Roots Of Characteristic Equation Research Articles

MATHEMATICAL MODELLING OF DYNAMIC PROCESSES IN A LAYERED INCOMPRESSIBLE HALF-SPACE WITH INITIAL STRESSES UNDER THE ACTION OF A MOVING LOAD

This work is devoted to the study of the influence of the protective coating, initial stresses, mechanical characteristics of materials, movement parameters of the surface load on the stress-deformed state of the elastic base. The relevance of the research results is related to the possibility of their use in the creation of qualitatively new materials, structures and building structures. Two models of a layered incompressible half-space are considered and compared: 1) an elastic plate on an elastic half-space; 2) the top layer (protective coating) is modelled by concentrated masses. The concentrated force moves along the free surface of the protective layer at a constant speed at a certain angle to the surface of the half-space. The solution of the problem was obtained using the method of integral Fourier transformations. Analytical results are given in a general form for materials with an arbitrary elastic potential, for cases of unequal and equal roots of characteristic equations, for various conditions of combination of elements of a layered medium and for any speed of movement of the load. The material with the Bartenev-Khazanovich potential was considered for numerical analysis. The calculations were carried out within the framework of the theory of finite initial deformations. The impact of the moving load, initial stresses and mechanical parameters of the elements of the layered base on the main characteristics of its stress-strain state was studied.

The stress-strain state of an incompressible prestressed half-space with a protective coating under the action of a moving load

This work is devoted to the study of the influence of the protective coating, initial stresses, mechanical characteristics of materials, movement parameters of the surface load on the stress-deformed state of the elastic base. The relevance of the research results is related to the possibility of their use in the creation of qualitatively new materials, structures and building structures. A prestressed incompressible half-space with inhomogeneity in the form of a thin surface layer is considered. The concentrated force moves along the free surface of the protective layer at a constant speed at a certain angle to the surface of the half-space. The solution of the problem was obtained using the method of integral Fourier transformations. Analytical results are given in a general form for materials with an arbitrary elastic potential, for cases of unequal and equal roots of characteristic equations, for various conditions of combination of elements of a layered medium and for any speed of movement of the load. The material with the Bartenev-Khazanovich potential was considered for numerical analysis. The calculations were carried out within the framework of the theory of finite initial deformations. The impact of the moving load, initial stresses and mechanical parameters of the elements of the layered base on the main characteristics of its stress-strain state was studied.

On the Feasibility of Dynamic Substructuring for Hybrid Testing of Vibrating Structures

AbstractDynamically substructured systems (DSS) are a typical technique to achieve real-time numerical simulations combined with physically tested components. However, a rigorous feasibility analysis before the implementation is missing. This paper is aimed to fill this gap by establishing rigorous conditions for when DSS is suitable for dynamic testing. The proposed method is based on novel symbolic recursive formulations for the transfer functions describing a generic lumped parameter vibrating structure, enabling the analysis of structural and other properties without requiring the computation of explicit symbolic expressions for the transfer functions involved, representing a significant breakthrough as it allows to perform feasibility analysis in analytical form, rather than solely relying on numerical approaches. The series of analytical conclusions presented in this paper, and future ones unlocked by the proposed approach, will significantly enrich the research in the community of DSS and structural vibrations. In particular, the proposed approach allows performing analysis of causality, controllability, and observability using much reduced knowledge of the structure, thus significantly simplifying such analysis. Analytical conclusions on stability can also be made with the help of novel recursive form, removing the need of repeatedly calculating the roots of characteristic equations, a task that can be performed only via numerical approaches and for which analytical results are not available. The proposed methodology can be applied to a whole class of vibration problems and is not linked to any specific structure, going beyond the specific examples available in the literature.

Stability and hopf bifurcation of fractional complex–valued BAM neural networks with multiple time delays

In this paper dynamical behaviors of a class of high–order fractional complex–valued bidirectional associative memory neural networks with multiple time delays are investigated. Firstly, they are reduced to real–valued systems by separating the real and imaginary parts. Then, stability criteria of fractional complex–valued bidirectional associative memory neural networks without delay are obtained. Concerning the delay case, the time delay is set as a bifurcation parameter and the condition of Hopf bifurcation is given by analyzing roots of characteristic equations. Finally, two numerical examples are presented and illustrate that Hopf bifurcation does happen when time delay exceeds the critical value.

