Search results for: Lyapunov equations

Authors: Maria Ahmadnezhad, Mohammad Reza Soltanpour

Abstract:

Electrohydraulic servo systems have been used in industry in a wide number of applications. Its dynamics ‎are highly nonlinear and also have large extent of model uncertainties and external disturbances. In this ‎thesis, a robust back-stepping control (RBSC) scheme is proposed to overcome the problem of ‎disturbances and system uncertainties effectively and to improve the tracking performance of EHS ‎systems. In order to implement the proposed control scheme, the system uncertainties in EHS systems ‎are considered as total leakage coefficient and effective oil volume. In addition, in order to obtain the ‎virtual controls for stabilizing system, the update rule for the system uncertainty term is induced by the ‎Lyapunov control function (LCF). To verify the performance and robustness of the proposed control ‎system, computer simulation of the proposed control system using Matlab/Simulink Software is ‎executed. From the computer simulation, it was found that the RBSC system produces the desired ‎tracking performance and has robustness to the disturbances and system uncertainties of EHS systems.‎

Keywords: electro hydraulic servo system, back-stepping control, robust back-stepping control, Lyapunov redesign

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Authors: Hassan Manouzi

Abstract:

We present in this paper a useful strategy to solve stochastic partial differential equations (SPDEs) involving stochastic coefficients. Using the Wick-product of higher order and the Wiener-Itˆo chaos expansion, the SPDEs is reformulated as a large system of deterministic partial differential equations. To reduce the computational complexity of this system, we shall use a decomposition-coordination method. To obtain the chaos coefficients in the corresponding deterministic equations, we use a least square formulation. Once this approximation is performed, the statistics of the numerical solution can be easily evaluated.

Keywords: least squares, wick product, SPDEs, finite element, wiener chaos expansion, gradient method

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Authors: Y. J. Zhou, G. Lu

Abstract:

In this paper, an analytical study is made for the dynamic behavior of human brain tissue under transient loading. In this analytical model the Mooney-Rivlin constitutive law is coupled with visco-elastic constitutive equations to take into account both the nonlinear and time-dependent mechanical behavior of brain tissue. Five ordinary differential equations representing the relationships of five main parameters (radial stress, circumferential stress, radial strain, circumferential strain, and particle velocity) are obtained by using the characteristic method to transform five partial differential equations (two continuity equations, one motion equation, and two constitutive equations). Analytical expressions of the attenuation properties for spherical wave in brain tissue are analytically derived. Numerical results are obtained based on the five ordinary differential equations. The mechanical responses (particle velocity and stress) of brain are compared at different radii including 5, 6, 10, 15 and 25 mm under four different input conditions. The results illustrate that loading curves types of the particle velocity significantly influences the stress in brain tissue. The understanding of the influence by the input loading cures can be used to reduce the potentially injury to brain under head impact by designing protective structures to control the loading curves types.

Keywords: analytical method, mechanical responses, spherical wave propagation, traumatic brain injury

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Authors: Abdel-Maksoud Abdel-Kader Soliman

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The General Regularized Long Wave (GRLW) equation is solved numerically by giving a new algorithm based on collocation method using quartic B-splines at the mid-knot points as element shape. Also, we use the Fourth Runge-Kutta method for solving the system of first order ordinary differential equations instead of finite difference method. Our test problems, including the migration and interaction of solitary waves, are used to validate the algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm.

Keywords: generalized RLW equation, solitons, quartic b-spline, nonlinear partial differential equations, difference equations

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Authors: Saurabh Rawat, Anushree Sah

Abstract:

K Maps are generally and ideally, thought to be simplest form for obtaining solution of Boolean equations. Cubical Representation of Boolean equations is an alternate pick to incur a solution, otherwise to be meted out with Truth Tables, Boolean Laws, and different traits of Karnaugh Maps. Largest possible k- cubes that exist for a given function are equivalent to its prime implicants. A technique of minimization of Logic functions is tried to be achieved through cubical methods. The main purpose is to make aware and utilise the advantages of cubical techniques in minimization of Logic functions. All this is done with an aim to achieve minimal cost solution.r

Keywords: K-maps, don’t care conditions, Boolean equations, cubes

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Authors: Maali Kachouri, Anis Jarboui

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We focus on the causal relationship between governance and information transparency as well as interrelation among the various governance mechanisms. This paper employs a simultaneous equations approach to show this relationship in the Tunisian context. Based on an 8-year dataset, our sample covers 28 listed companies over 2006-2013. Our findings suggest that internal and external governance mechanisms are interdependent. Moreover, in order to analyze the causal effect between information transparency and governance mechanisms, we found evidence that information transparency tends to increase good corporate governance practices.

