Linear Perturbation Method Research Articles

Cosmographic analysis of the dynamics of universe in higher dimensional compactified space with Tachyonic field

In this work, the considered model is contained in higher dimensional compactified space in the form of M4 × M6, where M6 is compact inner space. Since, for a higher dimensional cosmological model, Tachyon matter is an automatic and fantastic choice, we have supposed that the scalar field is driven by Tachyon together with the higher dimensional cosmological constant [Formula: see text]. Our observation is that the theory provides a good solution for the cosmological scale factor of the present universe. Here, it is important that the forms of the tachyonic field ϕ and potential V are simple trigonometric hyperbolic functions. We study the cosmographical parameters like equation of state parameter, statefinder parameters, jerk, snap, and lerk parameters. The graphical representations of cosmological parameters show the overall evolution of the universe under a tachyonic field in a higher dimensional model. The classical stability of the model due to small perturbations both for dark energy and metric are examined by the study of the square speed of sound and by the linear perturbation method. We also analyzed the model in ω−ω′ and r− s cosmological plane.

Osmotically driven beading instability in axons: Continuum theory, perturbation analysis and finite element implementation

Axonal beading or formation of multiple beads along an axon is characteristic of many brain pathological states like Alzheimer's, Parkinson's and traumatic injuries. Despite the many existing experimental studies, the underlying mechanisms of this shape instability remain still poorly understood. In this paper, we establish a combined theoretical and numerical framework to study the governing key factors of this morphological transformation. We develop a three-dimensional (3D) non-equilibrium large deformation thermodynamic model with two main parts: the central axoplasm which is considered as a polyelectrolyte hydrogel and the encapsulating cortical membrane which is modeled as an incompressible hyperelastic layer with surface energy and growing surface. The model constitutive and evolution equations are then extracted employing thermodynamic balance principles for both bulk and surface material points. It is shown that the second law of thermodynamics indicates that the axolemma growth rate is proportional to the membrane tension which is in perfect agreement with the available experimental findings. While the developed model is general and can be extended to cover other types of axonal beadings, for the sake of simplicity, here, we focus on osmotically driven axisymmetric beadings which are compressible viscoelastic periodic modulations. We solve the corresponding governing equations using the linear perturbation method. This perturbation analysis proves that: 1) the beading instability is a rate dependent phenomenon that is controlled by the axolemma growth, 2) the initially dominant beading waves (the fastest waves) might be replaced only by longer waves which are more stable and 3) the wavelength of the fastest beads should vary roughly linearly with the axonal radius. These main findings are all in good agreement with the existing experimental results. Finally, the finite element implementation of the model is also presented to verify the results of the linear stability analysis for slow waves. The obtained axisymmetric finite element results are in good agreement with the corresponding theoretical findings.

Comparative study on static and dynamic characteristics of pumping-out and pumping-in herringbone grooved journal bearings

The aim of this paper is to compare the static and dynamic performances of two types of herringbone grooved journal bearings (HGJBs), that is, pumping-out HGJB and pumping-in HGJB. Firstly, based on finite control volume method and the principle of flow balance, an equivalent Reynolds equation considering the turbulence effect is derived. Then, the journal motion considering angular displacement is analyzed with linear perturbation method. The equivalent Reynolds equation is linearized into static-form one and perturbed-form ones. These equations are solved by finite difference method and their solutions are integrated on the bearing surface to determinate the static characteristics and dynamic coefficients of HGJB. Additionally, the critical mass and critical whirl frequency are defined for evaluating the stability and rigidity of bearing by solving a motion equation with four degree of freedoms (DOFs). Finally, the static and dynamic characteristics of two types of HGJBS are analyzed and compared symmetrically in consideration of effects of eccentricity ratio, length to diameter ratio, groove angle, groove depth, the width of sealing damp, tilting degree of journal, and rotation speed. The results show that, within the interested range of investigated parameters, static performances of pumping-out HGJB are superior to that of pumping-in HGJB generally. Stability of pumping-in HGJB is better than pumping-out HGJB but bearing rigidity of pumping-in HGJB is inferior to pumping-out HGJB.

