“数轴模型”在小学低段数学拓展课中的构建与应用

数形结合是一种重要的数学思想方法,它将抽象的数学语言与直观的图像结合起来,通过数与形之间的相互转化来解决数学问题.“数轴模型”是一种数形结合方式.文章根据“数轴模型”既能帮助学生从直观层面上理解数的含义、理清数与数之间的关系,还能帮学生理解运算意义、明晰数量关系等四个维度分别拓展了四个案例,旨在通过这四个案例的剖析,让教师可以借鉴思路,创造更多能利用“数轴模型”解决问题的拓展课,从而将数形结合的思想有效渗透到学生的头脑中,提高学生的核心素养.

ANALYTIC STUDY ON POTENTIAL INSTABILITY AND SPIRAL STRUCTURE OF RAIN CLUSTERS

This paper presents a study on potential instability and spiral structure of unstable rain clusters.First,we develop a linearized non-axisymmetrical mathematic model for rain clusters in circular cylindrical coordinates and acquire its analytic solution.Second,we discuss the potential instability of non-axisymmetrical rain clusters.Finally,we conclude that spiral structures can exist in rain clusters.Our analysis indicates that potential instability occurs when humid stratification coefficient is less than zero.Unstable growth rate increases with the increase of the absolute value for humid stratification coefficient.The simpler the vertical structure of perturbation,the thicker the inversion layer;additionally,the smaller the radius of the rain clusters,the larger the unstable growth rate.Simulation results agree well with those from observation and forecast.The spiral structure simulated by our model is similar to a radar echo,suggesting that rain clusters with spiral structures can occur in the atmosphere.In addition,they are generally close to the model solution in this work.

Application of artificial neural networks and multivariate statistics to estimate UCS using textural characteristics

Before any rock engineering project,mechanical parameters of rocks such as uniaxial compressive strength and young modulus of intact rock get measured using laboratory or in-situ tests,but in some situations preparing the required specimens is impossible.By this time,several models have been established to evaluate UCS and E from rock substantial properties.Artificial neural networks are powerful tools which are employed to establish predictive models and results have shown the priority of this technique compared to classic statistical techniques.In this paper,ANN and multivariate statistical models considering rock textural characteristics have been established to estimate UCS of rock and to validate the responses of the established models,they were compared with laboratory results.For this purpose a data set for 44 samples of sandstone was prepared and for each sample some textural characteristics such as void,mineral content and grain size as well as UCS were determined.To select the best predictors as inputs of the UCS models,this data set was subjected to statistical analyses comprising basic descriptive statistics,bivariate correlation,curve fitting and principal component analyses.Results of such analyses have shown that void,ferroan calcitic cement,argillaceous cement and mica percentage have the most effect on USC.Two predictive models for UCS were developed using these variables by ANN and linear multivariate regression.Results have shown that by using simple textural characteristics such as mineral content,cement type and void,strength of studied sandstone can be estimated with acceptable accuracy.ANN and multivariate statistical UCS models,revealed responses with 0.87 and 0.76 regressions,respectively which proves higher potential of ANN model for predicting UCS compared to classic statistical models.

Use of axial dispersion model for determination of Sherwood number and mass transfer coefficients in a perforated rotating disc contactor

The mass transfer process in a perforated rotating disk contactor(PRDC) using a toluene-acetone-water system was investigated.The volumetric overall mass transfer coefficients are calculated in a PRDC column.Both mass transfer directions are considered in experiments.The influences of operating variables containing agitation rate,dispersed and continuous phase flow rates and mass transfer in the extraction column are studied.According to obtained results,mass transfer is significantly dependent on agitation rate,while the dispersed and continuous phase flow rates have a minor effect on mass transfer in the extraction column.Furthermore,a novel empirical correlation is developed for prediction of overall continuous phase Sherwood number based on dispersed phase holdup,Reynolds number and mass transfer direction.There has been great agreement between experimental data and predicted values using a proposed correlation for all operating conditions.

A Mathematical Model for Pulsating Flow of Ionic Fluid under an Axial Electric Field： EMF Effect on Blood Flow

The present work introduces a mathematical model for ionic fluid that flows under the effect of both pulsating pressure and axial electromagnetic field. The fluid is treated as a Newtonian fluid applying Navier-Stokes equation. The fluid is considered as a neutral mixture of positive and negative ions. The effect of axial electric field is investigated to determine velocity profiles. Hydroelectric equation of the flow is deduced under dc and ac external electric field. Hence the effect of applied frequency （0-1 GHz） and amplitude （10-350 V/m） is illustrated. The ultimate goal is to approach the problem of EMF field interaction with blood flow. The applied pressure waveform is represented as such to simulate the systolic-diastolic behavior. Simulation was carried out using Maple software using blood plasma parameters; hence velocity profiles under various conditions are reported.

