Averaged Equations Research Articles

Darcy's law survival from no-slip to perfect-slip flow in porous media

A macroscopic model for perfect-slip flow in porous media is derived in this work, starting from the pore-scale flow problem and making use of an upscaling technique based on the adjoint method and Green's formula. It is shown that the averaged momentum equation has a Darcy form in which the permeability tensor, $\boldsymbol{\mathsf{K}}_{ps}$ , is obtained from an associated adjoint (closure) problem that is to be solved on a (periodic) unit cell representative of the structure. Similarly to the classical permeability tensor, $\boldsymbol{\mathsf{K}}$ , in the no-slip regime, $\boldsymbol{\mathsf{K}}_{ps}$ is intrinsic to the porous medium under consideration and is shown to be symmetric and positive. Integral relationships between $\boldsymbol{\mathsf{K}}_{ps}$ , the partial-slip flow permeability tensor, $\boldsymbol{\mathsf{K}}_{s}$ , and $\boldsymbol{\mathsf{K}}$ are derived. Numerical simulations carried out on two-dimensional model porous structures, together with an approximate analytical solution and an empirical correlation for a particular configuration, confirm the validity of the macroscopic model and the relationship between $\boldsymbol{\mathsf{K}}_{ps}$ and $\boldsymbol{\mathsf{K}}$ .

The Creation of Value-added on Wood Carving Handicraft in Luang Prabang city, Luang Prabang Province

The study aims to investigate: the purchasing behavior of customers in Luang Prabang district and province, the perspectives of entrepreneurs and consumers who purchase goods to enhance the value of wood carving craft products in Luang Prabang district and province and Adding value to wood carving craft items that influences customers' decisions to buy in Luang Prabang district and province. employing interviews as a method to gather information from 3 merchants of wood carving craft products and 385 respondents' questionnaires. The frequency, percentage, average, standard deviation, and multiple regression equation. The study's findings revealed: Customer behavior in choosing to buy wood carving craft products as a gift or souvenir, ordering them from the store, and recognizing that information from online advertising media influences his decision to buy. Products that the customer may choose from include wood carving handicrafts for other people's daily lives, low-image elephant products, fish and gold fish products, high-image elephant products, and horse products. The product has the same features as before, but price is a deciding factor when making a purchase, so there's no need to mark the product with a purchase value of between 500,001 and 1,000,000 kip and require a cash payment. Entrepreneurs' thoughts on adding value to handcrafted wood-carving goods. Three Entrepreneurs believe that adding value to wood carving craft goods is important in six areas: designing the product, using wood as a raw material, branding, design, service, and production. Customers' perceptions of the entrepreneur's ability to create added value across six dimensions are influenced in part by the following factors: product quality; service quality; design and production; and modeling. Three factors creating additional value in terms of products, design, and production were involved in the manufacture of wood carving craft products that influence customers' purchase decisions.

On Average Optimality for Non-Stationary Markov Decision Processes in Borel Spaces

This paper studies Borel models of nonstationary Markov decision processes (MDPs) with average criteria. We establish a suitable fixed point theorem, which is used to show the existence of solutions to the average optimality equation (AOE), without using contractions. The existence of optimal policies follows from the obtained solutions to the AOE. Furthermore, we show that versions of the rolling horizon algorithm can be used to produce an optimal policy or an ϵ-optimal policy. Finally, we compare the optimality conditions imposed in this paper with the existing ones in the literature, and demonstrate that they can be satisfied while the previous ones are not. Funding: This research was supported by National Key Research and Development Program of China [2022YFA1004600], National Natural Science Foundation of China [No. 12301170], Guangdong Basic and Applied Research Foundation [2023A1515012644], and the Fundamental Research Funds for the Central Universities, Sun Yat-Sen University [No. 23qnpy39].