Keywords: simultaneous equations approach, transparency, causal relationship, corporate governance

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Authors: M. Shaban, A. Alibeigloo

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This paper studies free vibration behavior of single-walled carbon nanotubes (SWCNTs) embedded on elastic medium based on three-dimensional theory of elasticity. To accounting the size effect of carbon nanotubes, nonlocal theory is adopted to shell model. The nonlocal parameter is incorporated into all constitutive equations in three dimensions. The surrounding medium is modeled as two-parameter elastic foundation. By using Fourier series expansion in axial and circumferential direction, the set of coupled governing equations are reduced to the ordinary differential equations in thickness direction. Then, the state-space method as an efficient and accurate method is used to solve the resulting equations analytically. Comprehensive parametric studies are carried out to show the influences of the nonlocal parameter, radial and shear elastic stiffness, thickness-to-radius ratio and radius-to-length ratio.

Keywords: carbon nanotubes, embedded, nonlocal, free vibration

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Authors: Arash Jafari, Mehdi Taghaddosi, Azin Parvin

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In the paper, our purpose is to enhance the ability to solve a nonlinear differential equation which is about the motion of an incompressible fluid flow going down of an inclined plane without thermal effect with a simple and innovative approach which we have named it new method. Comparisons are made amongst the Numerical, new method, and HPM methods, and the results reveal that this method is very effective and simple and can be applied to other nonlinear problems. It is noteworthy that there are some valuable advantages in this way of solving differential equations, and also most of the sets of differential equations can be answered in this manner which in the other methods they do not have acceptable solutions up to now. A summary of the excellence of this method in comparison to the other manners is as follows: 1) Differential equations are directly solvable by this method. 2) Without any dimensionless procedure, we can solve equation(s). 3) It is not necessary to convert variables into new ones. According to the afore-mentioned assertions which will be proved in this case study, the process of solving nonlinear equation(s) will be very easy and convenient in comparison to the other methods.

Keywords: viscos fluid, incompressible fluid flow, inclined plane, nonlinear phenomena

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Authors: S. Srednyak

Abstract:

We study a class of functional partial differential equations(FPDEs). This class is suggested by Quantum Field Theory. We derive general properties of solutions to such equations. In particular, we demonstrate that they lead to systems of coupled integral equations with singular kernels. We show that solutions to such hierarchies can be sought among functions with regular singularities at a countable set of subvarieties of the physical space. We also develop a formal analogy of basic constructions of differential geometry on functional manifolds, as this is necessary for in depth study of FPDEs. We also consider the case of linear overdetermined systems of functional differential equations and show that it can be completely solved in terms of formal solutions of a functional equation that is a functional analogy of a system of determined algebraic equations. This development leads us to formally define the functional analogy of algebraic geometry, which we call functional algebraic geometry. We study basic properties of functional algebraic varieties. In particular, we investigate the case of a formally discrete set of solutions. We also define and study functional analogy of discriminants. In the case of fully determined systems such that the defining functionals have regular singularities, we demonstrate that formal solutions can be sought in the class of functions with regular singularities. This case provides a practical way to apply our results to physics problems.