Threshold dynamics of an age-space structure vector-borne disease model with multiple transmission pathways

This paper aims to study threshold dynamics of an age-space structured vector-borne epidemic model with multiple transmission pathways. We develop a model in a bounded space, and demonstrate the well-posedness by proving the global existence of the solution and the model admits a global attractor with using the theory of fixed point problem, Picard sequences and iteration. In the special case of homogenous space, the explicit formula of basic reproduction number $ R_{0} $ is established, which can be used to discuss whether a disease is persistent or extinct. The local and global stability of steady states are achieved by analyzing the distribution of characteristic roots of characteristic equations and designing Lyapunov functions. These theoretical results are illustrated by numerical simulations. Besides, sensitive analysis on $ R_{0} $ reveals that certain parameters are more effective to impact the $ R_{0} $. We find that the PRCC indexes of solutions can be time-dependent, and the relative importance of parameters can also depend on time. Relying on this analysis and exhaustive numerical simulations, we suggest that insecticides and mosquito nets should be used in a timely manner to disinfect vectors and reduce the contact between vectors and hosts, and that ignoring the horizontal transmission in the host underestimates the risk of disease transmission.

Stability analysis of some neutral delay-differential equations with a frequency-domain approach

<p style='text-indent:20px;'>Two general schemes of neutral delay differential equations are considered. They represent non delayed systems with delayed, nonlinear feedback control. The stability conditions in terms of system parameters are derived using an approach based on feedback systems, namely, the Nyquist stability criterion. It is also shown that those results coincide with some already found in the literature, and others derived here, which are based on classical tools, that investigate directly the roots of characteristic equation. Finally, some examples are given to illustrate the usefulness of this approach.</p>

НАСТРОЙКА И ИССЛЕДОВАНИЕ РЕЖИМОВ РАБОТЫ НАБЛЮДАТЕЛЯ УГЛОВОЙ ЧАСТОТЫ ВРАЩЕНИЯ РЕГУЛИРУЕМОГО ЭЛЕКТРОПРИВОДА ПОСТОЯННОГО ТОКА

In modern closed-loop controlled electric drives used in manufacturing, the use of speed, current, torque, and other technological variable sensors is required. In some cases, indirect estimation of the electric drive state vector is required, when the use of technological parameter sensors (primarily speed sensors) is difficult or impossible. One of the problems is the length of the information cable in the feedback channel from the actuator to the power converter, which makes it difficult to use direct measurement methods. There are also difficulties in using sensors in an aggressive environment. For example, in the coal mining industry, electric motors or the entire electric drive are often located in an explosion-proof enclosure, and installing additional sensors will require expensive modernization of the entire system. State observers in sensorless electric drives allow restoring unmeasurable state vector components based on measured electrical signals in the power converters of the electric drive. From the perspective of a closed-loop controlled electric drive system, the main feedback is the feedback on the angular rotation frequency. Electric drives based on separately excited DC motors (SEDC) are simpler to investigate and implement such systems. The linearized mathematical model of an SEDC contains a relatively smaller number of differential equations compared to variable frequency drives due to the presence of independent channels for forming magnetic flux in the air gap and electromagnetic torque, which simplifies the analysis and synthesis of sensorless control systems using speed observers. Researching the observer of angular rotation for DC electric drives allows for developing design skills on a simpler system before moving on to variable frequency drives. Research in this area for both DC and AC electric drives is promising because it allows for an increased range of regulation and accuracy of the output coordinate in sensorless electric drives at the required level. This article is dedicated to the study of the properties of speed observers for DC electric drives with independent excitation of various structures. Aim of research Analysis of rotation frequency observer structure in DC motor drives with error reduction based on the proportional principle. Research methods Theoretical and experimental research methods were used in this study. Theoretical methods included the theory of automatic control, state observer theory, theory of electric drives, theory of electric machines, mathematical modeling, systems of differential equations, Laplace transforms, and numerical methods for solving differential equations. Experimental studies were conducted on a software-hardware complex, where the genetic algorithm method was used to identify the parameters of the electric machine equivalent circuit. Results Analytical expression for the characteristic equation roots for the investigated structure of the DC motor drive's speed observer was obtained. Based on this expression, stability criteria and trends in changes of the quality indicators for the angular velocity estimation were determined with varying observer coefficients.

