Uncertainty Analysis Of Systems Research Articles

Uncertainty analysis of powertrain mounting systems with independent and correlated parameters based on multidimensional parallelepiped model

Uncertainty analysis of powertrain mounting systems with independent and correlated parameters based on multidimensional parallelepiped model

Uncertainty Analysis and Design of Air Suspension Systems for City Buses Based on Neural Network Model and True Probability Density

The accuracy of uncertainty analysis in suspension systems is closely tied to the precision of the probability distribution of sprung mass. Consequently, traditional assumptions regarding the probability distribution fail to guarantee the accuracy of uncertainty analyses results. To achieve more precise uncertainty analysis outcomes, this paper proposes a data-driven approach for analyzing the uncertainties in bus air suspension systems. Firstly, a bus vehicle dynamics model is established to investigate the influence of sprung mass on suspension system performance. Subsequently, a deep neural network model is trained using road test data, for the accurate identification of the sprung mass. The historical mass of the bus is then computed using vehicle network data to obtain the true probability density of the sprung mass. Lastly, the real probability distribution of the sprung mass is utilized to perform uncertainty analysis on the bus suspension system, and the results are compared with those obtained by assuming a probability distribution. Comparative analysis reveals substantial disparities in uncertainty response, with a maximum relative error of 9% observed for wheel dynamic loads, thus emphasizing the significance of precise probability distribution information concerning the sprung mass.

A review of uncertain factors and analytic methods in long-term energy system optimization models

A larger number of uncertain factors in energy systems influence their evolution. Owing to the complexity of energy system modeling, incorporating uncertainty analysis to energy system modeling is essential for future energy system planning and resource allocation. This study focusses on long-term energy system optimization model. The important uncertain parameters in the model are analyzed and divided into policy, economic, and technical factors. This study specifically addresses the challenges related to carbon emission reduction and energy transition. It involves collecting and organizing relevant research on uncertainty analysis of long-term energy systems. Various energy system uncertainty modeling methods and their applications from the literature are summarized in this review. Finally, important uncertainty factors and uncertainty modeling methods for long-term energy system modeling are discussed, and future research directions are proposed.

An information theoretic Bayesian uncertainty analysis of AEM systems over Menindee Lake, Australia

SUMMARY Long-range, active-source airborne electromagnetic (AEM) systems for near-surface conductivity imaging fall into two categories: helicopter (rotary-wing) borne or fixed-wing aircraft borne. A multitude of factors such as flying height, transmitter loop area and current, source waveforms, aerodynamic stability and data stacking times contribute to the geological resolvability of the subsurface. A comprehensive comparison of the relative merits of each system considering all such factors is difficult, but test flights over well-constrained subsurface geology with downhole induction logs are extremely useful for resolution studies. However, given the non-linear nature of the electromagnetic inverse problem, handling transmitter–receiver geometries in fixed-wing aircraft is especially challenging. As a consequence of this non-linearity, inspecting the closeness of downhole conductivities to deterministic inversion results is not sufficient for studying resolvability. A more comprehensive picture is provided by examining the variation in probability mass of the depth-wise Bayesian posterior conductivity distributions for each kind of AEM system within an information theoretic framework. For this purpose, probabilistic inversions of data must be carried out. Each acquiring system should fly over the same geology, survey noise levels must be measured and the same prior probabilities on conductivity must be used. With both synthetic models as well as real data from over the Menindee calibration range in New South Wales, Australia, we shed new light on the matter of AEM inverse model uncertainty. We do this using two information theoretic attributes derived from different Kullback–Leibler divergences—Bayesian information gain, and a strictly proper scoring rule, to assess posterior probabilities estimated by a novel Bayesian inversion scheme. The inversion marginalizes fixed-wing geometry attributes as generic nuisance parameters during Markov chain sampling. This is the first time-domain AEM study we know of, that compares nuisance marginalized subsurface posterior conductivities from a fixed-wing system, with rotary-wing derived posterior conductivities. We also compare field results with induction log data where available. Finally, we estimate the information gain in each case via a covariate shift adaptation technique that has not been used before in geophysical work. Our findings have useful implications in AEM system selection, as well as in the design of better deterministic AEM inversion algorithms.

