Offset Approximation of Rational Trigonometric Bezier ´ Curves

Offset curves are one of the crucial curves, but the presence of square root function in the representation is main hindrance towards their applications in CAD/CAM. The presented technique is based on offset approximation using rational trigonometric Bezier curves. The idea is ´ to construct a new control polygon parallel to original one. The two end points of the offset control polygon have been taken as exact offset end points, while the middle control points and weights have been computed using definition of parallel curves. As a result, offsets of rational and nonrational trigonometric Bezier curves have been approximated by rational ´ cubic trigonometric Bezier curve. An error between exact and approxi- ´ mated offset curves have also been computed to show the efficacy of the method.

Experimental Investigation of Film Cooling Performance on Blade Endwall With Diffusion Slot Holes and Stator-Rotor Purge Flow

This paper focuses on the influences of the discrete hole shape and layout on the blade endwall film cooling effectiveness. The effect of upstream purge slot injection on the film cooling performance of the discrete hole was also investigated. The diffusion slot hole was first applied to the blade endwall. As a comparison, the cooling performance of the fan-shaped hole was also measured. Totally, six discrete-hole cooling configurations (2 hole shapes × 3 layouts) were investigated. Experiments were performed in a seven-blade linear cascade with the exit Reynolds number of 2.64 × 105. The average blowing ratios (BR) of the discrete holes changed from 0.5 to 2.5, and the coolant mass flow ratio of the purge slot (MFR) was fixed at MFR = 1.5%. The distributions of the cooling effectiveness on the blade endwall were measured by the pressure sensitive paint technique. Results indicate that the diffusion slot hole significantly increases the film cooling effectiveness on the blade endwall compared to the fan-shaped hole, especially at high blowing ratio. The maximum relative increment of the cooling effectiveness is over 40%. The layout with the discrete holes arranged lining up with the tangent direction of the blade profile offset curves exhibits a comparable film cooling effectiveness with the layout with the discrete holes arranged according to the cross-flow direction. The film cooling effectiveness on the pressure surface corner is remarkably enhanced by deflecting the hole orientation angle towards the pressure surface. The combination of purge slot and diffusion slot holes supplies a full coverage film cooling for the entire blade endwall at MFR = 1.5% and BR = 2.5. In addition, the slot injection leads to a non-negligible influence on the cooling performance of the discrete holes near the separation line.

Analysis of Deformation Characteristics of Foundation-Pit Excavation and Circular Wall

The surrounding ground settlement and displacement control of an underground diaphragm wall during the excavation of a foundation pit are the main challenges for engineering safety. These factors are also an obstacle to the controllable and sustainable development of foundation-pit projects. In this study, monitoring data were analyzed to identify the deformation law and other characteristics of the support structure. A three-dimensional numerical simulation of the foundation-pit excavation process was performed in Midas/GTS NX. To overcome the theoretical shortcomings of parameter selection for finite-element simulation, a key data self-verification method was used. Results showed that the settlement of the surface surrounding the circular underground continuous wall was mainly affected by the depth of the foundation-pit excavation. In addition, wall deformation for each working condition showed linearity with clear staged characteristics. In particular, the deformation curve had obvious inflection points, most of which were located deeper than 2/3 of the overall excavation depth. The characteristics of the cantilever pile were not obvious in Working Conditions 3–9, but the distribution of the wall body offset in a D-shaped curve was evident. Deviation between the monitoring value of the maximal wall offset and the simulated value was only 4.31 %. The appropriate physical and mechanical parameters for key data self-verification were proposed. The concept of the circular-wall offset inflection point is proposed to determine the distribution of inflection-point positions and offset curves. The method provides new opportunities for the safety control and sustainable research of foundation-pit excavations.

Approximating offset Curves using Bezier curves with high accuracy

In this paper, a new method for the approximation of offset curves is presented using the idea of the parallel derivative curves. The best uniform approximation of degree 3 with order 6 is used to construct a method to find the approximation of the offset curves for Bezier curves. The proposed method is based on the best uniform approximation, and therefore; the proposed method for constructing the offset curves induces better outcomes than the existing methods.

Five-Axis Freeform Surface Color Printing Technology Based on Offset Curve Path Planning Method

Great progress has been made in 2D color printing with inkjet technology, and mature related products have come out, but there still exists great developmental space in 3D color printing. Therefore, a new path planning method based on the offset curve for 3D inkjet technology is proposed in this paper. Offset curves are generated on a freeform surface with geodesic equidistance, and then points for color printing are generated along the offset curves. In this paper, the principle of color printing technology with a 5-axis platform and the offset curve path planning (OCPP) method are presented. In addition, comparisons between the OCPP and adaptive filling algorithm based on the section method (AFSM) have been implemented. The OCPP significantly increased the rate of the theoretical filling area from 0.89 to 0.99 on a freeform surface.

Simulation of positioning of the tool axis in five-axis machining

The work analyzed the impact of the positioning of the tool axis on the dynamics of the rotary axis of the machine in the five-axis processing of elements with complex geometry. A strategy for smooth change of positioning of the tool axis based on offset curves has been proposed, and verified using a NC code.

The square-freeness of the offset equation to a rational planar curve, computed via resultants

It is well known [Algebraic properties of plane offset curves, Comput. Aided Geom. Des. 7 (1990) 101–127] that an implicit equation of the offset to a rational planar curve can be computed by removing the extraneous components of the resultant of two certain polynomials computed from the parametrization of the curve. Furthermore, it is also well known that the implicit equation provided by the nonextraneous component of this resultant has at most two irreducible factors [Algebraic analysis of offsets to hypersurfaces, Math. Z. 234 (2000) 697–719]. In this paper, we complete the algebraic description of this resultant by showing that the multiplicity of the factors corresponding to the offset can be computed in advance. In particular, when the parametrization is proper, i.e. when the curve is just traced once by the parametrization, we prove that any factor corresponding to a simple component of the offset has multiplicity 1, while the factor corresponding to the special component, if any, has multiplicity 2. Hence, if the parametrization is proper and there is no special component, the nonextraneous part of the resultant is square-free. In fact, this condition is proven to be also sufficient. Therefore, this result provides a simple test to check whether or not the offset of a given rational curve has a special component, and in turn, whether a given rational curve is the offset of another curve.