Keywords: functional equations, quantum field theory, holomorphic functions, Yang Mills mass gap problem, quantum chaos

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Authors: Aysegul Sener, Gonca Tuncel Memis, Mirac Kamislioglu

Abstract:

Chaos theory is one of the new paradigms of science since the last century. After determining chaos in the weather systems by Edward Lorenz the popularity of the theory was increased. Chaos is observed in many natural systems and studies continue to defect chaos to other natural systems. Acid rain is one of the environmental problems that have negative effects on environment and acid rains values are monitored continuously. In this study, we aim that analyze the chaotic behavior of acid rains in Turkey with the chaotic defecting approaches. The data of pH degree of rain waters, concentration of sulfate and nitrate data of wet rain water samples in the rain collecting stations which are located in different regions of Turkey are provided by Turkish State Meteorology Service. Lyapunov exponents, reconstruction of the phase space, power spectrums are used in this study to determine and predict the chaotic behaviors of acid rains. As a result of the analysis it is found that acid rain time series have positive Lyapunov exponents and wide power spectrums and chaotic behavior is observed in the acid rain time series.

Keywords: acid rains, chaos, chaotic analysis, Lypapunov exponents

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Authors: Y. M. Aiyesimi, S. O. Abah, G. T. Okedayo

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A general analysis has been developed to study the chemical reaction effects on unsteady MHD double-diffusive free convective flow over a vertical stretching plate. The governing nonlinear partial differential equations have been reduced to the coupled nonlinear ordinary differential equations by the similarity transformations. The resulting equations are solved numerically by using Runge-Kutta shooting technique. The effects of the chemical parameters are examined on the velocity, temperature and concentration profiles.

Keywords: chemical reaction, MHD, double-diffusive, stretching plate

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Authors: Haniye Dehestani, Yadollah Ordokhani

Abstract:

In this work, we present an efficient approach for solving variable-order time-fractional partial differential equations, which are based on Legendre and Laguerre polynomials. First, we introduced the pseudo-operational matrices of integer and variable fractional order of integration by use of some properties of Riemann-Liouville fractional integral. Then, applied together with collocation method and Legendre-Laguerre functions for solving variable-order time-fractional partial differential equations. Also, an estimation of the error is presented. At last, we investigate numerical examples which arise in physics to demonstrate the accuracy of the present method. In comparison results obtained by the present method with the exact solution and the other methods reveals that the method is very effective.

Keywords: collocation method, fractional partial differential equations, legendre-laguerre functions, pseudo-operational matrix of integration

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Authors: Daniil Karzanov

Abstract:

This paper studies the effect of quarterly earnings reports on the stock price. The profitability of the stock is modeled by geometric Brownian diffusion and the Constant Elasticity of Variance model. We fit several variations of stochastic differential equations to the pre-and after-report period using the Maximum Likelihood Estimation and Grid Search of parameters method. By examining the change in the model parameters after reports’ publication, the study reveals that the reports have enough evidence to be a structural breakpoint, meaning that all the forecast models exploited are not applicable for forecasting and should be refitted shortly.

Keywords: stock market, earnings reports, financial time series, structural breaks, stochastic differential equations

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Authors: Seyed Amin Vakili, Sahar Sadat Vakili, Seyed Ehsan Vakili, Nader Abdoli Yazdi

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The main purpose of this paper is to present the Modified Newmark Method as a method of non-linear frame analysis by considering the effect of the axial load (second order analysis). The discussion will be restricted to plane frameworks containing a constant cross-section for each element. In addition, it is assumed that the frames are prevented from out-of-plane deflection. This part of the investigation is performed to generalize the established method for the assemblage structures such as frameworks. As explained, the governing differential equations are non-linear and cannot be formulated easily due to unknown axial load of the struts in the frame. By the assumption of constant axial load, the governing equations are changed to linear ones in most methods. Since the modeling and the solutions of the non-linear form of the governing equations are cumbersome, the linear form of the equations would be used in the established method. However, according to the ability of the method to reconsider the minor omitted parameters in modeling during the solution procedure, the axial load in the elements at each stage of the iteration can be computed and applied in the next stage. Therefore, the ability of the method to present an accurate approach to the solutions of non-linear equations will be demonstrated again in this paper.

