Eigenvalue Analysis Research Articles

Dynamics analysis of a reaction-diffusion-advection benthic-drift model with logistic growth.

This paper aims to investigate the benthic-drift population model in both open and closed advective environments, focusing on the logistic growth of benthic populations. We obtain the threshold dynamics using the monotone iteration method, and show that the zero solution is globally attractive straightforward when linearly stable. When unstable, limits from monotonic iteration of upper and lower solutions are upper and lower semi-continuous, respectively. By employing a part metric, we prove these limits are equal and continuous, leading to a positive steady state. In the critical case, we establish that the limit function from the upper solution iteration must be the zero solution by analyzing an algebraic equation. Furthermore, we conduct a quantitative analysis of the principal eigenvalue for a non-self-adjoint eigenvalue problem to examine how the diffusion rate, advection rate, and population release rates influence the dynamics. The results suggest that the diffusion rate and advection rate have distinct effects on population dynamics in open and closed advective environments, depending on the population release rates.

Adaptive Controller Design for Improving Helicopter Flying Qualities

A comprehensive flight control law design method based on adaptive control is presented in this paper. The proposed method consists of three basic modules—model decoupling, online system identification and adaptive pole placement. The model decoupling module decouples the helicopter flight dynamics model based on dynamic inversion technique. This procedure helps to reduce the difficulties in online system identification and adaptive controller design. In online system identification module, a recursive extended least squares algorithm is established to identify the augmented linear flight dynamics model which is composed of helicopter model and unideal noise model. The helicopter model parameters and the noise parameters are identified simultaneously which improves the identification accuracy as well as robustness. Pole placement is implemented in the last module, and an optimization method is developed to help selecting ideal poles. The adaptive rule in this step is designed based on eigenvalue analysis of the model to remove all unnecessary oscillations of the control parameters. An adaptive controller is designed according to the developed method for the UH-60A helicopter based on a nonlinear simulation program. Typical response types are also implemented. The simulation results show that the designed adaptive controller has high performance as well as robustness in both hover and forward flight.

Ground Tests on BOLT at Mach 7: Cross-Facility Comparison and Stability Analysis

Transition measurements have been obtained through two experimental campaigns conducted independently by the German Aerospace Center and the French Aerospace Lab with the French Alternative Energies and Atomic Energy Commission on subscale models of the BOLT-1 flight experiment geometry. This paper details a cross-facility comparison of measurements obtained at Mach 7, as well as subsequent computational analysis. Infrared thermography measurements obtained by both campaigns have facilitated a global comparison of the transition front across facilities at analogous conditions, which are found to be in good agreement. High-frequency surface pressure fluctuation data demonstrate significant amplification of instabilities with Mack-mode characteristics in the outboard regions of the acreage. These measurements are compared to stability analyses of varying fidelity. The computational methods employed to characterize the boundary-layer transition phenomena include the traditional line-marching implementation of the parabolized stability equations (PSEs), 2D eigenvalue analysis coupled with PSE. Although the 2D eigenvalue analysis is found to predict instabilities, which correlate in terms of frequency and acreage location to the experimental measurements, the predicted amplification for these instabilities is lower than would typically be expected for transition. Line-marching results for traveling crossflow produce the best match to the experimental transition front, with a consistent transition N factor of approximately 3–3.5.

Dynamic and quasi-static evaluation of stiffness properties of CLT: longitudinal MoE and effective rolling shear modulus

Cross-Laminated Timber (CLT) is an engineered wood product composed of solid layers of glued sawn timber. In this study, essential material stiffness parameters for CLT made from Norway spruce and Scots pine are evaluated. Specifically, the longitudinal modulus of elasticity (MoE) for longitudinally oriented layers and the effective rolling shear modulus for transversely oriented layers are the focus. By combining finite element (FE) analysis with four-point, out-of-plane bending tests using digital image correlation (DIC), a robust assessment of the effective rolling shear modulus of CLT layers is achieved. Additionally, eigenvalue analysis, applied to an FE model, along with resonance frequencies obtained from dynamic excitation of CLT, enables stable and simultaneous assessment of the dynamic longitudinal MoE and effective rolling shear modulus. Notably, while the dynamic MoE of longitudinal CLT layers is only 4% higher than the quasi-static local MoE, the dynamic effective rolling shear modulus of CLT layers is 40% higher than the quasi-static effective rolling shear modulus. This finding indicates a tangible viscoelastic behavior of wood concerning rolling shear.

