A NECESSARY AND SUFFICIENT CONDITION FOR THE COEXISTENCE OF A CLASS OF CUBIC CURVE SEPARATRIX CYCLES AND LIMIT CYCLES TO THE CUBIC SYSTEM

In this paper, we give the necessary and sufficient condition for the coexistence of a class of cubic curve separatrix cycles and limit cycles to the cubic system, and study their topological structures.

A Finite Element Cable Model and Its Applications Based on the Cubic Spline Curve

For accurate prediction of the deformation of cable in the towed system, a new finite element model is presented that provides a representation of both the bending and torsional effects. In this paper, the cubic spline interpolation function is applied as the trial solution. By using a weighted residual approach, the discretized motion equations for the new finite element model are developed. The model is calculated with the computation program complier by Matlab. Several numerical examples are presented to illustrate the numerical schemes. The results of numerical simulation are stable and valid, and consistent with the mechanical properties of the cable. The model can be applied to kinematics analysis and the design of ocean cable, such as mooring lines, towing, and ROV umbilical cables.

Development of Cubic Bezier Curve and Curve-Plane Intersection Method for Parametric Submarine Hull Form Design to Optimize Hull Resistance Using CFD

Optimization analysis and computational fluid dynamics （CFDs） have been applied simultaneously, in which a parametric model plays an important role in finding the optimal solution. However, it is difficult to create a parametric model for a complex shape with irregular curves, such as a submarine hull form. In this study, the cubic Bezier curve and curve-plane intersection method are used to generate a solid model of a parametric submarine hull form taking three input parameters into account： nose radius, tail radius, and length-height hull ratio （L/H）. Application program interface （API） scripting is also used to write code in the ANSYS DesignModeler. The results show that the submarine shape can be generated with some variation of the input parameters. An example is given that shows how the proposed method can be applied successfully to a hull resistance optimization case. The parametric design of the middle submarine type was chosen to be modified. First, the original submarine model was analyzed, in advance, using CFD. Then, using the response surface graph, some candidate optimal designs with a minimum hull resistance coefficient were obtained. Further, the optimization method in goal-driven optimization （GDO） was implemented to find the submarine hull form with the minimum hull resistance coefficient （Ct）. The minimum C, was obtained. The calculated difference in （7, values between the initial submarine and the optimum submarine is around 0.26%, with the C, of the initial submarine and the optimum submarine being 0.001 508 26 and 0.001 504 29, respectively. The results show that the optimum submarine hull form shows a higher nose radius （rn） and higher L/H than those of the initial submarine shape, while the radius of the tail （r1） is smaller than that of the initial shape.

Fairing of Parametric Cubic B-spline Curves and Bicubic B-spline Surfaces

A method of fairing parametric cubic B_spline curves and bicubic B_spline surfaces is presented. The basic idea of the method is to reposition the control points by an optimization process.A new objectijve function presented is based on the variation of the third order derivatives of the cubic B_spline curves and bicubic B_spline surfaces at the nodes. The curves and surfaces faired using this method tend to possess curvature continuities. The numerical examples show that the effect of this method is acceptable.

Research on cubic polynomial acceleration and deceleration control model for high speed NC machining

To satisfy the need of high speed NC (numerical control) machining, an acceleration and deceleration (acc/dec) control model is proposed, and the speed curve is also constructed by the cubic polynomial. The proposed control model provides continuity of acceleration, which avoids the intense vibration in high speed NC machining. Based on the discrete characteristic of the data sampling interpolation, the acc/dec control discrete mathematical model is also set up and the discrete expression of the theoretical deceleration length is obtained furthermore. Aiming at the question of hardly predetermining the deceleration point in acc/dec control before interpolation, the adaptive acc/dec control algorithm is deduced from the expressions of the theoretical deceleration length. The experimental result proves that the acc/dec control model has the characteristic of easy implementation, stable movement and low impact. The model has been applied in multi-axes high speed micro fabrication machining successfully.

Discussion on relationship between minimal energy and curve shapes

Energy minimization has been widely used for constructing curve and surface in the fields such as computer-aided geometric design, computer graphics. However, our testing examples show that energy minimization does not optimize the shape of the curve sometimes. This paper studies the relationship between minimizing strain energy and curve shapes, the study is carried out by constructing a cubic Hermite curve with satisfactory shape. The cubic Hermite curve interpolates the positions and tangent vectors of two given endpoints. Computer simulation technique has become one of the methods of scientific discovery, the study process is carried out by numerical computation and computer simulation technique. Our result shows that： （1） cubic Hermite curves cannot be constructed by solely minimizing the strain energy; （2） by adoption of a local minimum value of the strain energy, the shapes of cubic Hermite curves could be determined for about 60 percent of all cases, some of which have unsatisfactory shapes, however. Based on strain energy model and analysis, a new model is presented for constructing cubic Hermite curves with satisfactory shapes, which is a modification of strain energy model. The new model uses an explicit formula to compute the magnitudes of the two tangent vectors, and has the properties： （1） it is easy to compute; （2） it makes the cubic Hermite curves have satisfactory shapes while holding the good property of minimizing strain energy for some cases in curve construction. The comparison of the new model with the minimum strain energy model is included.

