Mathematical Models of Tire-Lateral Road Adhesion for Use in Road Vehicle Dynamics Studies

Mathematical models of tire-lateral mad adhesion for use in mad vehicle dynamics studies are set up to express the relations of adhesion coefficients with slip ratio in lateral direction.The models of tire-lateral mad adhesion revolutionize the Pacejka's model in concept and therefore make it possible for applications in vehicle dynamics studies by the expression of lateral adhesion coefficient as a function of wheel slip ratio,instead of the wheel slip angle,taking into account in the mean time the influences of mad surface condition, vehicle velocity,vertical load,tire slip angle,and wheel camber angle.

Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler’s equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic assumptions can be derived using the conservation and balance laws of classical or non-classical continuum mechanics. The mathematical models based on Kirchhoff hypothesis (classical plate theory, CPT) and first order shear deformation theory (FSDT) that are representative of most mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models.

Mathematical Models for Vacuum Drying Characteristics of Pomegranate Aril

The aim of this study was to investigate and determine the best mathematical models in describing vacuum drying characteristic of pomegranate arils in the range of 55-75 ℃. The vacuum batch dryer used in the evaluation was successful in drying a thin layer of pomegranate arils from the initial moisture content of 464.02% （d.b.） to 6.18% （d.b.） within 6.5 to 13.5 hr of continuous drying at the above mentioned temperature range. The drying rates increased with an increase in temperature and drying time. Five of the well known semi-theoretical and empirical models were fitted to the vacuum drying of pomegranate arils. The semi-empirical Midilli model has shown an excellent fit to predict drying behavior of the pomegranate arils because this model gave the highest coefficient of determination （RE）, the least chi-square （X2）, and the lowest root mean square error （RMSE）. The total drying occurs during falling period, signifying the influence of moisture diffusion during the drying. The effective diffusivity varied from 1.25× 10^10 to 2.91 × 10^10 m^2/s over the temperature range. Temperature dependence of the diffusivity was well documented by Arrhenius models. The activation energy of moisture diffusion during drying was found to be 40.46 kJ/mol.

Mathematical Models for a Social Partitioning Problem

In this paper we develop modeling techniques for a social partitioning problem. Different social interaction regulations are imposed during pandemics to prevent the spread of diseases. We suggest partitioning a set of company employees as an effective way to curb the spread, and use integer programming techniques to model it. The goal of the model is to maximize the number of direct interactions between employees who are essential for company’s work subject to the constraint that all employees should be partitioned into components of no more than a certain size implied by the regulations. Then we further develop the basic model to take into account different restrictions and provisions. We also give heuristics for solving the problem. Our computational results include sensitivity analysis on some of the models and analysis of the heuristic performance.

PHYSICAL AND MATHEMATICAL MODELS OF POLLUTION DISPERSION IN THE YANGTZE ESTUARY

The method of combining a physical model with a mathematical model is described to study the concentration profile of pollutant dispersion in the Yangtze Estuary. the Experiments are described regarding a jet in a tidal physical model and two-dimensional calculations of diffusion using momentum and mass conservation equations of unsteady flow. The feature of dispersion in the tidal flow, which is different from that in the steady flow such as rivers, is explained. Dilution and dispersion mainly depend on the volume of runoff and tidal range. The results of the measurement and calculation are presented, and it can be seen that they are in good agreement.

Investigation on hot deformation behavior and mathematical models of flow stress of high-strength structural steel

The effect of deformation conditions on dynamic recrystallization behavior of Nb,V,Ti microalloyed high-strength structural steel was investigated via high-temperature single pass reduction tests on a MMS-300 thermomechanical simulator,with mathematical models established for flow stress during hot deformation.The results show that the deformation resistance decreases with the increase of temperature and is in power function relationship with the temperature.Meanwhile,it increases with the increase of strain rate and is in log-log relationship with the strain rate.The dynamic recrystallization activation energy of tested steel was determined to be about 329.55 kJ/mol,295.31 kJ/mol at peak and steady states.The prediction models developed for flow stress indicated that they are in good agreement with experimental results.

Application of Mathematical Models in the Research of Chinese Herbal Compound Prescriptions

With the interdisciplinary development of multiple disciplines,mathematical models play a role as a bridge in the research process of Chinese herbal compounds,which can make full use of the interrelationships of knowledge,making the research and use of Chinese herbal compounds from complex and cumbersome to simple.This paper reviewed the research and application of mathematical models in the production and preparation,pharmacodynamics,pharmacokinetics and other aspects of Chinese herbal compound prescriptions,providing a theoretical basis for the development of the modern Chinese medicine industry.