Research on the multi-source vibration identification method of distributed fiber sensing based on improved fast independent component analysis.

Distributed acoustic sensing (DAS) is a technology that uses optical fiber as a sensing unit to detect external vibration signals. Due to the high resolution and high sensitivity of DAS, it has great application potential in the detection of vibration events. However, high detection performance will bring limitations to DAS in multi-source detection. In order to solve the problem of poor multi-source and multi-target recognition ability of DAS in complex multi-source environments, a multi-target recognition processing method of DAS system is proposed. Firstly, the acquired signal is pretreated to suppress the influence of system noise. Secondly, based on the independence of different events, a multi-source vibration location identification method based on the principal eigenvalue analysis of the observation matrix is proposed to realize the identification of the number and location of the multi-source vibration. Finally, the dimension of the acquired signal is reduced according to the number of sound sources, and the multi-source vibration signal recognition method based on amplitude uncertainty suppression and fast independent component analysis is proposed to realize the multi-source vibration signal recognition. The experimental results show that the proposed method has achieved good processing effect in both simulation and actual measurement, and the relative error is kept below 0.3. In comparison experiments, the proposed method is superior to principal component analysis, EMD and EWT in location recognition and signal recognition effect. This method provides technical support for DAS application in complex multi-source environment and improves the practical application value of DAS.

An analytical model for estimating aerodynamic damping of wind turbines in the shutdown state

As wind farms are constantly being constructed, the risk of tower failure for wind turbines increases significantly under strong winds. Compared with the extensively concerned wind-induced behaviors during the operating state, those ones during the shutdown state attract little attention but may lead to serious problems of damages or even collapse. To clearly grasp the aero-structure interaction in the shutdown state, this paper develops an analytical model for estimating aerodynamic damping of wind turbines. In this method, an analytical expression of aerodynamic damping coupling matrix is derived via the combination of multibody dynamics and first-order Taylor expansion. This matrix is further quantified as the ratio of modal aerodynamic damping with the aid of state-space equation and complex eigenvalue analysis. This treatment can facilitate the straightforward application of efficient calculation methods, such as frequency domain analysis and uncoupled analysis. More importantly, the developed model is able to simultaneously consider multiple realistic factors, such as blade flexibility, tower top rotation, yaw error, wind shear, and pitch angle. This model may have the high calculation efficiency and accuracy, as well as strong applicability for estimating the aerodynamic damping. Numerical examples based on a typical 5 MW wind turbine are employed to validate the effectiveness of the developed model. Experimental analyses demonstrate that this model outperforms the existing formula and presents a high consistency with OpenFAST in the estimation of aerodynamic damping. Meanwhile, the influence of multiple realistic factors is quantitatively analyzed, which even makes the estimation error exceed 70%.

On the mechanism of frequency lock-in vibration of airfoils during pre-stall conditions

Potential frequency lock-in vibration can frequently occur in aircraft flying at separated flow conditions during take-off and landing stages, severely threatening the safety of the aircraft. A deeper understanding of the lock-in phenomenon in pre-stall (steady separated flow) conditions is necessary to improve aircraft reliability and safety. In this paper, a reduced-order model (ROM) for the pitching NACA0012 airfoil in steady separated flow is established. A linear aeroelastic model is then obtained by coupling the ROM with the structural dynamical equation with the pitching degree of freedom, and it is verified by the computational fluid dynamics/computational structural dynamics (CFD/CSD) simulation. Next, the mechanism of frequency lock-in vibration is revealed by the ROM-based aeroelastic model of different structural natural frequencies. Results from the complex eigenvalue analysis indicate that the instability can be divided into two patterns. At high frequencies, the flutter frequency locked onto the natural frequency of the structure, and it is dominated by the instability of structural mode. At low frequencies, the flutter frequency follows the fluid characteristic frequency, which is dominated by the instability of the fluid mode. Finally, the effects of the angle of attack and mass ratio are investigated. The damping of dominant fluid mode decreases with the increase of angle of attack, which affects the structural mode through coupling effects. Therefore, the angle of attack influences the upper boundary of the coupling system’s instability (high frequency boundary). On the contrary, the mass ratio mainly influences the lower boundary of instability (low frequency boundary), because fluid mode becomes unstable at low frequencies merely when the mass ratio is relatively low.

