Analytical Determination of Range of Number of Teeth in Generating Non-Involute Tooth Forms Using Fixed Reference Profiles

The standard ISO 17396:2017 specifies sizes and tolerances of pulleys for synchronous belt drives. The pulleys feature trapezoidal tooth profiles that can be machined with generating methods (such as hobbing). The profile is different for each pulley diameter. A reference profile, produced by a generating tool (such as a hob), can closely approximate the nominal tooth shape over a range of a number of teeth (Ref. 1).

The reference profiles for involute gears have straight-line sections that shape theoretically correct tooth flanks regardless of the number of teeth on the gear. On the other hand, a unique reference profile corresponds to the number of trapezoidal teeth on a belt pulley. That reference profile can be obtained for each pulley with methods described in Ref. 2, 3, 4. If imported into CAD software, it can be compared with other profiles, which were created for a different number of teeth on the pulleys. Tooth forms generated with fixed reference profiles can be found using the same methods, and the profiles can be compared with each other and against the specifications from the ISO standard. As these comparisons are usually done manually, they may take a significant amount of time and effort.

When a reference profile suitable for generating the subject toothing is obtained, a tool’s profile can be derived from it, which may include process-related modifications. In the case of hobbing, these modifications may be necessary for several reasons, including tool-workpiece kinematics of two bodies rotating around skew axes (Ref. 2), and local cutting conditions which change along the cutting edge having an influence on the final shape of the gap being cut. A single tool can typically produce a single reference profile, therefore the act of selecting the reference profile or profiles required for a given task is a very important step in designing the manufacturing process as it has a direct and decisive influence on the costs. The methods based on manual comparisons of profiles in CAD software might not be efficient enough in some cases, so an analytical method needed to be developed to streamline the reference profile development process.

The toothed belt and pulley system known by the designation T, which has been selected as an example within this paper, was developed in the 1950s and standardized first in DIN 7721 (1977) and then in ISO 17396:2014 (Ref. 5). Here, in this case study, we check if a single hob can properly cut T5 profile pulleys with 25 and 30 teeth, and if so—define the range of the number of teeth covered by this hob. Standard tool designs indicate a range from 21 to 36 teeth (Ref. 6), 21 to 50 teeth (Ref. 7), and 21 teeth and more (Ref. 8).

Pulley Geometry

As the number of teeth to be considered first are 25 and 30, a T5 28-tooth pulley profile has been selected from the ISO 17396:2017 standard, and is shown here (in millimeters):

The specified flank angle tolerance is ±1.5 degrees. If constant gap width at points where straight flank lines end and rounding R0.6 mm begins was maintained, changing the flank angle within that tolerance produces changes of specified gap width 3.345 mm. The magnitude of these changes for each side is of the same order as the gap width tolerance (Figure 1—details drawn red). Furthermore, the specified gap width is dependent also on the tolerance of major diameter ϕ43.695 mm. To address problems resulting from these changes, it has been assumed that nominal values in the middle of respective tolerances will be used within the scope of this paper.

The generating tool could machine the major diameter of the pulley. In such a case the tolerances on the major diameter would limit the possible positions of the generating tool. To keep the generating tool’s design as flexible as possible it has been assumed that the major diameter would be machined during the manufacture of the blank, and the generating tool would only machine the tooth gaps.

The tolerance of rounding R0.6 mm is specified two times wider than the tolerance of the major diameter, and the generating tool would machine this rounding. Even when the radius of rounding is kept within that tolerance, as the number of teeth generated on a pulley with a fixed reference profile changes, the radial height of rounding would also change. A generating tool could be designed to achieve a perfect transition of rounding into major diameter for one chosen number of teeth. Since different numbers of teeth are to be generated with that tool, it has been assumed in this paper that the radial height of rounding R0.6 mm (with a nominal value equal to 0.392 mm) must be monitored and maintained within the same tolerance as specified for the radius of rounding (±0.05 mm).

Reference Profile of Generating Tool

A reference profile of generating tool for a T5 28-tooth pulley has been obtained, using methods described in Refs. 2, 3, 4:

The local curvature of the reference profile at the point where rounding R0.6 ends and generating of flank begins is measured 8.68 mm, and the profile angle at that point is measured 21.06 degrees. When a small section of the profile is isolated, its deviation from a straight line is small enough to assume that it will shape an involute of a circle. In this study, a section with a length of 0.000536 mm (0.0005 mm high) has been used with a deviation from a straight line d calculated from Equation 1 as 4.1E-09 mm.

(1)

Calculating Parameters of Pulley Profile

The circular tooth thickness on pitch diameter for a 28-tooth pulley, generated with a reference profile shown on Figure 2, can be measured straight on the profile as no profile shift is required to achieve the required gap width of 3.345 mm. This circular tooth thickness is measured at 1.822 mm. When the number of teeth on the pulley is other than 28, the reference profile would also be different, but since the reference profile for 28 teeth will be used, profile shift needs to be applied to achieve the required gap width.