A mathematical analysis of anti-plane surface wave in a magneto-electro-elastic layered structure with non-perfect and locally perturbed interface

In the present work, an analytical method has been adapted in order to study dispersive behaviour of anti-plane surface wave, in particular, Love wave in a homogeneous fully-coupled magneto-electro-elastic layer/half-space configuration. The interface is considered mechanically, electrically and magnetically non-perfect as well as slightly non-parallel where a geometrically perturbed region of finite span is considered to be small compared to the wavelength of observed phases. Closed-form expressions of dispersion equations for rectangular-shaped irregularity and non-perfect interface existing in magneto-electro-elastic layered structure have been derived analytically for four different types of electromagnetic boundary conditions. Eringen’s linear perturbation method and Fourier transformation have been used and the obtained results have been validated with the solutions for simpler layered structures. Two laminated composites BaTiO3/CoFe2O4 of different volume fractions have been considered in order to plot the dispersion curves in distinct cases. The effect of irregularity and mechanical, electrical and magnetic non-perfect bonding on phase velocity and magneto-electromechanical coupling has been studied. The present study holds potential applications in multidisciplinary fields, such as designing acoustic devices, magnetic field sensors and actuators, in particular, Love-wave SAW magnetic-field sensors as well as in seismology when seismic signals propagate through Earth medium consisting of magnetic materials.

Robust optimization of MTMD systems for the control of vibrations

Multiple Tuned Mass Dampers (MTMD) is one of the simplest and most reliable solution to control the vibrations of a structure, e.g. allowing to deal with a wide distribution of structural natural frequencies and usually showing inherent stability. Nevertheless, uncertainties in the behavior of structures under random dynamic input may induce implications on the MTMD effectiveness, e.g. detuning or amplifications. Typically, uncertainties in the input are exclusively considered, while uncertainties affect as much also the mechanical parameters.This paper proposes a robust optimum design of the MTMD, considering uncertainties both in the structural parameters and in the earthquake input. At this aim, a random vibration analysis of the response is adopted, and a direct linear perturbation method is applied on the uncertain parameters. Results show that a deterministic optimization is inappropriate for sensible systems as TMDs, while a robust optimization aims to better control the response with low computational efforts. Furthermore, the selection of the objective functions significantly affects the optimal design parameters, as the number of TMDs influences the robust designs.

Complex dynamical properties of coupled Van der Pol–Duffing oscillators with balanced loss and gain

We consider a Hamiltonian system of coupled Van der Pol–Duffing (VdPD) oscillators with balanced loss and gain. The system is analyzed perturbatively by using renormalization group (RG) techniques as well as multiple scale analysis (MSA). Both the methods produce identical results in the leading order of the perturbation. The RG flow equation is exactly solvable and the slow variation of amplitudes and phases in time can be computed analytically. The system is analyzed numerically and shown to admit periodic solutions in regions of parameter-space, confirming the results of the linear stability analysis and perturbation methods. The complex dynamical behavior of the system is studied in detail by using time-series, Poincaré-sections, power-spectra, auto-correlation function and bifurcation diagrams. The Lyapunov exponents are computed numerically. The numerical analysis reveals chaotic behaviour in the system beyond a critical value of the parameter that couples the two VdPD oscillators through linear coupling, thereby providing yet another example of Hamiltonian chaos in a system with balanced loss and gain. Further, we modify the nonlinear terms of the model to make it a non-Hamiltonian system of coupled VdPD oscillators with balanced loss and gain. The non-Hamiltonian system is analyzed perturbatively as well as numerically and shown to posses regular periodic as well as chaotic solutions. It is seen that the -symmetry is not an essential requirement for the existence of regular periodic solutions in both the Hamiltonian as well as non-Hamiltonian systems.

Numerical analysis on the performance characteristics of a new gas journal bearing by using finite difference method

This paper proposes a novel gas journal bearing in which orifices are different in diameter and distribute unevenly. Finite Difference Method (FDM) combined with Linear Perturbation Method (LPM) is used to solve the unsteady-state Reynolds equation of the flow field in the bearing clearance. Moreover, four types of bearing structures are used to discuss the effects of orifices different in diameter and uneven distribution on the bearing performance. The results demonstrate that the new bearing has better static and dynamic performances compared with those of traditional bearing in which orifices are equal in diameter and distribute evenly. Moreover, thin gas film thickness, high supply pressure, and large eccentricity ratio are hopeful for improving load capacity of the new bearing. Furthermore, the stability of the novel bearing is improved if eccentricity ratio is 0.25–0.3.