Study on Dynamic Uncertainty and Sensitivity of Gear System Considering the Influence of Machining Accuracy

Due to the limitation of machining accuracy, the transmission performance uncertainty of mass gears must be evaluated quantitatively to provide the basis for its application in the whole machine. Based on the polynomial chaotic expansion (PCE) method, a dynamic uncertainty analysis method for gear systems with a specified precision was proposed in this paper. Combined with tooth surface contact analysis and load-bearing contact analysis, a dynamic model of the gear system was established to fully reflect the influence of typical manufacturing errors. Based on this, a PCE model was established to approximate the system dynamics model. The dynamic uncertainty of the gear system was quantified based on the PCE approximation model and the Monte Carlo method, respectively, and the computational accuracy and efficiency of the PCE model with different orders and numbers of sample points were compared and analyzed. Finally, Sobol′ sensitivity indices from the PCE model of the gear system to random errors were computed, and the primary and secondary relationships of influence on the dynamic performance of the gear system were determined. The results showed that the PCE method had good applicability to the quantification of dynamic uncertainty and error sensitivity analysis of gear systems, and it had both accuracy and high efficiency.

An Error-Pursuing Adaptive Uncertainty Analysis Method Based on Bayesian Support Vector Regression

The Bayesian support vector regression (BSVR) metamodel is widely used in various engineering fields to analyze the uncertainty arising from uncertain parameters. However, the accuracy of the BSVR metamodel based on the traditional one-shot sampling method fails to meet the requirements of the uncertainty analysis of complex systems. To this end, an error-pursing adaptive uncertainty analysis method based on the BSVR metamodel is presented by combining a new adaptive sampling scheme. This new sampling scheme was improved by a new error-pursuing active learning function that is named, herein, adjusted mean square error (AMSE), which guides the adaptive sampling of the BSVR metamodel’s design of experiments (DoE). During the sampling process, AMSE combines mean square error and leave-one-out cross-validation error to estimate the prediction error of the metamodel in the entire design space. Stepwise refinement of the metamodel was achieved by placing the sampled regions at locations with large prediction errors. Six benchmark analytical functions featuring different dimensions were used to validate the proposed method. The effectiveness of the method was then further illustrated by a more realistic application of an overhung rotor system.

A non-intrusive frequency normalisation approach for uncertain response analysis of nonlinear dynamic systems

This paper proposes a non-intrusive technique based on the frequency normalisation for non-intrusive propagations of parameter uncertainties in nonlinear mechanical systems. The multi-solution dilemma found in resonance areas, which prohibits successful applications of uncertainty quantification methods, is alleviated by the additional normalised frequency measure. The spurious peaks around the nonlinear peaks and discontinuous points, known as the Gibbs phenomenon, due to parameter variabilities are resolved as well. The proposed technique coupled with the non-intrusive surrogate modelling is applied to two deliberately constructed nonlinear dynamic systems, i.e., a mass-spring system with interactive cubic nonlinearity and a piece-wise rotor/stator contact problem in rotating machines. The two systems exhibit complex amplitude-frequency response characteristics featured by the softening and hardening effects. Numerous case investigations show the effectiveness of the proposed normalisation method for uncertainty analysis of complex nonlinear vibration systems and its working principle is demonstrated in detail via examples. Accuracy tests against the traditional sampling methods are carried out in both systems. The proposed technique can be easily generalised to other nonlinear systems for random and non-random uncertainty propagations because it works non-intrusively and permits users to choose arbitrary nonlinear tools.

