-Input Estimation for Structural Systems Subjected to Gamma-Distribution Noise

Simultaneous estimations of the states, parameters and unknown inputs of nonlinear structural systems subjected to non-Gaussian noise are essential, because the inputs are not always available and the system noise is not always Gaussian. However, filters developed for this task require the inter-story drift measurements to be “fused” with the acceleration measurements to avoid the “drift phenomenon”. Because inter-story drifts may be inconvenient to obtain in practice, it is more desirable to use acceleration measurements only. To this end, this study develops a novel iterative augmented unscented Kalman particle filter (IAUKPF) for the simultaneous state-parameter-input estimation for systems subjected to non-Gaussian noise. The iterative strategy proposed in this study effectively corrects the “drift phenomenon” when only acceleration measurements are incorporated, so that the accuracies of the estimated unknown inputs are promoted. The importance density of the particle filter at each time step is provided by the augmented unscented Kalman filter so that the filter takes the best known information into consideration. Numerical examples of nonlinear structural systems affected by non-Gaussian noise are adopted to examine the effectiveness of the proposed IAUKPF.

A Novel Coupled State/Load/Parameter Identification Method Based on the Improved Particle Filter Algorithm

There have been increasing research studies focusing on the identification of structural load with uncertain model in recent years. This paper presents a novel approach based on the improved Particle Filter (PF) algorithm for the nonlinear structural system. The main strategy of the approach is the fusion of the weighted least-square algorithm and the conventional PF algorithm. The weighted least-square algorithm is derived for the estimation of the unknown loads, and the PF algorithm is used for the identification of the augmented states which includes the structural displacement, velocity and unknown parameters. Additionally, the parallel algorithm is adopted for the improvement of computing speed, and a resampling step is also used for alleviating the degeneracy of the particles to improve the accuracy. A numerical example modelling as a four-story hysteretic shear-beam building is studied to validate the capability of the presented approach.

The Efficiency of Using Machine Learning Techniques in Fiber-Reinforced-Polymer Applications in Structural Engineering

Sustainable solutions in the building construction industry have emerged as a new method for retrofitting applications in the last two decades. Fiber-reinforced polymers (FRPs) have garnered much attention among researchers for improving reinforced concrete (RC) structures. The existing design guidelines for FRP-strengthened RC members were developed using empirical methods that are based on specific databases, limiting the accuracy of the predicted results. Therefore, the use of innovative and efficient prediction tools to predict the behavior of FRP-strengthened RC members has become essential. During the last few years, efforts have been progressively focused on the use of machine learning (ML) as a feasible and effective technique for solving various structural engineering problems. Its capability to predict the behavior of complex nonlinear structural systems while considering a wide range of parameters offers a distinctive opportunity to make the behavior of RC members more predictable and accurate. This paper aims to evaluate the current state of using various ML algorithms in RC members strengthened with FRP to enable researchers to determine the capabilities of current solutions as well as to find research gaps to carry out more research to bridge revealed knowledge and practice gaps. Scopus databases were searched using predefined standards. The search revealed ninety-six articles published between 2016 and 2023. Consequently, these articles were analyzed for ML applications in the field of FRP retrofitting, including flexural and shear strengthening of RC beams, flexural strengthening of slabs, confinement and compressive strength of columns, and FRP bond strength. The results reveal that 32% of the reviewed studies focused on the application of ML techniques to the flexural and shear strengthening of RC beams, 32% on the confinement and compressive strength of columns, 6.5% on the flexural strengthening of slabs, 22% on FRP bond strength, 6.5% on materials, and 1% on beam–column joints. This research also revealed that the application of various ML algorithms has shown a significant improvement in resistance prediction accuracy as compared with the existing empirical solutions. Supervised learning techniques were the most favorable learning method due to their good generalization, interpretability, adaptability, and predictive efficiency. In addition, the selection of suitable ML algorithms and optimization techniques is found to be mainly dictated by the nature of the problem and the characteristics of the dataset. Nonetheless, selecting the most appropriate ML model and optimization algorithm for each specific application remains a challenge, given that each algorithm is developed with different principles and methodologies.

