Analytical Stability Analysis Research Articles

Generalized extended (2+1)-dimensional Kadomtsev-Petviashvili equation in fluid dynamics: analytical solutions, sensitivity and stability analysis

This study discusses a new version of the (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-dimensional Kadomtsev-Petviashvili equation. This equation is used to model the behavior of nonlinear waves in various fields like ferromagnetic media, ion-acoustic waves in plasma physics, and fluid dynamics. It is instrumental in modeling surface and internal waves in straits or channels. The main goal of the research is to determine the exact solutions for this equation and analyze their physical characteristics. We obtain exact solutions using two improved techniques, namely the modified extended tanh-function and the modified generalized Kudryashov methods. These techniques investigate various exact solutions, such as exponential, rational, hyperbolic, and trigonometric. Besides, the sensitivity and the stability analysis of the model are presented. Additionally, three- and two-dimensional contour plots are created to present the physical behavior of the exact solutions.

Exploring dynamic behaviors of diverse electrical soliton pulses in lossy nonlinear electrical transmission lines: Insights from analytical method and linear stability analysis technique

Exploring dynamic behaviors of diverse electrical soliton pulses in lossy nonlinear electrical transmission lines: Insights from analytical method and linear stability analysis technique

An analytical solution and stability analysis of asymmetric non-actuator guided projectiles

PurposeOne of the challenges encountered in the design of guided projectiles is their prohibitive cost. To diminish it, an appropriate avenue many researchers have explored is the use of the non-actuator method for guiding the projectile to the target. In this method, biologically inspired by the flying concept of the single-winged seed, for instance, that of maple and ash trees, the projectile undergoes a helical motion to scan the region and meet the target in the descent phase. Indeed, the projectile is a decelerator device based on the autorotation flight while it attempts to resemble the seed’s motion using two wings of different spans. There exists a wealth of studies on the stability of the decelerators (e.g. the mono-wing, samara and pararotor), but all of them have assumed the body (exclusive of the wing) to be symmetric and paid no particular attention to the scanning quality of the region. In practice, however, the non-actuator-guided projectiles are asymmetric owing to the presence of detection sensors. This paper aims to present an analytical solution for stability analysis of asymmetric decelerators and apprise the effects of design parameters to improve the scanning quality.Design/methodology/approachThe approach of this study is to develop a theoretical model consisting of Euler equations and apply a set of non-dimensionalized equations to reduce the number of involved parameters. The obtained governing equations are readily applicable to other decelerator devices, such as the mono-wing, samara and pararotor.FindingsThe results show that the stability of the body can be preserved under certain conditions. Moreover, pertinent conclusions are outlined on the sensitivity of flight behavior to the variation of design parameters.Originality/valueThe analytical solution and sensitivity analysis presented here can efficiently reduce the design cost of the asymmetric decelerator.

Analytical and numerical stability analysis of a toroidal wheel with nonholonomic constraints

In this paper, a detailed and comprehensive linear stability analysis of a rolling toroidal wheel is performed. The wheel is modeled as a rigid toroid-shaped body rolling without slipping on a horizontal surface. The nonlinear equations of motion constitute a Differential-Algebraic Equations system, given by the dynamic equilibrium equations augmented with the nonholonomic constraints, which arise from the no-slip condition. The circular steady motion and the linearized equations of motion along this relative equilibrium are obtained, for both the solid and hollow tori. The expressions of the linearized equations and the corresponding eigenvalues are derived analytically as a function of the torus aspect ratio. The variation of the stability boundary with the torus aspect ratio is shown. A comparison of the results obtained in the solid and hollow scenarios is included, and all the results are validated with the rolling hoop, which corresponds to a degenerate torus with zero aspect ratio. In the particular case of the steady straight-line rolling and spinning about a vertical diameter, which constitute limit motions of the circular steady motion, the critical rotational and angular speeds required for stabilization are obtained.