Adaptive phase field method using novel physics based refinement criteria

Amongst the available methods to model fracture processes, the phase field approach has proved to be efficient and has received extensive attention. However, the approach is computationally demanding as it requires a very high resolution, both in space and time to resolve the fracture characteristics. In this paper, a novel adaptive phase field method is proposed for modelling quasi-static crack propagation in brittle materials. The adaptive refinement is based on the stability analysis which is combined with the quadtree decomposition. The stability analysis is based on a linear perturbation method that is used herein to determine the onset of fracture initiation on the fly. Three standard benchmark problems are solved to show the efficiency and the robustness of the proposed method. It is shown that the proposed framework does not require any post processing technique for adaptive refinement and yet is effective.

Head-on Collisional Effects of Ion Acoustic Waves in Magnetized Plasma: Linear and Nonlinear Analyses

Head-on collisional phenomena of ion acoustic (IA) waves in electron–positron–ion magnetized plasmas are investigated through linear and nonlinear analyses. The plasma system is considered collisionless and comprising cold ions, kappa distributed cold electrons, and positrons. Dispersion relation is derived with the aid of the linear perturbation method. The two sided nonlinear Korteweg-de Vries (KdV) equations are derived employing extended Poincare-Lighthill-Kuo (ePLK) method. The effects of wave frequency on linear group velocity with species density ratio are studied. Further, the production of ion acoustic solitons in terms of ion gyrofrequency ( $${\omega }_{\mathrm{g}}$$ ), superthermality parameters for electrons ( $${\kappa }_{\mathrm{e}}$$ ), and positrons ( $${\kappa }_{\mathrm{p}}$$ ) is investigated. Additionally, the effects of obliqueness (angle between magnetic field and wave propagation vector) on nonlinearity coefficient and phase shifts with density ratios (positron to electron and ion to electron) are demonstrated.

Flow stability in a wide vaneless diffuser

This work is concerned with the theoretical aspects of flow stability in a two dimensional vaneless diffuser. Specifically, the appearance of self-excited oscillations, also referred to as rotating stall, is investigated considering a two-dimensional inviscid flow in an annulus. We consider a linear perturbation method, taking as basic flow the steady potential velocity field whose radial and tangential components are inversely proportional to the radial coordinate. We show that such flow may become unstable to small two-dimensional perturbations provided that the ratio between the inlet tangential velocity and the radial one is sufficiently large and a certain amount of vorticity is injected in the flow field. Such an instability is purely kinematical, i.e. it does not involve any boundary layer effects, contrary to the classical hypothesis which ascribes the instability to a peculiar boundary layers interaction.

Fractional quantum couplers

Fractional quantum coupler, a new type of quantum couplers that is composed of arrays of two coupled waveguides or a dual-core waveguide with intermodal coupling, within which the light waves diffraction is of the fractional-order differentiation, is put forward in the territory of fractional quantum mechanics. The modelling equations of such fractional couplers are derived in the framework of coupled nonlinear fractional Schrödinger equations with the space derivative of fractional order denoted by Lévy index α, and localized wave solutions as spatial optical solitons of these equations are constructed and their nonlinear propagation properties are discussed. Linear perturbation method based on linear stability analysis, and direct simulations are conducted to identify the stability and instability regions of the predicted solitons.

Study on the integration effect of multiple vehicles’ delayed velocities on traffic stability in intelligent transportation system environment

PurposeIn an intelligent transportation system (for short, ITS) environment, a vehicle’s motion is affected by the information in a large scale. The purpose of this paper is to study the integration effect of multiple vehicles’ delayed velocities on traffic flow.Design/methodology/approachThis paper constructed a new car-following model to study the integration effect of multiple vehicles’ delayed velocities on traffic flow. The new model is analyzed by linear and nonlinear perturbation method theoretically and also verified by simulation.FindingsIt is found out that the integration of preceding vehicles’ delayed velocities affect the stability of traffic flow importantly, and three preceding vehicles’ delayed velocities information should be considered in real traffic.Originality/valueThe new car-following model by considering the integration effect of multiple vehicles’ delayed velocities is firstly proposed in this paper. The research result shows that three preceding vehicles’ delayed velocities information is the best choice to stabilizing traffic flow.

