Geometric calculation of non-circular gear segments of the planetary mechanism in rotary hydraulic machines

. The article discusses the improvement of the methods of geometric design of positive-displacement hydraulic machines with floating satellites. At the first stage, based on standard methods, an "average" round-link planetary mechanism, having the number of teeth as in the designed hydraulic machine, is calculated. At the second stage, the same-type cyclic functions characterizing the paths of the center of the satellite in the coordinates associated with each of the central gearwheels are specified. At the third stage, the coordinate arrays of the satellite in a variety of its positions are calculated. The angular position of the satellite relative to one of the profiled gearwheels (for example, the sun gearwheel) provides equidistance of the centroid of this gearwheel to the path of the center of the satellite. The angular position of the satellite relative to the other profiled wheel is determined by its running around the first wheel. At the final stage of the design, the satellite is constructed in a variety of positions, and the profile of the corresponding non-circular gear rim is obtained as an envelope of the family of satellite curves-profiles.


Introduction
One of the promising trends in pump and compressor engineering is the use of planetary rotary hydraulic machines (PRHM) with floating satellites. The issue of profiling of noncircular PRHM gears was first systematically researched in An I-Kan's doctoral dissertation [1]. However, the methodology used by this author is complex, not universally applicable, and therefore not suitable for engineering purposes. The topic of designing planetary mechanisms with non-circular gears was touched upon in publications [2,3,4,5,6,7,8]. Our articles [9,10,11], which consistently improve the design methodology, are devoted to the development of an engineering method of profiling the PRHM segments. Nonetheless, the matters of simplification, unification and improvement of the methods of geometric calculation of the planetary mechanism in rotary hydraulic machines, the solution of which this article is devoted to, remain topical.

Basic Method
Representative schemes of PRHM's are shown in Fig. 1. The PRHM schemes differ from each other in the number of waves of the epicycle N and the sun gearwheel M. Obligatory ratios: number of satellites V=N+M, the ratio of numbers of teeth Z2 of epicycle and Z1 of the sun gearwheel is equal to the ratio of their wave numbers The numbers of waves N and M are given before designing the PRHM segments. We will consider the N-M=3-2 scheme (Fig. 1b) as an example.
The first stage of the geometric design consists of choosing the numbers of teeth Z1, Z2, Z3 and the calculated displacement factors X1, X2, X3. The corresponding calculations are proposed to be carried out based on standard methods (GOST 16532-70, GOST 19274-73) for round-link planetary gears analogs of the PRHM (Fig. 2): -choose the number of teeth Z3 and displacement factor X3 of the satellite, -choose the numbers of teeth Z1, Z2, as well as the calculated displacement factors X1, X2 taking into account ratio (1) and the condition of coaxiality of the round-link planetary mechanism. For PRHM schemes where N-M>0 ( Fig. 1 a, b), the condition of coaxiality, in the first approximation, can be set without considering the displacement factors: -for the chosen numbers of teeth Z1, Z2, calculate the displacement factors X1 and X2 of the round-link planetary mechanism according to the standard method (GOST 16532-70, GOST 19274-73) or using the KOMPAS library.
At the second stage of the design, the interdependent (same-type) cyclic functions characterizing the paths of the center of the satellite in the coordinates associated with each of the gearwheels 1 and 2 are specified. The number of intersection points of these paths corresponds to the number of satellites V= N+M. When the mechanism is operating, the center of each satellite is located at the corresponding calculated intersection point of the center paths.
In the general case, the equations of these paths in polar coordinates associated with each of the gearwheels 1 and 2 are the following: where r1(φ1) and r2(φ2)radius vectors of the satellite center paths, φ1 and φ2current angles of rotation of the imaginary carrier in polar coordinates associated with the corresponding segments, r0 = аwthe radius of the calculated circle (into which both paths are degenerate at kH = 0), kHcoefficient of "non-circularity" of the paths (characterizing the "steepness" of the waves).
In the simplest cases, a cyclic function is used: The choice of the non-circularity coefficient kH depends on the angle λmax, the so-called satellite retention angle (see Fig. 1). In the first approximation, we can take [13]: where ka is an empirical coefficient, ka = 0.018 (1/degree). The angle λ°max, in turn, depends on the gearing angles αω1, αω2 and the maximum permissible angles a1, a2 of pressure in the gearings where index i takes the value 1 or 2.
For example, if we take the permissible pressure angle a=43°, and the gearing angle αω=21.5°, we get λ=43°, and the coefficient kH=0.155.
Another limitation imposed on the value of the coefficient kH is the condition of proximity (adjacency) of the central gearwheels. A gap must remain between the tops of the central gearwheels teeth in any phase of motion. It is advisable to check the observance of this condition in the process of designing the gear rims.
The third stage of the PRHM design involves calculating the values of the parameters that define the variety of satellite positions. The arrays of polar coordinates φi and ri of the satellite center on the paths given by equations (3), (4), and angles φci of rotation of the satellite relative to the sun gearwheel 1 and epicycle 2 are calculated.
In the fourth and final stage of design, graphic software, (e.g., KOMPAS) is used. The satellite is constructed in a variety of positions, and the profile of the corresponding noncircular gear rim is obtained as an envelope of the family of satellite curves-profiles.
Note that the values of the satellite position parameters calculated at the 3rd design stage will depend on the choice of the centroids of the non-circular PRHM segments. Let's consider two options.