Variable-structure PD controller for satellite attitude: Stability analysis and design for optimal performance

A novel variable structure controller is investigated for satellite attitude control. This novel controller does not lead to a sliding mode, which avoids the chattering problem, making it suitable for practical implementation. Detailed formulation of the controller is presented along with analytical stability analysis for a second order system. The system having this nonlinear controller is homogeneous, which leads to a similar response at large and small deviations from the equilibrium. A procedure for tuning homogeneous controllers for all initial conditions is proposed. A detailed model of the satellite attitude dynamics, actuator dynamics, and delays in the system is utilized to optimize controller gains numerically. A comparative analysis shows the superiority of the proposed controller over a conventional PID controller in terms of better transients and lower energy consumption.

Lévy-type solutions for the buckling analysis of unsymmetrically laminated plates with rotational restraints for various plate theories

The stability behaviour of unsymmetrical laminated structures made of fibre-reinforced plastics is significantly influenced by bending–extension coupling and the comparatively low transverse shear stiffnesses. The aim of this work is to improve the analytical stability analysis of unsymmetrically laminated structures. With the discrete plate theory, the stability of laminated structures can be reduced to single laminated plates. The structure is divided into individual segments, and the surrounding structure is modelled by rotational elastic restraints. The governing equations for single plates under specific boundary conditions can be solved exactly with Lévy-type solutions. In this study, Lévy-type solutions for the mentioned boundary conditions under biaxial compressive load is described for the classical laminated plate theory, the first-order shear deformation theory and the third-order shear deformation theory (TSDT). In addition to transverse shear, bending–extension couplings of unsymmetrical cross-ply and antisymmetrical angle-ply laminates are considered. For the implementation of boundary conditions for the rotational restraints in the context of TSDT, a new set of conditions is formulated. The investigation shows very good agreement of the buckling load with comparative finite element analyses for different layups.

Numerical Analysis on Shimmying Wheels with Dry Friction Damper

The dynamics of the 1.5-degree-of-freedom model of towed wheel is investigated. Dry friction at the king pin is considered, leading to a non-smooth dynamical system. Beyond analytical and numerical linear stability analysis, the nonlinear vibrations are investigated by numerical bifurcation analysis with smoothing and by numerical simulations with event handling. The effect of dry friction at the king pin on the birth of separated periodic branches is presented on bifurcation diagrams. The presence of bistable parameter domains is also shown. The effect of smoothing is investigated by comparing bifurcation diagrams of the smoothed and the original non-smooth systems.

Analytical Stability Analysis of Rainfall-Infiltrated Slopes Based on the Green–Ampt Model

Rainfall infiltration can alter the pore-water pressure and decrease matric suction, which can lead to slope instability and landslides. Most of the previous numerical studies conducted on rain infiltration have assumed a two-dimensional (2D) failure mechanism, which neglects the three-dimensional (3D) characteristics of slopes. An analytical assessment framework was developed using the upper-bound theorem of limit analysis to determine the factor of safety and failure pattern in the case of a 3D vertical cut slope under diverse rainfall situations combined with the unsaturated soil mechanics. The Green–Ampt model was adopted to depict the infiltration process and the progression of the wetting front into the soil. The slope factor of safety was determined using the energy balance equation by adopting the strength reduction approach. The validity of the framework was verified using the numerical simulation method. The effects of duration, intensity, and patterns of rainfall and the 3D traits of slope on the slope failure pattern and stability were determined. A set of design charts was included for the preliminary design. The present study quantitatively determines the stability of slopes susceptible to rainfall infiltration.

Shear Flows of Dilatant Fluids with Limited Shear Rates: Analytical Results and Linear Stability Analysis

In this paper, we study the simple shear flows of a class of dilatant fluids with a limited shear rate. This class of fluids is characterized by shear thickening behavior in which the apparent viscosity tends to infinity as the modulus of the stress approaches a finite threshold. The apparent viscosity function is a logarithmic type with two material parameters. We considered this specific form because it fits very well with the flow curves of some granular suspensions for specific values of the material parameters. Despite the nonlinearity of the constitutive law, it is possible to determine explicit steady-state solutions for a simple shear flow, namely (i) the channel flow; (ii) the flow between coaxial cylinders, and (iii) the flow down an inclined plane. We performed a two-dimensional linear stability analysis to investigate the onset of possible instabilities of the steady basic flow, putting into evidence the dependency of the critical Reynolds number on the material parameters.

