A Computer Algorithm to Design Compound Gear Trains for Arbitrary Ratio

"NomenclatureR A{ Bs " M FM
196
= = — = = =
/
arbitrary prescribed value for overall gear train ratio number of teeth of driving gear of ith gear pair number of teeth of driven gear of ith gear pair minimum permissible number of teeth on any gear maximum permissible number of teeth on any gear Farey series of order M
A Computer Algorithm to Design Compound Gear Trains for Arbitrary Ratio
A novel numerical method to determine the numbers of teeth in the driver and driven gears in a compound gear train is presented as an algorithm for computer-aided design applications. A brief review of available methods for double-pair gear sets is made, and compared to the proposed algorithm from the viewpoints of precision, computational efficiency, and ease of programming. In comparison, we now have for the first time completely eliminated dependency on tables of prime factors and conjugate fractions, and/or the ingenuity of the designer {albeit an interesting intellectual challenge), and reduced the problem to a direct, straightforward calculation as a self-contained algorithm easily programmed by the designer himself. The arbitrary irrational value of 1/TT is used in a numerical example; sixteen candidate gear sets are obtained having an overall ratio within ± 5 micro-units. The triple-pair gear set design problem is considered, and an extended form of the algorithm derived. This allows, also for the first time, the systematic design of a triplepair gear set, and obviates heuristic or cut and try methods. The same value of 1/ir is used in an illustrative example, and the results presented in tabidarform.
introduction
IN many precision gear train design problems, the gear designer seeks an overall gear ratio which is a close approximation to an arbitrary value, oftentimes irrational. For example, the quick-change gear box in a helical/spiral gear hobber must be capable of a full range of gear ratios, many involving ir/n or n/ir as factors and hence irrational. Alternatively, conversion from English to metric units involves the irrational factor 2.5400051 . . . cm/in. which is commonly approximated in a single gear pair by li = 127/50 = 2.54. In certain cases, it may be necessary to avoid a large 127-tooth gear by utilizing a compound (double-pair) gear train.
Consider a compound gear train comprising two gear pairs (Ai\Bi) and (Ai'.Bz), where gears Ai and BL are joined by a common shaft. The overall gear ratio is: R Ay Bi Ai B2
Manuscript received at ASME Headquarters, July 27, 1970. No.70-Mech-31.
Paper
where At and B; represent the numbers of teeth. For simplicity of discussion consider the gear design problem arranged so t h a t R < 1. Practical design considerations often impose constraints on the minimum and maximum number of teeth on any gear. Thus, we have conditions A{, B{ > J; and Ai: B{ < M, where typically J" « 12-15 teeth to avoid undercutting, and M ~ 6 0 100 teeth to minimize gear-box size. At this stage of the analysis we do not consider additional design constraints such as obtain from reverted gear train design (Ai + Bi = At + Bi) or from considerations of gear strengths or inertia. These may be incorporated at a later stage as part of an overall computer-aided design optimization problem. This paper presents an algorithm to systematically calculate
Contributed by the Design Engineering Division and presented at the Mechanisms Conference, Columbus, Ohio, November 2-4,
1970, of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS.
FEBRUARY 1 971
> Ft G G{ G' G/
= = = = = =
symbol for "greater than and conjugate t o " terms of the Farey series series of ascending integer-pair products terms of G series as defined series of ascending integer-triple-products terms of G' series as defined
than R, and try
1
Bi
for n ^ M;
or alternately try
Brief Review of Available Methods
If the prescribed ratio R = N/D = rational, then heuristic methods involving factoring, continued fractions, etc., are usually adequate [6] .x However, for R = irrational, more systematic procedures are necessary. Most of these are based on the properties of conjugate fractions as derived from the theory of numbers and diophantine equations [4, 7, 8, 9, 10, 13]; or the application of linear programming techniques [11]. Conjugate Fraction Methods. Conjugate fractions may be defined as a pair of irreducible (no common factors in Ait B/}, proper (A,- < B{) fractions where no intermediate fraction has a denominator less than either of the Bt. Then: (where the symbol > means greater t h a n and conjugate to): Ax j A 2 Bi> Ai A, Bi 1 BiB; • AiB 2 - A2Bt = 1