CONSTANT DISTANCE BOUNDARIES OF THE t-QUASICIRCLE AND THE KOCH SNOWFLAKE CURVE

LetΓbe a Jordan curve in the complex plane and let Γ_(λ) be the constant distance boundary ofΓ.Vellis and Wu[10]introduced the notion of a(ζ,r_(0))-chordal property which guarantees that,whenλis not too large, Γ_(λ) is a Jordan curve whenζ=1/2 and Γ_(λ) is a quasicircle when 0<ζ<1/2.We introduce the(ζ,r_(0),t)-chordal property,which generalizes the(ζ,r_(0))-chordal property,and we show that under the condition thatΓis(ζ,r_(0),√t)-chordal with 0<ζ<r_(0)^(1−√t)/2,there existsε>0 such that Γ_(λ) is a t-quasicircle once Γ_(λ) is a Jordan curve when 0<λ<ε.In the last part of this paper,we provide an example:Γis a kind of Koch snowflake curve which does not have the(ζ,r_(0))-chordal property for any 0<ζ<1/2,however Γ_(λ) is a Jordan curve whenλis small enough.Meanwhile,Γhas the(ζ,r_(0),√t)-chordal property with 0<ζ<r_(0)^(1−√t)/2 for any t∈(0,1/4).As a corollary of our main theorem, Γ_(λ) is a t-quasicircle for all 0<t<1/4 whenλis small enough.This means that our(ζ,r_(0),t)-chordal property is more general and applicable to more complicated curves.

Lift-Off Effect of Koch and Circular Differential Pickup Eddy Current Probes

A flexible or planar eddy current probe with a differential structure can suppress the lift-off noise during the inspection of defects.However,the extent of the lift-off effect on differential probes,including different coil structures,varies.In this study,two planar eddy current probes with differential pickup structures and the same size,Koch and circular probes,were used to compare lift-off effects.The eddy current distributions of the probes perturbed by 0°and 90°cracks were obtained by finite element analysis.The analysis results show that the 90°crack can impede the eddy current induced by the Koch probe even further at relatively low lift-off distance.The peak-to-peak values of the signal output from the two probes were compared at different lift-off distances using finite element analysis and experimental methods.In addition,the effects of different frequencies on the lift-off were studied experimentally.The results show that the signal peak-to-peak value of the Koch probe for the inspection of cracks in 90°orientation is larger than that of the circular probe when the lift-off distance is smaller than 1.2 mm.In addition,the influence of the lift-off distance on the peak-to-peak signal value of the two probes was studied via normalization.This indicates that the influence becomes more evident with an increase in excitation frequency.This research discloses the lift-off effect of differential planar eddy current probes with different coil shapes and proves the detection merit of the Koch probe for 90°cracks at low lift-off distances.

Describing Chaos of Continuous Time System Using Bounded Space Curve

The qualitative solutions of dynamical system expressed with nonlinear differential equation can be divided into two categories. One is that the motion of phase point may approach infinite or stable equilibrium point eventually. Neither periodic excited source nor self-excited oscillation exists in such nonlinear dynamic circuits, so its solution cannot be treated as the synthesis of multiharmonic. And the other is that the endless vibration of phase point is limited within certain range, moreover possesses character of sustained oscillation, namely the bounded nonlinear oscillation. It can persistently and repeatedly vibration after dynamic variable entering into steady state;moreover the motion of phase point will not approach infinite at last;system has not stable equilibrium point. The motional trajectory can be described by a bounded space curve. So far, the curve cannot be represented by concretely explicit parametric form in math. It cannot be expressed analytically by human. The chaos is a most universally common form of bounded nonlinear oscillation. A number of chaotic systems, such as Lorenz equation, Chua’s circuit and lossless system in modern times are some examples among thousands of chaotic equations. In this work, basic properties related to the bounded space curve will be comprehensively summarized by analyzing these examples.