Enhanced high-frequency continualization scheme for inertial beam-lattice metamaterials

A multifield continualization technique is introduced that offers a thermodynamically consistent description of the constitutive and dispersive properties of beam-lattice inertial metamaterials with periodic microstructures. The balance equations governing the mechanics of the discrete Lagrangian system are appropriately handled using an innovative continualization scheme to derive an equivalent integral-type continuum model. Based on formal Taylor series expansion of the integral kernels or the corresponding pseudo-differential functions incorporating shift operators and appropriate pseudo-differential downscaling laws, the proposed multifield enhanced continualization scheme allows the derivation of a gradient-type continuum model of given rank and equivalent to lattices. Two different resolution techniques are proposed. Firstly, the corresponding infinite-order average differential equations are tackled using a perturbative approach to describe the forced Bloch wave propagation in the metamaterial. Secondly, higher-order continuum models are employed through proper differential equation truncation to characterize the dispersive properties of the metamaterial in both high- and low-frequency regimes. Moreover, an energetically consistent generalized equivalent Micropolar continua, with non-local inertial terms, are here identified. The multifield continualization procedure is applied to two-dimensional periodic microstructures with tetrachiral, hexachiral, and hexa-tetrachiral topologies. Illustrative examples highlight the ability of the equivalent continuum model to accurately describe the effective constitutive properties of inertial metamaterials with periodic microstructures and to define a dynamic response consistent with the discrete Lagrangian model, validated and tested through virtual experimental verification under free and forced wave conditions.

Linear pressure waves in mono- and poly-disperse bubbly liquids: Attenuation and propagation speed in slow and fast and evanescent modes

Using volumetric averaged equations from a two-fluid model, this study theoretically investigates linear pressure wave propagation in a quiescent liquid with many spherical gas bubbles. The speed and attenuation of sound are evaluated using the derived linear dispersion. Mono- and poly-disperse bubbly liquids are treated. To precisely describe the attenuation effect, some forms of bubble dynamics equations and temperature gradient models are employed. Focusing on the dissipative effect, we analyze the stop band that occurs in the linear dispersion relation. In the two-fluid model, even if the dissipation effect is considered, the inconvenience that the wavenumber diverges to infinity in the resonance frequency cannot be resolved. Additionally, the validity of terminating that wavenumber value in the middle of the frequency is demonstrated. To determine a linear dispersion relation that can exactly predict thermal conduction and acoustic radiation, wave propagation velocities and attenuation coefficients are compared with some experimental data and existing models. The results show that thermal conduction and acoustic radiation should be set appropriately to accurately predict the propagation velocity and attenuation except in the high frequency range, the phase velocity in the resonance frequency range, or the attenuation in the high frequency range.

Formation and decay of oscillons after inflation in the presence of an external coupling. Part I. Lattice simulations

We investigate the formation and decay of oscillons during the post-inflationary reheating epoch from inflaton oscillations around asymptotically flat potentials V(φ) in the presence of an external coupling of the form 1/2 g 2 φ 2 χ 2. It is well-known that in the absence of such an external coupling, the attractive self-interaction term in the potential leads to the formation of copious amounts of long-lived oscillons both for symmetric and asymmetric plateau potentials. We perform a detailed numerical analysis to study the formation of oscillons in the α-attractor E- and T-model potentials using the publicly available lattice simulation codeCosmoLattice. We observe the formation of nonlinear oscillon-like structures with the average equation of state ⟨wφ ⟩ ≃ 0 for a range of values of the inflaton self-coupling λ and the external coupling g 2. Our results demonstrate that oscillons form even in the presence of an external coupling and we determine the upper bound on g 2 which facilitates oscillon formation. We also find that eventually, these oscillons decay into the scalar inflaton radiation as well as into the quanta of the offspring field χ. Thus, we establish the possibility that reheating could have proceeded through the channel of oscillon decay, along with the usual decay of the oscillating inflaton condensate into χ particles. For a given value of the self-coupling λ, we notice that the lifetime of a population of oscillons decreases with an increase in the strength of the external coupling, following an (approximately) inverse power-law dependence on g 2.

