Characteristic Equation Research Articles

Analyses of gas pulsation characteristics in the discharge pipeline of a hyper compressor

For the analysis of gas pulsation characteristics in the pipeline systems of low density polyethylene (LDPE) hyper compressors, a time-domain calculation method for compressor pipeline gas pulsation based on real gas properties is proposed. First the thermophysical property tables are prepared with gas state parameters as independent and dependent variables. Then one-dimensional unsteady flow equations and characteristic equations for pipeline flow are established based on real gas properties. After setting pipeline parameters and boundary conditions, the flow equations are discretized using a two-step method at the inner points of the pipeline. The characteristic equations are discretized using the trapezoidal integration method at the boundary points. Thus, the time-domain solving of pulsating flow field in the pipeline is realized. A self-developed numerical code was used to conduct the time-domain calculation and analysis of gas pulsation in the discharge pipeline of a two-stage hyper compressor. The calculated pressure pulsation curves based on real gas properties are in good agreement with experimental data, with a maximum difference lower than 4% of the local average pressure, which is smaller than the difference between the results calculated based on the plane wave theory and the measured data. The analysis of gas pulsation characteristics shows that the pressure pulsation in the main pipeline is mainly contributed by the fundamental and second harmonics. The pressure pulsation in the safety valve branch is higher than that in the main pipeline, and the harmonic components falling into the acoustic resonance frequency range of this branch have significant amplitudes. The acoustic resonance frequency of the branch pipeline is not affected by the main pipeline. Pressure pulsation varies with changes in rotational speed and back pressure, and the pulsation level is more significantly affected by rotational speed compared to back pressure. The renovation scheme of connecting the safety valve branch and expansion tube in series can effectively reduce the pressure pulsation levels of the main pipeline and branch pipeline.

Nonlinear dynamical modeling and response analysis of complex structures based on assumed mode weighting

The assumed mode method (AMM) is widely used for dynamical modeling but faces increasing challenges in developing low-dimensional and high-precision models of complex structures for investigations of nonlinear dynamical responses and active vibration control. This is mainly because the current complex structure has an increasing number of components, each of which needs to be discretized by multiple assumed modes in dynamical modeling, leading to increasing degrees of freedom for the system. To address this issue, this paper proposes a dimension-reduced method based on the assumed mode weighting method, referred to as Mode Weighting Method (MWM). The modeling process of a multi-beam structure is fully presented as an example to show the strategy of the MWM. During this process, the assumed mode (AM) model (the dynamic model derived in terms of a set of assumed modes) is developed first where the joints of the structure between its components are considered as artificial springs and each component is discretized by its vibration modes without constraints. The components of the eigenvector of the characteristic equation derived from the AM model are taken as the weighting coefficients to get the approximate analytical global modes of the structure by multiplying the corresponding assumed modes. Consequently, a nonlinear dynamic model with a few degree of freedom is obtained by discretizing the structure with the weighted modes. Moreover, a technique for simply obtaining the weighted mode (WM) models by directly processing the AM model through a transformation matrix is presented. To validate the proposed method, the simulations on the natural characteristics and the dynamical responses for both linear and nonlinear models, are performed respectively based on the WM and the finite element models, where both results are well matched each other. This work provides a new way of developing advanced dynamical modeling methods for structural dynamics design, optimization, and active vibration control.

On fractional ring neural networks with multiple time delays: Stability and Hopf bifurcation analysis

Stability and Hopf bifurcation analysis of neural networks have become popular topics in recent years. In this paper, a large-scale fractional ring neural network with multiple time delays is considered. The model comprises three rings that share a single node. The effects of time delay and fractional order on its stability and Hopf bifurcation have been investigated. Firstly, the characteristic equation is obtained by using the Laplace transform. Then, the time delay and fractional order are set as a bifurcation parameter respectively. Correspondingly, the local stability and Hopf bifurcation conditions are derived by analyzing the characteristic equations. Finally, two examples are given to support the theoretical analysis. In fact, this article provides an effective method for analyzing the stability and Hopf bifurcation of fractional ring neural networks.

Pattern Control of Neural Networks with Two-Dimensional Diffusion and Mixed Delays

In this paper, a two-neuron reaction–diffusion neural network with discrete and distributed delays is proposed, and the state feedback control strategy is adopted to achieve control of its spatiotemporal dynamical behaviours. Adding two virtual neurons, the original system is transformed into a neural network only containing the discrete delay. The conditions under which Hopf bifurcation and Turing instability arise are determined through analysis of the characteristic equation. Additionally, the amplitude equations are derived with the aid of weakly nonlinear analysis, and the selection of the Turing patterns is determined. The simulation results demonstrate that the state feedback controller can delay the onset of Hopf bifurcation and suppress the generation of Turing patterns.

