Sensitivity analysis and practical identifiability of the mathematical model for partial differential equations

A sensitivity-based identifiability analysis of mathematical model for partial differential equations is carried out using an orthogonal method and an eigenvalue method. These methods are used to study the properties of the sensitivity matrix and the effects of changes in the model coefficients on the simulation results. Practical identifiability is investigated to determine whether the coefficients can be reconstructed with noisy experimental data. The analysis is performed using correlation matrix method with allowance for Gaussian noise in the measurements. The results of numerical calculations to obtain identifiable sets of parameters for the mathematical model arising in social networks are presented and discussed.

On the Statistical Discrepancy and Affinity of Priority Vector Heuristics in Pairwise-Comparison-Based Methods

There are numerous priority deriving methods (PDMs) for pairwise-comparison-based (PCB) problems. They are often examined within the Analytic Hierarchy Process (AHP), which applies the Principal Right Eigenvalue Method (PREV) in the process of prioritizing alternatives. It is known that when decision makers (DMs) are consistent with their preferences when making evaluations concerning various decision options, all available PDMs result in the same priority vector (PV). However, when the evaluations of DMs are inconsistent and their preferences concerning alternative solutions to a particular problem are not transitive (cardinally), the outcomes are often different. This research study examines selected PDMs in relation to their ranking credibility, which is assessed by relevant statistical measures. These measures determine the approximation quality of the selected PDMs. The examined estimates refer to the inconsistency of various Pairwise Comparison Matrices (PCMs)—i.e., W = (wij), wij > 0, where i, j = 1,…, n—which are obtained during the pairwise comparison simulation process examined with the application of Wolfram’s Mathematica Software. Thus, theoretical considerations are accompanied by Monte Carlo simulations that apply various scenarios for the PCM perturbation process and are designed for hypothetical three-level AHP frameworks. The examination results show the similarities and discrepancies among the examined PDMs from the perspective of their quality, which enriches the state of knowledge about the examined PCB prioritization methodology and provides further prospective opportunities.

The Influence of Circulatory Forces on the Stability of Undamped Gyroscopic Systems

In this paper, the effects of circulatory forces on undamped gyroscopic systems have been studied. The stability or otherwise of arising MGKN systems have been analysed using eigenvalue method and Lyapunov method. Bottema-Lakhadanov-Karapetyan theorem is also employed to further analyze the systems. Examples are given to illustrate the use of these methods in determining the stability or otherwise of undamped gyroscopic circulatory systems.

Modelling and Stability Analysis of Articulated Vehicles

This study constructs a nonlinear dynamic model of articulated vehicles and a model of hydraulic steering system. The equations of state required for nonlinear vehicle dynamics models, stability analysis models, and corresponding eigenvalue analysis are obtained by constructing Newtonian mechanical equilibrium equations. The objective and subjective causes of the snake oscillation and relevant indicators for evaluating snake instability are analysed using several vehicle state parameters. The influencing factors of vehicle stability and specific action mechanism of the corresponding factors are analysed by combining the eigenvalue method with multiple vehicle state parameters. The centre of mass position and hydraulic system have a more substantial influence on the stability of vehicles than the other parameters. Vehicles can be in a complex state of snaking and deviating. Different eigenvalues have varying effects on different forms of instability. The critical velocity of the linear stability analysis model obtained through the eigenvalue method is relatively lower than the critical velocity of the nonlinear model.

Transient Flutter Stability Analysis of Structural Mistuned Blisk by Energy Growth Method

Blisks suffer from flutter, a self-sustained vibration caused by aerodynamic coupled forces. This instability could cause serious damage to the blades and the machine. Flutter stability is usually analyzed based on the eigenvalue method in the aspect of the linear structural dynamic system, which transforms a dynamics stability analysis into a point of equilibrium in an infinite time scale. However, in reality, most of the blisk vibrations arise on a finite time horizon. The transient vibration amplification may cause serious damage. This paper proposes a transient flutter stability analysis method in a finite time for structural mistuned blisk based on the energy growth method. Firstly, two common blisk models coupled aerodynamic force with different complexity are built, and are all expressed in the state space representation. A novel energy growth method is then employed to analyze the transient stability and to find the maximum energy growth of the models. The optimal initial condition which leads to the maximum energy growth is obtained. A new flutter stability criterion is developed to consider the transient stability based on the energy growth method and the infinite time stability based on the eigenvalue method. The new transient stability method is verified by two numerical studies. It is found that the structural mistuned blisk model which is traditionally predicted stable still has a transient instability in a finite time due to the non-normal property of the dynamic state matrix.

A Numerical Comparison of the Sensitivity of the Geometric Mean Method, Eigenvalue Method, and Best–Worst Method

In this paper, we compare three methods for deriving a priority vector in the theoretical framework of pairwise comparisons—the Geometric Mean Method (GMM), Eigenvalue Method (EVM) and Best–Worst Method (BWM)—with respect to two features: sensitivity and order violation. As the research method, we apply One-Factor-At-a-Time (OFAT) sensitivity analysis via Monte Carlo simulations; the number of compared objects ranges from 3 to 8, and the comparison scale coincides with Saaty’s fundamental scale from 1 to 9 with reciprocals. Our findings suggest that the BWM is, on average, significantly more sensitive statistically (and thus less robust) and more susceptible to order violation than the GMM and EVM for every examined matrix (vector) size, even after adjustment for the different numbers of pairwise comparisons required by each method. On the other hand, differences in sensitivity and order violation between the GMM and EMM were found to be mostly statistically insignificant.

Stability Assessment Approach Based on Trajectory Section Eigenvalue

Eigenvalue Method can analyze the operating features of the system. The trajectory section eigenvalue method combines the model and trajectory, and extends the equilibrium point eigenvalue to the unbalanced point of the cross section at any time on the disturbed trajectory, which well solves the non-linearity and the time-varying issues of the system. Based on the theoretical basis of clarifying the trajectory section eigenvalue (TSE) method, the paper establishes a stability evaluation framework based on the time-varying second-order resonant circuit, which extrates the damping feature and frequency feature information in the eigenvalue sequence of the trajectory section, then compares it with the results of signal processing. The influence mechanism of time-varying factors on system stability is clarified, and the advantages of the trajectory section eigenvalue method in stability evaluation are demonstrated.