Time‐Series Models for Nonlinear Systems

Discrete time‐series models can be used for the dynamic response prediction of linear structures. When structural nonlinearities are present, it may be possible to modify the form of the discrete time‐series model to account for the nonlinearities. One approach is to allow the model parameters to become functions of state. This paper explores some possible forms of the parameter functions for various nonlinear structures. Numerical case studies using both a Duffing oscillator and a combined viscous and coulomb damped oscillator are presented. Also, experimental data from a highly nonlinear aircraft landing gear strut are used to evaluate different model forms. The results from these studies show the potential for future applications of nonlinear time‐series models.     

Peak Response of a Nonlinear Beam

A model for estimating the peak dynamic response distribution of a nonlinear beam, based on a special class of non-Gaussian stochastic processes, is proposed in this paper. It is shown that the stochastic response of a cantilever beam with geometrically nonlinear behavior can be accurately calibrated with translation processes. Different models to describe the significant bimodal features in the marginal probability density functions of the response time histories are proposed. Finally, two of these models are used to estimate the response peak value distributions and the results are compared. This comparison demonstrates the effects of inaccurate models for the parent response processes on the peaks estimation.     

Genetic-Algorithm-Based Strategies for Dynamic Identification of Nonlinear Systems with Noise-Corrupted Response

The main objective of this paper is to investigate efficiency and correctness of different real-coded genetic algorithms and identification criteria in nonlinear system identification within the framework of non-classical identification techniques. Two conventional genetic algorithms have been used, standard genetic algorithm and microgenetic algorithm. Moreover, an advanced multispecies genetic algorithm has been proposed: it combines an adaptive rebirth operator, a migration strategy, and a search space reduction technique. Initially, a critical analysis has been conducted on these soft computing strategies to provide some guidelines for similar engineering and physical applications. Therefore, the hysteretic Bouc-Wen model has been numerically investigated to achieve three main results. First, the computational effectiveness and accuracy of the proposed strategy are checked to show that the proposed optimizer outperforms the aforementioned conventional genetic algorithms. Secondarily, a comparative study is performed to show that an improved performance can be obtained by using the Hilbert transform-based acceleration envelope as objective function in the optimization problem (instead of the pure acceleration response). Finally, system identification is conducted by making use of the proposed optimizer to verify its substantial noise-insensitive property also in the presence of high noise-to-signal ratio.     

Nonlinear Controller Synthesis for Single-Input Multiple-Output Systems with Application to Cruise Missile

The purpose of this paper is to present a new systematic procedure for synthesis of nonlinear controllers for single-input multiple-output systems. The procedure is based on several describing function models of the plant, and the nonlinear gains are obtained via an inverse describing function technique. The procedure and associated software are applied to control the bank angle of a cruise missile. The results are compared with a linear controller and another nonlinear controller that was previously reported in the open literature; it is shown that the developed procedure has resulted in a less conservative design (i.e., performance is not sacrificed to assure absolute stability).     

Stochastic Response and Stability of a Nonintegrable System

For determining the stochastic response and stability of a strongly nonlinear single-degree-of-freedom system using the stochastic averaging technique, the size of excitations should be small such that the response of the system converges weakly to a Markov process. This condition is not often met with practical problems, and therefore, application of this method for obtaining their responses becomes difficult. Further, for systems with nonlinearities that cannot be integrated in closed form, stability analysis by examining the conditions of the two boundaries of the problem is not possible. A semianalytical method along with a weighted residual technique is presented here to circumvent these difficulties and to determine the response and stability of a strongly nonlinear system subjected to sizable stochastic excitation. The weighted residual technique is employed to correct the errors in averaged drift and diffusion coefficients resulting due to the size of the stochastic excitation. Two example problems are solved as illustrations of the method.     

Linearized Stability and Accuracy for Step-by-Step Solutions of Certain Nonlinear Systems

In this work, stability and accuracy of the Newmark method for nonlinear systems are obtained from a linearized analysis. This analysis reveals that an unconditionally stable integration method for linear elastic systems is unconditionally stable for nonlinear systems and a conditionally stable integration method for linear elastic systems remains conditionally stable for nonlinear systems except that its upper stability limit might vary with the step degree of nonlinearity and step degree of convergence. A sufficient condition to have a stable computation for nonlinear systems in a whole step-by-step integration procedure is also developed in this study. Furthermore, it is also found that numerical accuracy in the solution of nonlinear systems is closely related to the step degree of nonlinearity and step degree of convergence although its characteristics are similar to those of the preceding works for linear elastic systems. Since these results are obtained from a linearized analysis, they can be applicable to the nonlinear systems that satisfied the simplifications for the analysis but may not be applicable to general nonlinear systems.     

