Routh-Hurwitz Criterion Research Articles

Nonlinear Dynamics of an Electromagnetically Actuated Cantilever Beam Under Harmonic External Excitation

The present work is devoted to the study of nonlinear vibrations of an electromagnetically actuated cantilever beam subject to harmonic external excitation. The soft actuator that controls the vibratory motion of such components of a robotic structure led to a strongly nonlinear governing differential equation, which was solved in this work by using a highly accurate technique, namely the Optimal Auxiliary Functions Method. Comparisons between the results obtained using our original approach with those of numerical integration show the efficiency and reliability of our procedure, which can be applied to give an explicit analytical approximate solution in two cases: the nonresonant case and the nearly primary resonance. Our technique is effective, simple, easy to use, and very accurate by means of only the first iteration. On the other hand, we present an analysis of the local stability of the model using Routh–Hurwitz criteria and the eigenvalues of the Jacobian matrix. Global stability is analyzed by means of Lyapunov’s direct method and LaSalle’s invariance principle. For the first time, the Lyapunov function depends on the approximate solution obtained using OAFM. Also, Pontryagin’s principle with respect to the control variable is applied in the construction of the Lyapunov function.

Transmission Dynamics of Typhoid Fever Outbreak: A Mathematical Modelling and Optimal Control Approach

In this paper, we propose and analyze a deterministic mathematical model for the control of typhoid fever. This model incorporates five effective control strategies, including vaccination, booster vaccination, personal hygiene, treatment, and bacteria sterilization, in order to effectively stop the spread of the disease in the population. The model consists of seven compartments, each representing a different stage in the disease’s progression: vaccinated, susceptible, carrier, infected, recovered, booster vaccinated, and bacterial. Using the next-generation matrix approach, we calculated the reproduction number , which measures the disease’s potential for spreading. Our stability analysis, using the Routh-Hurwitz criteria, revealed that the disease-free equilibrium point is locally asymptotically stable when is less than 1, among other conditions. This means that typhoid can be completely eradicated if the rate of secondary infection is kept below a certain threshold. Further sensitivity analyses were conducted to determine the key parameters that impact . Based on our findings, we formulated an optimal control problem and utilized Pontryagin’s Maximum Principle to determine the most effective strategy for controlling and eliminating the disease. Our results show that a combination of vaccination, booster vaccination, personal hygiene, treatment, and bacteria sterilization is the most optimal approach. With our comprehensive model and effective control strategies, we are one step closer to wiping out typhoid fever for good.

Analysis of COVID-19’s Dynamic Behavior Using a Modified SIR Model Characterized by a Nonlinear Function

This study develops a modified SIR model (Susceptible–Infected–Recovered) to analyze the dynamics of the COVID-19 pandemic. In this model, infected individuals are categorized into the following two classes: Ia, representing asymptomatic individuals, and Is, representing symptomatic individuals. Moreover, accounting for the psychological impacts of COVID-19, the incidence function is nonlinear and expressed as Sg(Ia,Is)=βS(Ia+Is)1+α(Ia+Is). Additionally, the model is based on a symmetry hypothesis, according to which individuals within the same compartment share common characteristics, and an asymmetry hypothesis, which highlights the diversity of symptoms and the possibility that some individuals may remain asymptomatic after exposure. Subsequently, using the next-generation matrix method, we compute the threshold value (R0), which estimates contagiousness. We establish local stability through the Routh–Hurwitz criterion for both disease-free and endemic equilibria. Furthermore, we demonstrate global stability in these equilibria by employing the direct Lyapunov method and La-Salle’s invariance principle. The sensitivity index is calculated to assess the variation of R0 with respect to the key parameters of the model. Finally, numerical simulations are conducted to illustrate and validate the analytical findings.

