Nonlinear Characteristic Equation Research Articles

Insightful inspection of the nonlinear instability of an azimuthal disturbance separating two rotating magnetic liquid columns

The nonlinear stability examination of two revolving magnetized liquid columns connecting two completely submerged fluids in a permeable region is the aim of the existing paper. Two endless vertical cylinders occupied with two magnetic fluids make up the present structure. Significantly, the disturbance at the border displays an azimuthal behavior. The entire structure is activated by an azimuthal unchanging magnetic field (MF). The increasing interest in the atmospheric and oceanic dynamics is the primary motivation in exploring this problem. To relax the complication of the mathematical processes, the viscous potential theory (VPT) is established. The motion is assessed using three basic coexistent field formulations: Maxwell's formula, Brinkman's formula, and the continuity condition, in the construction of the Coriolis force and centrifugal implications. The explanations of the linearized formula of motion produce a nonlinear categorizing diffusion structure because of the implications of the nonlinear boundary conditions (BCs). The non-perturbative approach (NPA) based on the He's frequency formulation (HFF) is employed to transform the nonlinear characteristic ordinary differential equation (ODE) into a linear one. A short description of the NPA is also presented. The nonlinear ODE with real and imaginary coefficients is exposed by the stability analysis. The stability requirements are implemented using only a nonlinear analysis. As demonstrated, as an unusual state, it is exposed that ignoring the Weber number removes all complex items of the nonlinear formulation. Physically, this means the absence of the angular velocities from the physical model. For both the real and complex situations of the original equation, the stability remains unchanged. It is found that the azimuthal MF, rotating parameter, and Darcy’s numeral have a maintenance impact. On the other hand, the azimuthal wave numeral has a destabilizing one. Several polar designs are drawn to agreement the stability situations.

Transient analysis of a bulk stream queue with arbitrarily distributed arrival intervals

We study the classical infinite buffer single server queueing model with renewal input of customers in batches of random size, having arbitrarily distributed arrival intervals and exponentially distributed service times. Using the technique of supplementary variable and shift operator we derive closed form expression of the time dependent system content distribution in terms of its Laplace transform. The analysis is mainly based on the root-finding technique of the non-linear characteristic equation in terms of the Laplace transform variable. Additionally, using asymptotic properties of Laplace transform, we deduce the corresponding steady-state distribution. We discuss some special cases of the model, thus providing an alternative approach in deriving the transient distribution. We further evaluate certain performance measures and present extensive numerical examples in tabular and graphical form to illustrate the applicability of our theoretical work. The effect of system parameters, interarrival time distribution and traffic intensity on the system behavior is also demonstrated.

Effects of uniform and periodic magnetic fields at the nonlinear stability of three magnetic fluids in porous media

The current manuscript tackles the interaction between three viscous magnetic fluids placed on three layers and saturated in porous media. Two of them fill half of the spaces above and below a thin layer that lies in the middle region. All layers are laterally extended to infinity in both horizontal directions. All fluids move in the same horizontal direction with different uniform velocities and are driven by pressure gradients. The system is stressed by tangential stationary/periodic magnetic fields. The viscous potential theory is used to simplify the mathematical procedure. The motion of the fluids is described by the Brinkman–Darcy equations, and Maxwell equations are used for the magnetic field. The work has been motivated in light of the engineering applications like petroleum products manufacturing and the electromagnetic field effect that can be used to control the growth of the perturbation and then the recovery of crude oil from the pores of reservoir rocks. The nonlinear technique typically relies on solving linear equations of motion and presenting the nonlinear boundary conditions. The novelty of the problem concerns the nonlinear stability of the double interface under the impact of periodic magnetic fields. Therefore, the approach has resulted in two nonlinear characteristic differential equations governing the surface displacements. Accordingly, the development amplitudes of surface waves are designated by two nonlinear Schrödinger equations. Stability is theoretically analyzed; the nonlinear stability criteria are derived, and the corresponding nonlinear stability conditions are explored in detail. The analysis reveals the resonance as well as the non-resonance situations. Approximate bounded solutions of the perturbed interfaces are estimated. It is found that the thickness of the intermediate layer is represented as a function of time and plotted. The impact of different parameters on the stability profile is investigated. For the middle layer, it is observed that magnetic permeability and viscosity have a stabilizing effect. By contrast, basic streaming and permeability have a destabilizing influence. Due to different saturation amplitude at each case, the analysis of the periodic case shows that the lower interface is much more stable than the upper one.