Contact problems for a transversely isotropic layer

Two spatial, one axisymmetric and two plane contact problems are considered for a transversely isotropic elastic layer with one face subjected to sliding support. In the spatial and plane contact problems, the planes of isotropy may be either parallel or perpendicular to the layer faces. In the case of axial symmetry, the planes of isotropy are parallel to the layer faces. By using Fourier integral transforms, the contact problems are reduced to integral equations with respect to the contact pressure, the limiting cases of which are the well-known equations of the corresponding problems for an isotropic layer. For solving the spatial problems with unknown contact domains, the nonlinear boundary integral equations method is used, which make it possible to determine the contact pressure and the contact domain simultaneously. To extract the kernel principal part of the spatial problem integral equation when the isotropy planes are perpendicular to the layer faces, it is used the kernel of the integral equation of the corresponding contact problem for a transversely isotropic half-space obtained earlier without quadratures. The integral equation of the axially symmetric problem is reduced to a Fredholm integral equation of the second kind with the help of the method of pair equations, and the method of mechanical quadratures is used for numerical solutions. Plane problems are solved in a closed form based on special approximations of the kernel symbols. The approximations accuracy grows as anisotropy increases. Here, the anisotropy level can be characterized by the difference between ratio of a characteristic equation roots and unit because the unit value corresponds to the isotropic case. Mechanical characteristics as well as errors of the approximations are calculated for well-known transversely isotropic materials.

Influence of ion structure and solvent electric dipole on ultrananoporous supercapacitor: a lattice model study

A five-state modified BEG model is proposed for describing electrolyte in polar solvent in cylindrical electrode pore in which only one row of particles can be stationed. Two properties the closest to the actual situation of the aqueous electrolyte are integrated into the model: cation is dimer comprised of two tangentially tethered hard spheres of different diameters: one is charged, while the other is neutral, and polar solvent is modeled as neutral hard sphere with an electric dipole. The system partition function is obtained by numerically solving maximum real root of characteristic equation of Kramers-Wannier transfer matrix. The model is used to investigate effects of ion structure and electric valence, solvent electric dipole, and cylindrical pore radius on specific differential capacitance and specific energy storage of the resulting supercapacitor. Main findings are briefly described below. (i) The dimer counter-ion electric valence does not exert obvious influence on the saturation value of the but reduces the threshold electrode potential strength beyond which the value is just reached. (ii) Whether the dimer counter-ion is monovalent or bivalent, the peak height and the value tend to rise with the dimer neutral site size decreasing, but influence of the value on the value is very small. Only when the value is small enough, the value rises a little with decrease of the value. (iii) Increase of solvent electric dipole moment (whether by increasing the dipole charge or dipole length) always raises the value, and enlarges width of the pure solvent domain or zero and zero domain around the zero electrode potential, but does not change the value obviously. (iv) With the pore size increasing, the value reduces and the value increases. So, using small size pore is favorable to improve the supercapacitor performance from the perspective of energy storage and electrode potential applied. (v) The three parameters: dipole charge, and dipole length, are cooperative in influencing the value.