Fractional-Order Modeling of COVID-19 Transmission Dynamics: A Study on Vaccine Immunization Failure

COVID-19 is an enveloped virus with a single-stranded RNA genome. The surface of the virus contains spike proteins, which enable the virus to attach to host cells and enter the interior of the cells. After entering the cell, the virus exploits the host cell’s mechanisms for replication and dissemination. Since the end of 2019, COVID-19 has spread rapidly around the world, leading to a large-scale epidemic. In response to the COVID-19 pandemic, the global scientific community quickly launched vaccine research and development. Vaccination is regarded as a crucial strategy for controlling viral transmission and mitigating severe cases. In this paper, we propose a novel mathematical model for COVID-19 infection incorporating vaccine-induced immunization failure. As a cornerstone of infectious disease prevention measures, vaccination stands as the most effective and efficient strategy for curtailing disease transmission. Nevertheless, even with vaccination, the occurrence of vaccine immunization failure is not uncommon. This necessitates a comprehensive understanding and consideration of vaccine effectiveness in epidemiological models and public health strategies. In this paper, the basic regeneration number is calculated by the next generation matrix method, and the local and global asymptotic stability of disease-free equilibrium point and endemic equilibrium point are proven by methods such as the Routh–Hurwitz criterion and Lyapunov functions. Additionally, we conduct fractional-order numerical simulations to verify that order 0.86 provides the best fit with COVID-19 data. This study sheds light on the roles of immunization failure and fractional-order control.

Synchronization characteristics and stable states of four co-rotating vibrators with balanced slanted elliptical trajectory

In the petroleum drilling and mining screening industries, our goal is to enhance excitation force, integrate circular screening with rectilinear transport to obtain the balanced slanted elliptical trajectory (BSET), and reduce screen mesh clogging. This work investigates the synchronization characteristics and stable states of four co-rotating vibrators. The novelty of this work lies in using a novel vibration model to achieve the BSET in the synchronization state and in further considering the influence of a new skewing angle between two eccentric rotors (ERs), which are few in previous vibration screening field. Initially, the vibration model is extracted from the designed prototype. Then, in accordance with Routh–Hurwitz criterion, the synchronization and stability are theoretically analyzed using small parameters of vibrator velocity and phase differences (PDs). The synchronization characteristics related to mechanical structures are discussed via numerical calculation and experimental verification. Results reveal that the synchronization implementation is adversely affected by the skewing angle, even leading to unordered behaviors. Achieving zero-phase or in-phase synchronization with four vibrators is only feasible with a larger symmetric distance and installation angle. This work can contribute a foundation for designing some new types of large vibration screening and material conveyance machines.

Method for Determining the Conditions for Ensuring Stable Rearward Movement of a Semi-trailer Combination with Non-turning Semi-trailer Wheels

The efficiency of organising freight transport and using motor transport in a Smart City depends on a set of its properties, which in the process of operation determine its suitability for use in given operating conditions. Vehicles have different overall dimensions and designs, which determine their maneuverability and directional stability, their ability to make a curve-line movement in the city, as well as to move safely in reverse. Nowadays, tractor-trailer combinations with non-turning semi-trailer wheels are widely used for freight transport. Such a scheme of vehicle construction does not ensure its directional stability when reversing, which can lead to the folding of the tractor and semi-trailer and loss of mobility of the combination. In the absence of additional special systems, the driver controls the rearward movement of the road train using the steering wheel and rear-view mirrors, and the accuracy of delivery to the object depends on the level of the driver's training. The research of driver's reaction time spent on situation assessment, decision making, and realisation has been carried out. It has been revealed that with manual control, there are difficulties connected with training of drivers to estimate the parameters of movement and control of the road train, in particular, with the process of parking and its accurate delivery to the object. With the help of the developed mathematical model of the rearward movement of the road train and the proposed new turn control law, the research of maneuvering of the road train during its delivery to the object has been carried out and the results confirming the possibilities of building an automated system allowing to reduce the influence of the human factor on the parking process, as well as to reduce the time required to perform the maneuver have been obtained. The conducted analysis of system stability with the help of Routh–Hurwitz criterion has determined the conditions of providing stability of the system moving in reverse. The method of controlling the backward movement of the road train and the design of the device realising it has been developed. The results of simulation modeling allowed us to find the necessary values for the control law for the movement of road trains of different lengths, as well as the preferred variants of the initial positions of the road train for the realisation of its precise delivery to the object.

