General Dynamic Equation Research Articles

Free fermions in the repeated interactions scheme

We discuss collision models made of fermions colliding repeatedly with an extended fermionic system whose hamiltonian is quadratic. After a review of the general theory of collision models, we derive a general Lyapunov dynamical equation that governs the covariance matrix of the fermionic extended system. We solve explicitly, in a wide range of physical parameters, several situations of interest such as the case of a boundary driven chain or the replication of entanglement leading to a rainbow like state.

One-order closed model of fluctuating particle coagulation term in the Reynolds averaged general dynamic equation for nanoparticles

One-order closed model of fluctuating particle coagulation term in the Reynolds averaged general dynamic equation for nanoparticles

A Technique for Efficient Estimation of Dynamic Structural Equation Models: A Case Study

Dynamic structural equation models (DSEM) are designed for time series analysis of latent structures. Inherent to the application of DSEM is model parameter estimation, which has to be addressed in many applications by a single time series. In this context, however, the methods currently available either lack estimation quality or are computationally inefficient. Given the era of big data, the necessity for a trade-off between these properties may be detrimental to the applicability of DSEM. The paper is aimed at tackling this trade-off by proposing a novel estimator recursioning technique (ER technique) that facilitates the development of computationally efficient raw-data maximum likelihood estimation algorithms through data transformation, covariance matrix block decomposition, likelihood function reduction, and estimator recursioning steps. The ER technique is introduced by applying it to a special case of the general dynamic structural equation model that encompasses a noisy Wiener-process-type structure with input from the factor-analytic model. The resulting algorithm has been verified through a number of numerical experiments as well as implemented in a brand new R package EMLI, which is available on CRAN.

Generalized dynamics equation with causal symmetry

ABSTRACT The particle dynamics equation’s formal logic is examined in the context of classical mechanics to investigate the nature of inertial forces. First, a fresh perspective on the dual dependence of the inertial reference frame and inertial force in Newtonian dynamics is presented. The causal relationship between the two sides of Newton’s second law is found to be asymmetrical and inconsistent. Secondly, a more general particle dynamics equation, applicable in any translational frame of reference without additional assumptions, is introduced. Essentially, Newton’s second law is only an extreme case of the newly generalized dynamics equation since in Newton’s second law an entire term of the forces acting on the reference object is omitted. The nature of inertial force is unveiled as the mass-ratio weighted real force acting on the reference object. This qualitative explanation is entirely naturalistic as the reference frame’s acceleration depends directly on the acceleration of its reference object(s), and the acceleration of every reference object depends directly on the forces acting on this reference object.

Important role of turbulence on the distribution of particle radioactivity in the nuclear explosions

The distribution of particle radioactivity is one of the most important source items of radioactive fallout prediction model for nuclear explosion. For radioactive particles with the diameter larger than 0.5 μm, the influence of turbulent coagulation cannot be ignored. However, few scholars have considered the role of turbulence in the study of the distribution of particle radioactivity. The General Dynamic Equation (GDE), which is solved using the Multi-Monte Carlo method, is used in this study to establish a new model of the distribution of particle radioactivity that takes the impact of turbulent coagulation into account. The results present that the surface and volume distributions of particle radioactivity are closer to those of the Defense Land Fallout Interpretative Code (DELFIC) model, and the relative error of the surface (volume) distribution reduces from 1.243 (0.687) to 0.945 (0.284) when compared to the previous model that simply takes Brownian coagulation into account. The geometric median diameter of total particle radioactivity increases as the solidification temperature increases and as the particle size range increases when considering the influence of various geological conditions. When considering the effects of turbulent coagulation, the differences in the particle radioactivity produced under different geological conditions are smaller than those only considering the Brownian coagulation. This study highlights the importance of turbulent coagulation on the distribution of particle radioactivity in the nuclear explosions.