Keywords: nonlinear, stability, buckling, modified newmark method

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Authors: Rafat Alshorman, Fadi Awawdeh

Abstract:

By employing a simplified bilinear method, a class of generalized fifth-order KdV (gfKdV) equations which arise in nonlinear lattice, plasma physics and ocean dynamics are investigated. With the aid of symbolic computation, both solitary wave solutions and multiple-soliton solutions are obtained. These new exact solutions will extend previous results and help us explain the properties of nonlinear solitary waves in many physical models in shallow water. Parametric analysis is carried out in order to illustrate that the soliton amplitude, width and velocity are affected by the coefficient parameters in the equation.

Keywords: multiple soliton solutions, fifth-order evolution equations, Cole-Hopf transformation, Hirota bilinear method

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Authors: Fatemeh Behbahani, Rubiyah Yusof

Abstract:

To avoid collision between following vehicles and vehicles in front, it is vital to keep appropriate, safe spacing between both vehicles over all speeds. Therefore, the following vehicle needs to have exact information regarding the speed and spacing between vehicles. This project is conducted to simulate the tuning of controller gain for a vehicle-following system through the selected control strategy, spacing control policy and fixed-time headway policy. In addition, the paper simulates and designs an adaptive gain controller for a road-vehicle-following system which uses information on the spacing, velocity and also acceleration of a preceding vehicle in the proposed one-vehicle look-ahead strategy. The mathematical model is implemented using Kirchhoff and Newton’s Laws, and stability simulated. The trial-error method was used to obtain a suitable value of controller gain. However, the adaptive-based controller system was able to optimize the gain value automatically. Model Reference Adaptive Control (MRAC) is designed and utilized and based on firstly the Gradient and secondly the Lyapunov approach. The Lyapunov approach considers stability. The Gradient approach was found to improve the best value of gain in the controller system with fixed-time headway.

Keywords: one-vehicle look-ahead, model reference adaptive, stability, tuning gain controller, MRAC

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Authors: Ahmed Elsayed Ahmed, Ashraf Hafez, A. N. Ouda, Hossam Eldin Hussein Ahmed, Hala Mohamed ABD-Elkader

Abstract:

Unmanned Aircraft Systems (UAS) are playing increasingly prominent roles in defense programs and defense strategies around the world. Technology advancements have enabled the development of it to do many excellent jobs as reconnaissance, surveillance, battle fighters, and communications relays. Simulating a small unmanned aerial vehicle (SUAV) dynamics and analyzing its behavior at the preflight stage is too important and more efficient. The first step in the UAV design is the mathematical modeling of the nonlinear equations of motion. In this paper, a survey with a standard method to obtain the full non-linear equations of motion is utilized,and then the linearization of the equations according to a steady state flight condition (trimming) is derived. This modeling technique is applied to an Ultrastick-25e fixed wing UAV to obtain the valued linear longitudinal and lateral models. At the end, the model is checked by matching between the behavior of the states of the non-linear UAV and the resulted linear model with doublet at the control surfaces.

Keywords: UAV, equations of motion, modeling, linearization

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Authors: Jorge Gonzalez Camus, Carlos Lizama

Abstract:

In this work is proved the existence of at least one globally attractive mild solution to the Cauchy problem, for fractional evolution equation of neutral type, involving the fractional derivate in Caputo sense. An almost sectorial operator on a Banach space X and a kernel belonging to a large class appears in the equation, which covers many relevant cases from physics applications, in particular, the important case of time - fractional evolution equations of neutral type. The main tool used in this work was the Hausdorff measure of noncompactness and fixed point theorems, specifically Darbo-type. Initially, the equation is a Cauchy problem, involving a fractional derivate in Caputo sense. Then, is formulated the equivalent integral version, and defining a convenient functional, using the analytic integral resolvent operator, and verifying the hypothesis of the fixed point theorem of Darbo type, give us the existence of mild solution for the initial problem. Furthermore, each mild solution is globally attractive, a property that is desired in asymptotic behavior for that solution.

