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Analytic Representation of Envelope Surfaces Generated by Motion of Surfaces of Revolution Ivana Linkeová

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Analytic Representation of Envelope Surfaces Generated by Motion of Surfaces of Revolution Ivana Linkeová International Journal of Scientific & Engineering Research Volume 8, Issue 8, August-2017 1577 ISSN 2229-5518 Analytic Representation of Envelope Surfaces Generated by Motion of Surfaces of Revolution Ivana Linkeová Abstract—A modified DG/K (Differential Geometry/Kinematics) approach to analytical solution of envelope surfaces generated by continuous motion of a generating surface – general surface of revolution – is presented in this paper. This approach is based on graphical representation of an envelope surface in parametric space of a solid generated by continuous motion of the generating surface. Based on graphical analysis, it is possible to decide whether the envelope surface exists and recognize the expected form of unknown analytical representation of the envelope surface. The obtained results can be used in application of envelope surfaces in mechanical engineering, especially in 3-axis and 5-axis point and flank milling of freeform surfaces. Index Terms—Characteristic curve, Eenvelope surface, Explicit surface, Flank milling, Parametric curve, Parametric surface, Surface of revolution, Tangent plane, Tangent vector. ———————————————————— 1INTRODUCTION ETERMINATION of analytic representation of an enve- analysis of the problem in parametric space of the generated D lope surface generated by continuous motion of a gener- solid. Based on graphical analysis, it is possible to decide ating surface in sufficiently general conceptionof the whether the envelope surface exists or not. If the envelope generating surface as well as the trajectory of motion is a very surface exists, it is possible to recognize expected form of ana- challenging problem of kinematic geometry and its applica- lytical representation of the envelope surface and determine tions in mechanical engineering [1], [2]. From application its concrete expression. point of view, the generating surface corresponds to the mill- The paper is organized as follows. The specification of mo- ing tool in CNC (Computer Numeric Control) milling, the tion is given in Section 2. Section 3 deals with the definition trajectory of motion corresponds to the milling path and the and basic properties of an envelope surface generated by uni- resulting envelope surface corresponds to the machined sur- variate motion of a generating surface. The method developed face. for determination of analytical representation of the envelope Academic textbooks describe primarily synthetical solution surface is described in Section 4. Examples of envelope sur- of envelope surface problem based on pointwise construction faces generated by motion of general surface of revolution are of characteristic curve of envelope surface, i.e. the curve along given in Section 5. which the generating and envelope surfaces are touching at each instant during the wholeIJSER motion. If an analytical solution OTION is published, it is either restricted to implicit expression of the 2 M figures or such a motion (linear motion, rotation, helical mo- Envelope surface is a figure generated by motion of another tion) and such a generating surface (plane [3] or spherical surface. To be able to investigate the generated figure, it is surface [4], [5]) are chosen to obtain envelope surface its ana- necessary to specify the motion, first. Let ( ), ( ), ( ), ( ) lytical representation is known in advance of solution. So be Cartesian local coordinate system of moving figure. Ana- called DG/K (Differential Geometry/Kinematics) approach to lytic representation of motion along a trajectory�Ω ( ) in ho-� find analytical representation of envelope surface in suffi- mogenous coordinates of projective extension of Euclidean ciently general conception can be find in [2], [6]. In DG/K three-dimensional space (3D space) given by approach, the envelope surface is considered to be the surface ( ) = ( ( ), ( ), ( ), 1), [ 1, 2] (1) which is touching every surface of the univariate family of is transformation matrix generating surfaces along a characteristic curve. Then, the ( ) ( ) () ∈ 0 envelope surface is represented by a set of characteristic ( ) ( ) ( ) 0 curves. ( ) = , [ 1, 2], (2) ( ) ( ) ( ) 0 In this paper, a modified DG/K approach is described. The ⎛ ⎞ ( ) ( ) ( ) 1 ∈ envelope surface is considered to be the surface in the solid ⎜ ⎟ generated by univariate motion of the generating surface at where ( ), ( ), ( ) and ( ), ( ), ( ) and ( ), ( ), ( ) are cosine of angles formed by axes ( ), ( ) and ( ) of whose every pointall three tangent vectors to all three para- ⎝ ⎠ local