Vibration analysis of graphene-reinforced porous aluminum-based variable-walled thickness sandwich joined conical-conical panel with elastic boundary conditions using differential quadrature method

In this paper, a unified solution is proposed to investigate the vibration characteristics of the variable-walled thickness graphene-reinforced porous aluminum-based (GRPA) sandwich joined conical-conical panel (JCCP) with the arbitrary elastic support boundary conditions by using the differential quadrature method (DQM). The two surfaces of the sandwich JCCP are made of metallic aluminum and the central core layer is the GRPA. The core thickness of each conical plate varies linearly with its generatrix. Three types of the graphene distributions and two types of porosity distributions are considered along the core thickness direction. Based on the first-order shear deformation theory (FSDT), von-Karman strain displacement relationship, constitutive relationship and Hamiltonian principle, the partial differential governing equations of motion are obtained for the variable wall thickness GRPA sandwich JCCP. Using the DQM, the dynamic equation is discretized into the ordinary differential equation. The matrix of the characteristic equation is analyzed to solve the frequencies and mode shapes of the variable wall thickness GRPA sandwich JCCP. The effects of the spring stiffness, boundary conditions, graphene distributions, porosity distributions and geometric parameters on the vibration properties are studied for the variable wall thickness GRPA sandwich JCCP with two interesting elastic supported boundary conditions. At the same time, this article provides a useful approach for studying the arbitrary boundary coupled plate and shell structures with the variable wall thickness.

Prediction Distribution Model of Moisture Content in Laminated Wood Components.

Shrinkage cracks are some of the most common defects in timber structures obtained from woods with an uneven distribution of moisture content and are subject to external dynamic environmental changes. To accurately predict the changes in the moisture content of wood components at any time and position, this study first applied the principles of food drying and established a moisture field model for laminated wood based on the analogy between heat and humidity transfer. A model for predicting the moisture content of wood that considers time and spatial distribution was then proposed. Second, by collecting relevant experimental data and establishing a finite element analysis model, three moisture absorption conditions (0-9.95%, 0-13.65%, and 0-17.91%) and four desorption conditions (34-5.5%, 28-8.3%, 31-11.8%, and 25.5-15.9%) were analyzed. In the moisture absorption comparison, the time needed to reach 95% equilibrium moisture content was 2.43 days, 4.07 days, and 6.32 days. The rate at which the internal components reached equilibrium moisture content exceeded 10 days. The temporal and spatial distribution of wood moisture content revealed the correctness of the proposed wood moisture field model. Finally, the moisture content prediction model was applied in the order of characteristic equation solutions, moisture content gradient difference, and laminated wood size. The results revealed that the established humidity field model can predict the wood moisture content and how it changes over time and in space. Notably, 1-2 orders for the solution of the characteristic equation are recommended when applying the prediction model. The greater the difference in moisture content, the faster the equilibrium moisture content is reached. The moisture content varies greatly based on the component size and position. Notably, the influence of moisture gradient and wood size on the average wood moisture content cannot be ignored.

Some infinite series summations involving linear recurrence relations of order 2 and 3

This paper extends known results of second and third order recursive sequences through extensive formulations of properties of the roots of their characteristic equations, some are old but most are new. They are applied to novel studies of $\sum_{n=0}^{\infty}{\frac{a_{mn}}{10^{n+1}}, \ m=1,2,3}$, including their convergence criteria, and applied to many standard sequences, as particular cases of a generic $\left\{a_n\right\}$. The detailed development of the algebra of the pertinent theorems, and their associated lemmas and corollaries, should open up new vistas for interested number theorists with the concluding results on series values.

Higher‐order effect of axial force on free vibration and buckling of functionally graded sandwich beams

AbstractThis paper presents a novel higher‐order shear deformation beam theory for analyzing the stability and free vibration of functionally graded (FG) sandwich beams with emphasis on the effect of higher‐order moment (HOM) and cross‐sectional warping. The governing equation of axially loaded FG sandwich beams is derived from three‐dimensional equations of the theory of elastic waves in bodies with homogeneous initial stresses. The characteristic equations for typical end conditions are obtained exactly. The numerical results of the natural frequencies and critical loads are calculated and verified for special cases by comparing them with the existing solutions. The effects of gradient index, core thickness, HOM, and warping shapes on the natural frequencies and critical buckling loads are elucidated for different slenderness ratios and end constraints.