Keywords: attractive mild solutions, integral Volterra equations, neutral type equations, non-local in time equations

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Authors: Mohammad Vahedi Torshizi

Abstract:

In this investigation, at first, bending stress, contact stress, Safety factor of bending and Safety factor of contact between sun gear and planet gear tooth was determined using mathematical equations. Also, The amount of Sun Revolution in, Speed carrier, power Transmitted of the sun, sun torque, sun peripheral speed, Enter the tangential force gears, was calculated using mathematical equations. According to the obtained results, maximum of bending stress and contact stress occurred in three plantary and low status of four plantary. Also, maximum of Speed carrier, sun peripheral speed, Safety factor of bending and Safety factor of contact obtained in four plantary and maximum of power Transmitted of the sun, Enter the tangential force gears, bending stress and contact stress was in three pantry and factors And other factors were equal in the two planets.

Keywords: bending stress, contact stress, plantary, mathematical equations

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Authors: Emad K. Jaradat, Ala’a Al-Faqih

Abstract:

Nonlinear Schrödinger equations are regularly experienced in numerous parts of science and designing. Varieties of analytical methods have been proposed for solving these equations. In this work, we construct an approximate solution for the nonlinear Schrodinger equations, with harmonic oscillator potential, by Elzaki Decomposition Method (EDM). To illustrate the effects of harmonic oscillator on the behavior wave function, nonlinear Schrodinger equation in one and two dimensions is provided. The results show that, it is more perfectly convenient and easy to apply the EDM in one- and two-dimensional Schrodinger equation.

Keywords: non-linear Schrodinger equation, Elzaki decomposition method, harmonic oscillator, one and two-dimensional Schrodinger equation

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Authors: Jatupon Em-Udom, Nirand Pisutha-Arnond

Abstract:

The phase-field-crystal, PFC, approach is a density-functional-type material model with an atomic resolution on a diffusive timescale. Spatially, the model incorporates periodic nature of crystal lattices and can naturally exhibit elasticity, plasticity and crystal defects such as grain boundaries and dislocations. Temporally, the model operates on a diffusive timescale which bypasses the need to resolve prohibitively small atomic-vibration time steps. The PFC model has been used to study many material phenomena such as grain growth, elastic and plastic deformations and solid-solid phase transformations. In this study, the pressure-controlled dynamic equation for the PFC model was developed to simulate a single-component system under externally applied pressure; these coupled equations are important for studies of deformable systems such as those under constant pressure. The formulation is based on the non-equilibrium thermodynamics and the thermodynamics of crystalline solids. To obtain the equations, the entropy variation around the equilibrium point was derived. Then the resulting driving forces and flux around the equilibrium were obtained and rewritten as conventional thermodynamic quantities. These dynamics equations are different from the recently-proposed equations; the equations in this study should provide more rigorous descriptions of the system dynamics under externally applied pressure.

Keywords: driving forces and flux, evolution equation, non equilibrium thermodynamics, Onsager’s reciprocal relation, phase field crystal model, thermodynamics of single-component solid

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Authors: Bakur Gulua

Abstract:

In this work, we consider some boundary value problems for the plate. The plate is the elastic material with voids. The state of plate equilibrium is described by the system of differential equations that is derived from three-dimensional equations of equilibrium of an elastic material with voids (Cowin-Nunziato model) by Vekua's reduction method. Its general solution is represented by means of analytic functions of a complex variable and solutions of Helmholtz equations. The problem is solved analytically by the method of the theory of functions of a complex variable.

Keywords: the elastic material with voids, boundary value problems, Vekua's reduction method, a complex variable

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Authors: Fernando Maass, Pablo Martin, Jorge Olivares

Abstract:

Fractional derivatives have become very important in several areas of Engineering, however, the solutions of simple differential equations are not known. Here we are considering the simplest first order nonhomogeneous differential equations with Bessel regular functions of the first kind, in this way the solutions have been found which are hypergeometric solutions for any fractional derivative of order α, where α is rational number α=m/p, between zero and one. The way to find this result is by using Laplace transform and the Caputo definitions of fractional derivatives. This method is for values longer than one. However for α entire number the hypergeometric functions are Kumer type, no integer values of alpha, the hypergeometric function is more complicated is type ₂F₃(a,b,c, t2/2). The argument of the hypergeometric changes sign when we go from the regular Bessel functions to the modified Bessel functions of the first kind, however it integer seems that using precise values of α and considering no integers values of α, a solution can be obtained in terms of two hypergeometric functions. Further research is required for future papers in order to obtain the general solution for any rational value of α.