Prescribed Performance Tracking Control Under Uncertain Initial Conditions: A Neuroadaptive Output Feedback Approach.

This work is concerned with the prescribed performance tracking control for a family of nonlinear nontriangular structure systems under uncertain initial conditions and partial measurable states. By combining neural network and variable separation technique, a state observer with a simple structure is constructed for output-based finite-time tracking control, wherein the issue of algebraic loop arising from a nontriangular structure is circumvented. Meanwhile, by using an error transformation, the developed control scheme is able to ensure tracking with a prescribed accuracy within a pregiven time at a preassigned convergence rate under any bounded initial condition, eliminating the long-standing initial condition dependence issue inherited with conventional prescribed performance control methods, and guaranteeing the predeterminability of convergence time simultaneously. Two simulation examples also demonstrate the effectiveness of the presented control strategy.

Synchronous identification of nonlinear structural parameters and unknown external excitation based on improved UKF

The traditional Unscented Kalman Filter (UKF) approach for nonlinear structural identification usually requires knowledge of external excitation. However, in real application, obtaining the input excitation of nonlinear structures can be challenging due to measurement error or difficulty in acquiring the accurate information. Consequently, the real-time application of the UKF method will be constrained. To address this issue, this paper develops an improved UKF-based synchronous identification method of nonlinear structural parameters and unknown external excitation. The proposed method starts with a preliminary estimate of the unknown excitation using the current predicted values of structural dynamic responses and parameters. Subsequently, the external excitation is further identified based on the updated state vector. In addition, to reduce the effect of measurement noise on the nonlinear structural identification and ensure the accuracy of the identified results, a Kalman Filter (KF) process is embedded in the UKF to optimize measurement noise covariance matrix in real time. The practicability and accuracy of the improved UKF method are confirmed by conducting numerical simulations on a single degree-of-freedom (S-DOF) and a six degree-of-freedom (six-DOF) nonlinear structural systems due to unknown seismic excitation. Moreover, a shake table test of a five-storey steel frame subjected to seismic excitation is employed to attest the feasibility of the proposed method. Both numerical and experimental results demonstrate that the improved UKF method can effectively achieve synchronous identification for nonlinear structural parameters and unknown excitation.

An iterative augmented unscented Kalman filter for simultaneous state-parameter-input estimation for systems with/without direct feedthrough

A few unscented Kalman filters (UKFs) have been developed for simultaneous state-parameter-input estimation, however, these UKFs often have at least one of these limitations: requiring ad hoc procedures; requiring displacement measurements, which may be inconvenient to obtain, to fuse with acceleration measurements to avoid the “drift phenomenon” in the estimated displacement responses and unknown inputs; requiring a (full-rank) feedthrough matrix; and undesirable computational efficiency. To overcome these limitations, a novel iterative augmented unscented Kalman filter (IAUKF) for simultaneous state-parameter-input estimation for systems either with or without direct feedthrough is developed and examined in this study. The IAUKF closely follows the Bayes’ theorem with no ad hoc procedure, featuring theoretical simplicity. An iterative strategy is introduced to improve the accuracy of the unknown input estimation at each time step, so that the “drift phenomenon” encountered by the existing augmented Kalman filter is avoided. Further, the IAUKF requires no prior models for the time histories of the unknown inputs and applies to systems with/without direct feedthrough. With realistic assumptions, the steps for computationally efficient implementations of the IAUKF are proposed. Numerical examples of nonlinear structural systems with/without direct feedthrough of the unknown inputs are adopted to examine the effectiveness and the computational efficiency of the IAUKF.