Algorithm for Smooth S-curve Feedrate Profiling Generation

The algorithms for feedrate profile generation,such as linear and S-curve profiles,have been widely used in machinery controllers,and these algorithms can greatly improve the smoothness of motion.However,most of the algorithms lead to the discontinuous acceleration/deceleration and jerk,or high jerk levels,which is very harmful to machine tool or robot in most occasions. This paper presents a smooth S-curve feedrate profiling generation algorithm that produces continuous feedrate,acceleration,and jerk profiles.Smooth jerk is obtained by imposing limits on the first and second time derivatives of acceleration,resulting in trapezoidal jerk profiles along the tool path.The discretization of smooth S-curve feedrate is realized with a novel approach that improves the efficiency without calculating the deceleration point in each sampled time.To ensure that the interpolation time is a multiple of the value of sampled time,the feedrate,acceleration,jerk,and jerk derivative are recalculated.Meantime,to improve the efficiency,the interpolation steps of all regions are computed before interpolation.According to the distance of trajectory,the smooth S-curve acceleration and decelerations are divided into three blocks：normal block,short block type-Ⅰ,and short block type-Ⅱ.Finally feedrate discretization of short block type-Ⅰand type-Ⅱis obtained with considering the efficiency.The proposed generation algorithm is tested in machining a part on a five axis milling machine,which is controlled with the CNC system for newly developed high-speed machine tools.The test result shows that the smooth S-curve approach has the smoother feedrate,acceleration,deceleration,and jerk profiles than S-curve.The proposed algorithm ensures the automated machinery motion smoothness,and improves the quality and efficiency of the automated machinery motion planning.

A novel type curve for estimating oil recovery factor of gas flooding

A novel type curve is presented for oil recovery factor prediction suitable for gas flooding by innovatively introducing the equivalent water-gas cut to replace the water cut,comprehensively considering the impact of three-phase flow(oil,gas,water),and deriving the theoretical equations of gas flooding type curve based on Tong’s type curve.The equivalent water-gas cut is the ratio of the cumulative underground volume of gas and water production to the total underground volume of produced fluids.Field production data and the numerical simulation results are used to demonstrate the feasibility of the new type curve and verify the accuracy of the prediction results with field cases.The new type curve is suitable for oil recovery factor prediction of both water flooding and gas flooding.When a reservoir has no gas injected or produced,the gas phase can be ignored and only the oil and water phases need to be considered,in this case,this gas flooding type curve returns to the Tong’s type curve,which can evaluate the oil recovery factor of water flooding.For reservoirs with equivalent water-gas cuts of 60%-80%,the regression method of the new type curve works well in predicting the oil recovery factor.For reservoirs with equivalent water-gas cuts higher than 80%,both the regression and assignment methods of the new type curve can accurately predict the oil recovery factor of gas flooding.

Scattering of SH-wave from interface cylindrical elastic inclusion with a semicircular disconnected curve

Scattering of SH wave from an interface cylindrical elastic inclusion with a semicircular disconnected curve is investigated. The solution of dynamic stress concentration factor is given using the Green＇s function and the method of complex variable functions. First, the space is divided into upper and lower parts along the interface. In the lower half space, a suitable Green＇s function for the problem is constructed. It is an essential solution of the displacement field for an elastic half space with a semi-cylindrical hill of cylindrical elastic inclusion while bearing out-plane harmonic line source load at the horizontal surface. Thus, the semicircular disconnected curve can be constructed when the two parts are bonded and continuous on the interface loading the undetermined anti-plane forces on the horizontal surfaces. Also, the expressions of displacement and stress fields are obtained in this situation. Finally, examples and results of dynamic stress concentration factor are given. Influences of the cylindrical inclusion and the difference parameters of the two mediators are discussed.

Analysis Approach to Durability Based on Material Initial Fatigue Quality and S-N Curve

Based on probabilistic fracture mechanics approach, a new concept of material initial fatigue quality （MIFQ） is developed. Then, the relation between S-N curve and crack propagation curve is studied. From the study, a new durability analysis method is presented. In this method, S-N curve is used to determine crack growth rate under constant amplitude loading and evaluate the effects of different factors on durability and then the structural durability is analyzed. The tests and analyses indicate that this method has lower dependence on testing, and higher accuracy, reliability and generality and is convenient for application.