Campbell diagrams, dynamics and instability zones of graphene-based spinning shafts

Carbon-based nanocomposite drive shafts have received considerable attention because of their excellent material and mechanical properties, with applications in modern aerospace, automotive industries and others. Therefore, for the first time, new Campbell diagrams, vibration and bifurcation points for multilayer shafts reinforced with three technologically justified distribution patterns of graphene nanoplatelets (GPLs) in the polymer matrix are presented and examined. A new formulation in the framework of a modified Timoshenko beam theory making use of the rotation angles of the cross sections of the shaft with five degrees of freedom is utilized. The modulus of elasticity of the nanocomposite material is estimated using the Halpin-Tsai micromechanical model, whereas the mass density and Poisson's ratio are computed by means of the law of mixtures. Hamilton's variational principle is used to derive weak and strong forms of motion equations of the shaft with classical and mixed boundary conditions. Assumptions for the derived model of the nanocomposite shaft are presented to clearly describe the mechanical state of the system. The Ritz-Chebyshev method is applied to discretize the governing equations and obtain the generalized nonlinear characteristic equations in matrix form including the gyroscopic effect and elastic stiffness caused by the rotation of the shaft. Verification of obtained results and convergence study were performed to show the correctness and advantages of the proposed solution method. The effect of different key parameters such as GPLs distribution and weight percentage, axial speed of the shaft, and boundary conditions, on its global damping, dynamics and instability zones are shown. For the first time, it is clearly shown how studied factors affect the shape of Campbell diagrams as well as the shift of natural frequency and bifurcation point of the graphene-based nanocomposite shafts.

NONLINEAR EHD STABILITY OF ENCLOSED COAXIAL JETS WITH MASS AND HEAT TRANSMISSION IN THE EXISTENCE OF AN OSCILLATORY GAS VELOCITY PROFILE

In this paper, the present work investigates a nonlinear temporal instability of coaxial cylinders in porous media in the existence of an oscillating gaseous velocity. Heat and mass transfer (MHT), as well as a uniform axial electric field were all taken into account. The viscous potential flow (VPF) was employed to calculate liquid and gas velocities to make the mathematical manipulations easier. As a result of the analysis of the boundary-values problem, the cylindrical interface displacement provided a nonlinear characteristic equation. To achieve the stability inspection, a novel approach was created. The article designates both the oscillatory and uniform streaming gas. The nonlinear analysis was completed using the homotopy perturbation method (HPM), which resulted in a Klein-Gordan equation, to arrive the stability maps. Additionally, the resonance and non-resonance cases were accomplished. In the previous situations, the stability standards were theoretically derived and numerically proven using regular diagrams. It was found that the unchanging flowing had a twofold effect. The linear MHT parameter was identified to perform a twofold character in the stability setup. On the other hand, nonlinear parameters have opposing effects. The outcomes of the homogeneous gas velocities were substantially conforming.

Modeling the Submergence Depth of Oil Well States and Its Applications

Obtaining the liquid storage state of oil wells in real time is very important for oilfield production. In this paper, under the premise of fully considering the transformation factors of full-pumping and nonfull-pumping states of oil wells, submergence depth models suitable for full- and nonfull-pumping wells are constructed. To reduce the application complexity of the models, parameter-reduction processing is performed to enhance the usability of the models. By analyzing the change trend of the submergence depth during the rising, maintaining, and falling of the oil well in the full-pumping state and nonfull-pumping state models, the judgment criteria for the transition of the oil well state are provided. On this basis, the application methods of nonlinear interpolation and least squares curve-fitting numerical solutions of submergence depth models are studied, and the unique existence of the solution of the corresponding one-variable nonlinear characteristic equation in the (0, 1) open interval is proven. Finally, the error estimation of the numerical solution is carried out, the calculation formula of the number of iterations for the numerical solution of the dichotomy is provided, and the error of the relevant numerical solution is verified.

