Moment Equations Research Articles

Application of Configuration Force to Dynamic Crack of Shale under Hydration

The natural microdefects of shale and the expansion of microcracks under hydration and overlying rock loadings are important for the wellbore stability. According to the conservation of energy, the force of the microdefects and microcracks under finite deformation is studied by the method of configuration force through the migrating control volume in the spatial observer. Under the hydration stress and rock pressure, the equation of hydration stress and its work in reference configuration has been obtained, and the equations of configuration forces and configuration moment have been established as a consequence of invariance under changes. The relationship between the configuration and deformation forces is determined by the second law. The energy dissipation equation of the crack tip has been deduced, which shows that the projection of the concentrated internal configuration body force at the crack tip in the opposite direction of the crack is equal to the energy dissipation of the crack tip per unit length. The inertial and internal parts of the concentrated configuration body force at the crack tip have been derived; it is indicated that the internal configuration force plays a leading role in the irreversible fracture process. Moreover, the energy release rate of shale under hydration is proved to depend on constitutive responses and hydration stress. In the theoretical system of configuration force, the migrating control volume at the crack tip contains inclusions, microcracks, microvoids, and heterogeneity of the rock itself. We use the configuration force theory to solve the problem of rock crack propagation and rock fracture. The factors considered are more comprehensive, which can better reflect the actual situation and provide a theoretical basis for the study of wellbore stability.

Mathematical model of an asynchronous machine in real coordinates of state

A mathematical description of an asynchronous machine with unreduced parameters in the space of real phase coordinates is given. When studying an asynchronous motor in a dual power supply system with controlled converters in rotor and stator circuits, there is a need for an AM model without parameters, in which the processes in the stator and rotor circuits will correspond to reality in magnitude. In addition, such a model is necessary when analyzing the energy parameters of the whole electric drive controlled by the rotor. To observe real processes in the stator and rotor, the model should be designed in separate spatial coordinate systems. In reference books for AM with a wound rotor (WR), as a rule, the real parameters (resistances, currents and voltages) of the stator and rotor and the voltage reduction coefficient ( ke ) are given. Bringing currents and resistances (inductances) is carried out by coefficients ki=ke and kr=ke2 respectively. Let's consider the description of the machine without reduction of parameters. For the convenience of further consideration, let's introduce the general value of the mutual inductance.
 We obtain an equation for the electromagnetic moment in real coordinates. Let's design a model in the MATLAB dynamic modeling environment using vector-matrix representation. Matrix algebraic operations with vector variables are implemented by Matlab Fn blocks, which are the calls to userdefined functions described in the form of M-files. The content of Matlab Fn functions is considered. On the model, the processes of starting of an AM with a phase rotor of AK-52-6 type were. Shows the graphs of start-up transients. The processes in the obtained model coincided with the results of modeling of this engine in the model with the coordinates reduced to the rotor. Thus, the energy processes described by this model correspond to the processes to the model with the given parameters, and the processes of currents and flux linkages changing of the stator and rotor are real.
 The model designed in the MATLAB/Simulink dynamic modeling environment can be used to study double-powered asynchronous electric drives.

Thick plate bending analysis using a single variable simple plate theory

A third order single variable simple plate theory has been used for the bending deflection analysis of thick plates with simply supported edges. The plate theory used herein contains third order terms in the displacement field. The displacement field of the theory utilizes only one unknown displacement variable for the complete formulation of thick plates. All the necessary equations connected with this plate theory can be expressed in terms of a single unknown displacement variable. The governing equation, bending moment, shear force and boundary condition equations of the theory have close similarity to those of Classical Plate Theory equations. The mathematical effort needed in formulation of plates using this theory is slightly more in comparison with Classical Plate Theory. To demonstrate the usefulness of the plate theory used herein bending deflection analysis of thick plates with simply supported edges is carried out. Bending deflection results predicted using the present theory are compared with deflection results of other first order and higher order thick plate theories. Bending deflection results predicted by thick plate theory used herein are observed to be accurate.

