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yueqianxing新虫 (小有名气)
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Equivolumetric tubular solids for volume-preserving bend of cylinders
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有没有中文版的文章，这是一篇英文的文献，看得有些吃力，烦请大家帮忙。Equivolumetric tubular solids for volume-preserving bend of cylinders 1、We present the equivolumetric tubular solid which is a model of volume-preserving bend of right cylinders. The equivolumetric tubular solid is a special class of tubular solids which are the generalization of pipe solids or normal ringed solid. For a given spine curve, the volume of the sectorial tube of a pipe solid suffers distortion due to the curvature of the spine curve. The radial distance function of the equivolumetric tubular solid is elaborately chosen so as to compensate for this curvature effect. We also investigate the algebraic property of the equivolumetric torus which is an equivolumetric tubular solid with circular spine curve. Finally, we prove that the equivolumetric tubular surface is a limit of an equivolumetric offset surface of the pipe surface along the spine curve。 2、The tubular surfaces, including their special instances such as pipe surfaces and normal ringed surface, are geometric objects which are important in designing a thin volume along a given space curve. The radius of a pipe surface is a constant, and the radius of the normal ringed surface is a function of the curve parameter. The tubular surface, as the generalization of the normal ringed surface, has the radius function which depends on both the curve parameter and the angular parameter. So we can have more flexibility with the tubular surfaces in designing a thin volume than with the normal ringed surfaces. Our main interest in this paper is the volume variation of the tubular solids depending on the curvature of the spine curve. As we will show later, it is easy to see that the solid generated by the pipe surface of radius r with the spine curve of length L has the same volume as the cylinder of radius r and of height L . However, if we cut this pipe solid along the envelope of the rectifying plane of the spine curve, then two separated pieces of solids have different volumes in general. If the spine curve has no inflection points, then the piece of solid located in the direction of the normal vector of the spine curve has the smaller volume than the other piece. So if we consider the pipe solid as the result of the bending process of a right solid, the local volume is distorted due to the curvature effect although the total volume remains the same. Our goal is to devise a construction scheme for a tubular solid which preserves the local volume of the original cylinderunder the bending process. In order to clarify the concept of the local volume, we will introduce the cylindrical angle to measure the size of the locality of the tubular solids. Then the equivolumetric tubular solid is defined as the tubular solid which exhibits the property that the volume of any sector of the solid is proportional to its cylindrical angle. So the equivolumetric tubular solid preserves the volume of any locality of the original cylinder under the bending process. To construct such tubular solid, its radial distance function should be chosen carefully. We will analyze the curvature effect on the local volume and derive the equation for the radial distance function of the equivolumetric tubular solid. This equation is a cubic equation which can be solved explicitly by employing the Chebyshev cube root.The concept of the equivolumetric offset was first introduced (Moon, 2008) in the context of 2-dimensional pocket machining. In the CNC machining practice, the material removal rate is an important machining parameter to control the cutting force. So one may need to control the feedrate to achieve the constant material removal rate (Farouki et al., 1998). In Moon (2008), we devised the machining method to maintain both the material removal rate and the feedrate as constants by using the equivolumetric offset curves. Then we showed this machining scheme can also be applied to control both the material removal rate and the contact trajectory speed. To do this, we need the equivolumetric offset curve with constant contact trajectory speed. This type of offset curve is a special class of variable distance offsets which has the property that the area between the original curve and its offset is proportional to the arc length of the original curve. This geometric concept has been extended to the higher dimension yielding the equivolumetric offset surface (Moon,2009). For a given surface, the equivolumetric offset surface is defined by the property that the volume between the original surface and its equivolumetric offset surface is proportional to the area of the original surface. The proportionality constant is called the volumetric ratio. The equivolumetric offset surface is a variable distance offset whose offset distance satisfies a cubic equation with the coefficient depending on the mean curvature and the Gaussian curvature of the original surface. We could derive the explicit expression of the offset distance function by using the Chebyshev cube root.The equivolumetric tubular solid can be regarded as a limiting case of the equivolumetric offset surface. For a given spine curve, we consider a pipe surface along the spine curve with very small radius. Then the equivolumetric offset surface of the pipe surface with appropriate volumetric ratio converges to the equivolumetric tubular surface as the radius goes to 0. We will prove this relation later in this paper.The remainder of the paper is structured as follows. In Section 2, we summarize the basics of tubular surfaces and frames, and fix the notations and the conventions. Then we analyze the curvature effect on the volume of tubular solids in Section 3.Section 4 defines the equivolumetric tubular solid and provides the explicit expression of the radial distance function of the equivolumetric tubular solid. We introduce the equivolumetric torus as a special instance of the equivolumetric tubular solid, and show that it is a surface of revolution whose cross section is a component of an algebraic curve of degree 3 in Section 5. In Section 6, we prove that the equivolumetric tubular solid is the limit of equivolumetric offset surface of the pipe surface with appropriate volumetric ratio. Finally, Section 7 concludes the paper. |
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2015-01-15 10:57:52, 588.6 K
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