Synergetic effect of absorber and condenser nano-coating on evaporation and thermal performance of solar distillation unit for clean water production

Two design modifications to increase the evaporation/condensation and freshwater yield of conventional solar stills (CSS) are presented – absorber coating with black paint (BP) in which 10 wt % reduced graphene oxide (rGO) was dispersed, and use of a nano-silicon-coated glass cover alongside the rGO coated absorber. Water contact angle analysis revealed that the hydrophilic glass/condenser hindered water droplet movement, whereas the hydrophobic silicon-coated glass enabled significantly faster water droplet movement. The rGO-coated absorber SS, with 99.2% solar absorptivity, improved the average water temperature by 11% leading to an augmented full-day freshwater yield of 3031 mL (versus 2160 mL for the CSS). The synergistic effect of the nano-silicon-coated glass and rGO-coated absorber further enhanced the full-day freshwater yield to 3410 mL. Energy/exergy analysis demonstrated enhanced efficiencies of the combined condenser and absorber coated SS of 37% (energy) and 112% (exergy), compared to the CSS. Non-linear characteristic equations described the instantaneous gain in energy efficiency of the combined absorber and condenser coated SS. Absorber/basin exhibited the highest rate of irreversibility in all three SS. The cost of water was 0.01 $/L for the dual-modified still. Water physicochemical quality analysis before and after desalination revealed that the purified water was suitable for drinking. • A solar still was fabricated with absorber and condenser coating for improved yield. • RGO coating improved the absorber average temperature by 10.9% compared to CSS. • Glass coated with nano-silicon altered condensation behavior to drop-wise from film-wise. • Modified SS showed augmentation in water yield compared to conventional SS. • Brackish water was converted into distilled water this is suitable for drinking.

Linear and nonlinear free vibration analysis of functionally graded porous nanobeam using stress-driven nonlocal integral model

Linear and nonlinear free vibration analysis of functionally graded porous (FGP) nanobeam with four different porous distribution patterns is performed on the basis of stress-driven two-phase local/nonlocal integral model. The strain is defined according to von Karman nonlinearity, and the differential governing equations as well as standard boundary conditions are derived through the Hamilton’s principle. The integral relation between nonlinear strain and nonlocal stress is transformed equivalently into differential form with constitutive boundary conditions. Neglecting the nonlinear terms, the bending deflection and moment are derived and expressed explicitly through the Laplace transformation with six unknown constants for linear free vibration, and the nonlinear characteristic equation for vibration frequency and linear vibration mode shape can be calculated through standard and constitutive boundary conditions. Two analytic methods including Ritz–Galerkin and perturbation methods based on both local and nonlocal linear vibration mode shape are applied to study the linear and nonlinear free vibration of nonlocal FGP nanobeams under different boundary conditions Meanwhile, direct numerical method based on general differential quadrature method and Newton’s iterative process are utilized to obtain the numerical solutions of nonlinear vibration frequencies. The influence of nonlocal parameters and vibration amplitude on the free vibration frequencies is investigated numerically. The numerical results show that influence of nonlocal parameters on linear vibration mode shape is not significant. Local linear vibration mode shape based Ritz–Galerkin​ and perturbation methods would overestimate and underestimate the linear free vibration of nonlocal simply-supported and clamped–clamped nanobeams. Nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods could provide more accurate prediction on nonlinear vibration frequencies than local mode shape based methods when W(0.5) is small. However, the relative error increases quickly with the increase of W(0.5). Based on the results obtained in this study, local and nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods cannot provide accurate prediction on nonlinear vibration frequencies when the vibration magnitude is big, since linear vibration mode shape assumption is not suitable for strong nonlinear vibration.

