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[ABSTRACT] Conical involute gears (beveloids) are used in transmissions with intersecting or skew axes and for backlash-free transmissions with parallel axes. Conical gears are spur or helical gears with variable addendum modification (tooth thickness) across the face width. The geometry of such gears is generally known, but applications in power transmissions are more or less exceptional. ZF has implemented beveloid gear sets in various applications: 4WD gear units for passenger cars, marine transmissions (mostly used in yachts), gear boxes for robotics, and industrial drives. The module of these beveloids varies between 0.7 mm and 8 mm in size, and the crossed axes angle varies between 0°and 25°. These boundary conditions require a deep understanding of the design, manufacturing, and quality assurance of beveloid gears. Flank modifications, which are necessary for achieving a high load capacity and a low noise emission in the conical gears, can be produced with the continuous generation grinding process. In order to reduce the manufacturing costs, the machine settings as well as the flank deviations caused by the grinding process can be calculated in the design phase using a manufacturing simulation. This presentation gives an overview of the development of conical gears for power transmissions: Basic geometry, design of macro and micro geometry, simulation, manufacturing, gear measurement, and testing.
1 Introduction
In transmissions with shafts that are not arranged parallel to the axis, torque transmission is
possible by means of various designs such as bevel or crown gears , universal shafts , or conical involute gears (beveloids). The use of conical involute gears is particularly ideal for small shaft angles (less than 15°), as they offer benefits with regard to ease of production, design features, and overall input. Conical involute gears can be used in transmissions with intersecting or skew axes or in transmissions with parallel axes for backlash-free operation. Due to the fact that selection of the cone angle does not depend on the crossed axes angle, pairing is also possible with cylindrical gears. As beveloids can be produced as external and internal gears, a whole matrix of pairing options results and the designer is provided with a high degree of flexibility;
Table 1.
Conical gears are spur or helical gears with variable addendum correction (tooth thickness)
across the face width. They can mesh with all gears made with a tool with the same basic rack. The geometry of beveloids is generally known, but they have so far rarely been used in power transmissions. Neither the load capacity nor the noise behavior of beveloids has been examined to any great extent in the past. Standards (such as ISO 6336 for cylindrical gears ), calculation methods, and strength values are not available. Therefore, it was necessary to develop the calculation method, obtain the load capacity values, and calculate specifications for production and quality assurance. In the last 15 years, ZF has developed various applications with conical gears:
? Marine transmissions with down-angle output shafts /1, 3/, Fig. 1
? Steering transmissions /1/
? Low-backlash planetary gears (crossed axes angle 1…3°) for robots /2/
? Transfer gears for commercial vehicles (dumper)
? Automatic car transmissions for AWD /4/, Fig. 2
2 GEAR GEOMETRY
2.1 MACRO GEOMETRY
To put it simply, a beveloid is a spur gear with continuously changing addendum modification across the face width, as shown in Fig. 3. To accomplish this, the tool is tilted towards the gear axis by the root cone angle ? /1/. This results in the basic gear dimensions:
Helix angle, right/left
tanβ=tanβ·cosδ (1)
Transverse pressure angle right/left
(2)
Base circle diameter right/left
(3)
The differing base circles for the left and right flanks lead to asymmetrical tooth profiles at helical gears, Fig. 3. Manufacturing with a rack-type cutter results in a tooth root cone with root cone angle δ. The addendum angle is designed so that tip edge interferences with the mating gear are avoided and a maximally large contact ratio is obtained. Thus, a differing tooth height results across the face width.Due to the geometric design limits for undercut and
tip formation, the possible face width decreases as the cone angle increases. Sufficiently well-proportioned gearing is possible up to a cone angle of approx. 15°.
2.2 MICRO GEOMETRY
The pairing of two conical gears generally leads to a point-shaped tooth contact. Out-side this contact, there is gaping between the tooth flanks , Fig. 7. The goal of the gearing correction design is to reduce this gaping in order to create a flat and uniform contact. An exact calculation of the tooth flank is possible with the step-by-step application of the gearing law /5/, Fig. 4. To that end , a point (P) with the radiusrP1and normal vectorn1is generated on the original flank. This generates the speed vector V with
(4)
For the point created on the mating flank, the radial vector rp:
(5)
and the speed vector apply
(6)
The angular velocities are generated from the gear ratio:
(7)
The angle γ is iterated until the gearing law in the form
(8)
is fulfilled. The meshing point Pa found is then rotated through the angle
(9)
around the gear axis, and this results in the conjugate flank point P.
3 GEARING DESIGN
3.1 UNDERCUT AND TIP FORMATION
The usable face width on the beveloid gearing is limited by tip formation on the heel and undercut on the toe as shown in Fig. 3. The greater the selected tooth height (in order to obtain a larger addendum modification), the smaller the theoretically useable face width is. Undercut on the toe and tip formation on the heel result from changing the addendum modification along the face width. The maximum usable face width is achieved when the cone angle on both gears of the pairing is selected to be approximately the same size. With pairs having a significantly smaller pinion, a smaller cone angle must be used on this pinion. Tip formation on the heel is less critical if the tip cone angle is smaller than the root cone angle, which often provides good use of the available involute on the toe and for sufficient tip clearance in the heel.