Updating Properties of Nonlinear Dynamical Systems with Uncertain Input

A spectral density approach for the identification of linear systems is extended to nonlinear dynamical systems using only incomplete noisy response measurements. A stochastic model is used for the uncertain input and a Bayesian probabilistic approach is used to quantify the uncertainties in the model parameters. The proposed spectral-based approach utilizes important statistical properties of the Fast Fourier Transform and their robustness with respect to the probability distribution of the response signal in order to calculate the updated probability density function for the parameters of a nonlinear model conditional on the measured response. This probabilistic approach is well suited for the identification of nonlinear systems and does not require huge amounts of dynamic data. The formulation is first presented for single-degree-of-freedom systems and then for multiple-degree-of freedom systems. Examples using simulated data for a Duffing oscillator, an elastoplastic system and a four-story inelastic structure are presented to illustrate the proposed approach.     

Modeling Blockage Failures in Sewer Systems to Support Maintenance Decision Making

The objective of this research is to develop and implement a stochastic method that can be applied to characterize random failures in critical infrastructure systems. We particularly focus on blockage failures in sewer systems that are nonmechanistic and result from combination of external factors, including deterioration in condition. The method was implemented using a data set consisting of sewer blockage failure records from a small municipality. Statistical tests were conducted to: (1) ensure that available data set is representative and (2) estimate parameters of distributions that appropriately characterize failure event arrival pattern. Failure trends were also analyzed to identify the influence of local factors and justify the choice of the distributions used to characterize interarrival times. Based on the analysis, we explored the challenges in developing a reliability model across the life cycle of a sewer system. In addition, specific examples were also presented to illustrate how the method can be applied to support system maintenance decisions. The results of this study illustrate how the memoryless property can be assumed in analyzing failure events, while explicitly considering context specific influences. Finally, the methods described in this paper are extensible and can be applied generally to analyzing random failures in other infrastructure systems as well.     

Stochastic Averaging of Preisach Hysteretic Systems

The sustained progress in the study of the hysteretic behavior of structural and mechanical systems has led to the adoption of increasingly sophisticated and reliable mathematical representations. Models based on the distributed elements (hysterons) appear to be quite versatile. Among these, the Preisach hysteretic model has received considerable attention in the field of engineering mechanics. In this paper, the stochastic response of a Preisach hysteretic system driven by a white noise process is investigated. In this regard, the method of stochastic averaging is modified to be applicable for the determination of the probability density of the stationary system response envelope. Remarkably, this probability density expression in conjunction with the response of an auxiliary linear system can also be used to determine the power spectrum of the system response. The approximate theoretical solutions are validated by data derived by a pertinent Monte Carlo study.     

Stochastic Analysis of a Single-Degree-of-Freedom Nonlinear Experimental Moored System Using an Independent-Flow-Field Model

Stochastic characteristics of the surge response of a nonlinear single-degree-of-freedom moored structure subjected to random wave excitations are examined in this paper. Sources of nonlinearity of the system include a complex geometric configuration and wave-induced quadratic drag. A Morison-type model with an independent-flow-field formulation and a three-term-polynomial approximation of the nonlinear restoring force is employed for its proven excellent prediction capability for the experimental results investigated. Wave excitations considered in this study include nearly periodic waves, which take into account the presence of tank noise, noisy periodic waves that have predominant periodic components with designed additive random perturbations, and narrow-band random waves. A unified wave excitation model is used to describe all the wave conditions. A modulating factor governing the degree of randomness in the wave excitations is introduced. The corresponding Fokker–Planck formulation is applied and numerically solved for the response probability density functions (PDFs). Experimental results and simulations are compared in detail via the PDFs in phase space. The PDFs portray coexisting multiple response attractors and indicate their relative strengths, and experimental response behaviors, including transitions and interactions, are accordingly interpreted from the ensemble perspective. Using time-averaged probability density functions as an invariant measure, probability distributions of large excursions in experimental and simulated responses to various random wave excitations are demonstrated and compared. Asymptotic long-term behaviors of the experimental responses are then inferred.     