Sparse Bayesian machine learning for the interpretable identification of nonlinear structural dynamics: Towards the experimental data-driven discovery of a quasi zero stiffness device

Data-driven discovery of governing laws for complex nonlinear structural dynamic systems remains a challenging issue of paramount importance. This work addresses the above issue by leveraging the available noisy data and integrating sparse Bayesian machine learning (ML) techniques to discover the governing equations. The problem of discovery is re-cast as the automatic relevance determination of models (model selection) from the library of potential candidate basis terms and their coefficients are determined (parameter identification) using sparse Bayesian linear regression. Two sparsity promoting ML algorithms based on relevance vector machines have been employed. Both these approaches use Bayesian statistics and quantify the uncertainty associated with the model predictions. Results from four representative numerical examples of nonlinear structural dynamics illustrate excellent performance of both proposed approaches. The results have been validated with the true governing equations and time response data. Comparison has also been made with a recent and popular sparse discovery approach. Finally, the proposed framework is applied to real datasets that were generated from an in-house designed experimental setup of a quasi zero stiffness device and good performance has been observed.

Novel Girsanov correction based Milstein schemes for analysis of nonlinear multi-dimensional stochastic dynamical systems

This work proposes Girsanov corrected explicit, semi-implicit, and implicit Milstein approximations for the solution of nonlinear stochastic differential equations. The solution trajectories provided by the Milstein schemes are corrected by employing the change of measures, aimed at removing the error associated with the diffusion process incurred due to the transformation between two probability measures. The change of measures invoked in the Milstein schemes ensures that the solution from the mapping is measurable with respect to the filtration generated by the error process. The proposed scheme incorporates the error between the approximated mapping and the exact representation as an innovation that is accounted in the Milstein trajectories as an additive term. Numerical demonstrations using parametrically and non-parametrically excited stochastic oscillators, a practical problem involving ring type gyroscope, and a 7-degree-of-freedom nonlinear structural system subjected to both stationary and non-stationary excitation demonstrate the improvement in the solution accuracy for the proposed schemes with much coarser time steps when compared with the classical Milstein approximation with finer time steps.

General partial safety factor theory for the assessment of the reliability of nonlinear structural systems

The paper develops a novel and general theory for characterizing the nonlinearity of structural systems and for applying partial safety factors to these systems. The theory establishes a key relationship between the partial safety factor concept and the reliability theory of nonlinear structural systems, using the degree of homogeneity as a measure of nonlinearity at the design point. This measure allows for an efficient mathematical decoupling of the reliability index into nonlinearity-invariant partial reliability indexes. This formulation enables the identification of critical safety situations in extreme cases of nonlinearities in complex nonlinear structural systems. The theory leads to two main outcomes based on the asymptotic behavior of the reliability index. First, the reliability index of any nonlinear structural system is always bounded between an upper and lower bound, which can be determined using the concept of nonlinearity-invariant partial reliability indexes. Second, nonlinearity-invariant critical partial safety factors ensure that the reliability index is greater than the target reliability index. The proposed theory provides a simple and easy-to-implement method for assessing the safety of nonlinear structural systems, which can be used in conjunction with existing codes and standards. The theory can be applied to a wide range of nonlinear systems in engineering practice, including structural and geotechnical problems. It is designed primarily to provide code writers with the necessary procedure for calibrating partial safety factors for nonlinear structural systems and to identify the over-safe or under-safe cases.

A harmonic-wavelet-based representation for non-stationary stochastic excitations

Representation of nonstationary stochastic excitations is crucial for stochastic response analyses of (time-varying) linear and nonlinear structural systems. This paper proposes a new representation method of non-stationary stochastic excitations based on the generalized harmonic wavelet (GHW) that takes the phase angles and frequencies as basic random variables. The orthogonal properties of the discrete-form spectral process increments describing non-stationary stochastic processes are formulated. Then the GHW-based representation is derived by using the orthogonal properties. This method can be used to accurately reproduce non-stationary stochastic excitations with the target asymptotic Gaussianity and evolutionary power spectrum density. The effectiveness and accuracy of the proposed method have been validated via numerical examples. This study provides a novel way for the representation of non-stationary processes and deserves to be applied in the stochastic response analyses of structures.