Nonlinear Dynamic Characteristics of Smart FG-GPLRC Sandwich Varying Thickness Truncated Conical Shell with Internal Resonance for First Three Order Modes

This paper examines the 1:1:1 internal resonant nonlinear dynamic characteristic of the simply supported varying thickness functionally graded graphene platelets reinforced composite (FG-GPLRC) smart truncated sandwich conical shell subject to the combined effects of transverse load and in-plane force. The truncated smart sandwich conical shell is composed of an FG-GPLRC varying thickness core and two magneto-electro-elastic face layers, whose material properties and constitutive relations are individually identified by the rule of mixture, improved Halpin-Tsai approach and generalized Hooke's law. Utilizing the first-order shear deformation theory (FSDT), von Karman's geometrical nonlinearity, Hamilton's principle and Galerkin technique, the 3DOF dimensionless nonlinear dynamic formulations for the truncated smart FG-GPLRC conical shell are established. The multiple-scale technique is applied to developing the averaged equations for the truncated smart FG-GPLRC conical shell under combined resonance. The frequency-response and force-response curves, Poincare maps, phase portraits, time history diagrams, bifurcation and maximum Lyapunov exponent diagrams can be portrayed by the nonlinear equation solver and Runge-Kutta approach. The effects of the damping and tuning parameters, transverse and in-plane forces on the 1:1:1 internal resonant nonlinear dynamic characteristic of truncated smart varying thickness FG-GPLRC sandwich conical shell are examined.

PERTURBED MOTIONS OF A NEARLY DYNAMICALLY SPHERICAL RIGID BODY WITH A MOVABLE MASS SUBJECT TO CONSTANT BODY-FIXED TORQUE

The problem of motion of a rigid body about a fixed point is one of the classical problems of mechanics. The interest to the problems of the rigid body dynamics has increased in the second half of the XX century in connection with the development of rocket and space technologies. A spacecraft or satellite, while orbiting about its center of mass, experiences torques from forces of diverse physical nature. This includes torques generated by the motion of internal masses, which can arise from factors such as presence of rotating components (like rotors or gyroscopes), and the activities of crew members aboard the crew vehicle. The dynamics of rigid body incorporated moving masses is a significant focal point in classical mechanics. Extensive research is dedicated to investigating the rotation of a rigid body featuring motion of internal masses. It is assumed that the body contains a viscoelastic element that is modeled by a moving mass connected to the body by a strong damper. The moving mass model loosely attached elements in a space vehicles, which can significantly affect the vehicle’s motion about its center of mass during a long period of time. Some cases are considered of the motion of a rigid body containing internal masses connected to the body by means of elastic and dissipative elements. A number of works are devoted to the analysis of various problems of the dynamics of space vehicles containing internal movable masses. The paper develops an approximate solution by means of averaging method to the system of Euler’s equation terms for a nearly dynamically spherical rigid body containing a viscoelastic element under the action of constant body-fixed torque. We obtained the system of motion equations in the standard form which refined in square-approximation by small parameter. Asymptotic approach permits to obtain some qualitative results and to describe evolution of angular motion using simplified averaged equations and numerical solution. The main objective of this paper is to extend the previous results for the problem of motion about a center of mass of a rigid body under the influence of small internal torque (cavity filled with a fluid of high viscosity) or external torque (resistive medium). The importance of the results is in progress of moving mass control motion of spinning projectiles.

Proportional-derivative control and motion stability analysis of a 16-pole legs rotor-active magnetic bearings system

This paper analyzes the motion stability of a 16-pole rotor-active magnetic bearings (rotor-AMB) system and investigates the complex vibrations under a proportional-derivative (PD) controller. First, electromagnetic theory and Newton’s second law are applied to derive the two-degree-of-freedom differential governing equations for the 16-pole rotor-AMB system, incorporating the PD control terms. The resulting differential equations include parametrically excited, quadratic nonlinear, and cubic nonlinear terms. Subsequently, the multiple time scales perturbation analysis method is performed on the obtained governing equations, yielding four-dimensional averaged equations in both Cartesian and polar coordinates. Finally, numerical simulations including the amplitude–frequency response characteristics, motion trajectories, energy–amplitude relationships, as well as bifurcation and chaotic motion of the system are studied. The results indicate that the PD controller affects the softening and hardening spring characteristics of the system and has significant control effects on the system’s amplitude, energy, and stability. Additionally, increasing the differential gain coefficient [Formula: see text] can change the system’s motion from chaotic to periodic.