Dynamical system analysis of Randall-Sundrum model with tachyon field on the brane

In this manuscript we use the dynamical system approach to study the linear dynamics of a Randall-Sundrum braneworld model with a tachyon scalar field confined to the brane. We recognize that the form of the tachyon potential plays a significant role in the evolution of the universe. For the case of an inverse square potential we find that one of our new variables, $\lambda$, is constant. We obtain critical points of the system in this situation and investigate their stability using the linear perturbation method. Then we turn to a Gaussian potential in which $\lambda$ is not constant. Using the idea of instantaneous critical points we study the behavior of the universe and its possible fates. One of the interesting results of this manuscript is that our universe will probably experience another phase transition from acceleration to deceleration in the future.

Uncoupled Modeling of the Effects of Coating Layer on the Growth Instability in Pure Metal Solidification

The mold coating is one of the factors to control the heat transfer rate and development of microstructure in metal solidification. In this study, effects of the coating layer on growth instability in solidification of pure metals are theoretically examined. Solidification process is modeled in two phases: In the first phase, the heat conduction with phase change between the molten metal and the solidified shell, and the heat transfer problems between the solidified shell and the coating layer and the coating layer and the mold need to be established. In the second phase, mechanical problem which includes deformations in the layers due to stress distributions should be modeled. Presented model extends previous studies by examining the effects of coating layer in the early stages of solidification under the assumption that the thermal and mechanical problems are uncoupled. In this model, the thermal problem affects the thermal stress distribution in the layers, but the mechanical problem does not affect the thermal problem. This assumption is valid only for the early stages of solidification since the coupling between the thermal and mechanical problems plays an important role in further stages of the process. The thermal capacitance of the solidified shell, the coating layer and the mold is assumed to be zero in order to have a predominantly analytical solution. These assumptions clearly place severe restrictions on the accuracy of the resulting predictions, but they can be justified on the grounds that the resulting analysis retains some generality and hence permits deductions to be made about the effect of changes of material properties and operating conditions on the stability of the system. The heat transfer part of the problem is solved analytically using a linear perturbation method. However, the mechanical part of the problem is solved numerically using a variable step variable order corrector and predictor algorithm which is suitable for stiff problems. Effects of process parameters such as thermal conductivities, coating thickness and thermal contact resistances at the surface between each layer on the growth of solidified shell thickness are investigated in detail.

Comparative Study on Dynamic Characteristics of Double-Pad Inwardly and Outwardly Pumping Water-Lubricated Spiral-Groove Thrust Bearings

AbstractThe dynamic characteristics of the double-pad inwardly pumping water-lubricated spiral-groove thrust bearing (SGTB) is different from that of the outwardly pumping version, especially under a high-speed condition. However, no literature on this topic can be found up to now. In this study, a comparative study on the dynamic characteristics of the two types of SGTBs is conducted. First, a turbulent lubrication model considering centrifugal and cavitating effects for the double-pad SGTB is developed based on gas–liquid two-phase flow. The steady-state and perturbed Reynolds equations of liquid phase are derived using linear perturbation method. Subsequently, the boundary fitted coordinate system and control volume method are applied to solve the Reynolds equations. Then, the axial stiffness coefficients of high-speed water-lubricated bearings are verified by an experimental study. Finally, the influence of operating conditions and configuration parameters on the dynamic behaviors for the two types of SGTBs is investigated. The results show that the double-pad outwardly pumping bearing is superior to the inwardly pumping version in all the dynamic coefficients when the radius ratio exceeds 0.65. When the tilt angle exceeds 1.2 × 10−4 as well as when the eccentricity ratio exceeds 0.7, the dynamic damping, angular and cross-coupled stiffness coefficients of the outwardly pumping bearing may be larger than those of the inwardly pumping one.

Analytical Framework for Precise Relative Motion in Low Earth Orbits

This work presents a practical and efficient analytical framework for the precise modeling of the relative motion in low Earth orbits. Developed to support the design and verification of relative guidance navigation and control algorithms devoted to spacecraft rendezvous for active debris removal applications, only the orbital perturbation due to nonspherically symmetric mass distribution is considered. The relative motion is modeled in mean relative orbital elements, revisiting the available formulations to include the first-order expansion of the effects due to any even zonal harmonics and the second-order expansion of the unperturbed and terms. Mean/osculating orbital element conversions are obtained by merging a second-order Hamiltonian approach applied to the problem with the Kaula linear perturbation method for the remaining terms of the geopotential. The paper describes the main building blocks of the framework as well as their interfaces because the key aspect to achieve precision is to set up a fully consistent environment. The results show the achievable accuracy under realistic operational conditions for possible guidance and navigation applications.