Nonlinear EHD Instability of Two-Superposed Walters’ B Fluids Moving through Porous Media

The current work examines the application of the viscous potential flow to the Kelvin-Helmholtz instability (KHI) of a planar interface between two visco-elastic Walters’ B fluids. The fluids are fully saturated in porous media in the presence of heat and mass transfer across the interface. Additionally, the structure is pervaded via a uniform, normal electrical field in the absence of superficial charges. The nonlinear scheme basically depends on analyzing the linear principal equation of motion, and then applying the appropriate nonlinear boundary-conditions. The current organization creates a nonlinear characteristic equation describing the amplitude performance of the surface waves. The classical Routh–Hrutwitz theory is employed to judge the linear stability criteria. Once more, the implication of the multiple time scale with the aid of Taylor theory yields a Ginzburg–Landau equation, which controls the nonlinear stability criteria. Furthermore, the Poincaré–Lindstedt technique is implemented to achieve an analytic estimated bounded solution for the surface deflection. Many special cases draw upon appropriate data selections. Finally, all theoretical findings are numerically confirmed in such a way that ensures the effectiveness of various physical parameters.

Elastic buckling and free vibration of nonlocal strain gradient Euler‐Bernoulli beams using Laplace transform

AbstractPrevious studies have shown that Lim's nonlocal strain gradient differential model would lead to some inconsistencies for the static bending of Bernoulli‐Euler and Timoshenko beams, though it is widely applied to address both softening and toughening size‐effect. In this work, nonlocal strain gradient integral model is applied to study the elastic buckling and free vibration of Bernoulli‐Euler beam under different boundary conditions. The differential governing equation and corresponding boundary conditions are derived via the Hamilton's principle, and the relation between strain and nonlocal stress is expressed in integral form. The Laplace transform technique is applied to solve the integro‐differential equations directly, and the explicit expression for bending deflections and moments is obtained with six unknown constants. The nonlinear characteristic equations to determine the characteristic buckling load and vibration frequency are derived explicitly in consideration of corresponding boundary conditions. The buckling load and vibration frequency from present model are validated against to the existed results in literature. Consistent softening and toughening responses respect to length‐scale parameters can be obtained for both elastic buckling and free vibration.

Nonlinear EHD stability of cylindrical walters B' fluids: effect of an axial time periodic electric field

The current article is concerned with a nonlinear stability analysis of cylindrical Walters B' fluids. The system is pervaded by an axial time periodic electric field. A cylindrical interface is supposed to be disconnected from two dielectric fluids. The fluids are fully saturated in porous media. The motivation to scrutinize this area is attributed to the great attention it receives in many practical situations in physics and engineering applications. The implementation of the appropriate nonlinear boundary conditions of the linearized equations of motion yields a nonlinear characteristic dispersion equation. This equation manages the surface deflection of the surface waves. The use of the non-dimensional analysis resulted in various well-known non-dimensional numbers. A new approach to the characteristic equation is inspected by employing the Homotopy perturbation method (HPM). This new methodology resulted in a Klein-Gordon equation. Utilizing a travelling-wave solution to the linear part of the characteristic equation, a new restriction of the stability analysis appears. The investigation reveals the non-resonance as well as the resonance cases, and the stability criteria are established in both arguments. A set of diagrams is plotted to display the influence of various non-dimensional physical numbers on the stability profile and shows interesting features.

Nonlinear instability analysis of a vertical cylindrical magnetic sheet

This paper concerns with the nonlinear instability analysis of double interfaces separated three perfect, incompressible cylindrical magnetic fluids. The cylindrical sheet is acted upon by an axial uniform magnetic field. The current nonlinear approach depends mainly on solving the linear governing equations of motion and subjected to the appropriate nonlinear boundary conditions. This procedure resulted in two nonlinear characteristic equations governed the behavior of the interfaces deflection. By means of the Taylor expansion, together with the multiple time scales, technique, the stability analysis of linear as well as the nonlinear is achieved. The linear stability analysis reveals a quadratic dispersion equation in the square of growth rate frequency of the surface wave. On the other hand, the nonlinear analysis is accomplished by a coupled nonlinear Schrodinger equation of the evolution amplitudes of the surface waves. The stability criteria resulted in a polynomial of the eleventh degree in the square of the magnetic field strength, together with resonance transition curves. Several special cases are reported upon appropriate data choices. The stability criteria are numerically discussed, at which regions of stability and instability are identified. In the stability profile, the magnetic field intensity is plotted versus the wave number. The influences of the parameters on the stability are addressed. The nonlinear stability approach divides the phase plane into several parts of stability/instability. The nonlinear stability shows an in contrast mechanism of the role of the sheet thickness.