Dynamic Modeling and Stability Analysis for Spinning Circular Solar Sail Considering Periodically Time-Varying Characteristics

The spinning circular solar sail is a promising spacecraft for long-duration missions. This work reveals its structural dynamic and stability behavior under the periodically time-varying solar radiation pressure and gravitational force. The geometric stiffness generated by the centrifugal force due to spinning and the coupling effect between the deformation and solar radiation pressure are taken into account. The von Kármán plate theory is adopted by neglecting the high-frequency in-plane vibrations and considering the effect of the in-plane internal force on the transverse vibration. The partial differential equation of the spinning solar sail is derived and further spatially discretized into periodically time-varying equations of motion. Effects of Poisson ratio and radius ratio on natural frequencies and mode shapes are analyzed, and curve veering phenomena are then observed. Steady-state periodic responses of the solar sail under the solar radiation pressure with different orbit distances, incident angles, and spinning angular velocities are analyzed. The stability analysis is rigorously performed by the Floquet theory rather than the commonly used approach of conducting the eigenvalue analysis at a series of specific discrete time nodes. Moreover, the stability boundary associated with transverse vibrations is determined, which contributes to the parameter design of the spinning solar sail.

A role of fear on diseased food web model with multiple functional response.

In this paper, we analyze the role of fear in a three-species non-delayed ecological model that examines the interactions among susceptible prey, infectious (diseased) prey, and predators within a food web. The prey population grows in a logistic manner until it achieves a carrying capacity, reflecting common population dynamics in the absence of predators. Diseased prey is assumed to transmit infection to healthful prey by the use of a Holling type II reaction. Predators, alternatively, are modeled to consume their prey using Beddington--DeAngelis and Crowley--Martin response features. This evaluation specializes in ensuring the non-negativity of solutions, practical constraints on population dynamics, and long-term stability of the system. Each biologically possible equilibrium point is tested to understand the environmental stable states. Local stability is assessed through eigenvalue analysis, while global stability of positive equilibria is evaluated by the use of Lyapunov features to determine the overall stability of the model. Furthermore, Hopf bifurcation is explored primarily based on infection rate $\varepsilon$. Numerical simulations are carried out to validate the theoretical effects and offer practical insights into the model behaviour under specific conditions.

Stability analysis of nonlinear synchronous vibration in an inclined rotor system supported by journal bearing with variational gravity

In vertical rotating shaft-bearing systems, synchronous vibration undergoes destabilization and stabilization owing to the self-excited vibration. Recently, these phenomena have been clarified numerically, experimentally, and analytically based on a newly proposed analytical method. In this method, a two-step linearization of the nonlinear journal bearing (JB) force and generalized eigenvalue analysis are proposed to directly determine the stability of the synchronous orbit. However, this method cannot be used directly for inclined-rotor systems with a large gravitational effect. Moreover, stability changes caused by gravity variations in inclined-rotor systems have not been fully investigated. This study extended the analytical method to one with weighted average dynamic coefficients to efficiently predict the stability changes of synchronous whirling vibrations in inclined rotor systems. The extended analytical method clearly clarifies gravity-induced stability changes in comparison with the numerical (shooting) method. Our analytical method enhances the ability of rotor dynamics software to calculate the stability thresholds of inclined rotor systems with gravity variations.

Aeroelastic analysis of non-conventional and slender suspension bridges

The aim of this study was to investigate the dynamic effects of wind actions on bridge cross sections, with a goal to propose design charts for non-conventional and slender bridge decks. An eigenvalue analysis was conducted first and the results were validated experimentally. Subsequently, aeroelastic analysis of the bridge deck section was performed. This analysis encompassed the analytical method adhering to the Eurocode and computational fluid dynamics simulations. Both methodologies indicated that the bridge was susceptible to vortex-induced vibrations. To mitigate these vibrations, wind noses provide better stability than the currently used guide vanes. The effect of changing deck width was also studied. Wider and more streamlined sections displayed better aeroelastic performance. The study also revealed that the Eurocode overestimates the derivatives of aerodynamic moment coefficient by 53% to 93%. Consequently, optimized charts are proposed to enhance the accuracy of torsional divergence and flutter speed calculations, facilitating the safe and efficient design of such bridges.