3.2 FIELD OF ACTION AND SLIDING VELOCITY
The field of action for the beveloid gearing is distorted by the radial conicity with a tendency towards the shape of a parallelogram. In addition, the field of action is twisted due to the working pressure angle change across the face width. Fig. 5 shows an example of this. There is a roll axis on the beveloid gearing with crossed axes; there is no sliding on this axis as there is on the roll point of cylindrical gear pairs. With a skewed axis arrangement, there is always yet another axial slide in the tooth engagement. Due to the working pressure angle that changes across the face width, there is varying distribution of the contact path to the tip and root contact. Thus, significantly differing sliding velocities can result on the tooth tip and the tooth root along the face width. In the center section, the selection of the addendum modification should be based on the specifications for the cylindrical gear pairs; the root contact path at the driver should be smaller than the tip contact path. Fig. 6 shows the distribution of the sliding velocity on the driver of a beveloid gear pair.
4 CONTACT ANALYSIS AND MODIFYCATIONS
4.1 POINT CONTACT AND EASE-OFF
At the uncorrected gearing, there is only one point in contact due to the tilting of the axes. The gaping that results along the potential contact line can be approximately described by helix crowning and flank line angle deviation. Crossed axes result in no difference between the gaps on the left and right flanks on spur gears. With helical gearing, the resulting gaping is almost equivalent when both beveloid gears show approximately the same cone angle. The difference between the gap values on the left and right flanks increases as the difference between the cone angles increases and as the helix angle increases. This process results in larger gap values on the flank with the smaller working pressure angle. Fig.7 shows the resulting gaping (ease-off) for a beveloid gear pair with crossed axes and beveloid gears with an identical cone angle. Fig.8 shows the differences in the gaping that results for the left and right flanks for the same crossed axes angle of 10° and a helical angle of approx. 30°. The mean gaping obtained from both flanks is, to a large extent, independent of the helix angle and the distribution of the cone angle to both gears.
The selection of the helical and cone angles only determines the distribution of the mean gaping to the left and right flanks. A skewed axis arrangement results in additional influence on the contact gaping. There is a significant reduction in the effective helix crowning on one flank. If the axis perpendicular is identical to the total of the base radii and the difference in the base helix angle is equivalent to the (projected) crossed axes angle, then the gaping decreases to zero and line contact appears. However, significant gaping remains on the opposite flank. If the axis perpendicular is further enlarged up to the point at which a cylindrical crossed helical gear pair is obtained, this results in equivalent minor helix crowning in the ease-off on both flanks. In addition to helix crowning, a notable profile twist (see Fig. 8) is also characteristic of the ease-off of helical beveloids. This profile twist grows significantly as the helix angle increases. Fig.9 shows how the profile twist on the example gear set from Fig.7 is changed depending on the helix angle. In order to compensate for the existing gaping in the tooth engagement, topological flank corrections are necessary; these corrections greatly compensate for the effective helix crowning as well as the profile twist. Without the compensation of the profile twist, only a diagonally patterned contact strip is obtained in the field of action, as shown in Fig. 10.
4.2 FLANK MODIFICATIONS
For a given degree of compensation, the necessary topography can be determined from the existing ease-off. Fig. 11 shows these types of typographies, which were produced on prototypes. The contact ratios have improved greatly with these corrections as can be seen in Fig.12. For use in series production, the target is always to manufacture such topographies on commonly used grinding machines. The options for this are described in Section 6. In addition to the gaping compensation, tip relief is also beneficial. This relief reduces the load at the start and at the end of meshing and can also provide lower noise excitation. However, tip relief manufactured at beveloid gears is not constant in amount and length across the face width. The problem primarily occurs on gearing with a large root cone angle and a tip cone angle deviating from this angle. The tip relief at the toe is significantly larger than that at the heel. This uneven tip relief must be accepted if relief of the start and end of meshing is required. The production of tip relief using another cone angle as the root cone angle is possible; however, this requires an additional grinding step only for the tip relief. Independently of the generating grinding process, targeted flank topography can be manufactured by coroning or honing; the application of this method on beveloids, however, is still in the early stages of development.