Bending and Buckling Analysis of Functionally Graded Euler–Bernoulli Beam Using Stress-Driven Nonlocal Integral Model with Bi-Helmholtz Kernel

Static bending and elastic buckling of Euler–Bernoulli beam made of functionally graded (FG) materials along thickness direction is studied theoretically using stress-driven integral model with bi-Helmholtz kernel, where the relation between nonlocal stress and strain is expressed as Fredholm type integral equation of the first kind. The differential governing equation and corresponding boundary conditions are derived with the principle of minimum potential energy. Several nominal variables are introduced to simplify differential governing equation, integral constitutive equation and boundary conditions. Laplace transform technique is applied directly to solve integro-differential equations, and the nominal bending deflection and moment are expressed with six unknown constants. The explicit expression for nominal deflection for static bending and nonlinear characteristic equation for the bucking load can be determined with two constitutive constraints and four boundary conditions. The results from this study are validated with those from the existing literature when two nonlocal parameters have same value. The influence of nonlocal parameters on the bending deflection and buckling loads for Euler–Bernoulli beam is investigated numerically. A consistent toughening effect is obtained for stress-driven nonlocal integral model with bi-Helmholtz kernel.

Elastic buckling and free vibration analysis of functionally graded Timoshenko beam with nonlocal strain gradient integral model

In this work, nonlocal strain gradient integral model is applied to study the elastic buckling and free vibration response of functionally graded (FG) Timoshenko beam which is made of two constituents varying along thickness direction. The differential governing equations and corresponding boundary conditions are derived via the Hamilton's principle, and the relations between nonlocal stresses and strains are expressed as integral equations. The Laplace transform technique is applied to solve directly the integro-differential equations, and the explicit expression for bending deflections and moments, as well as cross-sectional rotation and shear force is expressed with eight unknown constants. The nonlinear characteristic equations to determine the characteristic buckling load and vibration frequency are derived explicitly in consideration of corresponding boundary conditions and constitutive constraints. The buckling load and vibration frequency from present model are validated against to the existed results in literature. Consistent softening and toughening responses with respect to length-scale parameters can be obtained.

Nonlinear hydromagnetic instability of oscillatory rotating rigid-fluid columns

The current paper is concerned with the nonlinear stability analysis of rotating magnetic fluid columns. The rotation sources are a mixture of both uniform and oscillating behavior. The motivation behind tackling this topic is the increasing interest in atmospheric and oceanic motions. The system consists of two magnetic phase fluid that fills two infinite vertical cylinders. An azimuthal uniform magnetic field is penetrated on the system. The governing equations of motion, in terms of the Coriolis force and reduced pressure, along with Maxwell’s equation in the quasi-static approximations are considered. Consequently, the disturbance of the interface has an azimuthal behavior. The fluids are fully saturated in porous media. In light of the implication of the nonlinear boundary conditions, the solutions of the linearized equations of motion resulted in a nonlinear characteristic dispersion equation. Utilizing the homotopy perturbation technique, this equation is analyzed. A modification of the latter equation is made to seem like a nonlinear Klein–Gordon equation. The stability criteria are realized in linear as well as nonlinear approaches. A set of diagrams is graphed to illustrate the effects of several non-dimensional numbers on the stability profile in resonance as well as non-resonance cases.

The Lamb problem with a nonuniform string

The time evolution of an oscillator coupled to an infinite string with a discontinuous mass density is investigated. It is shown that the equation of motion of the oscillator leads to a nonlinear characteristic equation due to the frequency-dependent nature of the point impedance of the string. A graphical method is applied to discuss the physical and unphysical roots associated with the characteristic equation. Then, the relevant eigen-frequencies (physical roots) are evaluated numerically to explain the motion of oscillator. It is found that the oscillator executes a damped oscillatory motion. However, due to the loss of homogeneity the real and imaginary parts of the eigen-frequencies simultaneously depend on the distance of oscillator from the discontinuity point and the difference between the densities of the semi-finite strings.