A novel analysis method for vibration systems under time-varying uncertainties based on interval process model

Non-probabilistic models can quantify the uncertainties of engineering systems with limited data, which is significantly convenient and economical for engineering applications. However, a key issue in the application of non-probabilistic methods to vibration problems is how to address the time-varying uncertainties, especially when the correlation is involved. In this study, a novel analysis method based on the interval process model is proposed to obtain the dynamic response bounds of vibration systems under time-varying uncertainties, where uncertainties inherently existed in external excitations and system parameters are considered. Firstly, the interval process model is utilized for describing the uncertain-but-bounded variables, and the correlation function quantifies the correlation between the variables at different times. By solving two optimization problems, the dynamic response bounds concerning uncertain system parameters are obtained. Based on the response bounds, the first-order Taylor expansion is used to handle the uncertainties of system parameters, and a homogenization treatment is proposed to suppress the interval expansion. Thus the upper and lower bounds of the dynamic response under time-varying uncertainties can be determined. To quantify the reliability of vibration systems under time-varying uncertainties, the time-varying reliability indexes related to the proposed method are constructed. Finally, the proposed method is validated by the corresponding Monte Carlo simulation method and further applied in lunar soil coring. The findings from this study can provide an important tool for uncertainty and reliability analysis of vibration systems in practical engineering.

A state-of-the-art review on uncertainty analysis of rotor systems

Uncertainty handling and analysis of rotor systems have been active research areas during the last two decades. This paper provides a state-of-the-art review on the research progress of uncertainty treatment in rotor systems. Firstly, sources of uncertainties are identified and then classified according to their nature. Secondly, the most popular uncertainty analysis methods are summarized based on different types of uncertainties, including their basic principles, strengths and limitations. Thirdly, the dynamic characteristics of rotor systems under various uncertainties are described in detail, i.e., the natural characteristics and unbalanced responses, faulted rotor dynamics, stability problems and parameter identifications. Then, the reliability analysis, optimizations and vibration control for uncertain rotor systems are reviewed. Finally, comments on the latest research progress are made and outlooks on future research directions are highlighted. It is expected that this review will provide useful guidelines for designers and researchers towards the more efficient handling of uncertainty and robust dynamic analysis for rotor systems as well as insightful prospects, showing where extra efforts are needed.

Uncertainty and correlation propagation analysis of powertrain mounting systems based on multi-ellipsoid convex model

In engineering practice, uncertainty and correlation may coexist in both input parameters and output responses of the powertrain mounting system (PMS) of a vehicle. A methodology is developed for the uncertainty and correlation propagation analysis of the inherent characteristics of PMSs in this research. In the proposed methodology, the multi-ellipsoid convex model is introduced to quantify multiple groups of uncertain parameters with correlation, where dependent and independent uncertain parameters are considered simultaneously. In the uncertainty propagation analysis, it aims to calculate the interval bounds of system responses. To perform uncertainty propagation analysis, the Monte Carlo uncertainty analysis (MCUA) method is firstly presented based on Monte Carlo simulation, and the first-order perturbation-central difference-Lagrange multiplier (FPCDLM) method and the second-order perturbation-central difference-Lagrange multiplier (SPCDLM) method are then derived to promote the computational efficiency. In the correlation propagation analysis, it aims to compute the correlation between different system responses. To conduct the correlation propagation analysis, the Monte Carlo correlation analysis (MCCA) method is firstly proposed and the second-order perturbation correlation analysis (SPCA) method are then developed to enhance the computational efficiency. Next, the whole procedures for uncertainty and correlation propagation analysis of PMS are established by combining the above uncertainty analysis and correlation analysis methods, and the elliptical domain of any two system responses can be obtained. Finally, numerical examples of the PMS of an electric vehicle are provided to demonstrate the effectiveness of the proposed methodology.