5 LOAD CAPACITY AND NOISE EXCITATION
5. 1 APPLICATION OF THE CALCULATION STANDARDS
The flank and root load capacity of beveloid gearing can only approximately be deter-mined using the calculation standards (ISO6336, DIN3990,AGMA C95) for cylindrical gearing. A substitute cylindrical gear pair has to be used, which is defined by the gear parameters at the center of the face width. The profile of the beveloid tooth is asymmetrical; that can, however, be ignored on the substitute gears. The substitute center distance is obtained by adding up the operating pitch radii at the center of the face width.When viewed across the face width, individual parameters will change, which significantly influence the load capacity. Table 2 shows the main influences on the root and flank load capacities. The larger notch effect due to the decrease in the tooth root fillet radius towards the heel is in opposition to the increase in the root thickness. In addition, there is a smaller tangential force on the larger operating pitch circle at the heel; at the same time, however, the addendum modification on the heel is smaller. The primary influences are nearly well-balanced so that the load capacity can be calculated sufficiently approximate with the substitute gear pair. The load distribution across the face width can be considered with the width factors (e. g. Kand K in DIN/ISO) and should be determined from additional load pattern analyses.
5.2 USE OF THE TOOTH CONTACT ANALYSIS
A more precise calculation of the load capacity is possible with a three-dimensional tooth contact analysis, as used at cylindrical gear pairs. The substitute cylindrical gear pair can be used in this analysis and the contact conditions are considered very well with flank topography. This topography is obtained from the superimposition of the load-free contact ease-off with the flank corrections used on the gear. In this process, the contact lines are determined on the substitute cylindrical gear and they differ slightly from the contact at the beveloid gear. Fig. 13 shows the load distributions calculated in this manner as compared to the load patterns recorded, and a very goodcorrelation can be seen.
This tooth contact analysis also generates the transmission error resulting from the tooth mesh as vibrational excitation. It can, however, only be used as a rough guide. The impreciseness in the contact behavior calculated has a stronger effect on the transmission error than it does on the load distribution.
5.3 EXACT MODELING USING THE FINITE-ELEMENT METHOD
The stress at the beveloid gears can also be calculated using the finite-element method. Fig. 14 shows examples of the modeling of the transverse section on the gears. Fig. 15 shows the computer-generated model in the tooth mesh section and the stress distribution calculated with PERMAS /7/ on the driven gear in a mesh position. The calculation was carried out for multiple mesh positions and the transmission error can be determined from the rotation of the gears.
5.4 TESTS REGARDING LOAD CAPACITY AND NOISE
A back-to-back test bench with crossed axes, upon which gear pairs from AWD transmissions were tested, was used to determine the load capacity, Fig.16. Different corrections were produced on the test gears in order to ascertain their influence on the load capacity. There was good correlation between the load capacity in the test and the FE (finite element) results. Particularly noteworthy is an additional shift of the load pattern towards the heel due to the increased stiffness in this area. This shift is not discernable in the calculation with the substitute cylindrical gear pair. Simultaneous to the load capacity tests, measurements of the transmission error and rotational acceleration were conducted in a universal noise test box, Fig. 17. In addition to the load influence, the influence of additional axis tilt on the noise excitation was also examined in these tests. With regard to this axis tilt, no large amount of sensitivity in the tested gear sets was found.
6 MANUFACTURING SIMULATION
With the assistance of the manufacturing simulation, machine settings and movements with continuous generation grinding as well as the produced profile twist can be obtained. Production-constrained profile twist can be considered as early as the design phase of a transmission and can be incorporated into the load capacity and noise analyses. Simulation software for the manufacturing of beveloids was specially developed at ZF, which is comparable to /9/.
6.1 PRODUCTION METHODS THAT CAN BE USED FOR BEVELOIDS
Only generating methods can be used to produce the beveloid gearing, because the shape of the tooth profile changes significantly along the face width. Only very slightly conical beveloids can be manufactured with the acknowledgment that there is profile angle deviation even with the shaping process. Hobs are the easiest to use for pre-cutting. Gear planning would theoretically be useable as well; however, the kinematics required makes this not really feasible on existing machines. Internal conical gears can then only be precisely manufactured with pinion-type cutters if the cutter axis is parallel to the tool axis and the cone is created by changing the center distance. If the internal gear is manufactured with a tilted pinion cutter axis such as used for crown gears, this results in a hollow crowning and a profile twist without corrective movements. These deviations are small enough to be ignored for minor cone angles. For final processing, continuous generation grinding with a grinding worm appears to be the best option. If the workpiece or tool fixture can be additionally tilted， then partial generation methods are also applicable. Processing in a topological grinding process is also possible (e.g. 5-axis machines), but with great effort, when the cone angle of the gearing can be considered in the machine control. In principle, honing and coroning can also be used for the processing; however, the application of these methods in beveloids still needs extensive development. The targeted hollow crowning can be created in the generation grinding process in the dual-flank grinding process via a bowshaped reduction in the center distance. This method results in a profile twist, that is the reverse of the profile twist from the contact gaping. Thus, this method provides extensive compensation for the profile twist and a significantly more voluminous load pattern as is typical on cylindrical gears.
6.2 WORKPIECE GEOMETRY
The following workpiece descriptions are used in the simulation:
? initial gear (with stock allowance for the grind processing)
? ideal gear (from the gear data, without flank corrections)
? finished gear (with production-constrained deviations and flank corrections).
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