Surface Wave Attenuation in Multilayer Structures With Lossy Media and Impedance Surfaces

A methodology for analyzing the attenuation constant of surface wave propagating in general multilayer structures is proposed. The multilayer structure may include any number of layers, that can be made up of lossy dielectric, magnetic-material, and impedance surfaces. The non-linear characteristic equation (transverse resonance equation - TRE) of the multilayer structure is defined by utilizing transmission line (TL) theory. Consequently, the root(s) of the TRE is/are computed which indicates the complex propagation constant of that characteristic equation. The obtained results are validated over the completely shielded waveguide structures where the full wave simulation is possible. The proposed approach is employed to analyze and discuss the propagation and attenuation constant of surface wave modes in several representative examples. The analysis is carried out in the frequency range from 1 to 12 GHz.

Optimal robust fault detection of discrete‐time LPV systems with measurement error‐affected scheduling variables combining ZKF and pQP

SummaryOptimal robust state estimation (SE) and fault detection (FD) methods of discrete‐time linear parameter varying systems with measurement error‐affected scheduling variables are proposed under the boundedness assumption of system uncertainties. By using the weighted Frobenius norm of the generator matrix of SE zonotope to characterize the set size, the optimal observer gain can be computed by a zonotopic Kalman filter (ZKF) procedure for the purpose of observation. Meanwhile, by minimizing the influence of system uncertainties while maximizing that of faults on SE to enhance the sensitivity of FD, the optimal FD criterion is characterized based on an on‐line fractional programming problem, which can be equivalently transformed into a parametric quadratic programming (pQP) problem. The pQP problem can be efficiently solved by searching the root of its nonlinear characteristic equation using secant method. In general, as long as sensors with sufficiently high precision are equipped to measure the scheduling variables, the bounds of measurement errors of scheduling variables can be less conservative than those direct bounds of scheduling variables, which can reduce the conservatism of FD or SE in this way. At the end of this article, a case study based on a practical circuit model is used to illustrate the effectiveness of the proposed method.

One-dimensional stress-driven nonlocal integral model with bi-Helmholtz kernel: Close form solution and consistent size effect

In this paper, stress-driven nonlocal integral model with bi-Helmholtz kernel is applied to investigate the elastostatic tensile and free vibration analysis of microbar. The relation between nonlocal stress and strain is expressed as first type of Fredholm integral equation which is transformed to first type of Volterra integral equation. The general solution to the axial displacement of nonlocal microbar is obtained through the Laplace transformation with four unknown constants. Taking advantage of boundary and constitutive constraint equations, one can obtain the exact tensile displacements of microbar under different boundary and loading conditions, and the nonlinear characteristic equations about vibration frequency of clamped-free and clamped-clamped nonlocal microbars. Numerical results show that the nonlocal microbar model can be degraded to local bar model when the nonlocal parameters approach to 0, and a consistent toughening response for elastostatic tension and free vibration can be obtained for different boundary and loading conditions.

Nonlinear stability of electro-visco-elastic Walters’ B type in porous media

This paper investigates the nonlinear instability of a non-Newtonian fluid of the Walters’ B type. The fluids fill the regions inside and outside a vertical circular cylinder. An axial electric field of uniform strength is pervaded along the axis of the jet. The fluids are saturated in porous media. Typically, the nonlinear analysis is based on solving the linear governing equations of motion, then applying the convenient nonlinear boundary conditions. This methodology yields a nonlinear characteristic equation which governs the behavior of the interface deflection. As the nonlinear terms are omitted, a linear dispersion relation arises. Therefore, the stability criteria are analytically analyzed and numerically confirmed. The nonlinear approach depends on the multiple time scale technique together with the support of the Taylor theory. This approach resulted in a Ginzburg–Landau equation. Consequently, the stability criteria are achieved in both analytical and numerical analysis. Furthermore, by means of the expanded frequency analysis, a bounded approximate solution of the amplitude of the surface waves is accomplished. The homotopy perturbation method (MPM) is utilized to obtain an approximate distribution of the conducted artificial frequency. Additionally, the generating function of the interface is graphically represented. Several special cases are reported upon convenient data choices. Regions of stability and instability are addressed. In the stability profile, the electric field intensity is plotted versus the wave number. The influences of the parameters on the stability are identified. The nonlinear stability approach divides the phase plane into several parts of stability/instability.