Application of Polynomial Mathematical Models for the Extraction of Bioactive Compounds from Plant Sources

This article focuses on the mathematical modelling of the extraction process of bioactive compounds from grape marc and berries (Aronia, rosehip, rowan, and hawthorn). The composition of the extraction medium (the concentration of the ethyl alcohol) served as a factor of influence. Furthermore, 8 experimental measured parameters were used as variables. The experimental results were processed using Hermite polynomials. In order to adapt the degree of the polynomial, the following conditions were imposed: high precision of the mathematical model by appealing to models on interval;obtaining a nominal model and two uncertain models (upper and lower);deduction of two predictive models, one superior and one inferior. It was found that the mathematical models based on Hermite polynomials do not provide explicit analytical expressions, although they allow the establishment of parameter values for any concentration of the extraction medium. In some cases, only high-grade polynomial models ensure the modelling error below 2%. Uncertain models (upper and lower 95%) include all experimental data. Predictive mathematical models (upper and lower) were established for a high prediction. The analytical expressions of the mathematical models on intervals are non-gaps, the coefficients having non-zero values. Dependencies between the measured parameters and the composition of the extraction solvent were analyzed, the results being presented through the calculation of a surface, with all the experimental values and their average values. Thus, it was found that polynomial mathematical models provide complete information for modelling the extraction processes of bioactive compounds of plant origin.

New approach to assess sperm DNA fragmentation dynamics： Fine-tuning mathematical models

Background： Sperm DNA fragmentation（sDF） has been proved to be an important parameter in order to predict in vitro the potential fertility of a semen sample. Colloid centrifugation could be a suitable technique to select those donkey sperm more resistant to DNA fragmentation after thawing. Previous studies have shown that to elucidate the latent damage of the DNA molecule, sDF should be assessed dynamically, where the rate of fragmentation between treatments indicates how resistant the DNA is to iatrogenic damage. The rate of fragmentation is calculated using the slope of a linear regression equation. However, it has not been studied if s DF dynamics fit this model. The objectives of this study were to evaluate the effect of different after-thawing centrifugation protocols on sperm DNA fragmentation and elucidate the most accurate mathematical model（linear regression, exponential or polynomial） for DNA fragmentation over time in frozen-thawed donkey semen.Results： After submitting post-thaw semen samples to no centrifugation（UDC）, sperm washing（SW） or single layer centrifugation（SLC） protocols, sD F values after 6 h of incubation were significantly lower in SLC samples than in SW or UDC.Coefficient of determination（R-2） values were significantly higher for a second order polynomial model than for linear or exponential. The highest values for acceleration of fragmentation（aSDF） were obtained for SW, fol owed by SLC and UDC.Conclusion： SLC after thawing seems to preserve longer DNA longevity in comparison to UDC and SW. Moreover,the fine-tuning of models has shown that sDF dynamics in frozen-thawed donkey semen fit a second order polynomial model, which implies that fragmentation rate is not constant and fragmentation acceleration must be taken into account to elucidate hidden damage in the DNA molecule.

Flow field simulation and establishment for mathematical models of flow area of spool valve with sloping U-shape notch machined by different methods

Precise function expression of the flow area for the sloping U-shape notch orifice versus the spool stroke was derived. The computational fluid dynamics was used to analyze the flow features of the sloping U-shape notch on the spool, such as mass flow rates, flow coefficients, effiux angles and steady state flow forces under different operating conditions. At last, the reliability of the mathematical model of the flow area for the sloping U-shape notch orifice on the spool was demonstrated by the comparison between the orifice area curve derived and the corresponding experimental data provided by the test. It is presented that the bottom arc of sloping U-shape notch （ABU） should not be omitted when it is required to accurately calculate the orifice area of ABU. Although the theoretical flow area of plain bottom sloping U-shape notch （PBU） is larger than that of ABU at the same opening, the simulated mass flow and experimental flow area of ABU are both larger than these of PBU at the same opening, while the simulated flow force of PBU is larger than that of ABU at the same opening. Therefore, it should be prior to adapt the ABU when designing the spool with proportional character.

