Intersecting the characteristic K and the great S circles The local and reference frames of a cutter As mentioned in the introduction, throughout this chapter it is assumed that the cutter follows a rigid motion in E3. As shown in Figure 1. Answered Mar 21, Feasible Contact Surfaces R: This prototype system creates STL model of the in-process workpiece.
The discrete vectors with their orientations in the directions of x,y,z-axes of R 3 have been used. If the toolpath has self intersections then for obtaining the correct CWEs it may be necessary to decompose the given toolpath into non-intersecting smaller segments. For example a Flat-End mill has a cylindrical part at the side and a flat part at the bottom. This technique, called imprint or cross product method, uses vector algebra for obtaining the points on the swept envelope and thus eliminates the use of the complicated SEDE equations. Another drawback in this work is that all discrete vectors of the workpiece model lie in one direction regardless of surface normal directions, where the directions are along the vertical z-axis of a Cartesian coordinate system Figure 2. The discretization of the workpiece surfaces are done according to the computer graphics image of the workpiece surfaces. The method is based on discretizing the tool into pseudo-inserts Figure 2.
Want to get paid to write? Because of the complexity of the torus shape these calculations have been made by using the numerical root finding methods. First finding triangles that intersect the cutter, then intersecting those triangles with the cutter for obtaining the intersection curve segments and finally using those curve segments to form the closed engagement region.
CWEs are extracted in geometric modeling for supporting cutting force prediction in process modeling. B-rep models contain two parts: In-process Workpiece Modeling 82 Figure 4.
The most commonly used shapes are triangles and converting the mathematically precise models to the triangulated models is called tessellation. The mathematical formulation of the swept volume problem has been investigated using jacobian rank deficiency method dissrrtation, sweep differential equation SDE , envelope theory [53,61,82], implicit modeling  shaep Minkowski sums .
However challenges exist to making this approach both efficient and generally robust to handle degenerate geometric conditions that can occur when large numbers of tool paths need to be simulated. The chordal error in cutter facet intersections If the cutter surfaces parameterized in section 4.
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The parametric representations of canal surfaces in one-parameter family of spheres are presented in section 3. In disserhation following two subsections the analytical formulas will be derived for these surfaces.
Also the tool center point F u is the center of the lower base. What is an example fissertation a dissertation? The following remarks and the property motivate finding grazing points.
Literature Review 20 2. On the other hand for the Fillet-End mill two spheres are shown, green sphere is for the torus part and the red sphere is for the cylinder part respectively.
The formulas for the cylindrical and bottom flat parts can be used from section 4. In this chapter an analytical approach for determining the shape of the swept envelopes generated by a general surface of revolution has been presented.
How long does it take to write a typical PhD thesis? Producing proefficinents be careful not to manage my dissertation. The vector based workpiece model and the localization of the cutter envelope are presented disseration section 4.
These models have become pervasive in supporting engineering applications. The edges of facets deviate from the real surface Literature Review 31 Figure 2.
On the other hand if a sphere with a constant radius follows a spine curve trajectory, it envelopes a diissertation canal surface called a pipe surface. In process modeling, the cutting forces are predicted by utilizing CWEs obtained from geometric modeling. The dimensions of the bounding cylinder are defined by the height and the largest radius of the cutter. Intersecting a segment against a plane For example in a plunging motion of a Flat-End mill the bottom flat surface removes material.
Identifying these regions is critical to simplifying the CWE extraction calculations for a wide range of cutters performing multi-axes machining. Expanding this denominator yields Chapter4.
In this section by utilizing the two-parameter families of spheres, the closed form swept profile equations for the general surface of revolution will be derived. The step taken between each CWE calculation is the feed per revolution of the cutter.