Gound State Properties of Nuclei in ^(295)118 α-Decay Chain

The properties of nuclei belonging to the α-decay chain of superheavy element ^295118 have been studied in the framework of axially deformed relativistic mean field （RMF） theory with the parameter set of NL-Z2 in the blocked BCS approximation. Some ground state properties such as binding energies, deformations, and α-decay energies Qα have been obtained and agree well with those from finite-range droplet model （FRDM）. The single-particle spectra of nuclei in ^295118 α-decay chain show that the shell gaps present obviously nucleon number dependence. The root-mean-square （rms） radii of proton, neutron and matter distributions change slowly from ^283112 to ^295118 but dramatically from ^279110 to ^283112, which may be due to the subshell closure at Z = 110 in ^279110. The α-decay half-lives in 295118 decay chain are evaluated by employing the cluster model and the generalized liquid drop model （GLDM）, and the overall agreement is found when they are compared with the known experimental data. The α-decay lifetimes obtained from the cluster model are slightly larger than those of GLDM ones. Finally, we predict the α-decay half-lives of Z=118, 116, 114, 112 isotopes using the cluster model and GLDM, which also indicate these two models can corroborate each other in studies on superheavy nuclei. The results from GLDM are always lower than those obtained from the cluster model.

Numerical analysis of bump foil bearings without nominal radial clearance

Bump foil bearings without nominal radial clearance were analyzed. An air film thickness model and a bearing theoretical analytical model were developed accounting for air compressibility and foil deformation. To analyze hydrodynamic characteristics of bump foil bearings with different operating eccentricities, the air film thickness equation and Reynolds equation were coupled through pressure and solved by Newton-Raphson Method (NRM) and Finite Difference Method (FDM). The characteristics of an bump foil bearing model were discussed including load carrying capacity, film thickness and pressure distributions. The results of simulation show that bump foil bearing without nominal radial clearance can provide better stability and greater load capacity. This numerical analytical method also reveals a good convergence in numerical calculation.

SOLUTION TO THE STRESS DISTRIBUTION ON THE SURFACE LAYER OF BUSHING ALLOY BY COUPLING BEM WITH ELASTICITY METHOD

In the design of the fatigue strength of dynamically loaded bearing in the equipmentssuch as internal combustion engines and roimg mun, the solution to the stress distribution on thebushing alloy layer is an important and difficult problem. In this paper, a new method has beenproposed by coupling BEM with etheticity method, The algorithm and its implementation were deseribed in details The calculation results verify that this up-dated method can provide us a moresimple and effective tool for solvingthe fatigue stress of the bushing alloy with tangible benefit oftime-saving and high computation accuraey. It may open a new vista in bearing fatigue strength design.

Behavior of Reinforced Concrete Columns under Combined Axial Load and Bending in Accordance with a Nonlinear Numerical Model

A nonlinear numerical model was developed to analyze reinforced concrete columns under combined axial load and bending up to failure. Results of reinforced concrete columns under eccentric compression tested to failure are presented and compared to results from a numerical nonlinear model. The tests involved 10 columns with cross-section of 250 mm × 120 mm, geometrical reinforcement ratio of 1.57% and concrete with compression strength around 40 MPa, with 3,000 mm in length. The main variable was the load eccentricity in the direction of the smaller dimension of cross-section. Experimental results of ultimate load and of the evolution of transverse displacements and concrete strains are compared with the numerical results. The estimated results obtained by the numerical model are close to the experimental ones, being suitable for use in verification of elements under combined axial load and bending.

NC Machining of Spiral Bevel Gear and Hypoid Gear Based on Unity Transformation Model

A unity transformation model (UTM) was presented for flexible NC machining of spiral bevel gears and hypoid gears. The model can support various machining methods for Gleason spiral bevel gears and hypoid gears, including generation machining and formation machining for wheel or pinion on a universal five-axis machining center, and then directly produce NC codes for the selected machining method. Wheel machining and pinion machining under UTM were simulated in Vericut 6.0 and tested on a five-axis machining center TDNC-W2000 with NC unit TDNC-H8. The results from simulation and real-cut verify the feasibility of gear machining under UTM as well as the correctness of NC codes.