Mathematical models and expert system for grate-kiln process of iron ore oxide pellet production.Part Ⅱ:Rotary kiln process control

Rotary kiln process for iron ore oxide pellet production is hard to detect and control.Construction of one-dimensional model of temperature field in rotary kiln was described.And the results lay a solid foundation for online control.Establishment of kiln process control expert system was presented,with maximum temperature of pellet and gas temperature at the feed end as control cores,and interval estimate as control strategy.Software was developed and put into application in a pellet plant.The results show that control guidance of this system is accurate and effective.After production application for nearly one year,the compressive strength and first grade rate of pellet are increased by 86 N and 2.54%,respectively,while FeO content is 0.05% lowered.This system can reveal detailed information of real time kiln process,and provide a powerful tool for online control of pellet production.

Mathematical models and expert system for grate-kiln process of iron ore oxide pellet production(Part Ⅰ):Mathematical models of grate process

Grate process is an important step in grate-kiln pellet production.However,as a relatively closed system,the process on grate is inaccessible to direct detection,therefore,it is hard to control.As a result,mathematical models of temperature distribution,moisture distribution and oxidation degree distribution in pellet bed,with good universality,computation speed and calculation accuracy,are presented based on analysis of heat transfer and physical-chemical reactions during grate process.And real-time visualization of temperature,moisture and oxidation degree distribution in pellet bed during grate process is realized.Model validation is displayed,and the similarity of 91% is proved.The results can reveal real time status on grate,and provide a solid foundation for the subsequent study of artificial intelligence control system of pellet production.

Mathematical models for foam-diverted acidizing and their applications

Foam diversion can effectively solve the problem of uneven distribution of acid in layers of different permeabilities during matrix acidizing. Based on gas trapping theory and the mass conservation equation, mathematical models were developed for foam-diverted acidizing, which can be achieved by a foam slug followed by acid injection or by continuous injection of foamed acid. The design method for foam-diverted acidizing was also given. The mathematical models were solved by a computer program. Computed results show that the total formation skin factor, wellhead pressure and bottomhole pressure increase with foam injection, but decrease with acid injection. Volume flow rate in a highpermeability layer decreases, while that in a low-permeability layer increases, thus diverting acid to the low-permeability layer from the high-permeability layer. Under the same formation conditions, for foamed acid treatment the operation was longer, and wellhead and bottomhole pressures are higher. Field application shows that foam slug can effectively block high permeability layers, and improve intake profile noticeably.

Mathematical models for properties of mortars with admixtures and recycled fine aggregates from demolished concretes

In order to expand the engineering application of recycle aggregate mortars （RAM） with aggregates from demolished concretes, the models for the properties of RAM and the replacement rate of these recycled fine aggregates were proposed. First, different kinds of mathematical models for the basic properties （compressive strength, water retention rate, and consistency loss） of RAM with two kinds of admixtures, thickening powders （TP） and self-made powdery admixtures （SSCT） designed for RAM, and the replacement rates were established, while the average relative errors and relative standard errors of these models were calculated. Additionally, the models and their error analyses for the curves of drying shrinkage and curing time of RAM ＋ SSCT at different replacement rates were put forward. The results show that polynomial functions should be used to calculate the basic properties of RAM ＋ TP and RAM ＋ SSCT at different replacement rates. In addition, polynonfial functions are the most optimal models for the sharp shrinkage sections in the curves of drying shrinkage-curing time of RAM ＋ SSCT, while exponential functions should be used as the models for the slow shrinkage sections and steady shrinkage sections.

Foundation of three—dimensional mathematical models for glass furnace regenerator

This paper presents a practical three dimensional mathematical model of circulation and heat transfer in generator of glass melting furnaces. The model was based on the heat transfer between the smoke flow and the lattice units, and between the air flow and the lattice units. This model not only bypassed the difficulty of complicated computation of the heat transfer process in the regenerator of glass furnaces, but also avoided the irrationality of fixing the temperature distribution on the surfaces. Use of the model yielded very important data and also the method for the design of the regenerator of glass furnaces in practical production.