Analytical Analysis of Infinite Heterogeneous Slope Stability considering Suction Influence

Soil heterogeneity in an infinite unsaturated slope increases the difficulty of analysing its stability. A common method of modelling the spatial variability in the slope stability analysis is to use random fields together with Monte Carlo simulation, but this requires an impressive amount of computational efforts. To reduce the computing time, this paper develops an analytical approach to study the stability of infinite heterogeneous unsaturated slopes. The analytical method provides a newly time-efficient way of analysing such kinds of slopes. It has been used to investigate the stability of four different kinds of unsaturated soil slopes. Meanwhile, the influences of the correlation between the strength parameters and the autocorrelation length of the strength parameters are investigated. The analysed results show that the suction has influences on the distribution of the FOS for both homogeneous slopes and heterogeneous slopes. In addition, the correlation coefficient between the strength parameters and the autocorrelation length of the strength parameters affects the mean and standard deviation of FOS for homogeneous slopes and the probability of failure for heterogeneous slopes.

Stability analysis of stochastic second-order macroscopic continuum models and numerical simulations

Second-order macroscopic continuum models have been constantly improving for decades to reproduce the empirical observations. Recently, a series of experimental studies have suggested that the stochastic factors contribute significantly to destabilizing traffic flow. Nevertheless, the traffic flow stability of the stochastic second-order macroscopic continuum model has not received the attention it deserves in past studies. More importantly, we have found that the destabilizing aspect of stochasticity is still not correctly validated in the existing theoretical stability analysis. In this paper, we analytically study the impact of stochasticity on traffic flow stability for a general stochastic second-order macroscopic model by using the direct Lyapunov method. Numerical simulations have been carried out for different typical stochastic second-order macroscopic models. Our analytical stability analysis has been validated, and our methodology has been proved more efficient. Our study has theoretically revealed that the presence of stochasticity has a destabilizing effect in usual stochastic macroscopic models.

A Novel Asymmetric Car Following Model for Driver-Assist Enabled Vehicle Dynamics

Adaptive cruise control (ACC) vehicles are proving to be the first generation of driver-assist enabled vehicles. In order to study the impacts of ACC vehicles on string stability and traffic flow characteristics, accurately calibrating microscopic car following models is crucial to simulate inter-vehicle dynamics. While many car following models have been used to simulate car following behavior, a single, continuous function may not describe both acceleration and braking realistically. We propose an asymmetric model which is based on the symmetric optimal velocity relative velocity (OVRV) model and switch parameters under different conditions to realize and reproduce car following dynamics of ACC vehicles. We conduct an analytical string stability analysis and the string stability criterion is derived. The calibration and simulation results show that the proposed asymmetric ACC model reduces model spacing error by up to 38% compared with the symmetric OVRV model. Compared with other commonly used asymmetric car following models in the transportation community, the proposed asymmetric ACC model can reduce spacing error by 44.8%. Furthermore, we validate the derived string stability criterion with a numerical test simulating with a string of vehicles. We conclude that an asymmetric car following model shows more accurate performance in the capture of ACC car following behavior.

Controlling particle transitions with weak periodic perturbations

We investigate the particle transition dynamics in two-component Bose-Einstein condensates dominated by pair-particle transitions by introducing weak periodic perturbations to the initial states. We show that the transitions can be well controlled by varying the perturbation period while maintaining the external fields and nonlinear parameters during the transition process. In particular, there exists a critical period at which the transition efficiency is approximately $100%$, with a numerical error of 0.72%. Analytic Akhmediev breather solutions and linear stability analyses show that modulation instability is the underlying mechanism for these transition dynamics. Furthermore, we establish a quantitative relation between the transition efficiency and phase shifts induced by the modulation instability. This relation can be used to measure the phase shift of the plane-wave background in well-known Akhmediev breathers.