Stability and Optimality Criteria for an SVIR Epidemic Model with Numerical Simulation

The mathematical modeling of infectious diseases plays a vital role in understanding and predicting disease transmission, as underscored by recent global outbreaks; to delve deep into the dynamic of infectious disease considering latent period presciently is inevitable as it bridges the gap between realistic nature and mathematical modeling. This study extended the classical Susceptible–Infected–Recovered (SIR) model by incorporating vaccination strategies during incubation. We introduced multiple time delays to an account incubation period to capture realistic disease dynamics better. The model is formulated as a system of delay differential equations that describe the transmission dynamics of diseases such as polio or COVID-19, or diseases for which vaccination exists. Critical aspects of the study include proving the positivity of the model’s solutions, calculating the basic reproduction number (R0) using next-generation matrix theory, and identifying disease-free and endemic equilibrium points. The local stability of these equilibria is then analyzed using the Routh–Hurwitz criterion. Due to the complexity introduced by the delay components, we examine the stability by studying the roots of a fourth-degree exponential polynomial. The effects of educational campaigns and vaccination efficacy are also investigated as control measures. Furthermore, an optimization problem is formulated, based on Pontryagin’s maximum principle, to minimize the number of infections and associated intervention costs. Numerical simulations of the delay differential equations are conducted, and a modified Runge–Kutta method with delays is used to solve the optimal control problem. Finally, we present a few simulation results to illustrate the analytical findings.

Mathematical Modelling of the Impacts of Morning Fatigue on Academic Performance: A Case Study of Federal University Wukari, Taraba State

Several authors have researched the phenomenon of sleep deprivation among students. In their studies, we discovered that they did not consider morning fatigue. So, we decided to fill the gap in the literature by using mathematical models to study the impact of morning fatigue. The aim is to develop mathematical modelling for the impact of morning fatigues on the academic performances of the students, a case study of the Federal University Wukari in Taraba State. In this project work, we formulated a mathematical model based on a system of ordinary differential equations to study the impact of morning fatigue on academic performance. We tested the model for existence and uniqueness and discovered that the model exists and that it is unique. The basic reproduction number was computed using the next-generation matrix approach. Questionnaires were distributed to 400 students and the data for this research were collected from the responses of the students. The result of the basic reproduction number shows that morning fatigue can be controlled. Using the Routh-Hurwitz criterion for local stability, the fatigue-free equilibrium (FFE) states of the model were established and proved to be locally asymptotically stable. Sensitivity analysis was then carried out to determine which parameters should be targeted by control intervention strategies of which the result shows that an increase in the acceptance of the control measure rate leads to a reduction in the prevalence of fatigue. Finally, a numerical simulation of the model was carried out and the result shows that Morning fatigue has a great impact on the academic performance of students. This means that there is a need for students to avoid reading till daybreak which in turn brings morning fatigue.

Vibrational and stability analysis of planar double pendulum dynamics near resonance

AbstractThe focus of this paper is to examine the motion of a novel double pendulum (DP) system with two degrees of freedom (DOF). This system operates under specific constraints to follow a Lissajous curve, with its pivot point moving along this path in a plane. The nonlinear differential equations governing this system are derived using Lagrange's equations. Their analytical solutions (AS) are subsequently calculated using the multiple-scales method (MSM), which provides higher-order approximations. These solutions are considered new, as the traditional MSM has been applied to this novel system for the first time. Additionally, the accuracy of these solutions is validated through numerical results obtained using the fourth-order Runge–Kutta method. The solvability conditions and characteristic exponents are determined based on resonance cases. The Routh–Hurwitz criteria (RHC) are employed to assess the stability of the fixed points corresponding to the steady-state solutions. They are also used to demonstrate the frequency response curves. The nonlinear stability analysis is performed by examining the stability and instability ranges. Resonance curves and time history plots are presented to analyze the behavior of the system for specific parameter values. The investigation delves into a comprehensive analysis of bifurcation diagrams (BDs) and Lyapunov exponent spectra (LEs), aiming to uncover the various types of motion present within the system. Systematic examination of these charts reveals critical insights into transitions between stable, quasi-stable, and chaotic dynamical behaviors. This work has practical applications in various fields, such as robotics, pump compressors, rotor dynamics, and transportation devices. It can be used to study the vibrational motion of these systems.

