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Procedia Technology 19 (2015) 141 - 144
8th International Conference Interdisciplinarity in Engineering, INTER-ENG 2014,9-10 October
2014, Tirgu-Mures, Romania
New geometrical aspects regarding ZPA worms
Alexandra Pozdirca*
Petru Maior University of Targu Mures, Str. Nicolae Iorga nr. 1, 540088, Romania
Abstract
The article presents new geometrical aspects regarding the ZPA worms - introduced by the author (a worm generated on NC lathe with standard cylindrical milling tools). First, the article reminds the definition of the ZPA worm and then presents details regarding the root fillet of worm's flank. In this regard is determined the contact line of a torus end mill tools and the helicoidally generated surface. This process is assisted by a CAD software, developed by the author. Controlling of the root fillet area is important to protect the edge of the flank finishing milling tool. The advantages of the ZPA worm type are: (i) the use of standard tools with reduced costs and also short processing time; (ii) the possibilities to cover other standard worms, using the standard cylindrical milling tools and non-specialized NC machining tools.
© 2015 PublishedbyElsevierLtd.Thisis anopen access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of "Petru Maior" University of Tirgu Mures, Faculty of Engineering
Keywords: Cylindrical worm; NC Lathe; Technology; ZPA.
1. Introduction
According to DIN 3975 standard [l],the cylindrical worms are classified as:
• ZA - worm gear with straight-line tooth profile in axial section.
• ZN - worm with a straight-line tooth profile in normal tooth section.
• ZE - worm with a straight-line tooth profile in a plane tangent to the main cylinder.
• ZK - profile formed by a cone ground using a wheel and/or shank tool.
• ZC - concave tooth profile.
* Corresponding author. Tel.: +40-745-510491. E-mail address: alexandru.pozdirca@upm.ing.ro
2212-0173 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of "Petru Maior" University of Tirgu Mures, Faculty of Engineering doi: 10. 1016/j .protcy.2015.02.021
The geometry and generation of each listed worm type are well known in literature [2, 3]. The worms ZA, ZN and ZE can be generated using a lathe tool and the resulted flank of worm is a ruled surface. The worms ZK are generated using conical tools with straight-line generatrix; the resulted flank of the worm is a non-ruled surface. The worm ZC is similar with ZK2; the generating surface is a revolution surface with circular arc as axial profile.
The article [4] proposes a new type of worm, called ZPA, generated by a cylindrical milling tool. The tool position relative to worm is determined by three parameters: angle a, radius r; and eccentricity e - as relative displacement along the axis Z (see Fig. 1). We seek to presents details regarding the root fillet area.
2. Contact curve torus end mill - worm
Let consider the milling position for fillet manufacturing with a torus end mill. In the proposed position, the axial lines of the torus end mill tool intersect the axial line of the worm (see Fig. 2).
Fig. 2. The torus end mill position for root filleting of the ZPA worm
In order to determine the contact curve between a torus end mill tool and the generated worm, we write the parametrical equations of the torus, in own system, and also of the unit normal, as follows (see Fig. 3):
Fig. 1. The cylindrical tool relative to the processed worm, with eccentricity
xl - (Rl + R2 sin v) cos u
yl - -R2 cos v
zl - -(Rl + R2 sin v) sin u
nxl - sin V COS u
/respectively \nyl -- cos v
nzl - - sin v cos u
We apply a transformation of the coordinates in order to rapport the tool surface to the system of the worm: r2 = M 21 • r, (2)
As a result of the reference to the worm system, the equations of the tool surface and the projections of the unit normal, become:
x2 (u, v) = (Rl + R2 sin v) cos u cos a - R2 cos v sin a
y2{u, v) = -(Rl + R2 sin v)cos u sin a- ^2cos v cos a + ri (3)
z2 (u, v) = -(Rl + R2 sin v) sin u
nx2 - sin v cos u cos a-cos v sin a
ny2--sin v cos u sin a-cos v cos (4)
n22 = - sin v sin u
Fig. 3. Parameters of the torus surface
In the contact points between the tool and the worm, the following condition must be respected [5, 6]:
-«V2 Z2 + «z 2 72 + hnx2 = 0
in which h represents the helical parameter of the worm. Replacing the relations (3) and (4) in relation (5) we obtain a connection between the u and v parameters as follows:
tan w = -
hr sin v - hx cos v tan a
Rl cos v + rt sin v
The contact curve between the a cylindrical tool and the worm is obtained by giving values to the v parameter; for each value of the v parameter we determine, using the relation (6), a corresponding value for parameter u. The pair of values, u and v, replaced in relation (3), will give the position of a contact point in the worm system. Please note that the contact curve between the torus tool and the flank of worm is not a straight line.