Reconstruction of probabilistic S-N curves under fatigue life following lognormal distribution with given confidence

When the historic probabilistic S-N curves are given under special survival probability and confidence levels and there is no possible to re-test, fatigue reliability analysis at other levels can not be done except for the special levels. Therefore, the wide applied curves are expected. Monte Carlo reconstruction methods of the test data and the curves are investigated under fatigue life following lognormal distribution. To overcome the non-conservative assessment of existent man-made enlarging the sample size up to thousands, a simulation policy is employed to address the true production where the sample size is controlled less than 20 for material specimens, 10 for structural component specimens and the errors matching the statistical parameters are less than 5 percent. Availability and feasibility of the present methods have been indicated by the reconstruction practice of the test data and curves for 60Si2Mn high strength spring steel of railway industry.

A new method of testing pile using dynamic P-S-curve made by amplitude of wave train

A new method of detecting the vertical bearing capacity for single-pile with high strain is discussed in this paper. A heavy hammer or a small type of rocket is used to strike the pile top and the detectors are used to record vibra- tion graphs. An expression of higher degree of strain (deformation force) is introduced. It is testified theoretically that the displacement, velocity and acceleration cannot be obtained by simple integral acceleration and differential velocity when long displacement and high strain exist, namely when the pile phase generates a whole slip relative to the soil body. That is to say that there are non-linear relations between them. It is educed accordingly that the force P and displacement S are calculated from the amplitude of wave train and (dynamic) P-S curve is drew so as to determine the yield points. Further, a method of determining the vertical bearing capacity for single-pile is dis- cussed. A static load test is utilized to check the result of dynamic test and determine the correlative constants of dynamic-static P(Q)-S curve.

Hölder Derivative of the Koch Curve

In this paper, we introduce a K H&#246;lder p-adic derivative that can be applied to fractal curves with different H&#246;lder exponent K. We will show that the Koch curve satisfies the H&#246;lder condition with exponent and has a 4-adic arithmetic-analytic representation. We will prove that the Koch curve has exact -H&#246;lder 4-adic derivative.

Curve integral with path independent in orthogonal curvilinear coordinate system

It is explored that the line integral is a path independent in two or three arbitrary dimensional orthogonal curvilinear coordinate systems, which is based on the integral condition with the path independent in two or three dimensional rectangular coordinate systems. Firstly, according to the coordinate transformation, the condition that the line integral is the path independent in the polar coordinate system is obtained easily from the Green's theorem in two-dimensional rectangular coordinate system and the condition is extended to arbitrary two-dimension orthogonal curvilinear coordinates. Secondly, through the coordinate transformation relationship and the area projection method, the Stokes formula in three-dimensional rectangular coordinate system is promoted to the spherical coordinate system and cylindrical coordinate system, and the condition that the line integral is a path independent is obtained. Furthermore, the condition is extended to arbitrary three-dimension orthogonal curvilinear coordinates. Lastly, the conclusions are made.

Algorithms and Identities for(q, h)-Bernstein Polynomials and(q, h)-Bézier Curves–A Non-Blossoming Approach

We establish several fundamental identities, including recurrence relations, degree elevation formulas, partition of unity and Marsden identity, for quantum Bernstein bases and quantum Bezier curves. We also develop two term recurrence relations for quantum Bernstein bases and recursive evaluation algorithms for quantum Bezier curves. Our proofs use standard mathematical induction and other elementary techniques.

GRAPHICAL COMPUTING METHOD ON FOUR-BAR LINKAGE(1)——Inflexion Circle,Focal Axis,Ball's Point,Centering-point Curve and Circling-point Curve For Four ISPs,Coupler Curves Based on Ball's Points——

By means of programs GTMPAC based- on generalized triangle method,analysis and synthesis of mechanism design in accordance with absolutely graphicalmethod( absolutely germetrical method) are developed.In this paper,we make aspecial study about centering- point curve and circling- point curve and couplercurves based on Ball’s points.