Improved Scaling Relationships for Seismic Moment and Average Slip of Strike-Slip Earthquakes Incorporating Fault-Slip Rate, Fault Width, and Stress Drop

ABSTRACT We develop a self-consistent scaling model relating magnitude Mw to surface rupture length (LE), surface displacement DE, and rupture width WE, for strike-slip faults. Knowledge of the long-term fault-slip rate SF improves magnitude estimates. Data are collected for 55 ground-rupturing strike-slip earthquakes that have geological estimates of LE, DE, and SF, and geophysical estimates of WE. We begin with the model of Anderson et al. (2017), which uses a closed form equation for the seismic moment of a surface-rupturing strike-slip fault of arbitrary aspect ratio and given stress drop, ΔτC. Using WE estimates does not improve Mw estimates. However, measurements of DE plus the relationship between ΔτC and surface slip provide an alternate approach to study WE. A grid of plausible stress drop and width pairs were used to predict displacement and earthquake magnitude. A likelihood function was computed from within the uncertainty ranges of the corresponding observed Mw and DE values. After maximizing likelihoods over earthquakes in length bins, we found the most likely values of WE for constant stress drop; these depend on the rupture length. The best-fitting model has the surprising form WE∝logLE—a gentle increase in width with rupture length. Residuals from this model are convincingly correlated to the fault-slip rate and also show a weak correlation with the crustal thickness. The resulting model thus supports a constant stress drop for ruptures of all lengths, consistent with teleseismic observation. The approach can be extended to test other observable factors that might improve the predictability of magnitude from a mapped fault for seismic hazard analyses.

Consistent, explicit, and accessible Boltzmann collision operator for polyatomic gases.

Based on a continuous internal energy state variable, we propose an explicit, fully nonlinear Boltzmann collision operator for the evolution of the distribution function describing a polyatomic gas with a constant heat capacity. The particle interaction is a polyatomic generalization of the variable hard-sphere model, used in a recent rigorous mathematical analysis, and includes frozen collisions. The model is consistent with the monatomic case and allows easy evaluations for moment equations and the Chapman-Enskog expansion. Using a publicly available computer algebra code we can explicitly compute nonlinear production terms for macroscopic systems of moments. The range of Prandtl number values recovers the Eucken formula for a specific choice of frozen collisions.

An approximate mathematical model for estimating the influence of vibrations on attached elements of a mobile robot video system

The problem of constructing and applying of mathematical model for assessing the influence of vibrations on the attached elements of the video system of a mobile robot is solved. This robot is considered as a mobile platform for placing of the special equipment for various purposes. The analysis of mathematical models and the results of experimental studies of vibrations of multi-support wheeled machine and its structural elements has been conducted. On the basis of this analysis, the form of a stochastic mathematical model of vibrations of the structure of a mobile robot has been substantiated. The vibration model is specified as a correlation function or spectral density of a random process for various conditions of the robot’s movement. Differential equations of the shaping filter for modeling а random vibration process with given characteristics are presented. Differential equations for the probabilistic moments of the vibration process are presented. Based on the equations for the probabilistic moments and the proposed formula, an assessment of events, which consist in exceeding the vibration amplitude of a given level was carried out. These events are named as outliers of a random process and characterize the range within the basic properties of the equipment installed on the mobile platform are preserved. An analytical study of a mathematical model to determine the intensity of emissions of a random vibration process has been carried out. The probabilistic characteristics of the vibration process and the intensity of emissions are determined. The obtained theoretical results make it possible to assess the performance of the elements of a video system installed on a mobile platform and to formulate the basic requirements for tolerances for their parameters. The results of computer modeling clearly showed the efficiency of the proposed mathematical model for assessing the influence of vibrations.

Compact formulation of the statistical moment method for the solution of the Fokker–Planck equation for two coupled macrospins

The statistical moment method hitherto formulated for single-macrospin relaxation problems is extended to the Fokker–Planck equation for two coupled macrospins describing the time evolution of the joint distribution function of magnetization orientations of an ensemble of such nanoparticles. The method is based on the reduction of that equation via expansion of the joint distribution function in an appropriate orthogonal basis to the solution of a hierarchy of multi-term differential-recurrence equations for the statistical moments (averaged products of spherical harmonics alias the Fourier coefficients), which may then be solved via standard matrix methods. Two particular examples of circularly symmetric potentials are mentioned, which simplify the hierarchy of statistical moment equations. The simplified equations, almost wholly expressed in terms of Clebsch-Gordan coefficients, which may be automatically calculated, agree with those previously derived via direct averaging of the governing Langevin equations with change of the random variables to products of spherical harmonics however leading to cumbersome expressions as originally formulated.

Uncertainty propagation for deterministic models of biochemical networks using moment equations and the extended Kalman filter.