Size-dependent effect on the interaction of surface waves in micropolar thermoelastic medium with dual pore connectivity

This research tackles a critical knowledge gap in Rayleigh surface wave propagation. It offers a comprehensive analysis that surpasses previous limitations. A size-dependent micropolar medium with unique void distributions and thermal effects is considered in this work. The constitutive relations and equations of motion for a nonlocal micropolar thermoelastic medium with double voids (MTMWDV) have been established by using Eringen’s nonlocal elasticity theory. Employing the three-phase-lag thermoelasticity theory (TPLTE), the study utilizes a wave-mode method to derive analytical solutions for Rayleigh waves in a nonlocal MTMWDV. To gain a comprehensive understanding of wave behavior, we solve the characteristic equation and analyze its roots, applying a filter based on the surface wave decay condition. A medium with stress-free and isothermal boundaries is explored through computational simulations to determine the attenuation coefficient and phase velocity. Furthermore, particle motion analysis is conducted to complement the analytical and computational approaches. Moreover, the influence of the nonlocal parameter and various thermoelastic models on these wave phenomena is investigated. The validity of the current mathematical model is confirmed through the derivation of particular scenarios.

Free and forced vibrations of elastically restrained cantilever with lumped oscillator

The efficiency assessment of cantilever-based energy harvesters relies on vibrational analysis, which necessitates modifications aimed at enhancing efficiency. These modifications involve manipulating the fundamental frequency to lower values and encompassing a wider range of resonances within a specified bandwidth. Consequently, this paper introduces an original analytical-numerical exploration into the vibratory response of a cantilever with a novel boundary condition involving an elastically restrained oscillator-spring arrangement. At the microbeam's tip, an oscillator is elastically confined by a linear spring, resulting in a novel set of coupled governing equations and a distinct shearing boundary condition. Microbeam equations is derived from the modified couple stress theory to capture size dependency. During free vibration analysis, a previously unreported characteristic equation is derived. This nonlinear transcendental equation is numerically solved utilizing root-solver algorithms, such as those available in MATLAB. Significantly, it is discovered that the inclusion of a lumped oscillator with an elastic support induces a minimal (new) natural frequency. Applying the extended Hamilton's principle, the effect of the lumped oscillator emerges both on the governing equations of motion and boundary conditions of the microbeam. Novelty of the paper focuses on the both characteristic equation and transmissibility by adopting the Galerkin’s modal decomposition technique. This finding carries vital implications as the efficiency of cantilever-based energy harvesters is directly contingent upon the resonance frequency. Notably, the oscillator mass and spring constant are two parameters that directly influence the vibratory response of the microbeam. In the context of forced vibrations, harmonic base excitation is considered as the input excitation, and the mechanical frequency response function is provided. The proposed system offers two distinct advantages for energy harvester systems: the creation of minimal resonance at lower values and the potential to manipulate the system's resonance toward a desired frequency spectrum.Article HighlightsModifying the boundary conditions of a cantilever beam with lumped-parameter system, can significantly change the behavior of the vibratory response.The boundary condition directly impact the resonance frequencies; which influences the maximum amount of harvestable voltage in vibration-based energy harvesters.Spring constant and mass of the lumped oscillator, are the key factors to alter the vibratory behavior and bandwidth of frequencies. Optimizing such mentioned parameters can help reaching to the maximum harvesting of energy.

Stability of the twisted states in a ring of oscillators interacting with distance-dependent delays

We consider a ring of phase oscillators interacting with distance-dependent time delays in general forms. It is found that the distance-dependent interaction delay is an essential driving mechanism for the twisted state to occur as a generic state. We carry out rigorous stability analysis for the existence and stability of the twisted states along the Ott–Antonsen invariant manifold and derive the exact characteristic equations for the stability conditions for arbitrarily chosen distance-dependent delay function. The complete stability diagrams are illustrated for two types of distance-dependent delay schemes, the propagational and step-wise interaction delays. Our theoretical results are verified using the direct numerical simulations of the model system.