Analytical Delay Model and Stability Analysis for MLGNR Interconnects

This paper analyzes the signal delay and relative stability of multilayer graphene nanoribbon (MLGNR) interconnects dependent on temperature at global lengths. A thermally aware driver interconnect load (DIL) system, constituted by equivalent single conductor (ESC) model along with mathematical equations, is proposed to convert multi-conductor transmission circuit into single transmission line of MLGNR. The temperature-dependent analytical delay model of MLGNR is evaluated and the obtained outcomes are compared with the simulated results. The analytical and simulated results are obtained at global interconnect length (2000-[Formula: see text]m) for 32[Formula: see text]nm, 22[Formula: see text]nm and 16[Formula: see text]nm nodes of technology under 200–500[Formula: see text]K temperature range of MLGNR. The simulation and analytical results reveal that the outcomes of the two models correspond well. The trend of the models shows the increase in delay with the rising temperature levels (200–500[Formula: see text]K) for three different nodes of technology. Further, relative stability analysis of MLGNR as interconnect line at three different technological nodes from 500–2000-[Formula: see text]m lengths is examined.

The physical origin and the properties of arm spurs/feathers in local simulations of the wiggle instability

ABSTRACT Gaseous substructures such as feathers and spurs dot the landscape of spiral arms in disc galaxies. One of the candidates to explain their formation is the wiggle instability of galactic spiral shocks. We study the wiggle instability using local 2D hydrodynamical isothermal non-self-gravitating simulations. We find that: (1) simulations agree with analytic linear stability analysis only under stringent conditions. They display surprisingly strong non-linear coupling between the different modes, even for small mode amplitudes (${\sim}1{{\ \rm per\ cent}}$). (2) We demonstrate that the wiggle instability originates from a combination of two physically distinct mechanisms: the first is the Kelvin–Helmholtz instability, and the second is the amplification of infinitesimal perturbations from repeated shock passages. These two mechanisms can operate simultaneously, and which mechanism dominates depends on the underlying parameters. (3) We explore the parameter space and study the properties of spurs/feathers generated by the wiggle instability. The wiggle instability is highly sensitive to the underlying parameters. The feather separation decreases, and the growth rate increases, with decreasing sound speed, increasing potential strength and decreasing interarm distance. (4) We compare our simulations with a sample of 20 galaxies in the HST Archival Survey of Spiral Arm Substructure of La Vigne et al. and find that the wiggle instability is able to reproduce the typical range of feather separations seen in observations. It remains unclear how the wiggle instability relates to competing mechanisms for spur/feather formation such as the magneto-jeans instability and the stochastic accumulation of gas due to correlated supernova feedback.

Coupled Numerical and Analytical Stability Analysis Charts for an Earth-Fill Dam under Rapid Drawdown Conditions

The presented article provides a comprehensive study on the stability analysis of earth-fill dams under rapid drawdown and transient flow conditions used to prepare stability analysis charts by conducting coupled finite-element numerical and analytical limit equilibrium procedures. In this regard, the impacts of different rapid drawdown conditions on the safety factor of the Alavian earth-fill dam are determined. The slope stability charts present for both shallow and deep slip surfaces with various permeabilities are verified by ground information obtained with extensive instrumentation on the dam’s site. The results showed that by decreasing the permeability of the core’s material, despite preventing seepage, the instability risk of the upstream slope as a result of rapid drawdown intensifies. Also, as stability charts can be stated, with increasing the slip surface’s depth and decreasing the hydraulic hydration, the reliability decreases, and the sliding surfaces’ sensitivity increases based on the drawdown rates, which have been revealed to be from 0.2 to 0.6, the most critical state for safety factors, showing significant declines.