Application and Comparison of SST Model in Numerical Simulation of the Axial Compressors

The shear-stress transport(SST)turbulence model is incorporated into Navier-Stokes equations to simulate a turbomachinery flowfield.A staggered finite volume method is used to make the mean flow equations and turbulence model equations strongly coupled and enhance the stability of the numerical computation.The implicit treatment of the source terms is applied to the SST model.A steady state solution is obtained using five-stage Runge-Kutta time-stepping scheme with local time stepping and residual smoothing to accelerate convergence. The wall distance d,a key parameter in the SST model,is solved by a partial differential equation.The validations of the code are conducted on rotor 37,wp11 at design and off-design conditions by comparison with measurements and the Spalart-Allmaras(SA)turbulence model.The flow within the tip is calculated with a multi-block grid.

Numerical Simulation of Rock Fracturing under Uniaxial Compression Using Virtual Internal Bond Model

A multi-scale virtual internal bond (VIB) model for the isotropic materials has been recently proposed to describe the material deformation and fracturing. During the simulation process of material fracturing using VIB, the fracture criterion is directly built into the constitutive formulation of the material using the cohesive force law. Enlightened by the similarity of the damage constitutive model of rock under uniaxial compression and the cohesive force law of VIB, a VIB density function of rock under uniaxial compression is suggested. The elastic modulus tensor is formulated on the basis of the density function. Thus the complete deformation process of rock under the uniaxial compression is simulated.

Projected total energy surface description for axial shape asymmetry in ^(172)W

The projected total energy surface(PTES)approach has been developed based on the triaxial projected shell model(TPSM)hybridized with the macroscopic–microscopic method.The total energy of an atomic nucleus is decomposed into macroscopic,microscopic and rotational terms.The macroscopic and microscopic components are described with the liquid drop model and Strutinsky method,respectively,and the rotational energy is given by the TPSM,the term beyond the mean field.To test theory,the PTES calculations have been carried out for the yrast states of the well deformed rare earth nucleus172W,and the theoretical results are in good agreement with the experimental data.By using the equilibrium quardrupole deformations(ε2andγ)determined by the PTES,the calculation of the transition quardrupole moment(Qt)in function of spin also reproduces the experimental data.A comparison between the PTES and TRS methods has been made for theoretical and application uses.

A miniature pump with a fluid dynamic bearing

This paper describes the development of a miniature pump having an impeller with an exit diameter of 24 mm supported with the motor rotor by a fluid dynamic beating. Tests verify that the miniature pump is stable and quiet for rotational speeds larger than 4000 rain-1. The three-dimensional turbulent flow in the entire pump flow passage and the laminar flow in the fluid dynamic bearing were then simulated numerically. The average pump performance was well predicted by the simulations. Both the tests and the simulations show that there is no obvious Reynolds effect for the miniature pump within the tested range of rotational speeds. The numerical results also show that the beating capacity of the fluid dynamic bearing increases with the pump rotor rotational speed and the eccentricity ratio of the journal to the bushing. This pump is very compact, so it is a prom- ising device for surgical use.

A novel flexure-based uniaxial force sensor with large range and high resolution

High performance force sensors often encounter the conflicting requirements of high resolution and large measurement range.To address this problem,this paper presents a conceptual design of a novel uniaxial force sensor with large range and dual-stage force resolutions which enables us to measure forces within a wide range with satisfied resolutions.The newly developed force sensor features an aluminum alloy body with a probe to transfer external forces into the sensing element.It employs an optical linear encoder to detect the displacement of the sensing body.This sensing scheme may immunize outside electromagnetic noises and therefore enhance the performance of the sensor thanks to its digital signal output.In this paper,an accurate,analytical model for calculating the static stiffness and dynamics of the system was developed by using pseudo-rigid-body-model(PRBM)methodology.To optimize the design,finite element simulations were conducted.After a prototype sensor was fabricated,preliminary characterization tests were carried out to verify the accuracy of the theoretical model and demonstrate the effectiveness of the design.The experiment results indicate that the structure of the new sensor is compact,and it has the ability to measure both micro range and macro range forces within one setup,meanwhile keeps very fine resolutions.

Axial Breakthrough in Ancient Greece. New Considerations： Plato, Confucius, and the Buddha

Here I will examine the resemblances and similarities between Plato and two other personages, Confucius and the Buddha, who were his near-contempo- raries, and who enabled a breakthrough. The Platonic break with the world of bodies and the sphere of belief, in favor of a level of intelligible realities, the models of the sensible realities explained by mathematics and moved by the soul, explains the specificity of the Western world, in the field of science, politics, ethics, and even religion.