Nonlinear Modeling of Flexible Cable Loads on Large Sheaves

Stress analysis of the components of a sheave used to transfer loads between the lift span and counterweight in a movable span bridge is investigated. Stress analysis is a requirement for properly designing such sheaves. Modeling of the mechanism of load transfer from the wire ropes to the sheave is accomplished in three ways: (1) the traditional manner using a uniform pressure distribution; (2) using a varying pressure distribution developed from belt/pulley theory; and (3) using the finite-element method with nonlinear contact elements between the wire rope and the sheave. Internal stresses in the sheave are calculated using uniform distributed pressure and a varying pressure distribution. It is determined that the load distribution on the sheave from the wire ropes is precisely the same for the nonlinear contact analysis and the belt/pulley analysis. The internal stress analysis results show that the traditional, uniformly distributed load representation is less conservative than the more realistic belt/pulley load representation. A methodology is developed that can be utilized to more accurately model the load transfer representation without the complexity of nonlinear analyses.     

Multilevel Modeling of Two Cyclical Processes: Extending Differential Structural Equation Modeling to Nonlinear Coupled Systems.

The authors present a dynamical multilevel model that captures changes over time in the bidirectional, potentially asymmetric influence of 2 cyclical processes. S. M. Boker and J. Graham's (1998) differential structural equation modeling approach was expanded to the case of a nonlinear coupled oscillator that is common in bimanual coordination studies in which participants swing hand-held pendulums but is also applicable to social systems in general. The authors' nonlinear coupled oscillator model decomposed the fluctuations into a competitive component, unique to each individual variable, and a cooperative component that captured bidirectional influence. The authors' model also generated an index of the symmetry/asymmetry of bidirectional influence. Together, the models are useful quantitative tools for the study of interacting, changing processes. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Capturing Nonlinear Vibratory Roller Compactor Behavior through Lumped Parameter Modeling

Continuous monitoring of soil properties using an instrumented roller compactor requires models that can capture the essential features observed during drum/soil vibration. This paper presents the results of lumped parameter modeling of the drum/soil system together with data from complex nonlinear behavior observed experimentally during operation on sandy soil. Model parameters and response were developed using experimental data collected over a wide range of operating frequencies. Three and four-degree-of-freedom (DOF) models with linear and nonlinear soil elements were investigated. The results showed that a 3DOF model incorporating the soil, drum, and frame of the roller was successful in capturing behavior during coupled drum/soil vibration and during decoupling (i.e., loss of contact between drum and soil). Modeling the drum/soil decoupling accounted for most of the experimentally observed nonlinearity. The addition of nonlinear soil stiffness due to the curved drum effect and due to strain hardening soil behavior accounted for additional nonlinearity observed experimentally. Experimentally observed drum rocking during coupled drum/soil vibration was successfully modeled with a 4DOF drum-frame model. The analysis also revealed that commonly observed heterogeneous soil conditions give rise to a transient response that can have a significant influence on vibration behavior.     

Linking Nonlinear System Identification with Nonlinear Dynamic Simulation under OpenSees: Some Justifications and Implementations

This study seeks to bridge the gap between nonlinear system identification and nonlinear dynamic finite-element analysis. Motivated by the needs in earthquake simulation, it is first investigated under which conditions and to what degree the prediction of maximum lateral drift and base shear requires accurate nonlinear hysteretic moment-rotation joint models. A series of simulations is carried out using a simple but typical steel frame under two different earthquake ground motion time histories scaled up to various levels. As one of the two major classes of models in nonlinear system identification, nonparametric models are proposed to be implemented into OpenSees. A methodology with details is provided to effectively implement feedforward neural networks with one hidden layer as a new one-dimensional nonlinear smooth material model directly from a MATLAB environment to OpenSees. The same methodology can be applied to benefit the implementation of other parametric and nonparametric models with linear parameterization. Numerical examples are provided. Challenges are discussed and future work is identified.     

Adaptive Fuzzy Backstepping Output Feedback Control of Nonlinear Time-delay Systems with Unknown High-frequency Gain Sign

In this paper, an adaptive fuzzy robust feedback control approach is proposed for a class of single-input and singleoutput (SISO) strict-feedback nonlinear systems with unknown nonlinear functions, time delays, unknown high-frequency gain sign, and without the measurements of the states. In the bazkstepping recursive design, fuzzy logic systems are employed to approximate the unknown smooth nonlinear functions, K-filters is dcsigncd to estimate the unmeasured states, and Nussbaum gain functions are introduced to solve the problem of unknown sign of high-frequency gain. By combining adaptive fuzzy control theory and adaptive backstepping design, a stable adaptive fuzzy output feedback control scheme is developed. It has been proven that the proposed adaptive fuzzy robust control approach can guarantee that all the signals of the closed-loop system are uniformly ultimately bounded and the tracking error can converge to a small neighborhood of the origin by appropriately choosing design parameters. Simulation results have shown the effectiveness of the proposed method.     