Averaging principle for reflected stochastic evolution equations

An averaging principle for reflected stochastic evolution equations is established in this paper. To this end, we firstly construct the averaged equations corresponding to the original equations and then demonstrate, by utilizing the time discretization method, that the original equations converge to the corresponding averaged equations in probability, as the parameter goes to zero. Our model includes stochastic Navier–Stokes equations as a special example.

Chaos and resonance of nonlinear FG shallow shells reinforced by oblique stiffeners

This paper presents an analysis of the nonlinear dynamics of functionally graded (FG) shallow shells that are simply supported, reinforced with oblique stiffeners, and subjected to external excitation. The properties of these oblique stiffened functionally graded (OSFG) shallow shells vary across their thickness, following a power law distribution based on the volume fractions. Employing the first-order shear deformation theory (FSDT) coupled with the von Kármán nonlinear strain-displacement relations, the derivation of the nonlinear equations of motion is accomplished through the application of Hamilton's principle. The model of FG shallow shells reinforced with oblique two-series stiffeners is developed, allowing for the angles of the two stiffeners to be similar or different from each other. In this context, the employment of the first-order shear deformation theory (FSDT) results in the enhancement of the complexity in modeling the oblique stiffeners. Galerkin's method is applied to discretize the given equations, converting them into a nonlinear dynamical system with two degrees of freedom. This system incorporates both quadratic and cubic nonlinearities and considers the effects of external excitation. The analysis of this research particularly focuses on the resonant scenario involving principal resonance and 1:1 internal resonance. Through the asymptotic perturbation method, a four-dimensional nonlinear averaged equation is obtained. For the first time, a numerical investigation of the research reveals critical insights into the behavior of the system, such as waveforms, phase diagrams, and Poincaré maps, highlighting the effects of different angles of the stiffeners and variations of material parameters on the nonlinear dynamics of these shells.

Motivasi Siswa terhadap Pembelajaran Seni Budaya (Seni Musik) pada Kelas VIII C di SMPN 2 Batang Gasan

Cultural arts learning is one of the compulsory subjects taught at the Junior High School (SMP) level. Through cultural arts at school, students are able to express themselves and channel their creativity. In learning, student motivation is an important factor in determining student success in achieving understanding. Because motivation will influence a person's desire to do something. This study was conducted to describe student motivation towards learning cultural arts (music) in class VIII C at SMPN 2 Batang Gasan District. In conducting this study, the researchers employed a quantitative approach with a descriptive method. The study focused on grade VIII students of SMPN 1 Batang Gasang, with the sample group being determined using the cluster random sampling technique. The selected sample consisted of students from class VIII C. To gather data, a questionnaire was distributed to the students, which was designed based on indicators of intrinsic and extrinsic learning motivation. The collected data was then analyzed using the average equation and subjected to descriptive analysis. Based on the results of the research obtained, it can be concluded that student motivation towards learning cultural arts (music) in class VIII C at SMPN 2 Batang Gasan District is obtained for internal motivation with an average value of 75.7% in the high category. As for extrinsic motivation, the average value is 65.32% in the high category. Overall, student motivation obtained an average value of 70.51% in the high category. Therefore, student motivation at SMPN 1 Batang Gasan can be categorised as high towards learning cultural arts, especially in music.