General solutions of plane problem for power function curved cracks

A new exact and universal conformal mapping is proposed. Using Muskhelishvili＇s complex potential method, the plane elasticity problem of power function curved cracks is investigated with an arbitrary power of a natural number, and the general solutions of the stress intensity factors （SIFs） for mode I and mode II at the crack tip are obtained under the remotely uniform tensile loads. The present results can be reduced to the well-known solutions when the power of the function takes different natural numbers. Numerical examples are conducted to reveal the effects of the coefficient, the power, and the projected length along the x-axis of the power function curved crack on the SIFs for mode I and mode II.

GRAPHICAL COMPUTING METHOD ON FOUR—BAR LINKAGE(2)——Burmester's Center-point Curve,Or Rather Burmester's Pole-locus Curve——

By means of programs GTMPAC based on generalized triangle method,analysisand synthesis of mechanism design in accordance with absolutely graphical method( absolutely germetrical method) are developed.In this paper,we make a specialstudy about Burmester’s curve.

The Assessment of Heavy Rains in the Region of Annaba （NE Algeria） to Improve Extreme Flood Forecasting ,, Use of Depth-Duration-Frequency Curves

The knowledge of space and time distribution of short duration rainfall depths is of primary interest in hydrotechnical studies and in extreme floods modelling. This work is carried out to establish a ＂DDF （Depth-Duration-Frequency）＂ relationship for the region of Annaba （Northeast of Algeria） through the examples of Pont Bouchet （El Hadjar）, Ain Berda and Chafia-Dam rainfall gauges. The results of the frequency study of stochastically generated annual series （Pearson＇s distribution type III model） and the regression analysis （least square method） permitted to develop a master curve that well describes heavy rainfalls distribution in the Annaba region. This 2-parameter model is used to predict, with sufficient accuracy, the amount of rain that could be recorded over a shorter duration from daily rainfall data in basins that lack recording rain gauges.

A New Shape Adjustment Method of S-λ Curve

In this paper, we study a new method in shape adjustment ofS-λcurves. By examples of finite and infinite degree generation function, we get the conclusion thatS-λcurves can be adjusted by perturbation of generation functionS（t）while the control points and parameters hold.

Study on Project Progress Curve Model

Because of the discrepancy of project types,the project progress curves present different characteristics.Studying project progress curves can reduce management risk of project and overall grasp the enforcement condition of the project.Combining project characteristic,this paper reaches 4 kinds of project progress curve patterns.If the front of the progress curve is concave,and its rear is protruding in the break point,it is named as S model.If its front is protruding and its rear is concave in the break point,it is named as the reverse side of S model.If the front and rear are concave in the break point,it is named as J model,and two half sections are both upward protruding,it is named as the reverse side of J model.Through a case study,it shows that application project progress curve model can better raise project management.

A new method for testing pile by sin- gle-impact energy and P-S curve

By studying the pile-formula and stress-wave methods (e.g., CASE method), the authors propose a new method for testing piles using the single-impact energy and P-S curves. The vibration and wave figures are recorded, and the dynamic and static displacements are measured by different transducers near the top of piles when the pile is im- pacted by a heavy hammer or micro-rocket. By observing the transformation coefficient of driving energy (total energy), the consumed energy of wave motion and vibration and so on, the vertical bearing capacity for single pile is measured and calculated. Then, using the vibration wave diagram, the dynamic relation curves between the force (P) and the displacement (S) is calculated and the yield points are determined. Using the static-loading test, the dynamic results are checked and the relative constants of dynamic-static P-S curves are determined. Then the sub- sidence quantity corresponding to the bearing capacity is determined. Moreover, the shaped quality of the pile body can be judged from the formation of P-S curves.