Application of the finite element method to the multicomponent general dynamic equation of aerosols

This paper focuses on the numerical approximation of the multicomponent dynamic equation of aerosols (MCGDE), an integro-partial differential equation that describes the temporal evolution of the aerosol number distribution containing several compounds. More specifically, we compare the performance of the finite element method (FEM) and a commonly used sectional method. In addition to the standard Galerkin FEM, we introduce the Petrov–Galerkin Finite Element Method (PGFEM) to the MCGDE. These methods are compared to analytical solutions of the MCGDE and to accurate solutions given by the discrete multicomponent GDE which models the condensation process numerically at the monomer level. Results show that in cases where condensation is the dominating process, the accuracy of the FEM and PGFEM can be significantly better than the accuracy of the sectional method. They also generally achieve a given accuracy of the solution with a significantly lower number of discretization points than the sectional method. Moreover, the sparser the discretization is, the more beneficial is the use of the PGFEM, because it stabilizes the solution of condensation dominated MCGDEs. In the case of pure coagulation, all three numerical methods yield a similar accuracy.

Event-driven sorting algorithm-based Monte Carlo method with neighbour merging method for solving aerosol dynamics

A new event-driven sorting algorithm-based merging Monte Carlo (SAMMC) method is proposed and developed for solving the general dynamic equation in aerosol dynamics. A neighbour merging method is proposed to maintain a constant-volume and constant-number scheme with minimal interference to the numerical particle population, where absolute volume difference (AVD) and relative volume difference (RVD) are used as the crucial merging criteria. The SAMMC method can be used for simulating all aerosol dynamic processes with very high computational accuracy, especially effective in those aerosol dynamic processes generating additional numerical particles. In the present study, comprehensive computational conditions are used to study their impacts on computational accuracy and efficiency by comparing the SAMMC method to previous MC methods and analytical solutions. Numerical results show that the SAMMC method has excellent agreement with analytical solutions for all specified cases of different aerosol dynamic processes and shows higher computational accuracy than equal-weight-based MC methods. In addition, the computational accuracy of the SAMMC method in the total particle number concentration is much higher than those of the weighted fraction Monte Carlo (WFMC) method and sorting algorithm-based merging weighted fraction Monte Carlo (SAMWFMC) method in non-homogeneous coagulation. The SAMMC method can also achieve the same computational precision as the multi-Monte Carlo (MMC) method at only slightly higher computational cost in homogeneous coagulation. More importantly, the SAMMC method can deal with breakage-related processes and simultaneous coagulation and nucleation with very high computational accuracy and efficiency, while the numerical results of the MMC method may significantly deviate from analytical solutions due to the introduction of systematic errors. Furthermore, the RVD can achieve higher computational accuracy in multi-breakage modelling than AVD, but AVD and RVD have almost the same computational efficiency and accuracy in other aerosol dynamic processes.

ВИЯВЛЕННЯ НЕВРАХОВАННОГО ВПЛИВУ СТИСНУТОЇ ШИНИ НА ЗМІНЕННЯ ЗНАЧЕННЯ КОЕФІЦІЄНТА ЗЧЕПЛЕННЯ ШИНИ КОЛІСНОГО РУШІЯ З ДОРОГОЮ ТА ПРИСКОРЕННЯ КОЛІСНОГО РУШІЯ