Keywords: Caputo, fractional calculation, hypergeometric, linear differential equations

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Authors: Ghodbane Azeddine, Saad Maarouf, Boland Jean-Francois, Thibeault Claude

Abstract:

This paper proposes a new design of an active fault-tolerant flight control system against abrupt actuator faults. This overall system combines a multiple time scale sliding mode controller for fault compensation and a geometric approach for fault detection and diagnosis. The proposed control system is able to accommodate several kinds of partial and total actuator failures, by using available healthy redundancy actuators. The overall system first estimates the correct fault information using the geometric approach. Then, and based on that, a new reconfigurable control law is designed based on the multiple time scale sliding mode technique for on-line compensating the effect of such faults. This approach takes advantages of the fact that there are significant difference between the time scales of aircraft states that have a slow dynamics and those that have a fast dynamics. The closed-loop stability of the overall system is proved using Lyapunov technique. A case study of the non-linear model of the F16 fighter, subject to the rudder total loss of control confirms the effectiveness of the proposed approach.

Keywords: actuator faults, fault detection and diagnosis, fault tolerant flight control, sliding mode control, multiple time scale approximation, geometric approach for fault reconstruction, lyapunov stability

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Authors: Palwinder Singh, Munish Sandhir, Tejinder Singh

Abstract:

The ordinary differential equations (ODE) represent a natural framework for mathematical modeling of many real-life situations in the field of engineering, control systems, physics, chemistry and astronomy etc. Such type of differential equations can be solved by analytical methods or by numerical methods. If the solution is calculated using analytical methods, it is done through calculus theories, and thus requires a longer time to solve. In this paper, we compare the numerical accuracy of the solutions given by the three main types of one-step initial value solvers: Taylor’s Series Method, Euler’s Method and Runge-Kutta Fourth Order Method (RK4). The comparison of accuracy is obtained through comparing the solutions of ordinary differential equation given by these three methods. Furthermore, to verify the accuracy; we compare these numerical solutions with the exact solutions.

Keywords: Ordinary differential equations (ODE), Taylor’s Series Method, Euler’s Method, Runge-Kutta Fourth Order Method

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Authors: A. Ghodbane, M. Saad, J. F. Boland, C. Thibeault

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Historically, actuators’ redundancy was used to deal with faults occurring suddenly in flight systems. This technique was generally expensive, time consuming and involves increased weight and space in the system. Therefore, nowadays, the on-line fault diagnosis of actuators and accommodation plays a major role in the design of avionic systems. These approaches, known as Fault Tolerant Flight Control systems (FTFCs) are able to adapt to such sudden faults while keeping avionics systems lighter and less expensive. In this paper, a (FTFC) system based on the Geometric Approach and a Reconfigurable Flight Control (RFC) are presented. The Geometric approach is used for cosmic ray fault reconstruction, while Sliding Mode Control (SMC) based on Lyapunov stability theory is designed for the reconfiguration of the controller in order to compensate the fault effect. Matlab®/Simulink® simulations are performed to illustrate the effectiveness and robustness of the proposed flight control system against actuators’ faulty signal caused by cosmic rays. The results demonstrate the successful real-time implementation of the proposed FTFC system on a non-linear 6 DOF aircraft model.

Keywords: actuators’ faults, fault detection and diagnosis, fault tolerant flight control, sliding mode control, geometric approach for fault reconstruction, Lyapunov stability

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Authors: Serifenur Cebesoy, Yelda Aygar, Elgiz Bairamov

Abstract:

In this study, we examine some spectral properties of non-selfadjoint matrix-valued difference equations consisting of a polynomial type Jost solution. The aim of this study is to investigate the eigenvalues and spectral singularities of the difference operator L which is expressed by the above-mentioned difference equation. Firstly, thanks to the representation of polynomial type Jost solution of this equation, we obtain asymptotics and some analytical properties. Then, using the uniqueness theorems of analytic functions, we guarantee that the operator L has a finite number of eigenvalues and spectral singularities.