Modeling Controlled Water Systems

The particular challenges of modeling controlled water systems are discussed. The high degree of freedom due to the control structures increases the risk of producing the right output for the wrong reasons. On the other hand, many controlled water systems are (partly) manually operated or at least supervised by an operational water manager. The decisions of these managers are not as rigid as a computer simulated control strategy. Therefore, getting a very close fit with a water-system control model is mostly not possible. A modeling framework is proposed that takes advantage of the vast availability of measurement data in controlled water systems. The water level and flow data at control structures allow for intensive validation and subsystem calibration to reduce the degree of modeling freedom and to model separately the natural rainfall-runoff and hydrodynamic processes. The framework is successfully applied to improve a simulation model of the controlled water system of Rijnland, The Netherlands. The yearly volume error was reduced from 11% to less than 1% and as a consequence, the short-term peak events were modeled more accurately as well. The resulting water-system control model is more reliable for both design studies and operational decision support. The framework will contribute to prepare more reliable simulation models of controlled water systems.     

Arrival Sampling without Replacement for Simulating Fixed Entity Streams in Civil Engineering Systems

The need for sampling arrivals without replacement for simulating certain types of civil engineering systems is presented. Specifically, sampling requirements in the form of arriving entities such as a stream of automobiles and trucks, aircraft, shipping vessels, etc., and their scheduling on transportation systems are discussed. A methodology for random sampling, without replacement, of entities from a known discrete distribution is presented. Computational complexities associated with organization and randomization of large input streams are addressed. In addition to addressing issues related to the generation of discrete and continuous event occurrences in the entity stream, issues related to incorporating scheduling overrides into the final entity streams are discussed. The methods discussed have successfully been applied for generating streams of shipping vessels for a simulation study of the Panama Canal.     

Hydrostatic versus Nonhydrostatic Euler-Equation Modeling of Nonlinear Internal Waves

Basin-scale internal waves are inherently nonhydrostatic; however, they are frequently resolved features in three-dimensional hydrostatic lake and coastal ocean models. Comparison of hydrostatic and nonhydrostatic formulations of the Centre for Water Research Estuary and Lake Computer Model provides insight into the similarities and differences between these representations of internal waves. Comparisons to prior laboratory experiments are used to demonstrate the expected wave evolution. The hydrostatic model cannot replicate basin-scale wave degeneration into a solitary wave train, whereas a nonhydrostatic model does represent the downscaling of energy. However, the hydrostatic model produces a nonlinear traveling borelike feature that has similarities to the mean evolution of the nonhydrostatic wave.     

Error Propagation in Implicit Pseudodynamic Testing of Nonlinear Systems

Error propagation analysis of an implicit pseudodynamic algorithm has been developed for linear elastic systems. However, these error propagation results might not be applicable to nonlinear systems. Since the pseudodynamic testing method aims at investigating the nonlinear behavior of a seismically loaded structure it is important to perform the error propagation analysis of the implicit pseudodynamic algorithm for nonlinear systems. A technique to conduct the nonlinear error propagation analysis of an implicit pseudodynamic algorithm is constructed and is illustrated by analyzing the constant average acceleration method. Theoretical results reveal that the numerical and error propagation properties for nonlinear systems are generally inherited from those of linear elastic systems although some properties are affected by the step degree of nonlinearity. It is verified that the most important property of unconditional stability is preserved for nonlinear systems for a complete pseudodynamic test procedure even if the convergence error is present. It is concluded that error propagation for the improved implementation is superior to that of the direct implementation and is consistent with that developed for linear elastic systems.     

Stochastic Modeling of Load-Induced Settlement in Loose Granular Materials

The response of loose cohesionless granular material to surface applied loads is investigated from the viewpoint of probabilistic mechanics of particulate media. A model is proposed that is based on the combined propagation of intergranular forces and an excess volume of voids. In this regard, it provides a bridge between earlier theories developed independently for the diffusion of stresses and for the propagation of settlements. In its general formulation, the theory can model three-dimensional, transient effects. However, the model is believed to be limited to normally consolidated or noncompacted, fully drained or dry, granular materials that do not exhibit dilatancy effects. The derived numerical modeling of steady state deflection patterns under a rigid footing is found to be in good agreement with x-ray images of laboratory model tests using noncompacted silt. The proposed theory recognizes the discrete and inherently random nature of natural granular materials such as cohesionless soils and builds upon these fundamental characteristics to predict responses of such materials to boundary applied load. This is achieved by modeling intergranular force and excess pore volume propagation as Markovian diffusion-advection processes. This approach, which departs from traditional continuum mechanics models, seems to have potential for addressing some of the challenging aspects of granular material mechanics in lunar or Martian environments.