A harmonic balance method combined with AFT and model reduction for parameter sensitivity analysis of shaft-hull-bearing systems

Comparison of harmonic balance and multi-scale method in characterizing the response of monostable energy harvesters

Chapter 2 - Harmonic balance method and time domain collocation method

The method of unrestricted harmonic balance (UHB) which is a generalization of the old method of harmonic balance and that was developed in preceding papers, is mathematically refined and applied to the evaluation of unstable limit cycles. The method is demonstrated for the case of the best investigated chaotic system, namely the Lorenz system. Some representative results are given

In this paper, the approximate analytical solutions obtained by using the constrained parameter-splitting-multiple-scales (C-PSMS) method to the primary and [Formula: see text] sub-harmonic resonances responses of a cantilever-type energy harvester are presented. The C-PSMS method combines the multiple-scales (MS) method with the harmonic balance (HB) method. Different from the erroneous stability results obtained by using the Floquet theory and the classical HB method, accurate stability results are obtained by using the C-PSMS method. It is found that the correction to the erroneous solution when the HB method and Floquet theory are adopted in the stability analysis of the primary and [Formula: see text] sub-harmonic resonances of a largely deflected cantilever-type energy harvester is necessary. On the contrary, the C-PSMS method gives much improved results compared to those obtained by using Floquet theory and HB method when the numbers of terms in each response expression are the same. The frequency response curves of the primary resonance and the [Formula: see text] sub-harmonic resonance of the harvester obtained by the C-PSMS method are compared to those obtained by the HB method and verified by those obtained by the fourth-order Runge–Kutta method. Moreover, the basin of attraction based on the fourth-order Runge–Kutta method is presented to confirm the inaccurate stability results obtained by using the HB method and Floquet theory. The convergence examinations on the stability analysis carried out by the HB method and Floquet theory show that enough terms in the response assumption are needed to achieve relatively accurate stability results when studying the stability of the primary and sub-harmonic resonances of a cantilever by using the HB method and the Floquet theory. However, the low-order C-PSMS method is able to give an accurate frequency-amplitude response and accurate stability results of the primary and sub-harmonic resonances of a largely deflected cantilever-type energy harvester.

A twice harmonic balance (THB) method is proposed to compute and analyze quasi-periodic (QP) responses of nonlinear dynamical systems, with emphasis on the stability and bifurcation of QP responses. In the first harmonic balancing, the original system is transformed into a truncated system via harmonic balance method with variable-coefficients. The truncated system is further solved via the second harmonic balancing, more specifically the incremental harmonic balance (IHB) method. The equivalence is addressed between the periodic solutions of the truncated system and the QP responses of the original system. According to the relationship, the presented method is in essence to convert the problem of solving the original system for QP responses into a truncated system for periodic solutions. Numerical examples show that the semi-analytical QP solutions obtained by the THB method are in well consistence with the solutions obtained by the Runge–Kutta (RK) method and the IHB method with two time scales, respectively. More importantly, the stability of the attained QP solutions can be analyzed by just applying the Floquet theory to the periodic response of the truncated system. The continuation of the QP responses is generated by the presented method, on which the possible bifurcations resulted from the stability reversal are analyzed in detail. In addition, the evolution of QP responses can also be tracked from periodic solutions, such as that due to the onset of a Neimark–Sacker bifurcation.

In recent years, the necessity for fast, accurate and less memory-intensive techniques to analyze nonlinear circuits has grown as technology has advanced. The harmonic balance (HB) method is a powerful tool and it has been used for some time in nonlinear circuit analysis. In order to keep up with the vast requirements of circuit design, the harmonic balance method is modified to make it fast, more accurate and require less memory. In the modified harmonic balance (MHB) method, circuits are analyzed by calculating voltages and currents of nonlinear components in the time domain and those of linear components in the frequency domain. After that, an iteration scheme is performed in which the voltage and current values have to be transformed from one domain to the other for each single iteration. A key point to reduce the analysis time and minimize the memory required is to use an efficient way to transform from one domain to another. One-dimensional Fourier transformations are used to convert from the time domain to the frequency domain and visa versa. The current and voltage values are handled using a vector matrix for each nonlinear element instead of using Jacobian matrices.