Nonlinear Vibrations of the Piezoelectric-Laminated Composite Cantilever Rectangular Plate Subjected to the Transverse and Parametric Excitation

The intelligent structure will be the key to the industrial manufacturing field in the future, piezoelectric materials are the most commonly used active materials in many mechatronic and vibration control systems. In this paper, we illustrated the nonlinear vibration behaviors of the piezoelectric-laminated composite cantilever rectangular plate subjected to transverse and parametric excitation. Two piezoelectric layers are attached to the upper and lower surfaces of graphite/epoxy composite laminated plate. The nonlinear dynamic equations are derived via the classical laminated plate theory, von Karman large deformation theory, the constitutive equation of the piezoelectric materials, and the Hamilton principle. Considering the Galerkin method, the nonlinear ordinary differential equations for the two-order vibration mode of the piezoelectric laminated composite cantilever rectangular plate are achieved. The multiple scale method is applied to obtain the average equations to study the nonlinear vibration characteristic of the stable motion. The primary resonance and the 1:1 internal resonance of the piezoelectric laminated composite cantilever rectangular plate are investigated. The amplitude-frequency response curves, force-amplitude response curves, time histories, phase curves, and bifurcation diagrams are given to investigate the nonlinear dynamic characteristic of the piezoelectric laminated composite cantilever rectangular plate. The detailed parametric analyses show that bubble response curves are separated from the amplitude-frequency response curves with the continuous increase of the external excitation due to the internal resonance. In addition, the results reveal the energy transform between the various order vibration modes of the system. This work is expected to provide theoretical guidance for the nonlinear vibration of the smart piezoelectric laminated composite structure.

Firing rate models for gamma oscillations in I-I and E-I networks.

Firing rate models for describing the mean-field activities of neuronal ensembles can be used effectively to study network function and dynamics, including synchronization and rhythmicity of excitatory-inhibitory populations. However, traditional Wilson-Cowan-like models, even when extended to include an explicit dynamic synaptic activation variable, are found unable to capture some dynamics such as Interneuronal Network Gamma oscillations (ING). Use of an explicit delay is helpful in simulations at the expense of complicating mathematical analysis. We resolve this issue by introducing a dynamic variable, u, that acts as an effective delay in the negative feedback loop between firing rate (r) and synaptic gating of inhibition (s). In effect, u endows synaptic activation with second order dynamics. With linear stability analysis, numerical branch-tracking and simulations, we show that our r-u-s rate model captures some key qualitative features of spiking network models for ING. We also propose an alternative formulation, a v-u-s model, in which mean membrane potential v satisfies an averaged current-balance equation. Furthermore, we extend the framework to E-I networks. With our six-variable v-u-s model, we demonstrate in firing rate models the transition from Pyramidal-Interneuronal Network Gamma (PING) to ING by increasing the external drive to the inhibitory population without adjusting synaptic weights. Having PING and ING available in a single network, without invoking synaptic blockers, is plausible and natural for explaining the emergence and transition of two different types of gamma oscillations.

Конвективные режимы псевдопластической жидкости в квадратной полости при воздействии высокочастотных вибраций в условиях пониженной гравитации

In this paper, we investigate convective modes of pseudoplastic fluid in a square cavity with solid perfectly heat-conducting boundaries under reduced gravity. The cavity performs vertical linearly polarized high-frequency vibrations. Between the vertical walls of the cavity a temperature drop is prescribed in the direction perpendicular to vibrations and the field of gravity. The fluid rheology is described using the Williamson model. The problem is solved based on the averaged thermovibration convection equations for nonlinear viscous fluids. The vibration intensity is determined by the vibration parameter V, which is proportional to the ratio of the amplitude of the vibration acceleration to the free-fall acceleration and does not depend on the temperature difference. The problem is found to have two types of solutions, which we call Newtonian and non-Newtonian modes. These solutions correspond to different convective structures, for which the dependences of the maximum of the stream function and the Nusselt number on the Grashof number are derived. We use these dependences for determining at different values of rheological parameters the threshold values of the Grasgof numbers, corresponding to the change in the modes of stationary averaged convection and the critical Grasgof numbers, corresponding to the loss of stability of stationary averaged flow and the occurrence of oscillatory modes of averaged convection. The structures of different modes of averaged stationary and oscillatory convection are studied. The results obtained for the Newtonian mode have shown that at small values of the Grasgoff number, a slow one-vortex stationary convective flow is realized in the cavity. With increasing the Grasgoff number it transforms into a three-vortex flow. A further increase in the Grasgoff number results in the appearance of a four-vortex convective flow, which eventually transforms again into a three-vortex motion. At even greater Gr, the stationary averaged motion becomes unstable and the oscillatory modes appear in the cavity. For the non-Newtonian mode, a stationary convective flow is the basic flow pattern detected in the cavity; the oscillatory modes are not observed in the whole investigated range of the Grasgoff numbers.