The Impact of the Nonlinear Integral Positive Position Feedback (NIPPF) Controller on the Forced and Self-Excited Nonlinear Beam Flutter Phenomenon

This article presents a novel approach to impact regulation of nonlinear vibrational responses in a beam flutter system subjected to harmonic excitation. This study introduces the use of a Nonlinear Integral Positive Position Feedback (NIPPF) controller for this purpose. This technique models the system as a three-degree-of-freedom nonlinear system representing the beam flutter, coupled with a first-order and a second-order filter representing the NIPPF controller. By applying perturbation analysis to the linearized system model, the authors obtain analytical solutions for the autonomous system with the controller. This study aims to reduce vibration amplitudes in a nonlinear dynamic system, specifically when 1:1 internal resonance occurs. The Routh–Hurwitz criterion is utilized to evaluate the system’s stability. Furthermore, the frequency–response curves (FRCs) exhibit symmetry across a range of parameter values. The findings highlight that the effectiveness of vibration suppression is directly related to the product of the NIPPF control signal after comparing with different controllers. Numerical simulations, conducted using the fourth-order Runge–Kutta method, validate the analytical solutions and demonstrate the system’s amplitude response. The strong correlation between the analytical and numerical results highlights the accuracy and dependability of the proposed method.

Bifurcation boundaries analysis of the thin-walled internal resonance energy harvester

The dynamic instability of the thin-walled 1: 2 internal resonant piezoelectric vibration energy harvester is investigated. Previously work (Nie et al., 2019) derived the nonlinear electromechanical-coupled governing equations and validated them experimentally. The modulation equations are performed by the method of multiple scales. The system stability is obtained from the eigenvalues of the Jacobi matrix. Then, three types of bifurcation boundaries of the system are determined analytically and simulated numerically via Routh–Hurwitz criterion. Results show that internal resonance complicates the dynamic properties of the bifurcation region. For a large external excitation, new saddle node bifurcation regions appear in the stability diagram. Moreover, the new saddle node and Hopf bifurcation regions may intertwine. Furthermore, as the acceleration excitation increases, the system can be observed to transition to stable–unstable–stable states. The system’s high and low branch responses are fed back by the system’s large and small initial conditions. The system’s periodic and quasi-periodic motions are characterized by time response, FFT, phase trajectory, and Poincaré map. The study of the stability boundary of the piezoelectric energy harvester contributes to the harvesting of energy from ambient vibrations over a broader frequency range.

Mathematical modeling of Ebola using delay differential equations

Nonlinear delay differential equations (NDDEs) are essential in mathematical epidemiology, computational mathematics, sciences, etc. In this research paper, we have presented a delayed mathematical model of the Ebola virus to analyze its transmission dynamics in the human population. The delayed Ebola model is based on the four human compartments susceptible, exposed, infected, and recovered (SEIR). A time-delayed technique is used to slow down the dynamics of the host population. Two significant stages are analyzed in the said model: Ebola-free equilibrium (EFE) and Ebola-existing equilibrium (EEE). Also, the reproduction number of a model with the sensitivity of parameters is studied. Furthermore, the local asymptotical stability (LAS) and global asymptotical stability (GAS) around the two stages are studied rigorously using the Jacobian matrix Routh–Hurwitz criterion strategies for stability and Lyapunov function stability. The delay effect has been observed in the model in inverse relation of susceptible and infected humans (it means the increase of delay tactics that the susceptibility of humans increases and the infectivity of humans decreases eventually approaches zero which means that Ebola has been controlled into the population). For the numerical results, the Euler method is designed for the system of delay differential equations (DDEs) to verify the results with an analytical model analysis.