The working process of the car is accompanied by the coupling of the wheel drive with the road, which is regulated by the coefficient of adhesion, which leads to the movement of the compressed part of the tire (stretching of the tire and its compression, which leads to the dynamic movement of the rolling arm of the wheel drive of the car. The article deals with the study of the movement of the deformed part of the wheel tire engine of a car using the theorem on the change of kinetic energy of this system, the general equation of dynamics. The purpose of the research is to improve the theory of the technological scheme of the wheel drive load with the detection of the movement of the deformed part of the movement of the car wheel drive and the connection of this movement with the movement of the rolling arm of the car wheel drive and the influence of the output parameters on this connection. The scientific and practical direction of the work consists in the fact that for the first time a technology was considered in which, during the rotation of the wheel drive, the points of the deformed part of the tire of the wheel drive of the car are displaced and the Coriolis force appears, which significantly affects the coefficient of adhesion of the wheel drive to the road. The methodology of the study was to establish a mathematical relationship between the coefficient of traction of the wheel drive tire and the road. As a result of the study, it was established: the influence of the initial parameters and the surrounding environment on the deformed part of the tire of the car's wheel drive is reflected in graphic dependencies; a change in the speed of the deformed part of the tire of the wheel drive causes the appearance of Coriolis acceleration, and for the appearance of Coriolis acceleration, the authors proposed a formula for calculating the acceleration of the deformed part of the car tire; the degree of dependence of the size of the deformed part of the tire on the size of the shoulder of the wheel of the wheel drive was established, and this degree is negative. Keywords: Coriolis forces, Coriolis acceleration, rolling shoulder, deformed tire part, wheel drive.

Understanding the effect of ambient gas pressure on the nanoparticle formation in electrically exploding wires

In this paper, a computational model characterizing the preparation of metallic nanoparticles by electrically exploding wires from the onset of current flowing through the wire to the final moment of nanoparticle formation in a gaseous environment is constructed. The computational model consists of a 1D magnetohydrodynamic model, a simplified magnetohydrodynamic model with two-temperature approximation, and a set of general dynamic equations based on the nodal approach, corresponding to the phase transition stage, plasma evolution stage, and nanoparticle growth stage, respectively. The numerical investigation on the formation of nanoparticles is performed with “cold-start” conditions. The computational predictions for the dependence of nanoparticle size on proportion under argon gas pressure of 10 kPa demonstrate that the nanoparticles of 21 nm in diameter account for the maximum proportion of 4.3%. It coincides with the experimental measurements for nanoparticles of 19 nm in diameter with the maximum proportion of 3.5%. The computational model is employed to reveal the influence of ambient gas pressures on the process of nanoparticle formation. The variation trends for parameters of exploding products, cooling rate, and nanoparticle diameter with the largest proportion on ambient gas pressures are discussed. The size distribution of nanoparticles under different argon gas pressures matches relatively well with relevant experimental data. This computational model bridges the gap between the electrically exploding wires and the growth of nanoparticles, providing theoretical support for the regulation and control technology in nanoparticle synthesization by electrically exploding wires.

Formulation and Numerical Solution of the Stability Loss Problem of Elastic-Plastic Shells of Revolution with an Elastic Filler under Combined Axisymmetric and Torsional Loadings

A dynamic statement and a method for numerically solving the buckling problems of elastoplastic shells of revolution with filler in axisymmetric and non-axisymmetric shapes under quasi-static and dynamic loading are presented within the framework of two approaches. In the first approach, the problem of elastic-plastic deformation and buckling of shells of revolution with an elastic filler under combined axisymmetric loading with torsion is formulated in a two-dimensional (generalized axisymmetric) formulation based on the hypotheses of the shells theory of the Timoshenko type and the Winkler foundation. The constitutive relations are written in the cylindrical system of Euler coordinates. For each shell element, a local Lagrangian coordinate system is introduced. Kinematic relations are recorded in the current state metric. The distribution of the displacement velocity components over the shell thickness and strain rate tensors in the local basis is written as the sum of the momentless and moment components, which, in turn, are written as the sum of the symmetric and asymmetric parts in the local and in the general basis. The elastoplastic properties of the shell material are taken into account within the framework of the theory of flow with nonlinear isotropic hardening. To take into account non-axisymmetric forms of buckling, the desired functions (both displacements and forces, moments, contact pressure) are expanded into a Fourier series in the circumferential direction. The variational equations of shell motion are derived from the general equation of dynamics. The contact between the shell and the deformable filler is modeled based on the conditions of non-penetration along the normal and free slip along the tangent. The variational equations of shell dynamics for axisymmetric and nonaxisymmetric processes are interconnected through the physical relations of the theory of plasticity. They take into account large axisymmetric shape changes and the instantaneous stress-strain state of the shell. At the initial stage of the nonaxisymmetric buckling process, the deflections are small; therefore, the equations of nonaxisymmetric buckling are obtained as linearized with respect to nonaxisymmetric forms. To initiate nonaxisymmetric buckling modes, initial nonaxisymmetric deflections are introduced. To solve the defining system of equations, a finite-difference method and an explicit time integration scheme of the “cross” type are used. The second approach is based on continuum mechanics hypotheses and implemented in a three-dimensional setting. Both approaches make it possible to simulate the nonlinear subcritical deformation of shells of revolution with an elastic filler, to determine the ultimate (critical) loads in a wide range of loading rates, taking into account geometric shape imperfections, to study the processes of buckling in axisymmetric and non-axisymmetric shapes under dynamic and quasi-static complex loading by tension, compression, torsion, internal and external pressure. The results of numerical simulation are compared with experimental data on the torsion of steel cylindrical elastoplastic shells ( R / h = 1.45) with an elastic filler.