Dynamic Analysis and PD Control in a 12-Pole Active Magnetic Bearing System

This paper conducts an in-depth study on the dynamic stability and complex vibration behavior of a 12-pole active magnetic bearing (AMB) system considering gravitational effects under a PD controller. Firstly, based on electromagnetic theory and Newton’s second law, a two-degree-of-freedom control equation of the system, including PD control terms and gravitational effects, is constructed. This equation involves not only parametric excitation, quadratic nonlinearity, and cubic nonlinearity but also a more pronounced coupling effect between the magnetic poles due to the presence of gravity. Secondly, using the multi-scale method, a four-dimensional averaged equation of the system in Cartesian and polar coordinates is derived. Finally, through numerical analysis, the system’s amplitude–frequency response, motion trajectory, the relationship between energy and amplitude, and global dynamic behaviors such as bifurcation and chaos are discussed in detail. The results show that the PD controller significantly affects the system’s spring hardening/softening characteristics, excitation, amplitude, energy, and stability. Specifically, increasing the proportional gain can quickly suppress the rotor’s motion, but it also increases the system’s instability. Adjusting the differential gain can transition the system from a chaotic state to a stable periodic motion.

Effects of Lévy noise and impulsive action on the averaging principle of Atangana–Baleanu fractional stochastic delay differential equations

As delays are common, persistent, and ingrained in daily life, it is imperative to take them into account. In this work, we explore the averaging principle for impulsive Atangana–Baleanu fractional stochastic delay differential equations driven by Lévy noise. The link between the averaged equation solutions and the equivalent solutions of the original equations is shown in the sense of mean square. To achieve the intended outcomes, fractional calculus, semigroup properties, and stochastic analysis theory are used. We also provide an example to demonstrate the practicality and relevance of our research.

Dynamics of large oscillator populations with random interactions.

We explore large populations of phase oscillators interacting via random coupling functions. Two types of coupling terms, the Kuramoto-Daido coupling and the Winfree coupling, are considered. Under the assumption of statistical independence of the phases and the couplings, we derive reduced averaged equations with effective non-random coupling terms. As a particular example, we study interactions defined via the coupling functions that have the same shape but possess random coupling strengths and random phase shifts. While randomness in coupling strengths just renormalizes the interaction, a distribution of the phase shifts in coupling reshapes the coupling function.

Bifurcations of a Laminated Circular Cylindrical Shell

In this paper, the stability and local bifurcations of a composite laminated circular cylindrical shell with radially pre-stretched membranes are explored by using analytical and numerical methods. On the basis of the four-dimensional averaged equation in the case of 1:1 internal resonance, three types of critical points are studied in detail. They are characterized as (1) one pair of purely imaginary eigenvalues and two negative eigenvalues; (2) a simple zero and one pair of purely imaginary eigenvalues; (3) two pairs of purely imaginary eigenvalues in nonresonant case. With the aid of normal form theory and Maple software, the steady-state solutions and the stability regions of the initial equilibrium solutions are obtained. The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are also presented. The presence of Hopf bifurcation indicates that the circular cylindrical shell will flutter. The results contribute to the design of reasonable structure parameters to avoid flutter. Finally, numerical simulations are also presented to demonstrate the good agreement with the analytical predictions.

Stochastic averaging principle for neutral stochastic functional differential equations driven by G-Lévy process

In this paper, we aim to establish an averaging principle for neutral stochastic functional differential equations driven by G-Lévy processes with non-Lipschitz coefficients. Utilizing the theory of sub-linear expectations, we show that the solutions of the considered stochastic systems can be approximated by the solutions of averaged neutral stochastic functional differential equations driven by G-Lévy processes in the mean square convergence and convergence in the sense of capacity. Finally, we give an example to support our main results.