coordinate system of moving figure and axes of world metric curves of this solid are coplanar. The main contribution of this approach consists in implementation of a graphical coordinate system , and in the given order (provided that ( ), ( ) and ( ) are univariate real functions of real variable ———————————————— , defined, continuous and at least once differentiable on • Ivana Linkeová is an Associate Professor in Department of Technical [ 1, 2]). Mathematics of Faculty of Mechanical Engineering at Czech Technical If the trajectory is a freeform curve, the motion is called University in Prague, Czech Republic, e-mail: [email protected] curvilinear. An example of curvilinear motion is motion of IJSER © 2017 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 8, August-2017 1578 ISSN 2229-5518 milling tool in CNC 5-axis milling. Two types of motion can be of tangent vectors ( , , ), ( , , ) and ( , , ) can be distinguished: general motion when the mutual position of the calculated as the determinant ( , , ) of matrixhaving the local coordinate system of moving figure with respect to the three vectors as its rows (or columns), i.e. natural coordinate system of trajectory (given by tangent, ( , , ) ( , , ) ( , , ) normal and binormal line of the trajectory) is preserved dur- ( , , ) = ( , , ) ( , , ) ( , , ) . (8) ing the whole motion and translational motion when the mu- (, , ) (, , ) (, , ) tual position of the local coordinate system of moving figure If there exists� a solution of equation � with respect to the world coordinate system is preserved dur- ( , , ) = 0, (9) ing the whole motion. then this solution is analytical representation of unknown envelope surface in parametric space ( , , ) of the solid 3ENVELOE SURFACE ( , , ) generated by motion of the generating surface. Example of the whole procedure of envelope surface gen- To derive generally analytical representation of an envelope eration is given in fig. 1. The generating surface is a general surface generated by motion of a generating surface, let us surface of revolution ( , ) given by axis of revolution and consider the generating surface in 3D space with analytic rep- meridian , see fig. 1 (a). In the case of general surface of resentation revolution the meridian is a freeform curve. To simplify the ( , ) = ( ( , ), ( , ), ( , ), 1), picture, the trajectory of motion is planar freeform curve ( ) [ , ], [ , ], (3) 1 2 1 2 located in the plane perpendicular to the axis . Several posi- and transformation matrix (2) of continuous motion. Then a tions of the generating surface along the trajectory is drawn in solid ∈ ∈ fig. 1 (b). The generating surface moves by general motion ( , , ) = ( , ) ( ) , because its position with respect to the trajectory is preserved. [ ] [ ] [ ] 1, 2 , 1, 2 , 1, 2 (4) In particular, the meridian plane of the generating surface and ∙ is generated by motion of the generating surface (3) along the the normal plane of the trajectory are identical. The solid ∈ ∈ ∈ trajectory (1). A part ( , , ) generated by motion of generating surface along the ( , ), [ 1, 2], [ 1, 2] (5) trajectory is drawn in fig. 1 (c). The resulting envelope surface of superficies of this solid is called envelope surface if the (, ) together with characteristic curve ( ) is shown in fig. 1 following conditions ∈ are ∈ satisfied. (d). Note that in practical applications, this type of motion and 1. Surface (5) and any -parametric surface envelope surface generation corresponds to 2.5-axis flank ( , , ), [ 1, 2], = 0,1, , (6) milling. of solid (4) are tangent along -parametric curve ( , ) of surface (5) called∈ characteristic ⋯ curve of envelope surface. 2. At each point of surface (5) exist s common tangent plane and normal line of surface (5) and one -parametric sur- face (6) of solid (4). 3. A surface which is simultaneouslyIJSER a part of surface (5) and any -parametric surface (6) does not exist. As the characteristic curve is located on the generating sur- face, each point of characteristic curve moves together with the generating surface. To satisfy the contact of the generating and envelope surfaces along the characteristic curve, the common tangent plane of the generating and envelope surface has to exist. The tangent plane at any point of the generating surface is given by tangent vectors to the parametric - and - curves of the generating surface. Considering the motion of the generating surface along the trajectory, it is necessary to exclude points of the generating surface located inside the solid (4). To exclude these points, the tangent plane has to Fig. 1: Process of envelope surface generation contain the tangent vector of parametric -curve of the solid, too. Obviously, this condition is fulfilled in case the tangent vectors of all three parametric curves of solid (4) are
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