Domain of Attraction’s Estimation for Grid Connected Converters With Phase-Locked Loop

A large number of non-linear hardware and control units exists in power electronic system used in grid connected devices. The analytical transient stability analysis of grid-connected converters presents numerous difficulties. A common method to tackle this problem is the stability analysis using Lyapunov’s method. By applying this method, difficulties arise not only from finding a suitable Lyapunov function, but also from checking the constraint of Lyapunov stability. If the appropriate Lyapunov function is a high-order polynomial, it is very challenging to test if it meets the constraints of Lyapunov stability in certain regions. In this paper, the sum-of-squares programming method is used to obtain the estimation of a converter’s domain of attraction with a relatively small number of iterations compared to classically applied methods, such as the Monte Carlo method. The estimation of the domain of attraction are verified by time-domain simulations and StarSim’s controller hardware-in-the-loop tests in this paper.

Lyapunov stability of grid-connected wind turbines with permanent magnet synchronous generator

Dynamic stability across a range of operating conditions and disturbances is an essential requirement in the control of wind turbines. To analyse and design for closed-loop stability, existing methods typically rely on the numerical evaluation of a small-signal model of the system around each operating condition. This approach, however, is an inefficient way to capture and shape dynamic behaviour over the wide ranges of operating conditions and parameter values arising in practical implementations. This work presents an analytic stability analysis for presents an analytic stability analysis for (PMSG) wind turbines. The proposed methodology, based on a Lyapunov function constructed from an analytic expression of the system Jacobian, enables the identification of regions of operation and parameter values for the wind turbine system within which its stability can be guaranteed. A feedback linearisation control strategy is adopted to deal with the nonlinear relationship between the generator speed and the DC-link voltage in the turbine back-to-back converter. The closed-loop response performance of the design based on the proposed methodology is compared to that of a conventional PI control design via simulation tests conducted on a 2 MW turbine model and a 5 MW (PMSG) reference turbine model on the FAST physics-based simulation tool developed by NREL.

Synchronization in the presence of time delays and inertia: Stability criteria.

Linear stability of synchronized states in networks of delay-coupled oscillators depends on the type of interaction, the network, and oscillator properties. For inert oscillator response, found ubiquitously from biology to engineering, states with time-dependent frequencies can arise. These generate side bands in the frequency spectrum or lead to chaotic dynamics. The time delay introduces multistability of synchronized states and an exponential term in the characteristic equation. Stability analysis using the resulting transcendental characteristic equation is a difficult task and is usually carried out numerically. We derive criteria and conditions that enable fast and robust analytical linear stability analysis based on the system parameters. These apply to arbitrary network topologies, identical oscillators, and delays.

Buckling of shear-deformable unsymmetrically laminated plates

The couplings of unsymmetric laminated structures and the comparatively low transverse shear stiffnesses significantly influence their stability behaviour. The aim of this work is to improve the purely analytical stability analysis of unsymmetric laminated structures. This problem can be reduced to one single plate with the use of the discrete plate theory. In this study, laminates under uniaxial compressive load are considered that are simply supported at all edges and have rotational restraints on the unloaded edges (SRSR) or have a free edge (SFSR). Therefore, the present study provides explicit solutions for the buckling load in two ways. Firstly, the first explicit solution for the buckling load of unsymmetric laminates with the mentioned boundary conditions is developed, which takes the bending–extension couplings into account in a direct way. The method is an energy-based method in the form of the Rayleigh-quotient for classical laminated plate theory (CLPT), first order shear deformation theory (FSDT) and third order shear deformation theory (TSDT). Secondly, the reduced bending (RBS) method is investigated for the first time in the framework of TSDT. The comparison of unsymmetric cross-ply and angle-ply laminates shows a good agreement with finite element analysis (FEA). Due to the high computational efficiency of the presented methods, they are excellently suited for pre-design and optimization of composite structures, with simultaneous high usability in practical application.