Differential equation models of biochemical networks are frequently associated with a large degree of uncertainty in parameters and/or initial conditions. However, estimating the impact of this uncertainty on model predictions via Monte Carlo simulation is computationally demanding. A more efficient approach could be to track a system of low-order statistical moments of the state. Unfortunately, when the underlying model is nonlinear, the system of moment equations is infinite-dimensional and cannot be solved without a moment closure approximation which may introduce bias in the moment dynamics. Here, we present a new method to study the time evolution of the desired moments for nonlinear systems with polynomial rate laws. Our approach is based on solving a system of low-order moment equations by substituting the higher-order moments with Monte Carlo-based estimates from a small number of simulations, and using an extended Kalman filter to counteract Monte Carlo noise. Our algorithm provides more accurate and robust results compared to traditional Monte Carlo and moment closure techniques, and we expect that it will be widely useful for the quantification of uncertainty in biochemical model predictions.

Micro Satellite Orbital Boost by Electrodynamic Tethers.

In this manuscript, a method for maneuvering a spacecraft using electrically charged tethers is explored. The spacecraft’s velocity vector can be modified by interacting with Earth’s magnetic field. Through this method, a spacecraft can maintain an orbit indefinitely by reboosting without the constraint of limited propellant. The spacecraft-tether system dynamics in low Earth orbit are simulated to evaluate the effects of Lorentz force and torques on translational motion. With 500-meter tethers charged with a 1-amp current, a 100-kg spacecraft can gain 250 m of altitude in one orbit. By evaluating the combined effects of Lorenz force and the coupled effects of Lorentz torque propagation through Euler’s moment equation and Newton’s translational motion equations, the simulated spacecraft-tether system can orbit indefinitely at altitudes as low as 275 km. Through a rare evaluation of the nonlinear coupling of the six differential equations of motion, the one finding is that an electrodynamic tether can be used to maintain a spacecraft’s orbit height indefinitely for very low Earth orbits. However, the reboost maneuver is inefficient for high inclination orbits and has high electrical power requirement. To overcome greater aerodynamic drag at lower altitudes, longer tethers with higher power draw are required.

Defect capturing and charging dynamics and their effects on magneto-transport of electrons in quantum wells

The calculated defect corrections to the polarization and dielectric functions for Bloch electrons in quantum wells are presented. These results were employed to derive the first two moment equations from the Boltzmann transport theory and then applied to explore the role played by defects on the magneto-transport of Bloch electrons. Additionally, we have derived analytically the inverse momentum-relaxation time and mobility tensor for Bloch electrons by making use of the screened defect-corrected polarization function. Based on quantum-statistical theory, we have investigated the defect capture and charging dynamics by employing a parameterized physics-based model for defects to obtain defect wave functions. Both capture and relaxation rates, as well as the density for captured Bloch electrons, were calculated self-consistently as functions of temperature, doping density and chosen defect parameters. By applying the energy-balance equation, the number of occupied energy levels and the chemical potential of defects were determined, with which the transition rate for defect capturing was obtained. By applying these results, the defect energy-relaxation, capture and escape rates, and Bloch-electron chemical potential were calculated self-consistently for a non-canonical subsystem of Bloch electrons. At the same time, the energy- and momentum-relaxation rates of Bloch electrons, as well as the current suppression factor, were also investigated quantitatively. By combining all these results, the temperature dependence of the Hall and longitudinal mobilities was presented for Bloch electrons in either single- or multi-quantum wells.

Nonlinear iterative projection methods with multigrid in photon frequency for thermal radiative transfer

This paper presents nonlinear iterative methods for the fundamental thermal radiative transfer (TRT) model defined by the time-dependent multifrequency radiative transfer (RT) equation and the material energy balance (MEB) equation. The iterative methods are based on the nonlinear projection approach and use multiple grids in photon frequency. They are formulated by the high-order RT equation on a given grid in photon frequency and low-order moment equations on a hierarchy of frequency grids. The material temperature is evaluated in the subspace of lowest dimensionality from the MEB equation coupled to the effective grey low-order equations. The algorithms apply various multigrid cycles to visit frequency grids. Numerical results are presented to demonstrate convergence of the multigrid iterative algorithms in TRT problems with large number of photon frequency groups.

Moment Theory of Chromatography for the Analysis of Reaction Kinetics of Intermolecular Interactions.