Zero Stability Well Predicts Performance of Convolutional Neural Networks

The question of what kind of convolutional neural network (CNN) structure performs well is fascinating. In this work, we move toward the answer with one more step by connecting zero stability and model performance. Specifically, we found that if a discrete solver of an ordinary differential equation is zero stable, the CNN corresponding to that solver performs well. We first give the interpretation of zero stability in the context of deep learning and then investigate the performance of existing first- and second-order CNNs under different zero-stable circumstances. Based on the preliminary observation, we provide a higher-order discretization to construct CNNs and then propose a zero-stable network (ZeroSNet). To guarantee zero stability of the ZeroSNet, we first deduce a structure that meets consistency conditions and then give a zero stable region of a training-free parameter. By analyzing the roots of a characteristic equation, we theoretically obtain the optimal coefficients of feature maps. Empirically, we present our results from three aspects: We provide extensive empirical evidence of different depth on different datasets to show that the moduli of the characteristic equation's roots are the keys for the performance of CNNs that require historical features; Our experiments show that ZeroSNet outperforms existing CNNs which is based on high-order discretization; ZeroSNets show better robustness against noises on the input. The source code is available at https://github.com/logichen/ZeroSNet.

THE IDENTIFICATION OF INDUCTION MOTOR INTERNAL QUANTITIES

In the presented work a new identification method of difficult measured internal quantities of IM, such as components of magnetic flux vector and electromagnetic torque, is proposed. Commonly measurable quantities of IM like stator currents, stator voltage frequency and mechanical angular speed are used for identification to determine a feedback effect of the rotor flux vector on vector of stator currents of IM. Based on this feedback it is also possible to identify actual value of the rotor resistance, which can alter during IM operation. This has a significant impact on precision of identified quantities as well as on master control of IM. Stability of the identification structure is guaranteed by position of roots of characteristic equation of its linear transfer function. Results obtained from simulation measurements confirm quality, effectivity, feasibility, and robustness of the proposed identification method.

Semi-rational vector rogon–soliton solutions of the five-component Manakov/NLS system with mixed backgrounds

The five-component Manakov system (alias 5-Manakov system) appears in many fields such as nonlinear optics, Bose–Einstein condensates (alias the spin-2 Gross–Pitaevskii equation), and ocean. In this Letter, we investigate the semi-rational vector rogon–soliton and soliton of the 5-Manakov system with mixed zero and non-zero backgrounds by using the modified Darboux transform. The key point is to explicitly solve the multiple roots of a sixth degree characteristic polynomial equation. In particular, we exhibit the wave structures of the fundamental semi-rational vector rogon with grey solitons and soliton solutions, and their higher-degree forms. Moreover, the semi-rational vector rogon with grey solitons corresponding to the components with non-zero backgrounds are PT-symmetric. These semi-rational wave structures will be useful to further understand some related physical phenomena and to design the relative experiments.

Stability analysis of functionally graded plates based on the three-dimensional theory of elasticity

Models available in publications for studying the stability of functionally graded plates are usually tested for some special cases available in publications, and then they are used to study a wide range of issues. This approach is fraught with the danger of making serious mistakes, especially when calculating structures on an elastic foundation. This report will develop two approaches to studying the stability of functionally graded plates in the 3D formulation, which minimize the probability of calculation errors.In the first approach, to construct differential equations of stability, a polynomial approximation of sought-for functions across the structure thickness is used. Its salient feature is assignment of the functions to the outer surfaces of layers, which allows splitting the layers into sublayers with a corresponding increase in the accuracy of calculation results. In the second approach, using the Reissner variational principle, a system of integrodifferential equilibrium equations and the corresponding boundary conditions are obtained without introducing an approximation. For the particular case of hinge-supported plates with a thermal load distributed according to a trigonometric law, the system of integrodifferential equilibrium equations obtained for the first approach and the differential equations of the second approach allow an analytical implementation. In the first approach, the system of differential equations is converted to a system of algebraic equations. The assignment of sought-for functions to the outer surfaces of layers allows one to split the layers into sublayers and thus to reduce the approximation error. Equations of the second approach are transformed into a system of ordinary differential equations for the distribution of required functions across the plate thickness, with an analytical search for the roots of characteristic equations and the corresponding eigenvectors. The same result obtained by the two methods may point to its reliability.In this message, for the particular case of hinged support, the proposed models are implemented analytically. The studies carried out have shown that the proposed applied model in the analysis of the stability of plates made of a functionally graded material with free outer surfaces provided high calculation accuracy without separating the layers into sublayers, even without the transverse compression. We also analysed the possibility of using the proposed approach with a polynomial approximation of the required functions by the thickness of the structure to study stability on an elastic foundation in the form of a half-infinite layer. The half-infinite layer in the calculation by the model (P) was modeled by a layer of finitely thickness. Its thickness was chosen in such a way that the calculation with a fixed and a free lower surface did not differ. The calculation with consideration of each layer within one layer in such task can be considered only as a calculation of the first rude approximation. The separation of the layers into ten sublayers gives the accuracy of the calculation, that is almost equivalent to the approach with the analytical search for the required functions by the thickness of the structure. Further fractination of the base into one hundred and sixty sub-layers gives the coincidence of the calculation results with an error of less than 1%. The presented results can be utilized as a benchmark for further studies of the stability of FGM plates. Further, it is assumed the numerical-analytical implementation of the proposed approaches .