Research on the prediction model of energy separation effect of vortex tube based on trajectory deflection characteristics and its numerical solution method

In this paper, in order to establish the energy separation mechanism of the vortex tube, the hydrodynamic behavior of the compressible fluid in the asymmetric cavity space is investigated, and a numerical model of the trajectory deflection behavior is deduced and established; in order to form the optimal design method of the structural parameters of the vortex tube, the force situation of the fluid microelements entering different regions of the vortex chamber of the vortex tube is analyzed, and the trajectory deflection equations are corrected by combining with the expansion behavior of the fluid and the characterizing equations of vortex strength, transportability, and vortex initiation characteristics are given. The characterization equations of vortex strength, transportability and vortex initiation characteristics are given, and the numerical simulation of their influence parameters is carried out; in order to realize the prediction of the vortex tube performance of a given structure, the multifactor Pearson thermodynamic map is used to correlate and analyze the experimental data of vortex tubes reported publicly in the past years, and the polynomial regression equations are designed and established for the prediction of the vortex tube's energy separation effect and the confidence level and the degree of coincidence of the prediction results are examined. The confidence level and degree of agreement of the prediction results were examined. It is found that: the trajectory deflection motion of the compressible fluid in the asymmetric cavity space is the result of the combined effect of structural air pressure bias and the expansion behavior of the incident fluid; in order to improve the vortex strength in the vortex tube, the vortex initiation chamber space should be as small as possible; the increase of the diameters of the hot-end pipe and the cold-end pipe is conducive to the enhancement of vortex strength, but at the same time, it weakens the vortex transport in the heat pipe; the vortex initiation chamber size has a negative correlation with the hot-end temperature rise, and the inlet fluid pressure has a The negative correlation between the size of the vortex chamber and the temperature rise at the hot end, the positive correlation between the increase of inlet fluid pressure and the resulting temperature rise, and the strong correlation between the inlet fluid pressure and the friction coefficient on the effect of energy separation; the predictive equations for the effect of energy separation obtained by the fitting are in good agreement with the real situation.

Analytical modelling and design of linear controlled dynamic systems

A generalized canonical mathematical model of multidimensional controlled mining transportation complexes and other technical systems in form by A.M. Letov is considered. A principle of symmetry and an algebraic concept are in the basis of the developed analytical methods of design. A principle of symmetry is realized on a set of indexes of roots of a characteristic equation of the system and on a set of indexes of phase coordinates of the mathematical model. The problem of quality of dynamic processes in time is reduced to an algebraic problem of distribution of roots of a corresponding characteristic equation in a complex plane. An analogy in a procedure of transformation of a characteristic determinant into a polynomial and a structure of elementary symmetrical polynomials of roots is established. A new formulation of an analytical representation of change of phase coordinates in time in a form of ordered determinants with respect to indexes of roots and indexes of phase coordinates is obtained based on residue theorem. An illustration of the developed analytical method of design is performed on a special case of a well-known controlled technical system of the fourth order.

Bifurcation analysis and chaos for a double-strains HIV coinfection model with intracellular delays, saturated incidence and Logistic growth

In this paper, a class of virus-to-cell HIV model with intracellular delays, saturated incidence and Logistic growth is proposed to characterize the interaction between two types HIV strains, i.e., wild-type and drug-resistant strains. First, a series of threshold criteria on the locally and globally asymptotic stability of (infection-free, dominant, coexistence) equilibria are discussed based on the basic reproduction number R0. Furthermore, a detailed Hopf bifurcation analysis is performed on the coexistence equilibrium using two delays as bifurcation parameters. We find that the Hopf bifurcations induced by double-strains are evidently different and more complicated than that of single strain, the former switches from stability (periodic branches) to un-stability (chaos) more frequently and earlier than the latter since double-strains would yield more pairs of imaginary roots in the characteristic equations. Meanwhile, the total viral load of double-strains would be higher than that of single-strain as well. The emergence of drug resistance imposes either negative or positive influences on the survival of wild-type strain, which would further facilitate the transmission of HIV.

Investigation and Control of Fluidized Bed Catalytic Cracking Unit

The fluidized catalytic cracking (FCC) unit is one the main unit of the oil refinery, its objective is upgraded heavy low value petroleum streams into higher value products, mainly gasoline and olefin. In this process the catalyst is circulated into the regenerator, the coke is burned, and the catalyst is regenerated. Many parameters are used to control FCC, the most importantare temperature, which should be kept at (505? C ) and the catalyst temperature (655? C). Other parameters are the level and flow rate of the catalyst. In this work, control strategy was build and the transfer functions were detected, so, the characteristic equations were calculated as well as the open- loop transfer functions. Tuning, stability analysis and simulation response of the system were achieved by using Routh, direct substitution, RootLocus and bode methods

Analysis of an age‐space structured tuberculosis model with treatment and relapse