Keywords: asymptotics, continuous spectrum, difference equations, eigenvalues, jost functions, spectral singularities

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Authors: Abdelhak Djoudi, Hachemi Chekireb, El Madjid Berkouk

Abstract:

The doubly fed induction generator (DFIG) received recently an important consideration in medium and high power wind energy conversion systems integration, due to its advantages compared to other generators types. The stator power sliding mode control (SPSMC) proves a great efficiency judge against other control laws and schemes. In the SPSMC laws elaborated by several authors, only the slide surface tracking conditions are elaborated using Lyapunov functions, and the boundedness of the DFIG states is never treated. Some works have validated theirs approaches by experiments results in the case of specified machines, but these verifications stay insufficient to generalize to other machines range. Adding to this argument, the DFIG states boundedness demonstration is widely suggested in goal to ensure that in the application of the SPSMC, the states evaluates within theirs tolerable bounds. Our objective in the present paper is to highlight the efficiency of the SPSMC by stability analysis. The boundedness of the DFIG states such as the stator current and rotor flux is discussed. Moreover, the states trajectories are finding using analytical proves taking into consideration the SPSMC gains.

Keywords: Doubly Fed Induction Generator (DFIG), Stator Powers Sliding Mode Control (SPSMC), lyapunov function, stability, states boundedness, trajectories mathematical proves

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Authors: Ebru Dural, M. Zulfu Asık

Abstract:

Laminated glass is a kind of safety glass, which is made by 'sandwiching' two glass sheets and a polyvinyl butyral (PVB) interlayer in between them. When the glass is broken, the interlayer in between the glass sheets can stick them together. Because of this property, the hazards of sharp projectiles during natural and man-made disasters reduces. They can be widely applied in building, architecture, automotive, transport industries. Laminated glass can easily undergo large displacements even under their own weight. In order to explain their true behavior, they should be analyzed by using large deflection theory to represent nonlinear behavior. In this study, a nonlinear mathematical model is developed for the analysis of laminated glass cylindrical shell which is free in radial directions and restrained in axial directions. The results will be verified by using the results of the experiment, carried out on laminated glass cylindrical shells. The behavior of laminated composite cylindrical shell can be represented by five partial differential equations. Four of the five equations are used to represent axial displacements and radial displacements and the fifth one for the transverse deflection of the unit. Governing partial differential equations are derived by employing variational principles and minimum potential energy concept. Finite difference method is employed to solve the coupled differential equations. First, they are converted into a system of matrix equations and then iterative procedure is employed. Iterative procedure is necessary since equations are coupled. Problems occurred in getting convergent sequence generated by the employed procedure are overcome by employing variable underrelaxation factor. The procedure developed to solve the differential equations provides not only less storage but also less calculation time, which is a substantial advantage in computational mechanics problems.

Keywords: laminated glass, mathematical model, nonlinear behavior, PVB

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Authors: Tan Zhang, Zhongjian Wang, Jack Xin, Zhiwen Zhang

Abstract:

We aim to efficiently compute the spreading speeds of reaction-diffusion-advection (RDA) fronts in divergence-free random flows under the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We study a stochastic interacting particle method (IPM) for the reduced principal eigenvalue (Lyapunov exponent) problem of an associated linear advection-diffusion operator with spatially random coefficients. The Fourier representation of the random advection field and the Feynman-Kac (FK) formula of the principal eigenvalue (Lyapunov exponent) form the foundation of our method implemented as a genetic evolution algorithm. The particles undergo advection-diffusion and mutation/selection through a fitness function originated in the FK semigroup. We analyze the convergence of the algorithm based on operator splitting and present numerical results on representative flows such as 2D cellular flow and 3D Arnold-Beltrami-Childress (ABC) flow under random perturbations. The 2D examples serve as a consistency check with semi-Lagrangian computation. The 3D results demonstrate that IPM, being mesh-free and self-adaptive, is simple to implement and efficient for computing front spreading speeds in the advection-dominated regime for high-dimensional random flows on unbounded domains where no truncation is needed.

Keywords: KPP front speeds, random flows, Feynman-Kac semigroups, interacting particle method, convergence analysis

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