Intracavity optical parametric oscillator: Model of dynamic system with different values of time delay for pump and signal radiation

Most of intracavity pumped optical parametric oscillators (OPO) are made nowaday according to a scheme with a single-resonance OPO located in the cavity of a pump laser. Usually the cavities of the pump and OPO (signal) emission have different values of round-trip time (delay). Aim of the study is therefore to build up the mathematical model of intracavity optical parametric oscillator (ICOPO) considered as a time-delay dynamic system with two values of delay in both cavities (the pump and signal). Methods. The model allows to analyze the steady state (equilibrium point) of the dynamic system and its stability with the help of characteristic equation’s solution. Results. Countless set of the characteristic equation roots is shown to consist of complex-conjugate pairs with imaginary parts which are nearly multiples of intermode beat frequencies in the pump and signal cavities. The diagram of stability depending on the position of nonlinear crystal in the resonator was built on the parameter plane. Features of the plane partition into the areas of stability/instabilty vs behavior of the characteristic equation roots are examined. Discussion. The results of the study allow to consider an ICOPO as the time-delay dynamic system thus adding to the physical picture of intracavity parametric oscillators.

Singular points classification of first order differential equations system not solved for derivatives

The article considers the singular system points’ classification of first-order differential equations that are not resolved with respect to derivatives. By displaying using some ratios that considered integral curves locations by special point in space taking into account the mutually located integral parabolas we obtain the corresponding pictures of the integral curves location by the singular system point. The types of singular equation points can be classified according the characteristic equation roots type, which we call a saddle, node, focus, center, or saddle focus

Analysis of a malaria epidemic model with age structure and spatial diffusion

This paper aims to provide the complete analysis on the threshold dynamics of an age-space structured malaria epidemic model. We formulate the model in a spatially bounded domain by assuming that: (i) the density of susceptible humans at space x stabilizes at H(x); (ii) the force of infection between human population and mosquitoes is given by the mass action incidence. By appealing to the theory of fixed point problem and Picard sequences and iteration, the well-posedness of the model is shown by verifying that the solution exists globally and the model admits a global attractor. In the spatially homogeneous case, we establish the explicit formula for the basic reproduction number, which governs the malaria extinction and persistence. The local and global stability of equilibria is achieved by studying the distribution of characteristic roots of characteristic equation and constructing the suitable Lyapunov functions, respectively.

Epidemiological Modelling and Analysis of COVID-19 Pandemic with Treatment

In this paper, Mathematical Model of COVID-19 Pandemic is formulated and discussed. The positivity, boundedness, and existence of the solutions of the model equations are stated and proved. The Disease-free equilibrium point & endemic equilibrium points are identified. Stability Analysis of the model is done with the concept of Next generation matrix. we have investigated that Disease-free equilibrium point (DFEP) of the model is locally asymptotically stable if α≤β+δ+μ & unstable if α>β+δ+μ, The basic reproduction number (threshold value) R₀ is the largest eigen value in spectral radius matrix ρ. Thus, eigen values of spectral radius Matrix ρ are determined from the roots of characteristic polynomial equation, det[ρ-λI]=0, Hence, the basic reproduction number is R₀=α / β. It is shown that if reproduction number is less than one, then COVID-19 cases will be reduced in the community. However, if reproduction number is greater than one, then covid-19 continue to persist in the Community. Lastly, numerical simulations are done with DEDiscover 2.6.4. Software. It is observed that with Constant treatment, increase or decrease contact rate among persons leads great variation on the basic reproduction number which is directly implies that infection rate plays a vital role on decline or persistence of COVID-19 pandemic.