Surrogate modeling for dynamic analysis of an uncertain notched rotor system and roles of Chebyshev parameters

The non-intrusive Chebyshev surrogate coupled with the harmonic balance method is developed to calculate harmonics and super-harmonic responses of an uncertain notched rotor, i.e., rotor with an open crack. The deterministic rotor model is formulated by the finite element method and an open crack in the elastic shaft introduces the asymmetry fault. A large size crack in the middle span of the shaft is considered, which makes the system susceptible to variations of parameters. The scanning approach is used to validate the accuracy of the surrogate function. Importantly, roles of Chebyshev parameters on performance are studied and proved. It is found that the Chebyshev order is the crucial factor to reduce the spurious peaks and overestimation near critical speeds of the rotor model. An improved surrogate function with fewer collocations is formulated to solve uncertain problems with many uncertainties and high orders are required in surrogate models. It is found that the surrogate function can efficiently and accurately estimate the uncertain responses of the notched rotor with multi-interval variables. The results will benefit interval uncertainty analysis of general large-scale and sensitive mechanical systems.

A framework for uncertainty and sensitivity analysis of district energy systems considering different parameter types

A variety of renewable energy sources, demand side loads, and other factors bring a lot of uncertainties to the operation of district energy systems. However, there is no strict standard for the selection of uncertainty factors. Too much uncertainty, or not enough of it, is not appropriate for the system optimization. In this paper, a method of uncertainty and sensitivity analysis for district energy system is proposed to identify the important uncertainties in the system optimization process. Firstly, the mathematical model of the system is established and the uncertain parameters are screened from technological, economic, environmental and demand-side perspectives. In addition, the selected parameters are divided into two categories according to whether they are time-related, and two methods are adopted to collect samples. Further, a global sensitivity analysis technique is introduced to calculate the sensitivity index of each parameter simultaneously. Case study is performed for a classic district energy system in a typical day of transition season. The sensitivity indexes under different objective functions are also discussed on this basis. The results show that the proposed framework can effectively analyze the uncertain parameters of the system, which can further contribute to the optimal operation decision.

A sequential radial basis function method for interval uncertainty analysis of multidisciplinary systems based on trust region updating scheme

A sequential radial basis function method for interval uncertainty analysis of multidisciplinary systems based on trust region updating scheme

Uncertainty and multi-criteria global sensitivity analysis of structural systems using acceleration algorithm and sparse polynomial chaos expansion

Sparse polynomial chaos expansion (PCE) can be used to emulate the stochastic model output where the original model is computationally expensive. It is a powerful tool in efficient uncertainty quantification and sensitivity analysis. Structural systems are usually associated with high dimensional and probabilistic input. The number of candidate basis functions increases significantly with input dimension, resulting in high computational burden for establishing sparse PCE. In this study, acceleration techniques are integrated to formulate an algorithm for efficient computation of sparse PCE (ASPCE). The integrated algorithm can improve efficiency of computational process compared with conventional greedy algorithm while ensuring the satisfying predictive performance. Once the sparse PCE model is obtained, the statistic moments, probability density function of stochastic output, and global sensitivity index could be computed efficiently. Traditional PCE based global sensitivity analysis only assesses the sensitivity on individual structural performance criterion. Assessing the global sensitivity considering multiple criteria is challenging as the sensitive parameters may not be consistent for different performance criteria. To address this issue, a two-stage multi-criteria global sensitivity analysis algorithm is proposed by coupling ASPCE and the technique for order preference by similarity to ideal solution (TOPSIS). A holistic global sensitivity index is proposed to identify the sensitive parameters incorporating multiple performance criteria. In order to illustrate the efficiency, accuracy, and applicability of the proposed approach, two illustrative cases are presented.