Radial Basis Function Method for Predicting the Evolution of Aerosol Size Distributions for Coagulation Problems

The dynamic evolution of particle size distributions (PSDs) during coagulation is of great importance in many atmospheric and engineering applications. To date, various numerical methods have been developed for solving the general dynamic equation under different scenarios. In this study, a radial basis function (RBF) method was proposed to solve particle coagulation evolution. This method uses a Gaussian function as the basis function to approximate the size distribution function. The original governing equation was then converted to ordinary differential equations (ODEs), along with numerical quadratures. The RBF method was compared with the analytical solutions and sectional method to validate its accuracy. The comparison results showed that the RBF method provided almost accurate predictions of the PSDs for different coagulation kernels. This method was also verified to be reliable in predicting the self-preserving distributions reached over long periods and for describing the temporal evolution of moments. For multimodal coagulation, the RBF method also accurately predicted the temporal evolution of a bimodal distribution owing to scavenging effects. Moreover, the computational times of the RBF method for these cases were usually of the order of seconds. Thus, the RBF method is verified as a reliable and efficient tool for predicting PSD evolution during coagulation.

Local regularity estimates for general discrete dynamic programming equations

We obtain an analytic proof for asymptotic Hölder estimate and Harnack's inequality for solutions to a discrete dynamic programming equation. The results also generalize to functions satisfying Pucci-type inequalities for discrete extremal operators. Thus the results cover a quite general class of equations.

Computational Fluid Dynamics (CFD) Modelling of ZnO-SiO2 Composite Through a Consecutive Electrospray and Spray Drying Method

Modelling of the droplet formation and drying process in the synthesis of Zinc Oxide-Silicon Dioxide (ZnO-SiO2) composite have been conducted through a CFD modelling. In general, modelling of spray drying only focused on exploring the drying chamber section. The phenomenon builds in a consecutive electrospray and spray drying method has been successfully studied in this paper. The influence of carrier gas flow rate (2 to 10 L/min), precursor flow rate (1 to 10 ml/h), drying chamber temperature and applied voltage (12 to 14 kV) were investigated systematically. Numerical modelling was conducted to describe the mechanism of the composite droplet formation through the jet shape of the electrospray. Evaporation of the initial composite droplet was considered in the modelling, accounting for its size evolution. Simultaneous mass transfer modelling due to the composite evaporation was solved in combination with a general dynamic equation solution. The modelling results show that the applied voltage and the precursor flow rate effectively affected the composite droplet size. While the carrier gas flow rate and the drying chamber temperature, influenced the effectiveness of the composite particle formation in the spray drying synthesis.