3. The axial profile ofthe worm's root fillet
For each P point of the contact curve between the tool and worm, we make a rotation movement, with a y angle, around the X2 axis in order to bring it to the axial plan XiYi. Fig. 4a represents the front view of the worm regarding this movement of the contact point P along the flank until this point reach the axial plane. Corresponding to this rotation, in order to keep the point on the flank, the point must go forward on the X2 axis, with a value equal to the helical parameter h. As a result, the coordinates of the points of the axial profile Pa will be determined as follows [5]:
xa - x2 - h -y y a - -Jy^+^2 ' where tan y = — = O
Using Z displacement of the torus end mill, a CAD application developed by the author allow the determination of the height H of root filleting curve according to the ZPA axial section after a flank roughing processing (Fig. 4b).
Fig. 4 (a) Front view of the worm
Flank ¡ finishion tool
Fillet in axial section
Worm /
Fig. 4 (b) Detail of the worm's root fillet in axial section
The purpose of Z displacement is to protect the edge of the flank finishing tool. The flank finishing processing follows the root filleting one, and for the finishing tool protection his edge must work "in the air".
4. Conclusions
The ZPA worms can be processed on recent NC lathes, with 4 or 5 controlled axes, using cylindro-frontal standard milling tools in some phases of roughing and finishing. The processing can be done in treated steel, and the surface roughness can be Ra= 0.8 ^m. Searching of the right correction can be optimizing using CAD applications. The authors have developed an application in AutoCAD environment in order to optimize the corrections, and evaluate the errors. This method use simple tools for worm processing with a reduced processing time. All these considerations make the presented technology very interesting from a practical point of view. The idea of this method can be used in order to process a new type of worm (ZPA) or other types of worms - as Archimedes by example - using controlled settings of a cylindrical standard milling tool
The article presents geometrical details regarding the root fillet curve determination in axial section of the ZPA worm, considering as generation surface a torus end mill. Using controlled Z displacement of this tool can fitted the position of the cylindrical milling tool for flank finishing - in order to protect its edge.
References
[1] DIN 3975 (2002). Definitions and parameters on cylindrical worm gear pairs with rectangular crossing shafts - Part 1: Worm and worm wheel, July 2002.
[2] Litvin FL, Fuentes A. Gear Geometry and Applied Theory, Second Ed., Cambridge University Press, 2004.
[3] Dudas I. The Theory & Practice of Worm Gears Drives, Penton Press, Kogan Page Ltd., London, 2000.
[4] Pozdirca A, Olteanu A, Albu SC, New worm technologies manufacturing on the NC lathe,The 4th International Conference on Power Transmissions - PT12, Springer Proceedings - Mechanisms and Machine Science 2012, 13:563-570.
[5] Pozdirca A. Curves and Surfaces Calculation and Representation, "Petru Maior" University of Tirgu Mures, 2010, pp. 166-175.
[6] Pelecudi C, Maros D, Mecanisme, Editura Didactica si Pedagogica, Bucuresti, 1985, pp. 164-176.
[7] Okuma Multus B300, OSP-P200LL, Programming Manual, Pub. No. 5238-E-R5 (LE33-013-R01a), 1st Edition, May, 2005.
[8] Emuge Franken Milling Technology, Tool Catalogue 240, FRANKEN GmbH&Co., 2012, www.emuge-franken.de.