AbstractThis paper concerns with the global threshold dynamics of a Tuberculosis (TB) model incorporating age‐space structure, treatment, and relapse. The original model is converted into a hybrid system comprising two Volterra integral equations and two partial differential equations by integrating along the characteristic line. The well‐posedness for the model is demonstrated by using the fixed point theory in conjunction with the induction method. In order to discuss whether a disease is persistent or extinct, we provide the explicit formulation of the basic reproduction number. By analyzing the distribution of the characteristic roots of the characteristic equations and constructing the proper Lyapunov functionals, the local and global stability for the steady states are addressed. Numerical simulations are conducted to confirm the conclusions of our analytical results and reveal that reduction of the TB transmission coefficient, reduction of infectiousness of treated individuals infected with TB, and increasing the treatment rate of infectious class are three feasible measures to control the transmission of TB.

МОДЕЛЮВАННЯ РОБОЧОГО ПРОЦЕСУ СТРУМИННОГО НАСОСА ПРИ ЙОГО АСИМЕТРИЧНОМУ ОБЕРТАННІ В СВЕРДЛОВИНІ

The structure of the equations of the pressure-flow characteristics of the high-pressure jet pump is supplemented with a component that determines the additional pressure caused by the influence of eddy currents during the asymmetric rotation of the ejection system around the axis of the well. On the basis of the joint solution of the Bernoulli equations, the preservation of the integrity of the moving medium, the momentum and moment of movement of the mixed flows, a relationship was obtained for the analytical determination of the additional dynamic pressure created by the rotation of an eccentrically placed jet pump in the well. In the process of researching the resulting equation, the extreme nature of the growth of the pressure-flow and energy characteristics of the overbite ejection system was established. The increase in pressure and efficiency caused by the asymmetric rotation of the over-chisel ejection system is inversely proportional to the value of the main geometric parameter of the jet pump.

Development of characteristic equations of conical solar still by incorporating N identical concentrator collectors with/without photovoltaic panel: A comparative investigation

This paper focuses on the development of characteristic equations and performance analysis of a conical solar still (CSS) connected to N identical compound parabolic concentrator (CPC) collectors, with or without photovoltaic thermal (PVT) integration. The objective is to achieve sustainable solar distillation in line with the United Nations' sustainability goals. The thermal model is developed by formulating energy balance equations for each part followed by its experimental validation considering N=0and1. A fair agreement between analytical and experimental results is observed having correlation coefficients as 0.996, 0.993, 0.992, and 0.97, 0.98, 0.99 for water temperature, yield, and glass temperature for N = 0 and 1, respectively. The study considers three cases: CSS integrated with N identical partially covered PVTCPCs (case (i)), CSS integrated with N identical fully covered PVTCPCs (case (ii)), and CSS integrated with N identical CPCs (case (iii)). The performance of these cases is compared using climate conditions in New Delhi on a typical day in June. The findings indicate that the mean value of instantaneous efficiency for case (i) is higher by 67.75 % than case (ii), however lower by 22.22 % than case (iii). The study also investigates the influence of variations in mass-flow rate and N on the output of the active CSS.

Mathematical analysis of stability and Hopf bifurcation in a delayed HIV infection model with saturated immune response

This paper explores the dynamics analysis of a human immunodeficiency virus (HIV) model with saturated cytotoxic T lymphocyte (CTL) immune response and Beddington–DeAngelis infection rate. There are two time delays in the model to describe the time needed for infection of cell and CTL immune response generation, respectively. We obtain two thresholds and three possible equilibria from the model. By analyzing the corresponding characteristic equations, we study the stabilities of equilibrium and the effect of delays on CTL immune response. The results indicate that when immune delay is present, the steady state of equilibrium is disrupted and leads to a Hopf bifurcation. Finally, we use sensitivity analyses to show the effect of parameters on thresholds and numerical simulations to illustrate the theoretical results.

Tricomi problem and integral equations

Formulas for inverting integral equations that arise when studying the Tricomi problem for the Lavrentyev–Bitsadze equation were derived. Solvability conditions of an auxiliary overdetermined problem in the elliptic part of the mixed domain were found using the Green function method. A connection was established between the Green functions of the Dirichlet problem and problem N for the Laplace equation in the form of integral equations mutually inverting each other. Various integral equations were considered, including explicitly solvable ones, to which the Tricomi problem can be reduced. An explicit solution of the characteristic singular equation with a Cauchy kernel was obtained without involving the theory of boundary value problems for analytic functions.