ЕЛЕМЕНТИ ТЕОРІЇ ПОДОЛАННЯ РУХЛИВОЇ ПЕРЕШКОДИ ВІЙСЬКОВИМ АВТОМОБІЛЕМ

The working process of the car is accompanied by the loading of the wheel drive by gravitational force, which leads to compression and stretching of the tire during its deformation, as well as hitting various obstacles and moving through them. The article considers the study of the mechanical system "car" using the theorem on the change of kinetic energy of this system, the general equation of dynamics. The aim of the study is to improve the technological scheme of loading the wheel drive when moving the movable support, converting the energy supplied to the wheel drive into controlled relative to the obstacle movement with the addition of traction of the car with transmission force, which is an auxiliary factor to innovative technology. The scientific and practical direction of work is that for the first time the technology is considered in which the law of change of mechanical energy is applied when rotating the wheel drive on a moving obstacle, and this allows a more appropriate approach to the implementation of torque on the wheel drive. The methodology of the study was to establish a mathematical relationship between the speed of the car wheel from the moving obstacle and the parameter of the obstacle it creates, as well as the dynamic mobility of the car itself. The result of the study is the development of elements of the theory of overcoming a moving obstacle that works with "physical discomfort of the support surface." In disclosing the concept of "physical discomfort of the support surface" differential equations were used, which mathematically confirm the overcoming of such an obstacle and such a support surface in certain operating conditions of the car. The value of the study, the results of this work will contribute to the automotive industry. The proposed car model is suitable for use in order to increase the ability to overcome obstacles by vehicles. Keywords: physical-mathematical model, force, reaction, wheel, moving obstacle.

Droplet dynamics driven by electrowetting

Even though electrowetting-on-dielectric (EWOD) is a useful strategy in a wide array of biological and engineering processes with numerous droplet-manipulation applications, there is still a lack of complete theoretical interpretation on the dynamics of electrowetting. In this paper we present an effective theoretical model and use the Onsager variational principle to successfully derive general dynamic shape equationsfor electrowetting droplets in both the overdamped and underdamped regimes. It is found that the spreading and retraction dynamics of a droplet on EWOD substrates can be fairly well captured by our model, which agrees with previous experimental results very well in the overdamped regime. We also confirm that the transient dynamics of EW can be characterized by a timescale independent of liquid viscosity, droplet size, and applied voltage. Our model provides a complete fundamental explanation of EW-driven spreading dynamics, which is important for a wide range of applications, from self-cleaning to novel optical and digital microfluidic devices.

Experimental and Numerical Study on the Gravitational Deposition and Coagulation of Aerosols

The gravitational deposition and coagulation of aerosols were studied experimentally based on the COSTTHES (Containment Source-Term and Thermal Hydraulics Experiment System) facility and numerically investigated based on a continuous general dynamic equation; the experimental and the numerical results were in good agreement. In the present study, the applicability of the zero-dimensional model for deposition and coagulation of aerosols was verified. A considerable distortion can occur due to the coagulation of the aerosol, which significantly influenced the removal rate of the aerosol with a diameter greater than 0.76 µm, and the dynamic shape factor value of the aerosol used in the present experiment was obtained as 4. In the experimental and numerical studies, it was found that the combined effect of the gravitational deposition and coagulation of the aerosol can increase the number density of the aerosol with specific diameters, and the concentration-increment size would increase as settling continued.