Options for choosing a centroid of non-circular PRGM links
Option 1. The centroids of both non-circular gearwheels are the equidistant of the corresponding center paths of the satellite [9,10,11]. In this case, the satellite coordinate arrays (φ1, r1, φc1) and (φ2, r2, φc2) for finding the contours of gearwheels 1 and 2 are calculated independently of each other. The angle φci of rotation of the satellites relative to the corresponding gearwheel (1 or 2) is determined by the following equation: where r'i(φi))the derivative of the corresponding function r1(φ1) or r2(φ2), ithe coefficient taking into account the change in the length of the corresponding center path of the satellite in comparison to the length of the center circle of the original round-link mechanism. (10) In the particular case, where the cyclic function F(φ)=cos(φ), the equations (9), (10) take the following form:    (14) In this example, the coefficients are 1=0.97681, 2=0.95028. The results using equations (11) and (12) are shown in Table 1. The profiles of the gear rims of segments 1 and 2 are envelopes to the families of satellite profiles shown in Fig. 3. After having assembled the mechanism from the resulting gearwheels 1 and 2, (Fig. 4) we see that in the positions corresponding to the inclined sections of the non-circular gearwheels rims, the satellite does not fit between them. For the PRHM with small numbers of waves, including N-M = 3-2 and a high value of the coefficient kn, this effect can cause "wedging" of the mechanism. For the PRHM with large numbers of waves, for example, N-M = 6-4 (see Fig. 1a) or low kH coefficient (Fig. 1c), the first option of centroids choice is acceptable, the errors of resulting profiles are minor and are compensated by radial gaps present in the gearings.
Option 2. The problem of "wedging" is solved fundamentally correctly and radically if the equidistant of the center path of the satellite is the centroid of only one non-circular gearwheel, for example, the sun gearwheel. In this case, the profile of the epicycle rim will be an envelope to the family of profiles of the satellite running around the sun gearwheel. Then, the left part of Table 1 (relating to the sun gearwheel) moves to the new table (Table  2) without any changes. In the right part of Table 2 (relating to the epicycle), the angle φ2 of rotation of the imaginary carrier relative to the epicycle is expressed by the angle φ1 of rotation of the carrier relative to the sun gearwheel φ2 = φ1 M/N. (15) The step Δφ2 of the variation of the parameter φ2 in Table 2 will change in the same proportion. At the same time, for correct calculation in CAD systems, the step Δφ1 must be chosen so that the step Δφ2 is a rational number (i.e., the division φ1M/N is done without a remainder).
The angle φc1 of rotation of the satellite relative to the imaginary carrier is already given at each position of the carrier. Using the kinematic relations characterizing the planetary mechanism under consideration, we can pass to the angle φc2 of rotation of the satellite relative to the epicycle φс2 = -(φс1 -φ1(1+M/N)). (16) In order to complete the full cycle (360°) along the angle φ2 of rotation of the imaginary carrier relating to the epicycle and to have the same number of rows in all columns of Table  2, we continue column φ1 to φ1=360°N/M. The profile of non-circular gear rims constructed based on Option 2 ( Fig. 5) completely eliminates the "wedging" of the satellites.

Conclusion
The suggested method of geometric calculation is available to a wide range of engineers. It allows you to obtain accurate contours of non-circular gear segments of the planetary mechanism in rotary hydraulic machines, using widespread computer software systems.