Exact parametric representations of orbits defined by cubic Hamiltonian

In this paper,we show that for any given planar cubic algebraic curves defined by a quadratic Hamiltonian vector field,we can always have their exact explicit parametric representations. We use a model of micro-structured solid to show an application of our conclusions.

A Hybrid Path Planning Method Based on Articulated Vehicle Model

Due to the unique steering mechanism and driving characteristics of the articulated vehicle,a hybrid path planning method based on the articulated vehicle model is proposed to meet the demand of obstacle avoidance and searching the path back and forth of the articulated vehicle.First,Support Vector Machine(SVM)theory is used to obtain the two-dimensional optimal zero potential curve and the maximum margin,and then,several key points are selected from the optimal zero potential curves by using Longest Accessible Path(LAP)method.Next,the Cubic Bezier(CB)curve is adopted to connect the curve that satisfies the curvature constraint of the articulated vehicle between every two key points.Finally,Back and Forth Rapidly-exploring Random Tree with Course Correction(BFRRT-CC)is designed to connect paths that do not meet articulated vehicle curvature requirements.Simulation results show that the proposed hybrid path planning method can search a feasible path with a 90-degree turn,which meets the demand for obstacle avoidance and articulated vehicle back-and-forth movement.

A New Method to Design Cubic Pythagorean-Hodograph Spline Curves with Control Polygon

A new method to design a cubic Pythagorean-hodograph(PH)spline curve from any given control polygon is proposed.The key idea is to suitably choose a set of auxiliary points associated with the edges of the given control polygon to guarantee the constructed PH spline has G1 continuity or curvature continuity.The method facilitates intuitive and efficient construction of open and closed cubic PH spline curves that typically agrees closely with the same friendly interface and properties as B-splines,for example,the convex hull and variation-diminishing properties.

The Location of Singular and Inflection Points for Planar Cubic B-Spline Curve

Using vectors between control points(a_i=P_(i+1)-P_i),parameters λ and μ(such that a_(i+1)=λ_(ai+μ_(a_i+2))are used to study the shape classification of planar parametric cubic B-spline curves. The regiosn of λμ space corresponding to different geometric features on the curves are investigated.These results are useful for curve design.

TIME-OPTIMAL INTERPOLATION FOR CNC MACHINING ALONG CURVED TOOL PATHES WITH CONFINED CHORD ERROR

In this paper, two new interpolation algorithms lot CNC machining along curve^l tom pathes are proposed： a time-optimal interpolation algorithm under chord error, feedrate, and tangential acceleration bounds, and a greedy interpolation algorithm under the chord error and tangential jerk bounds. The key idea is to reduce the chord error bound to a centripetal acceleration bound which leads to a velocity limit curve, called the chord error velocity limit curve. Then, the velocity planning is to find the proper velocity curve governed by the acceleration or jerk bounds ＇~under＂ the chord error velocity limit curve. For two types of simple tool pathes, explicit formulas for the velocity curve are given and the methods are implemented in commercial CNC controllers.

Hilbert Problem 15 and Ritt-Wu Method(Ⅱ)

This paper proves three statements of Schubert about cuspal cubic curves in a plane by using the concept of generic point of Van der Waerden and Weil and Ritt-Wu methods.They are relations of some special lines:1)For a given point,all the curves containing this point are considered.For any such curve,there are five lines.Two of them are the tangent lines of the curve passing through the given point.The other three are the lines connecting the given point with the cusp,the inflexion point and the intersection point of the tangent line at the cusp and the inflexion line.2)For a given point,the curves whose tangent line at the cusp passes through this point are considered.For any such curve,there are four lines.Three of them are the tangent lines passing through this point and the other is the line connect the given point and the inflexion point.3)For a given point,the curves whose cusp,inflexion point and the given point are collinear are considered.For any such curve,there are five lines.Three of them are tangent lines passing through the given point.The other two are the lines connecting the given point with the cusp and the intersection point of the tangent line at the cusp and the inflexion line.