Methods of determination of wormwheel tooth surface.
Key words: wormwheel, worm gear, hob,
Cylindrical worm gears are widely used in the driving mechanisms of machinery and equipment [Marciniak, 2001]. In order to make the analysis of the toothing of a gear [Litvin, 1997] the surface areas of the teeth of mating elements need to be determined. Wide coverage is given to the design and engineering of worms in relevant literature, and the helical surfaces of worm toothing are known [Dudas, 2000; Kang et al, 1996; Marciniak, 2001].
A wormwheel should be machined using a special conical hob (or a single cutter). In practice, the radial method of wormwheel machining with modular cylindrical hobs is also used. In such a case, the tool action surface is generally different from the surface of the worm that is to mate with the wormwheel being cut. While the worm surface is defined accurately, the surface of the wormwheel is often described only approximately. Therefore, methods have been given for the determination of the surface of a wormwheel being cut by the tangential or radial method using a special conical hob or a cylindrical hob, according to the technology.
2. WORM--WORMWHEEL ENGINEERING GEAR
In the machining process, after the hob has "cut" to the full depth by the tangential or radial method, the tool (its action surface) together with the wormwheel being machined form a technological gear of worm-wormwheel type [Nieszporek, 2004].
The worm-wormwheel gear, in which rotary motion around the axis of both gear links takes place, can be substituted with an equivalent gear, in which worm axial motion occurs without rotation--Figure 1 [Nieszporek, 2004].
Thus, in order to determine the wormwheel surface as the envelope of the tool action surface, transition must be made from the worm (tool) system to the wormwheel system (Fig. 1), which can be written with the following equation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where: [xi]--parameter of tool and wormwheel relative turning motion; [gamma]--angle between the tool and wormwheel axis of rotation; p--parameter of the helical tool action surface; a--distance of the tool and wormwheel axes for machining by the tangential method; [DELTA]a--axes distance error for the tangential method or the quantity accounting for tool cutting into the wormwheel machined by the radial method; [DELTA][gamma], [DELTA]z--tool positioning errors; i--worm gear ratio.
[FIGURE 1 OMITTED]
The parameter [DELTA]a defines the variable distance between the tool and wormwheel axes, and for the radial method it reflects tool "cutting" into the material being machined. This motion is continuous, but for the same wormwheel tooth it takes on a discrete value every one rotation. Thus, after each wormwheel rotation, the wormwheel tooth surface should be evaluated and, by comparing so obtained surfaces correspondingly, the tooth surface, as shaped by the radial method, should be determined. If the case of turning the tool and the wormwheel machined for the full cut-in depth ([DELTA]a = 0) is considered, then the wormwheel tool surface, as shaped by the tangential method, will be obtained. The algorithm described herein enables the determination of the wormwheel tooth surface by both the tangential and the radial methods.
3. MACHINING OF THE WORMWHEEL WITH THE SPECIAL HOB
If the tool action surface is to be consistent with the surface of the worm (which will mate with the wormwheel being machined), then it can be described with the system of equations defining the worm surface (as the envelope of the grinding wheels family)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]--parameter of the axial profile of the worm surface shaping grinding wheel; [phi]--parameter of the grinding wheel action surface; v--parameter of the family of grinding wheels. If the parameter [phi] is determined from the envelope condition (3) (in which the parameter v does not appear), then the worm surface equation will take on the general form of
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
In that case, the family of tool action surfaces (2) in the wormwheel system can generally be written with the equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
and in order to determine the wormwheel tooth surface, the envelope condition should be added to this equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
In cutting the toothing of a wormwheel, the circumferential clearance occurring in the worm gear should be allowed for (it is sufficient to add the respective worm cut width). In order to allow the hob to be reground several times, the increase of its pitch diameter in the range of (0, 1/ 0,05)[m.sub.o] is assumed, which can also be accounted for in the worm computation input data. The wormwheel tooth profile in the frontal section has been accepted to be determined from the following condition
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the third component of the vector (2), and the parameter s defines the position of the cutting plane relative to the frontal plane of symmetry.
The computation algorithm is as follows (the radial method). For a given value [DELTA]a of tool shift from the wormwheel, in a given cutting plane defined by the parameter s, for successive parameter values, for example [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the values of the parameters [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are determined, respectively, which, after being substituted in Equation (2), will determine the successive points of the wormwheel tooth profile. Next, the value of cutting plane position is changed and the computation cycle is repeated, which gives as a result the wormwheel tooth surface in the form of a set of points for a specified tool "cut" into the wormwheel being machined. After changing the [DELTA]a, the computation cycle is repeated for the same cutting planes (up to the full machining depth, [DELTA]a = 0). For the tangential method, only [DELTA]a = 0 is taken.
4. MACHINING OF THE WORMWHEEL WITH THE MODULAR CYLINDRICAL HOB
In this case, the radial method of wormwheel machining is applied. Usually, the hob diameter does not correspond to the diameter of the worm which will mate with the wormwheel being machined (the module and the pitch must agree), so the lead angle of hob coils on the pitch diameter will be different from the lead angle of worm coils. The angle [DELTA][gamma] will not be interpreted as an error, but as a correction of hob positioning. The profiles of hob blade cutting edges are different from the worm profile ([Piotrowski, 2002]). The modular hob action surface axial profile is given and the hob action surface is described by the following equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
where: [bar.x]--hob action surface axial profile; v--hob action surface parameter.
For the determination of the wormwheel tooth surface, the algorithm described above, with a slight modification, can be used.
The wormwheel surface has been determined for the cases of machining by the tangential and radial methods. A (special) wormwheel machining hob and a modular hob have been considered as the tool.
The determination of the wormwheel tooth surface in the case of machining by the radial method makes it possible to determine the truncation of the wormwheel tooth surface vertices resulting from the specificity of the machining method.
In considerations concerning the constructional gear it is generally assumed that the contact of worm and wormwheel teeth is linear (the surfaces of mating teeth are mutually enveloping). Meanwhile, even for special hobs, considering the necessity of allowing for the gear circumferential clearance and change in the pitch diameter of the hob as it wears, no strict consistency usually exists between the action surface of the hob and the surface of the worm that was the basis for the construction of the hob.
In the case of the radial method and use of modular hobs for the machining of wormwheels, machining errors are significant and a punctual tooth contact may occur in the gear in practice.
Dudas I. (2000). The theory and practice of worm gear drives. Penton Press, ISBN 1 8571 8027 5, London
Kang S.K., Ehmann K.F. & Lin C. (1996). A Cad Approach to Helical Groove Machining--I. Mathematical Model and Model Solution. Pergamon, Machine Tools & Manufacture, Vol. 36, No. 1 January 1996, ISSN 0890-6955, pp. 141-153
Litvin F.L (1997). Development of Gear Technology and Theory of Gearing. NASA Lewis Research Center, NASA RP-1406, Clevelend
Marciniak T. (2001). Cylindrical worm gears. The Polish Scientific Publishing House, PWN, ISBN 83-01-13474-7, Warsaw
Nieszporek T. (2004). Rudiments of the design of cutting tools for external cylindrical toothing. The Publishing House of the Czestochowa University of Technology, ISBN 83-7193-252-9, Czestochowa
Piotrowski A. (2002). Increasing the accuracy of modular hobs. The Doctor's Thesis. Czestochowa
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|Author:||Nieszporek, T.; Golebski, R.|
|Publication:||Annals of DAAAM & Proceedings|
|Date:||Jan 1, 2005|
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