Consistency and Validity of the Mathematical Models and the Solution Methods for BVPs and IVPs Based on Energy Methods and Principle of Virtual Work for Homogeneous Isotropic and Non-Homogeneous Non-Isotropic Solid Continua

Energy methods and the principle of virtual work are commonly used for obtaining solutions of boundary value problems (BVPs) and initial value problems (IVPs) associated with homogeneous, isotropic and non-homogeneous, non-isotropic matter without using (or in the absence of) the mathematical models of the BVPs and the IVPs. These methods are also used for deriving mathematical models for BVPs and IVPs associated with isotropic, homogeneous as well as non-homogeneous, non-isotropic continuous matter. In energy methods when applied to IVPs, one constructs energy functional (I) consisting of kinetic energy, strain energy and the potential energy of loads. The first variation of this energy functional (δI) set to zero is a necessary condition for an extremum of I. In this approach one could use δI = 0 directly in constructing computational processes such as the finite element method or could derive Euler’s equations (differential or partial differential equations) from δI = 0, which is also satisfied by a solution obtained from δI = 0. The Euler’s equations obtained from δI = 0 indeed are the mathematical model associated with the energy functional I. In case of BVPs we follow the same approach except in this case, the energy functional I consists of strain energy and the potential energy of loads. In using the principle of virtual work for BVPs and the IVPs, we can also accomplish the same as described above using energy methods. In this paper we investigate consistency and validity of the mathematical models for isotropic, homogeneous and non-isotropic, non-homogeneous continuous matter for BVPs that are derived using energy functional consisting of strain energy and the potential energy of loads. Similar investigation is also presented for IVPs using energy functional consisting of kinetic energy, strain energy and the potential energy of loads. The computational approaches for BVPs and the IVPs designed using energy functional and principle of virtual work, their consistency and validity are also investigated. Classical continuum mechanics (CCM) principles i.e. conservation and balance laws of CCM with consistent constitutive theories and the elements of calculus of variations are employed in the investigations presented in this paper.

MATHEMATICAL MODELS ARISING FROM A SURVEY OF FERROMAGNETIC MATERIALS UNDER MAGNETISATION

This paper focuses on mathematical models describing the mechanical behavior of ferromagnetic materials under magnetization. Through combination of the electromagnetic field theory with the theory of elastic mechanics, several nonlinear systems of fourth order partial differential equations were deduced. By making further assumption, the first-order approximation of the above equations was established, of which the solutions are good enough for engineering application.

Comparison on Mathematical Models for Description of Flow Curves of Stable Austenitic Steels

The flow curves were measured for the stable austenitic steels 304L and 304LN by means of tensile test at room temperature,which are described by the models σ=K1εn1 + exp(K2 + n2ε), σ=Kεn1+n2lnε and σ=σ0+Kεn (where, K1, K2, n1 andn2; K, n1 and n2; σ0, K and n are constant). The comparison of the maximum deviations and the consideration of thevariation of the work hardening rate with true strain show that the flow curves for the austenitic steels 304L and 304LN canbe described by the model σ=Kεn1+n2 lnε at higher precision.The derivatives of the models σ=K1εn1 + exp(K2 + n2ε) and σ=Kεn1+n2lnε with respect to true strain, exhibit theextreme at low true strain. This inherent character indicates that both models are unsuitable to describe the part of the workhardening rate curve at low true strain.

A Study on System Availability Vs System Administration Efforts with Mathematical Models

Two mathematical models are developed in this paper to study the effectiveness of system administration efforts on the improvement of system availability, based on the assumption that there exists a transitional state for a computer system in operation before it is brought down by some hardware or software problems and with intensified system administration efforts, it is possible to discover and fix the problems in time to bring the system back to normal state before it is down. Markov chain is used to simulate the transition of system states. A conclusion is made that increasing system administration efforts may be a cost-effective way to meet the requirements for moderate improvement on system availability, but higher demand on this aspect still has to be met by advanced technologies.

SCALE AND TIME EFFECTS ON MATHEMATICAL MODELS FOR TRANSPORT IN THE ENVIRONMENT

The purpose of this paper is to analyse mathematical models used in environmental modelling. Following a brief survey of the development in modelling scale- and time-dependent dispersion processes in the environment, this paper compares three similarity solutions, one of which is a solution of the generalized Feller equation （GF） with fractal parameters, and the other two for the newly-developed generalized Fokker-Planck equation （GFP）. The three solutions are derived with parameters having physical significance. Data from field experiments are used to verify the solutions. The analyses indicate that the solutions of both GF and GFP represent the physically meaningful natural processes, and simulate the realistic shapes of tracer breakthrough curves.