Rotor–stator interaction (RSI) is an inherent phenomenon in multi-row turbomachinery. Unsteady reduced-order methods, such as the harmonic balance (HB) method and the space-time gradient (STG) method, have been proposed to capture RSI with fewer computational resources compared to fully unsteady simulation. In this study, the steady mixing-plane method, the HB method, and the STG method are implemented into the open-source external computational fluid laboratory three-dimensional (CFL3D) flow solver to gain the ability to predict turbomachinery flows based on solving Reynolds-averaged Navier–Stokes equations. Additionally, a rotation interpolation approach for adjacent blades is implemented for the unsteady multi-row turbomachinery simulation. For the HB method, the phase-lag periodic conditions and the temporal interpolation approach between two adjacent blade rows are integrated into CFL3D. Then, the steady mixing-plane method, the HB method, the STG method, and the fully unsteady simulation method are conducted on a quasi-three-dimensional radial slice and a three-dimensional geometry of the National Aeronautics and Space Administration Stage-35 compressor. Both the transient and time-averaged flowfield predicted by the reduced-order methods are compared with the unsteady simulations. Results indicate that the STG method and the HB method can accurately simulate the unsteady flow with better predictions of RSI impact. For the HB method, accurate prediction of transient unsteady flow requires a minimum of seven harmonics, whereas the time-averaged flow requires only five harmonics. Additionally, a quantitative assessment of computational speed is conducted, revealing that the HB method with seven harmonics achieved a speed 28 times faster than the fully unsteady simulation.

Parametric sensitivity analysis (SA) aims to select the sensitive parameters that most significantly affect the model output variables, which helps to improve model optimization efficiency by adjusting a small number of sensitive parameters instead of all adjustable parameters. The qualitative and quantitative SA methods have been commonly used to quantify the sensitive parameters of the models. However, the response surface model based quantitative SA method was rarely used. Taking the simulation of a quasi twodimensional (quasi-2D) groundwater model as an example, this study systematically assess eight SA methods divided into three categories (qualitative SA, quantitative SA, and the response surface model-based quantitative SA). The study validates the effectiveness of these methods by comparing the parameter sensitivity results, and also demonstrates the efficiency of these methods by determining the minimum sample size required. Using the minimum samples means the least number of model runs. The results show that P1 and P2 are the most sensitive parameters of the quasi-2D model for simulating groundwater table elevation. Except for local method, four global qualitative SA methods obtain reasonable parameter sensitivity rankings using 200 samples, but the parameter sensitivity scores fail. For obtaining accurate sensitivity scores, at least 2000 samples are required by the quantitative SA methods. However, for the response surface model-based quantitative SA method, 60 samples are sufficient to obtain accurate sensitivity scores, demonstrating that the method is an effective and highly efficient, and should be recommended as the primary parametric SA method, especially for the complex models with large computational demand.

Bifurcation in a 3-DOF Airfoil with Cubic Structural Nonlinearity

This paper used both methods of single parameter and Multi-parameter combination sensitivity analysis in groundwater numerical model parameter sensitivity analysis. With the examples of Jiaozuo, Henan province, choosing coefficient of permeability, porosity and longitudinal dispersion degree as factor, the paper analysis the main controlling factors of contaminant migration of Hexavalent chromium in groundwater. Single parameter sensitivity analysis results show that the permeability coefficient gives largest impact to the migration of hexavalent chromium. It is positively correlates with groundwater numerical model. The rest of the parameters assume the inverse correlation. Multi-parameter combination sensitivity analysis results show that the permeability coefficient has absolute advantage among various parameter combinations. Another conclusion is that longitudinal dispersion degree affects smaller on the process of hexavalent chromium pollutants migration, but the impact on the final density is bigger.

Sensitivity analysis of urban flood model parameters is important for urban flood simulation. Efficient and accurate acquisition of sensitive parameters is the key to real-time model calibration. In order to quickly obtain the sensitive runoff parameters of the urban flood simulation model, this study proposes an artificial neural network-based identification method for sensitive parameters. Artificial neural network (ANN) models were constructed with the binary classification and multi-classification methods, and used environmental indicators that affect the parameter sensitivity of different hydrological response units as the input, with the sensitivity parameters of the Storm water management model (SWMM) being the output. The optimization of the ANN was realized by adjusting the number of nodes in the hidden layer and the maximum number of iterations. An example application was conducted in Zhengzhou, China. The results show that the binary classification ANN quickly identified sensitive parameters, and the prediction accuracy of all parameters exceeded 96%. Convergence can be achieved when the number of nodes in the hidden layer does not exceed twice the number of input nodes, and the maximum number of iterations does not exceed 200. Rapid and accurate identification of the sensitive runoff parameters of the urban flood simulation model was achieved, which reduced the time required for parameter sensitivity analysis.

An interactive graphical interface tool for parameter calibration, sensitivity analysis, uncertainty analysis, and visualization for the Soil and Water Assessment Tool

