Method of identification of parameters of closed planetary train based on exploratory design

. Practicability of usage of gears with closed planetary trains in high-powered drives having weight and dimensions limitations is represented. Method that makes it possible to perform evaluation of mass-dimensional characteristics of planetary double-flow gears at the exploratory design stage is proposed. The developed approach makes it possible to determine basic geometric parameters of gear-tooth system: diameters and width of tooth-wheels reasonably. Besides the obligatory fulfillment of contact tooth strength conditions, providing of the required transmission ratio and condition of coincidence of axes of the gear input and output shafts, the condition of its minimum weight and dimensions is taken into account either. The last requirement is achieved by determination of conditional factor that is square of kinematic scheme contour expressed through projected parameters. The minimum value of contour square corresponds to the best combination of diameters and width values of tooth wheels if all functional limitations are maintained. Then the need of arbitrary choice of any of projected parameters and further verification of the choice made at the closing designing stage is excepted. The proposed approach makes it possible to evaluate the efficiency of chosen gear structural scheme at the initial design stage without design study of workpieces and units.

The usage of planetary gears is more preferable in high-powered drives. Its constructions are continually-improving, new variations of toothing constructions are developed [4,5]. Known factors mainly connected with accuracy of manufacturing and influence of wear problem on efficiency coefficient [6] restrain its wide-spread occurrence.
Increasing of power transferred by a gear can be obtained by means of development of constructions that realize traditional construction schemes, in particular through providing of higher precision of power distribution through parallel flows [7,8].
Capability to obtain high transmission ratio and rotational torque simultaneously is reached by usage of double-flow schemes with nonparallel flows of transferred power [9,10,11].
Development of working project includes stage of choice of structural scheme and calculation of main parameters of kinematic scheme. In this case performance evaluation of chosen gear structural scheme is performed according to the results of finished working project. Particularly according to the power-weight ratio coefficient where Tout -torque at the output shaft; Ωout -angular velocity of output shaft; m -gear weight. However, the gear weight is determined at the conclusion of project. In this regard preliminary analysis of possible variations of structural schemes according to indirect index of power-weight ratio becomes essential. In this regard it is reasonable to have a criterion that makes it possible to evaluate the accepted working kinematic scheme at the initial design stage.
The expression (1) contains parameter m, its value is determined in a great measure by the volume of the gear internal space which is full of wheels. In this regard before engineering development the evaluation of the gear weight corresponding to the accepted scheme is possible to perform according to the conditional index that is square of kinematic scheme contour Кsc. Its value depends on the projected parameters such as reference circle diameter of wheels di and width bj (figure 1). In this case index Кsc can be a criterion of optimality determined by the set of projected parameters (2) Parameters that determine the gear scheme are connected with each other by obligatory correlations that are functional limitations placed on criterion Кsc: -maintenance of the required transmission ratio u [9] ( ) -fulfillment of conditions of tooth strength [12] ( where Кd = 840 for spur gear pairs; Кd = 780 for helical-gear set; КС -coefficient of irregularity in the distribution of load between satellite gears; КС = 1,1…1,2; Т2 and Т5 -rotational torques on annular rings 2 and 5; К1 and К2 -number of satellite gears of the first and second flows of power transmission correspondingly; ψbd1 and ψbd2 -coefficients of toothing width: -condition of coincidence of axes In the equations (4) and (5) the calculation is performed through rotational torques in nonparallel flows Т2 and Т5. Distribution of pull-in torque Т1 to flows is determined by geometric parameters based on force analysis (figure 2).
The width of rim of each wheel depends on applied loads determined during force analysis.
The loading in toothing (point E) between links 4 and 5 is determined by condition of equality of torques applied to link h (4) (figure 2, c) Then the load in point D is determined by the condition of equality of torques applied to satellite gear 5 about axis F ( figure 2, d) As a result, loads applied to each of fixedly connected toothed rims 3 and 6 (forces in points C and D correspondingly) are known during consideration of condition of equilibrium of output link 3(6) at the concluding stage of force analysis. This makes it possible to determine the rotational torque at the gear output shaft.
The results of the accomplishment of force analysis in the presented consequence are expressions of rotational torques: ( ) The correlation of rotational torques transferred by nonparallel power flows expressed by geometric parameters of a mechanism ( ) 6 2 1 4 3 The width of wheels is expressed during the solution of equations (6) and (7) including (4) and (5) The insertion of correspondences (7), (8), (9) into expression (2) makes it possible to introduce functional limitations into criterion of optimality Кsc. As a result, one functional limitation in the form of equation (3) is applied to criterion Кsc. In some cases, to simplify the procedure of a function extremum seeking it is reasonable to change the condition of equation to inequation where Δu -acceptable deviation of transmission ratio u. Projected parameters determined by the condition of minimum criterion of optimality and considering functional limitation correspond to better mass-dimensional characteristics of a gear. In this case the obligatory limitations to contact tooth strength, required transmission ratio of a gear and coincidence of axes of its input and output shafts are performed.
This method can be used either for exploratory design of double planetary gears. In this case modules of toothing for each step should be included in the number of projected parameters.

Conclusion
The decision to use planetary gears with two nonparallel flows of transferred power especially with small size and weight requirements in high-powered gears is rational.
The issue of rational and reasonable choice of main geometric parameters of toothing system is decided even on an exploratory design stage on the basis of evaluation of conditional factor that is square of kinematic scheme contour. This makes it possible to provide the best variant of combination of parameters according to the power-weight ratio coefficient even at the initial design stage.