Numerical analyses of seismic performance of underground and aboveground structures with friction pendulum bearings

The seismic design concept of base isolation is widely used in the aboveground structures, and the typical isolation device is friction pendulum bearing (FPB). However, there are significant differences in the seismic response characteristics between the underground structures and aboveground structures. Whether the base isolation is suitable for the underground structures remains to be further discussed. In order to shed light on the mechanism of FPB applied in the underground structures, two different seismic mitigation schemes are compared in this paper. One is the structure with FPB under the bottom of the structure and the other is the structure with FPB on the top of central column. Firstly, the FPB is equivalent to two linear springs in the horizontal and vertical directions. The influences on the natural vibration characteristic of the structures and soil-structure systems are discussed based on the linear perturbation method from the perspective of modal analysis. Secondly, the influences on the dynamic responses of aboveground and underground structures are compared based on the dynamic time history analysis of the 2D numerical model of soil-structure system. The results show that the FPB under the bottom of the aboveground structure can extend the natural vibration period of the structure and reduce the seismic response, while the mechanism is not suitable for the underground structure. The FPB under the bottom of the underground structure can only reduce the transmission of the soil shear effect at the bottom of the structure, and there is no obvious change in the seismic responses of the new structure. However, the shear force, bending moment and horizontal deformation of the key vertical support component are greatly reduced in the underground structure with FPB on the top of the central column, which can be considered as an effective seismic mitigation measure for the underground structure to avoid the shear failure of the central column under high axial compression.

Size Effect on Onset and Subsequent Evolution of Adiabatic Shear Band: Theoretical and Numerical Analysis

The adiabatic shear instability of ductile materials has attracted more and more attentions of researchers and groups, who have been sparing no effort in further understanding of the underlying mechanism since the first experimental depiction of adiabatic shear instability by Zener and Hollomon. As for the adiabatic shear instability, many factors account for its occurrence, including heat conduction, inertia effect, microstructure effect and so on. However, lots of experimental evidence has shown that metal materials display a strong size effect when the characteristic length scale is in the order of microns. The size effect has also been observed in the analysis of shear band in the ductile materials because the order of the bandwidth stays within the microscale range. However, a comprehensive understanding of the whole process of adiabatic shear banding (ASB), including the early onset and the subsequent evolution, is still lacking. In this work, a gradient plasticity model based on the Taylor-based nonlocal theory feasible for the linear perturbation analysis and convenient for numerical calculation is proposed to investigate the strain gradient on the onset of ASB and the coupling effect of heat conduction, inertia effect and strain gradient at the early stage, as well as on the subsequent evolution process at later stages. As for the onset of ASB, the linear perturbation method is used to consider the effect on the initial formation of ASB. After the investigation of the onset of ASB, the characteristic line method is applied to describe the subsequent nonlinear evolution process of ASB. Three stages of ASB evolution are clearly depicted during the evolution process, and the significance of size effect on the ASB nonlinear evolution process of ASB at different stages is analyzed. With the help of linear perturbation analysis and characteristic line method, a comprehensive description of the role of strain gradient in the ASB from the early onset to the end of the evolution is provided.

Dynamic analysis of a high speed motorized spindle for internal grinding of slender holes

The purpose of this paper is to investigate the dynamic characteristics of a high speed motorized spindle for internal grinding of slender holes. The modified transfer matrix method is used to establish the dynamic model of the motorized spindle rotor system including the additional stiffness of the non-contact rotary union. The linear perturbation method is adopted to derive the perturbation equations for the flow passing through the gap in the non-contact rotary union. The finite difference method is used to solve the Reynolds equation and the perturbation equations to obtain the four dimensional stiffness matrix of the non-contact rotary union. Finally, the effect of the non-contact rotary union on the dynamic behaviours of the motorized spindle are studied.

Analysis of static and dynamic characteristics of spiral-grooved gas journal bearings in high speed

The spiral grooves structures could promote load capacity and improve stability of the gas journal bearings working in high-speed condition. In this study, the unsteady Reynolds equation is solved by linear perturbation method and finite difference method in which the mesh of the groove region is specially treated. The static and dynamic characteristics of spiral grooves journal gas bearings are investigated in different working conditions and the pumping effect caused by spiral-groove structure is revealed and analyzed. Further, the influences of groove structural parameters on the dynamic stiffness and damping coefficients are studied and discussed, which provides guidelines for the design of the journal gas bearings with spiral grooves.