ANALYTICAL SOLUTION OF THE PROBLEM ON THE THERMALLY STRESSED STATE OF FUNCTIONALLY GRADED PLATES BASED ON THE 3D ELASTICITY THEORY

Two approaches to studying the thermally stressed state of layered functionally graded plates in the 3D formulation have been developed. In the first approach, using the Reissner variational principle, a system of integrodifferential equilibrium equations and the corresponding boundary conditions are obtained without introducing an approximation. In the second approach, to construct differential equations of 3D thermoelastic equilibrium, a polynomial approximation of sought-for functions across the structure thickness is used. Its salient feature is assignment of the functions to the outer surfaces of layers, which allows splitting the layers into sublayers with a corresponding increase in the accuracy of calculation results. For the particular case of hinge-supported plates with a thermal load distributed according to a trigonometric law, the system of integrodifferential equilibrium equations obtained for the first approach and the differential equations of the second approach allow an analytical implementation. Then, equations of the first approach are transformed into a system of ordinary differential equations for the distribution of required functions across the plate thickness, with an analytical search for the roots of characteristic equations and the corresponding eigenvectors. In the second approach, the system of differential equations is converted to a system of algebraic equations. The assignment of sought-for functions to the outer surfaces of layers allows one to split the layers into sublayers and thus to reduce the approximation error. The same result obtained by the two methods may point to its reliability.

Determination of the stress concentration in the corner point of the wedge-shaped region reinforced by a more rigid thin coating

We analysed the problem of determining the exponents in the asymptotic solution of the isotropic theory of elasticity problem at the top of the wedge-shaped region where its sides (or one of them) are supported by a thin coating and lean without friction on the rigid bases. On the other side of the wedge-shaped region, it is assumed that there are various boundary conditions, including when there is a thin coating. Mathematically, the problem reduces to the problem of determining the roots of transcendental characteristic equations arising from the condition for the existence of a nontrivial solution of a system of the linear homogeneous equations. The characteristics of the stress tensor components have been determined for the various combinations of boundary conditions and physical and geometric parameters. The qualitative conclusions are made. In particular, we have established the combinations of the values of these parameters at which the singular behaviour of stresses arises.

New Type Super Singular Integro-Differential Equation and Its Conjugate Equation

In this paper for a class of model partial integro-differential equation with super singularity in the kernels, is obtained an integral representation of manifold solutions by arbitrary constants. The conjugate equation for the above-mentioned type of equations is also investigated. Such types of integro-differential equations are different from Cauchy-type singular integro-differential equations. Cauchy-type singular integro-differential equations are studied by the methods of theory of analytical functions. However, the method of analytical functions is not applicable for our case of super singular equations with integrals understanding in Riemann–Stieltjes sense. Here, we have used the method of representation the considering equation as a product of two one-dimensional singular first order integro-differential operators. Further, a complete integro-differential equation and its conjugate equation have been investigated. It is shown that in every cases of characteristic equation roots the homogeneous integro-differential equation can have a nontrivial solutions. Non-model equation is investigated by the regularization method. Regularization of non-model equation is based on selecting a model part of equation. On the basis of the analysis of a model part of equation the solution of non-model equation reduced to the solution of a second kind Volterra integral equations with super singular kernel. It is important to emphasize that in contrast to the usual theory of Volterra integral equations, the studied homogeneous integral equation has nontrivial solutions. It is easy to see that the presence of a non-model part in the equation does not affect to the general structure of the obtained solutions. From here investigation of the model equations for given class of the integro-differential equations becomes important.