Bayesian Surrogate Learning for Uncertainty Analysis of Coupled Multidisciplinary Systems

AbstractEngineering systems are often composed of many subsystems that interact with each other. These subsystems, referred to as disciplines, contain many types of uncertainty and in many cases are feedback-coupled with each other. In designing these complex systems, one needs to assess the stationary behavior of these systems for the sake of stability and reliability. This requires the system level uncertainty analysis of the multidisciplinary systems, which is often computationally intractable. To overcome this issue, techniques have been developed for capturing the stationary behavior of the coupled multidisciplinary systems through available data of individual disciplines. The accuracy and convergence of the existing techniques depend on a large amount of data from all disciplines, which are not available in many practical problems. Toward this, we have developed an adaptive methodology that adds the minimum possible number of samples from individual disciplines to achieve an accurate and reliable uncertainty propagation in coupled multidisciplinary systems. The proposed method models each discipline function via Gaussian process (GP) regression to derive a closed-form policy. This policy sequentially selects a new sample point that results in the highest uncertainty reduction over the distribution of the coupling design variables. The effectiveness of the proposed method is demonstrated in the uncertainty analysis of an aerostructural system and a coupled numerical example.

The influence of mechanical uncertainties on the free vibration of functionally graded graphene-reinforced porous nanocomposite shells of revolution

This paper is concerned with the analysis of uncertainty propagation in the free vibration response of functionally graded porous (FGP) nanocomposite shells of revolution reinforced with graphene platelet (GPL). In this regard, the effects of uncertainties in the material properties as well as the geometry of the reinforcements on the free vibration responses of the FGP-GPL shells with different types of Gaussian curvature are assessed. The propagation of uncertainty and frequency sensitivity of the shells of revolution are examined for various GPL dispersions and porosity distributions. For this purpose, the interval analysis method which is an appropriate technique for the uncertainty analysis of systems with bounded uncertainties is utilized. In order to solve the equations of motion based on the higher-order shear deformation theory and to obtain the free vibration responses of the shells of revolution, the Fourier Differential Quadrature (FDQ) technique is employed. According to the analysis results, the shells’ type of curvature has a significant effect on the trend of uncertainty propagation and the sensitivity of the vibration frequencies in terms of various sources of uncertainty. Moreover, the results show that the uncertainty propagation and sensitivity are different in the frequencies corresponding to different modes of vibration.

The propagation of uncertainty in the geometrically nonlinear responses of smart sandwich porous cylindrical shells reinforced with graphene platelets

This paper is concerned with the analysis of uncertainty propagation in the nonlinear dynamic responses of smart sandwich cylindrical shells under rectangular, sine, and exponential loads. The sandwich shells are composed of a functionally graded porous (FGP) core reinforced with nanocomposite graphene platelets (GPLs) and surrounded by two piezoelectric layers. Various sources of uncertainties related to the material properties, the geometry of the reinforcements, and piezoelectric parameters are considered in this study. The uncertainty propagations in the nonlinear dynamic responses of the shells are examined for different variations of GPL dispersions and porosity distributions in the core. For this purpose, three different distributions for porosity and three different dispersions for graphene platelets have been considered in the thickness direction. In order to investigate the uncertainty propagation in the responses of the shells, the interval analysis method which is an appropriate technique for the uncertainty analysis of systems with bounded uncertainties is utilized. The Halpin-Tsai model is used to find the effective properties of the GPL-reinforced materials. The equations of motion are derived based on the higher-order shear deformation theory and Sanders’ nonlinear theory. In order to solve the equations of motion and to obtain the dynamic responses of the shells, the Fourier differential quadrature (FDQ) technique is employed. According to the analysis results, the sensitivities of the responses of internal moment, displacement, and internal force have the highest values relative to the uncertainty sources, respectively, whereas the uncertainty percentages for the responses of displacement, internal moment, and internal force respectively have the highest values for the examined loads.