A Highly-integrated valve-controlled cylinder (HIVC) that has the advantage of large power weight ratio and fast response is widely applied in many fields. To ensure high performance of HIVC, control strategy selection and structure parameters optimization are essential; thus, performance-influenced main parameters and secondary parameters should be well-grasped to target control compensation and structure optimization. Trajectory sensitivity analysis (TSA), a branch of sensitivity analysis, can be adapted to research on the effects of control and structure parameter variation on system performance; the analysis conclusions can be used to improve system performance and the analysis has been applied in many applications in various fields. In this paper, based on the mathematical model of the First-order trajectory sensitivity analysis (FOTSA), the mathematical model of Second trajectory sensitivity analysis (SOTSA) is further derived; the general expression of the second-order trajectory sensitivity equations and the special expression that is applicable to each system parameter sensitivity analysis of HIVC are built respectively. Furthermore, based on the position control system of the nonlinear mathematical model of HIVC involved with servo valve dynamic characteristics, flow-pressure nonlinearity, initial piston position of servo cylinder, and friction nonlinearity, the coefficient items and the free items matrices of the special expression are calculated. Moreover, the First-order trajectory sensitivity function (FOTSF) and the Second trajectory sensitivity function (SOTSF) of the 17 main parameters in HIVC are computed on the MATLAB/Simulink platform under nine typical working conditions of the displacement step response. Then, the dynamic change rules and the differences and similarities of each parameter sensitivity analysis results are obtained through FOTSA and SOTSA under different working conditions with different parameter variations; each parameter sensitivity variation histogram based on the two kinds of sensitivity indexes, including the maximum value of the percentage absolute value and the absolute value summation within the sampling time, are given. Furthermore, differences and similarities of each parameter sensitivity index change rule and value are quantitatively analyzed. Finally, the sensitivity index change rules and values of the four parameters in the position control system of HIVC, including system supply oil pressure, proportion gain, initial piston position of servo cylinder, and load are experimentally verified. All above studies indicate that each parameter SOTSA result of HIVC vary under different working conditions. When parameter variation is significant, the differences in certain parameter sensitivity analysis between FOSTA and SOTSA are relatively significant; and SOTSA results are more similar to the test results, which illustrates that SOTSA results are more accurate.

BackgroundStochastic modeling and simulation provide powerful predictive methods for the intrinsic understanding of fundamental mechanisms in complex biochemical networks. Typically, such mathematical models involve networks of coupled jump stochastic processes with a large number of parameters that need to be suitably calibrated against experimental data. In this direction, the parameter sensitivity analysis of reaction networks is an essential mathematical and computational tool, yielding information regarding the robustness and the identifiability of model parameters. However, existing sensitivity analysis approaches such as variants of the finite difference method can have an overwhelming computational cost in models with a high-dimensional parameter space.ResultsWe develop a sensitivity analysis methodology suitable for complex stochastic reaction networks with a large number of parameters. The proposed approach is based on Information Theory methods and relies on the quantification of information loss due to parameter perturbations between time-series distributions. For this reason, we need to work on path-space, i.e., the set consisting of all stochastic trajectories, hence the proposed approach is referred to as “pathwise”. The pathwise sensitivity analysis method is realized by employing the rigorously-derived Relative Entropy Rate, which is directly computable from the propensity functions. A key aspect of the method is that an associated pathwise Fisher Information Matrix (FIM) is defined, which in turn constitutes a gradient-free approach to quantifying parameter sensitivities. The structure of the FIM turns out to be block-diagonal, revealing hidden parameter dependencies and sensitivities in reaction networks.ConclusionsAs a gradient-free method, the proposed sensitivity analysis provides a significant advantage when dealing with complex stochastic systems with a large number of parameters. In addition, the knowledge of the structure of the FIM can allow to efficiently address questions on parameter identifiability, estimation and robustness. The proposed method is tested and validated on three biochemical systems, namely: (a) a protein production/degradation model where explicit solutions are available, permitting a careful assessment of the method, (b) the p53 reaction network where quasi-steady stochastic oscillations of the concentrations are observed, and for which continuum approximations (e.g. mean field, stochastic Langevin, etc.) break down due to persistent oscillations between high and low populations, and (c) an Epidermal Growth Factor Receptor model which is an example of a high-dimensional stochastic reaction network with more than 200 reactions and a corresponding number of parameters.

Implementation of sensitivity analysis (SA) procedures is helpful in calibration of models and also for their transposition to different watersheds. The reported studies on SA of Soil and Water Assessment Tool (SWAT) model were mostly focused on identifying parameters for pruning or modifying during the calibration process. This paper presents a sensitivity and identifiability analysis of model parameters that influence stream flow generation in SWAT. The analysis was focused on evaluating the sensitivity of the parameters in different climatic settings, temporal scales and flow regimes. The global sensitivity analysis (GSA) technique based on classical decomposition of variance, Sobol', was employed in this study. The results of the study indicate that modeled stream flow show varying sensitivity to parameters in different climatic settings. The results also suggest that the identifiability of a parameter for a given watershed is a major concern in calibrating the model for the specific watershed, as it might lead to equifinality of parameters. The SWAT model parameters show varying sensitivity in different years of simulation suggesting the requirement for dynamic updation of parameters during the simulation. The sensitivity of parameters during various flow regimes (low, medium and high flow) is also found to be uneven, which suggests the significance of a multi‐criteria approach for the calibration of models. Copyright © 2010 John Wiley & Sons, Ltd.