Studies on multi-frequency synchronization of two pairs of counter-rotating exciters in a split drive double-body vibrating system

In this article, the multi-frequency synchronization characteristics and stability of a split drive double-body vibrating system with two rigid frames (RFs), driven by two pairs of counter-rotating exciters symmetrically mounted on two different RFs, are investigated in detail by theory, numeric, and simulation. Based on Lagrange’s equations, the differential equations of motion of the system are deduced. The theoretical criteria for implementing foundation frequency, double-frequency, and triple-frequency synchronization of two pairs of exciters are obtained by applying the asymptotic method. The stability criteria of the synchronous states are derived according to the Routh–Hurwitz criterion. Then, the synchronous and stable regions and stability ability coefficients of the system are analyzed qualitatively by numeric. The simulations are conducted to verify the validity of the theoretical methods used and the correctness of corresponding theoretical results. It is shown that, under the precondition of realizing the multi-frequency synchronization, the ideal working point of the system should be set in Region I to obtain the larger and more stable superposition vibration amplitude between two RFs in engineering. The present work can provide theory guidance for designing new types of vibrating equipment with two different driving frequencies.

Global Analysis of a Fractional‐Order Hepatitis B Virus Model Under Immune Response in the Presence of Cytokines

AbstractThis research proposes and investigates an epidemiological model to study the dynamic behaviors of the Hepatitis B virus (HBV) under immune response and cytokine influence. The model's stability, positivity, boundedness, and equilibria are analyzed using Lyapunov functional methods and the Routh–Hurwitz criterion under Caputo fractional derivative. The study evaluates nucleoside analogues and interferon treatments, determining critical drug efficiencies. Equilibria, including infection‐free and endemic states, are analyzed using the fundamental reproduction number, , to predict disease elimination. Numerical simulations utilize the fractional Adams method and the L1 scheme, capturing memory traces as the fractional order changes. Results show the L1 scheme effectively captures memory traces, providing empirical support for the theoretical findings. Furthermore, Ulam–Hyers stability is treated according to the equilibrium point, which describes relationships between functions. Notably, the findings of the study yielded profound insights. They revealed that the HBV system remains locally asymptotic stable at disease‐free and the endemic point when . At the same time, the simulations illustrated a correlation between the rate of infection and the rise in infected individuals, indicating the feasibility of eradicating and effectively managing HBV infections through a multifaceted approach and various measures such as vaccination and effective drug administration protocols. The proposed framework can guide medical professionals and decision‐makers in developing effective strategies to limit and eliminate the spread of HBV in the population.

Stability of few-cycle light bullets in nonlinear metamaterials beyond the slowly varying envelope approximation

This letter explores stable few-cycle light bullets in nonlinear metamaterials beyond the slowly varying envelope approximation. Using a (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation, the letter explores solitonic behaviors of light bullets in materials with a negative refractive index under the few-cycle regime. The analysis involves a Lagrangian variational approach designed for dissipative nonlinear systems and utilizes direct numerical simulations to demonstrate the possibility of stable light bullets forming in metamaterials and corroborate the stability analysis. Based on the stability conditions for the system parameters using the Routh-Hurwitz criterion, optimal values of the cubic and quintic self-steepening effects promote stable propagation of light bullets, while an imbalance may lead to collapse. These findings suggest that combining few-cycle light bullets with metamaterial-based optics may enable new possibilities for ultrafast and nonlinear optics.