Comparison of KMO Results, Eigen Value, Reliability, and Standard Error of Measurement: Original & Rescaling Through Summated Rating Scaling

This study aims to compare the results of KMO MSA analysis, Eigen Value, reliability, and Standard Error Measurement (SEm) between raw scores (original) and standardized scores (rescaling) through the summated rating scaling method on critical reasoning attitudes of vocational students in Yogyakarta City. This study used a quantitative descriptive approach involving 204 private vocational students as subjects. The instrument used to measure critical reasoning attitudes has gone through thorough validity and reliability testing before the research was carried out. The analysis process was carried out by calculating the KMO MSA, Eigen Value, reliability, and SEm values on both raw and standardized scores. The results of the two types of scores were then compared to identify any differences. Based on the results of the analysis, it was found that the raw score had a KMO MSA of 0.87, reliability of 0.823, and SEm of 0.337. After rescaling, the KMO MSA value decreased slightly to 0.86, the reliability also decreased slightly to 0.821, while the SEm increased to 0.406. Eigenvalue analysis showed that both the raw and standardized scores yielded seven factors with Eigenvalues greater than 1. The differences found between these two types of scores, namely 0.01 for KMO MSA, 0.002 for reliability, and -0.069 for SEm, indicate small but significant changes, especially in terms of the increase in SEm after rescaling, which impacts the level of measurement accuracy.

Multi-timescale modeling and order reduction towards stability analysis of isolated microgrids

Microgrids incorporate a significant proportion of renewable energy sources and power electronic converters in the energy conversion process, creating a sustainable and clean energy infrastructure. However, the multi-timescale dynamics of microgrids are interactively coupled under a nonlinear structure, which makes it difficult to gain insight into the instability mechanisms without a high-fidelity reduced-order model that preserves the main dynamic behaviors of the system. For the isolated AC microgrid dominated by voltage source inverters (VSI), a detailed state-space model of the system, including the inverter, network, and load, is first developed. Based on this model, the eigenvalue analysis is carried out, and a participation factor analysis tool is also utilized to identify the relevant dynamics that have a strong impact on the system's dominant mode. Furthermore, to simplify the system modeling process without losing essential dynamic interactions, a novel multi-timescale coupled reduced-order model is proposed using a transfer function-based order reduction method, which retains the open-loop gain characteristics to preserve the critical couplings between fast inner loop dynamics and slow droop control dynamics. Finally, the accuracy of the reduced-order model is verified by comparing it with the detailed model and the conventional singular perturbation reduced-order model through eigenvalue distribution and time-domain simulation analysis.

On the optimal multi-objective design of fractional order PID controller with antlion optimization

This study introduces an optimal multi-objective design approach for a robust multimachine Fractional Order PID Controller (FOPID) using the Antlion algorithm. The research focuses on the need for effective stabilizers in multimachine power systems by employing traditional speed-based lead-lag FOPID controllers. The study formulates a multi-objective problem, optimizing the damping factor and damping ratio of lightly damped electromechanical modes to maximize a composite set of objective functions, tackled through the Antlion algorithm. Stability analysis of Single-Machine Infinite-Bus (SMIB) and multimachine power systems is conducted based on rotor speed and power deviation minimization in the time domain response, along with damping ratio and eigenvalue analysis. The proposed approach is implemented and tested on three IEEE test cases, showcasing significant improvements in stability through the reduction of maximum overshoot (Mp) and settling time (ts) of speed deviation. Comparative analysis with other optimization-based FOPID controllers underscores the superiority of the proposed approach in enhancing stability in multimachine power systems. The main impact of this research lies in its contribution to the advancement of stability enhancement techniques in multimachine power systems, offering a systematic framework for optimal FOPID controller design and empowering decision-making processes in power engineering.

On some spectral properties of nonlocal boundary-value problems for nonlinear differential inclusion

UDC 517.9 We study the solutions to the Sturm–Liouville boundary-value problem for a nonlinear differential inclusion with nonlocal conditions. The maximal and minimal solutions are demonstrated. The analysis of eigenvalues and eigenfunctions is performed. It is discussed whether multiple solutions may exist for the inhomogeneous Sturm–Liouville boundary-value problem for differential equation with nonlocal conditions.

Improved General Relativity Search Algorithm (IGRSA) for Designing Power System Stabilizers

This paper proposes a novel optimization algorithm for designing power system stabilizers (PSSs). The prospect of attaining higher stability motivated the authors to creat a new optimization algorithm for this study. A novel algorithm named the Improved General. Relativity Search Algorithm (IGRSA) was also developed by cloud theory to improve the performance of GRSA. The supremacy of the proposed approach is tested by comparing it with an introduced objective function in a medium multi-machine power system. The nonlinear simulation results and eigenvalues analysis has demonstrated that the proposed approach in this study is highly effective in enhancing the dynamic stability of the power system.