Moment theory was applied to the kinetic study of intermolecular interactions. The association equilibrium constant (KA) and association (ka) and dissociation (kd) rate constants of chemical reactions were analytically determined on the basis of the moment theory from elution peak profiles measured by high-performance liquid chromatography (HPLC). The HPLC data were measured under the conditions that neither immobilization nor fluorescence labeling of solute and ligand molecules is required. These are the advantages of the moment analysis method for determining accurate values of KA, ka, and kd. Moment equations were developed on the basis of the Einstein equation for diffusion, the random walk model, and the general rate model of chromatography. The moment analysis method was applied to the inclusion complex formation system between dibenzo-18-crown-6 or dibenzo-15-crown-5 and alkali metal cations. It was demonstrated that the values of KA, ka, and kd can be determined on the assumption that the stoichiometry between crown ethers and cations is 1:1 or 2:1. The influence of the difference in the size between the inner cavity of crown ethers and cations on the association and dissociation of the inclusion complex was considered. The moment analysis method using HPLC is effective for analyzing intermolecular interactions from various perspectives because it is based on the separation technique and has different characteristics from other methods such as spectroscopy. The results of this study contribute to the dissemination of an opportunity for studying intermolecular interactions from equilibrium and kinetic points of view to many researchers because HPLC is widespread.

Semi-Dilute Dumbbells: Solutions of the Fokker–Planck Equation

Spring bead models are commonly used in the constitutive equations for polymer melts. One such model based on kinetic theory—the finitely extensible nonlinear elastic dumbbell model incorporating a Peterlin closure approximation (FENE-P)—has previously been applied to study concentration-dependent anisotropy with the inclusion of a mean-field term to account for intermolecular forces in dilute polymer solutions for background profiles of weak shear and elongation. These investigations involved the solution of the Fokker–Planck equation incorporating a constitutive equation for the second moment. In this paper, we extend this analysis to include the effects of large background shear and elongation beyond the Hookean regime. Further, the constitutive equation is solved for the probability density function which permits the computation of any macroscopic variable, allowing direct comparison of the model predictions with molecular dynamics simulations. It was found that if the concentration effects at equilibrium are taken into account, the FENE-P model gives qualitatively the correct predictions, although the over-shoot in extension in comparison to the infinitely dilute case is significantly underpredicted.

Double/debiased machine learning for logistic partially linear model.

We propose double/debiased machine learning approaches to infer a parametric component of a logistic partially linear model. Our framework is based on a Neyman orthogonal score equation consisting of two nuisance models for the nonparametric component of the logistic model and conditional mean of the exposure with the control group. To estimate the nuisance models, we separately consider the use of high dimensional (HD) sparse regression and (nonparametric) machine learning (ML) methods. In the HD case, we derive certain moment equations to calibrate the first order bias of the nuisance models, which preserves the model double robustness property. In the ML case, we handle the nonlinearity of the logit link through a novel and easy-to-implement 'full model refitting' procedure. We evaluate our methods through simulation and apply them in assessing the effect of the emergency contraceptive pill on early gestation and new births based on a 2008 policy reform in Chile.

Multi-scale fluid dynamics simulation based on MP-PIC-PBE method for PMMA suspension polymerization

This research presents an advanced multi-scale computational fluid dynamics (CFD) model based on the ‘multi-phase particle-in-cell coupled with the population balance equation (MP-PIC-PBE)’ method to predict the stationary continuous stirred tank reactor for methyl methacrylate suspension polymerization. The developed CFD model can realistically simulate the flow patterns of the free-flowing particles and the continuous carrier phase based on the Euler-Lagrangian frame and can track the change in particle size based on PBE. In particular, the model can predict the polymer properties by free-radical polymerization in a parcel through the method of moment equations. To validate the suggested CFD model, the simulation results are compared with the reported experimental data in the literature. Various case-studies are then conducted to investigate the effect of different blade angles (pitched blade angles of 30˚, 45˚, and 60˚) of the impeller on the mixing, the particle size change, particulate flow pattern, and polymer properties. The simulation results demonstrate the phenomena that the low-density particles rise in the larger density solvent by buoyancy and that the higher the blade angle, the smaller the resulting particles due to a higher rate of breakage. It is also found that the particulate flow is well mixed with a 45˚ blade angle.