Flux-based modeling of heat and mass transfer in multicomponent systems

In the present work, the macroscopic governing equations governing the heat and mass transfer for a general multicomponent system are derived via a systematic nonequilibrium thermodynamics framework. In contrast to previous approaches, the relative (with respect to the mass average velocity) component mass fluxes (relative species momenta) and the heat flux are treated explicitly, in complete analogy with the momentum flux. The framework followed here, in addition to allowing for the description of relaxation phenomena in heat and mass transfer, establishes to the fullest the analogy between all transport processes, momentum, heat, and mass transfer, toward which R. B. Bird contributed so much with his work. The inclusion of heat flux-based momentum as an additional variable allows for the description of relaxation phenomena in heat transfer as well as of mixed (Soret and Dufour) effects, coupling heat and mass transfer. The resulting models are Galilean invariant, thereby resolving a conundrum in the field, and always respect the second law of thermodynamics, for appropriate selection of transport parameters. The general flux-based dynamic equations reduce to the traditional transport equations in the limit when mass species and heat relaxation effects are negligible and are fully consistent with the equations established from the application of kinetic theory in the limit of dilute gases. As an added benefit, for the particular example case of hyperbolic diffusion we illustrate the application of the proposed models as a method to allow the use of powerful numerical solvers normally not available for solving mass transfer models more generally.

A methodology for formulating dynamical equations in analytical mechanics based on the principle of energy conservation

In this work, a methodology is proposed for formulating general dynamical equations in mechanics under the umbrella of the principle of energy conservation. It is shown that Lagrange’s equation, Hamilton’s canonical equations, and Hamilton-Jacobi’s equation are all formulated based on the principle of energy conservation with a simple energy conservation equation, i.e., the rate of kinetic and potential energy with time is equal to the rate of work with time done by external forces; while D’Alembert’s principle is a special case of the law of the conservation of energy, with either the virtual displacements (‘frozen’ time) or the virtual displacement (‘frozen’ generalized coordinates). It is argued that all of the formulations for characterizing the dynamical behaviors of a system can be derived from the principle of energy conservation, and the principle of energy conservation is an underlying guide for constructing mechanics in a broad sense. The proposed methodology provides an efficient way to tackle the dynamical problems in general mechanics, including dissipation continuum systems, especially for those with multi-physical field interactions and couplings. It is pointed out that, on the contrary to the classical analytical mechanics, especially to existing Hamiltonian mechanics, the physics essences of Hamilton’s variational principle, Lagrange’s equation, and the Newtonian second law of motion, including their derivatives such as momentum and angular momentum conservations, are the consequences of the law of conservation of energy. In addition, our proposed methodology is easier to understand with clear physical meanings and can be used for explaining the existing mechanical principles or theorems. Finally, as an application example, the methodology is applied in fluid mechanics to derive Cauchy’s first law of motion.

Adaptive proportional-integral-derivative control for surgical plane cable-driven robots

Robotic technology has greatly improved the success rate of the surgeries by transmitting complicated operations from the surgeon to the tool. In this way, cable-driven robots operate in narrow spaces and take high loads with low weights and costs. This research work presents an adaptive Proportional-Integral-Derivative (PID) controller for surgical plane cable-driven robots and optimizes the gains by employing a Particle Swarm Optimization (PSO) approach. In this way, the general equation of kinematics and dynamics of the robot are investigated regarding a closed chain mechanism and the actuators adjoined to the base. An adaptive PID controller based on the robust sliding surfaces and gradient descent rules is applied to handle the end-effector (tool) to different desired conditions. The proper values of the control gains are found by using the PSO algorithm as a swarm-based technique for minimizing the output errors. The simulation results clearly indicate the capability of the introduced control approach to tracking the constant and time-dependent desired trajectories in the presence of external disturbances.

Understanding Micropolar Theory in the Earth Sciences I: The Eigenfrequency omega _r

Even though micropolar theories are widely applied for engineering applications such as the design of metamaterials, applications in the study of the Earth’s interior still remain limited and in particular in seismology. This is due to the lack of understanding of the required elastic material parameters present in the theory as well as the eigenfrequency omega _r which is not observed in seismic data. By showing that the general dynamic equations of the Timoshenko’s beam is a particular case of the micropolar theory we are able to connect micropolar elastic parameters to physically measurable quantities. We then present an alternative micropolar model that, based on the same physical basis as the original model, circumvents the problem of the original eigenfrequency omega _r laking in seismological data. We finally validate our model with a seismic experiment and show it is relevant to explain observed seismic dispersion curves.