Modeling Rabies Dynamics: The Impact of Vaccination and Infectious Immigrants on Public Health

Public health is still seriously threatened by rabies in many parts of the world, particularly in poorer nations. In order to address this issue, this work suggests an equation describing the mechanisms of rabies transmission between animals, taking into account infectious immigrants and vaccination as possible preventive measures. The next-generation matrix (NGM) Method method was used to calculate the effective reproduction number (R0). Using the Routh-Hurwitz Criterion, a disease-free equilibrium point (DFE) was discovered and It was demonstrated to have local asymptotic stability if (R0 < 1), and unstable in every other case. Additionally, DFE was discovered to be quadratic Lyapunov stable and globally asymptotically stable. Additionally, the model parameters’ sensitivity analysis on the (R0) was carried out using the central manifold theory for the bifurcation analysis, and the analysis’s normalized forward sensitivity index method. MATLAB software was utilized to do numerical analysis for simulation analysis. The findings of the simulated data showed that a higher vaccination rate and fewer infectious immigrants would slow the development of the decline, as shown visually.

Balancing riderless electric scooters at zero speed in the presence of a feedback delay

AbstractThe nonlinear dynamics of electric scooters are investigated using a spatial mechanical model. The equations of motion are derived with the help of Kane’s method. Two control algorithms are designed in order to balance the e-scooter in a vertical position at zero forward speed. Hierarchical, linear state feedback controllers with feedback delay are considered. In the case of a delay-free controller, the linear stability properties are analyzed analytically, with the help of the Routh–Hurwitz criteria. The linear stability charts of the delayed controllers are constructed with the help of the D-subdivision method and semi-discretization. The control gains of the controllers are optimized with respect to the robustness against perturbations. The effects of the feedback delay of the controllers, the rake angle, the trail, and the center of gravity of the handlebar on the linear stability are shown. The performance of the control algorithms is verified by means of numerical simulations.

Stability analysis of a [formula omitted] epidemic compartmental model with saturated incidence rate, vaccination and elimination strategies

In this study, we developed and analyzed a compartmental epidemic model and a system model that incorporate vaccination, elimination, and quarantine techniques, with a saturated incidence rate, by theoretical and numerical means. The model is described by a system of four nonlinear differential equations. Our analysis included determining the reproduction number Rq and examining equilibrium solutions. The outcomes of the disease are identified through the threshold Rq. When Rq<1, the disease-free equilibrium is globally asymptotically stable, as proved by the LaSalle invariance principle and disease extinction analysis. However, using the Routh–Hurwitz criterion, we have proved that the disease-free equilibrium is not stable, and when Rq>1, the unique endemic equilibrium is asymptotically stable locally. The stability of the endemic and disease-free equilibria globally has been examined using the Routh–Hurwitz and Dulac criteria. Subsequently, numerical simulations were utilized to illustrate the theoretical findings effectively.

Stability and control analysis of an unemployment model incorporating integration programs effect

A new mathematical model is proposed to investigate the mechanisms of unemployment with government interventions such as internships programs. The proposed model consists of a compartmental system of four components, which are the unemployment U, the employment E, the unemployed beneficiaries of government programs Up and the cumulative density of programs P. The qualitative analysis of differential equations is used to analyze the model. First the equilibrium’s states are calculated, then the analysis of their stability is discussed with to help of Routh-Hurwitz criterion and Lyapunov’s method. Furthermore,in order to achieve the effectiveness and performance of government programs an optimal control strategy is introduced, in which two control variables are introduced. Finally, to validate the main analytical results, some numerical investigations are performed. Our findings suggest that, a government’s efforts to increase employment opportunities and assist unemployed people in starting their own businesses are more effective at lowering the unemployment rate. Further, companies should be encouraged to take on the unemployed who have benefited from internship programs

A new kind of Robe’s problem with charged bodies

This paper studies the motion of charged test particle, moving in the outermost layer of a heterogeneous body (taken as first primary) filled with incompressible homogeneous viscous fluid and second primary is taken as point mass, whereas both the primaries are assumed to be charged. We compute the equations of motion of test particle (the third body) and stationary points (circular, axial and out-of-plane stationary points). And also, the stability of stationary points is examined utilizing characteristics equations and Routh–Hurwitz criterion. In addition to this, the numerical analysis of stationary points and their stability are worked out for different values of parameters.