Determination of collapse moment of different angled pipe bends with initial geometric imperfection subjected to in-plane bending moments

From the recent literature, it is revealed that pipe bend geometry deviates from the circular cross-section due to pipe bending process for any bend angle, and this deviation in the cross-section is defined as the initial geometric imperfection. This paper focuses on the determination of collapse moment of different angled pipe bends incorporated with initial geometric imperfection subjected to in-plane closing and opening bending moments. The three-dimensional finite element analysis is accounted for geometric as well as material nonlinearities. Python scripting is implemented for modeling the pipe bends with initial geometry imperfection. The twice-elastic-slope method is adopted to determine the collapse moments. From the results, it is observed that initial imperfection has significant impact on the collapse moment of pipe bends. It can be concluded that the effect of initial imperfection decreases with the decrease in bend angle from 150∘ to 45∘. Based on the finite element results, a simple collapse moment equation is proposed to predict the collapse moment for more accurate cross-section of the different angled pipe bends.

Study on torque of tangential fan in three operating mode

This paper is dedicated to the load moment calculation on the shaft of the tangential fan that is used with an electric motor to control the flight of an aircraft. The tangential fan can act as a propulsor and as an actuator for a gas-dynamic drive. In the latter case the tangential fan can work in three operating modes: compressor, generator and mixed mode because of different constructive compositions. In compressor mode the torque of an electric drive is transmitted to the tangential fan, which generates the thrust. In generator mode the tangential fan spins because of the incoming flow and the electric drive induces a current. In mixed mode the tangential fan spins because of the incoming flow and the electric fan additionally slows or fastens the flow. Load moment equations for the said modes are obtained using vector diagrams and the law of energy conservation. Solving the equations showed the load moment characteristics. It was observed that the increment of the load moment with increasing velocity of the incoming flow is non-linear. The obtained results enable the developers to estimate the power characteristics of an actuator based on a tangential fan at the design stage.

Coupled physical and magnetodynamic rotational diffusion of a single-domain ferromagnetic nanoparticle suspended in a liquid.

A concise operator form of the Fokker-Planck equation agreeing with that proposed by Weizenecker [Phys. Med. Biol. 63, 035004 (2018)10.1088/1361-6560/aaa186] for the joint orientational distribution of the coupled physical and magnetodynamic rotational diffusion of a single-domain ferromagnetic nanoparticle suspended in a liquid is written from the postulated Langevin equations for the stochastic dynamics. Series expansion of its solution in a complete set yields, using the theory of angular momentum, differential-recurrence equations for statistical moments for coupled motion with uniaxial symmetry of the internal anisotropy-Zeeman energy of a nanoparticle. The numerical results via the matrix iteration method suggest that the susceptibility is adequately approximated by a single Lorentzian with peak frequency given by the inverse integral relaxation time and are discussed in relation to those of the well-known "egg model".

Simulation of Aerosol Evolution within Background Pollution for Nucleated Vehicle Exhaust via TEMOM

This work is intended to study the effect of background particles on vehicle emissions in representative realistic atmospheric environments. The coupling of Reynolds-Averaged Navier–Stokes equation (RANS) and Taylor-series Expansion Method Of Moments (TEMOM) is performed to track the emissions of the vehicle and simulating the evolution of the matters. The transport equation of mass, momentum, heat, and the first three orders of moments are taken into account with the effect of binary homogeneous nucleation, Brownian coagulation, condensation, and thermophoresis. The parameterization model is utilized for nucleation. The measured data for Beijing’s particle size distribution under both polluted and nonpolluted conditions are utilized as background particles. The relationship between the macroscopic measurement results and the microscopic dynamic process is analyzed by comparing the variation trend of several physical quantities in the process of aerosol evolution. It is found with an increase of background particle concentration, the nucleation is inhibited, which is consistent with the existing studies.

A method for predicting lateral deflection of large-diameter monopile near clay slope based on soil-pile interaction

Seven soil-pile interactions and four change processes of soil-pile interaction are proposed to predict the deflection of large-diameter monopile near undrained clay slope under a lateral load in this paper. The equations of force and moment are established in each soil-pile interaction, and the predictive load-deflection curves are obtained by solving those equations. The effect of slope and near-slope distance on the p–y curves is considered. The relationship between ultimate soil resistance and depth is divided into two forms according to the near-slope distance, and the ultimate soil resistance varies nonlinearly with the depth. The modulus of horizontal subgrade reaction in front of pile and behind the pile vary bi-linear and constant with the depth, respectively. To evaluate the accuracy of the new method, it is compared with the results of field tests and the finite element analysis. Finally, the method is used to analyze the effect of slope angle, near-slope distance, pile-soil adhesion and base resistance on lateral deflection.