A diffusive plant-sulphide model: spatio-temporal dynamics contrast between discrete and distributed delay

Abstract This paper studies the spatio-temporal dynamics of a diffusive plant-sulphide model with toxicity delay. More specifically, the effects of discrete delay and distributed delay on the dynamics are explored, respectively. The deep analysis of eigenvalues indicates that both diffusion and delay can induce Hopf bifurcations. The normal form theory is used to set up an exact formula that determines the properties of Hopf bifurcation in a diffusive plant-sulphide model. A sufficiently small discrete delay does not affect the stability and a sufficiently large discrete delay destabilizes the system. Nonetheless, a sufficiently small or large distributed delay does not affect the stability. Both delays cause instability by inducing Hopf bifurcation rather than Turing bifurcation.

Stochastic modelling of uncertainty in Chilean radiata pine wood beams subject to lateral torsional buckling

ABSTRACT This paper aims to investigate the effect of the variability in the mechanical and cross-section properties of Chilean radiata pine on the lateral buckling strength of slender wooden beams. The study is conducted numerically through finite element modeling and using linear eigenvalue analysis. The numerical model is validated using experimental results available in the literature. Two approaches are employed in the analysis, firstly a deterministic approach using design of experiments (DOE) techniques to establish the relevance of the parameters, and secondly a stochastic one, through Monte-Carlo simulation to investigate the impact of each parameter on the lateral buckling resistance. Various distribution functions are considered to investigate the effect of uncertainties in the material parameters and their impact on the critical buckling moments. Moreover, Sobol indices indicated that from the total variability of the distribution of the critical moment, an average of 50% of the variability is attributed to the shear modulus G LR , 37% to the longitudinal elastic modulus E L , and a negligible percentage of the grain angle F .

Observation and Prediction of Instability due to RD Fluid Force in Rotating Machinery by Operational Modal Analysis

In recent years, as rotating machinery has become smaller and more efficient, various types of shaft vibration problems have arisen. Failure of rotating machinery may lead to major accidents and infrastructure shutdowns. Therefore, to prevent failures of rotating machinery, there is a growing need for the vibration analysis technology at the design stage and condition monitoring during operation stage. One of causes of the shaft vibration problems in rotating machinery is the rotordynamic (RD) fluid force acting on fluid elements such as journal bearings, seals, turbine blades, and so on. RD fluid force has a significant effect on the stability of rotating machinery and can destabilize the system. In recent years, operational modal analysis (OMA) methods, which identify modal parameters based on the measured data of a machine's operational condition, have been investigated in the condition monitoring. In this paper, the estimation of the modal parameters of rotating machinery using OMA from only the time history response of displacement data and, in particular, the prediction of the destabilization of rotating machinery caused by RD fluid force are investigated. As a result, the modal parameters are well estimated and, in particular, the destabilization of one mode due to RD fluid force is predicted and explained. The results are in good agreement with the results of the eigenvalue analysis of the original system, and the method is validated. Furthermore, the proposed method is applied to experimental data of the system destabilized by fluid force. The change in stability with rotational speed is observed, and the characteristics of the mode toward destabilization are confirmed. The results show the validity of OMA's predictions of destabilization in the experiments.

Non-Bloch band theory for time-modulated discrete mechanical systems

This study establishes a non-Bloch band theory for time-modulated discrete mechanical systems. We consider simple mass–spring chains whose stiffness is periodically modulated in time. Using the temporal Floquet theory, the system is characterized by linear algebraic equations in terms of Fourier coefficients. This allows us to employ a standard linear eigenvalue analysis. Unlike non-modulated linear systems, the time modulation makes the coefficient matrix non-Hermitian, which gives rise to, for example, parametric resonance, non-reciprocal wave transmission, and non-Hermitian skin effects. In particular, we study finite-length chains consisting of spatially periodic mass–spring units and show that the standard Bloch band theory is not valid for estimating their eigenvalue distribution. To remedy this, we propose a non-Bloch band theory based on a generalized Brillouin zone. This novel approach, the combination of the temporal Floquet theory for time-modulated systems and generalized Brillouin zone, enables the prediction of eigenvalue distribution under open boundary conditions and also quantitative characterization of non-Hermitian skin modes. The proposed theory is verified by some numerical experiments.