Linear substitute model-based uncertainty analysis of complicated non-linear energy system performance (case study of an adaptive cycle engine)

The design of energy systems is inevitably affected by uncertainty. Sources of uncertainty include user demands, system components and system operating conditions. Ignoring these uncertainties and developing deterministic designs can render such designs suboptimal and result in system failure. To optimize design under conditions of uncertainty necessitates the quantification of its effects. Monte Carlo Simulation is commonly used to this end because it is simple and easy to understand. However, for complex non-linear systems, it is difficult to directly apply Monte Carlo Simulation due to its high computational cost. Through the study of an innovative gas turbine (an adaptive cycle engine), this paper presents a method for the rapid quantification of uncertainty based on a linear substitute model. There are two basic types of linear substitute model: one based upon the Taylor series method; the other upon the least squares method. Out of the two, the least squares method offers better accuracy at an affordable computational cost. Using this method, it takes 500 s to estimate the means and standard deviation of a system’s performance. This is substantially less than the 100,000 s needed for direct Monte Carlo Simulation. The approximation error is typically less than 1% for the standard deviation and less than 5% for the mean under most conditions, far less than a comparable use of the Taylor series method. The same approach can also be adopted for the uncertainty analysis of other complex non-linear energy systems.

Robustness Analysis of Flight Controllers for Fixed-Wing Unmanned Aircraft Systems Using Integral Quadratic Constraints

A robust analysis framework for mathematical validation of flight control systems is presented. The framework is specifically developed for the complete uncertainty characterization, quantification, and analysis of small fixed-wing unmanned aircraft systems (UAS). The analytical approach presented within is based on integral quadratic constraint (IQC) analysis and uses linear fractional transformations on uncertainties to represent system models. While IQC analysis has a sound theoretical foundation, there remains a lack of literature on applying IQCs to complex engineering systems and evaluating their effectiveness with physical data. One difficulty lies in appropriately characterizing and quantifying the uncertainties such that the resulting uncertain model is representative of the physical system while forming a computationally tractable problem that is not overly conservative. This paper addresses these challenges by applying IQC analysis tools to analyze the robustness of a UAS flight control system. Uncertainties are characterized and quantified based on mathematical models and flight test data obtained in-house for a small commercial off-the-shelf platform with a custom autopilot. Furthermore, an established gradient-based minimization routine is implemented with IQC analysis to demonstrate how IQC analysis can guide the control design process. This approach also reveals the controlled system’s sensitivities to uncertainties, thereby assisting the designer in determining how much uncertainty allowance ought to exist in certain aspects of the UAS. Finally, these methods are tested in physical flight to showcase the effectiveness of IQC analysis and assess the conservativeness of the approach. The proposed framework is also transferable to other fixed-wing UAS platforms, effectively taking IQC analysis beyond academic examples to practical application in UAS system design, control design, and airworthiness certification.

A sparse surrogate model for structural reliability analysis based on the generalized polynomial chaos expansion

The reliability analysis of a structural system is typically evaluated based on a multivariate model that describes the underlying mechanistic relationship between the system’s input and output random variables. This is the need to develop an effective surrogate model to mimic the input–output relationship as the Monte Carlo simulation–based on the mechanistic model might be computationally intensive. In this regard, the article presents a sparse regression method for structural reliability analysis based on the generalized polynomial chaos expansion. However, results from the global sensitivity analysis have justified that it is unnecessary to contain all polynomial terms in the surrogate model, instead of comprising a rather small number of principle components only. One direct benefit of the sparse approximation allows utilizing a small number of training samples to calibrate the surrogate model, bearing in mind that the required sample size is positively proportional to the number of unknowns in regression analysis. Therefore, by utilizing the standard polynomial chaos basis functions to constitute an explanatory dictionary, an adaptive sparse regression approach characterized by introducing the most significant explanatory variable in each iteration is presented. A statistical approach for detecting and excluding spuriously explanatory polynomials is also introduced to maintain the high sparsity of the meta-modeling result. Combined with a variety of low-discrepancy schemes in generating training samples, structural reliability and global sensitivity analysis of originally true but computationally demanding models are alternatively realized based on the sparse surrogate method in conjunction with the brutal Monte Carlo simulation method. Numerical examples are carried out to demonstrate the applicability of the sparse regression approach to structural reliability problems. Results have shown that the proposed method is an effective, non-intrusive approach to deal with uncertainty analysis of various structural systems.