separate constraints).

In the interest of conciseness we shall not dwell on the details of their calculation (the interested reader may find all the details of the gearing calculations in the relevant industrial standard document (DIN 1987) and the catalogs and calculation methods used for bearings and seals in the SKF online catalog (http://tinyurl.com/p7u4yx, Accessed on 17 August 2009)); we do note though that they are mostly based on a combination of lookup tables and simple analytical models, their computational cost therefore being minimal.

Additionally, it is worth noting that to enable the calculation of some of the constraints (such as some of the geometrical inequalities, as well as those related to lubrication and operating temperature), the housing was also designed automatically once the virtual design of each two-stage gearing was generated.

The following list of constraints should be viewed with

reference to the sketch in Fig. 1.

Constraint group 1 The relative dif ference between the

required and the actual gearing ratio must be within the

range [−2.5%, 2.5%] on both stages.

Constraint group 2 The Hertzian contact pressure on the

teeth of both stages must not exceed a specif ied value.

Constraint group 3 The bending stress on the teeth of

gears 1 through 4 must not exceed a specif ied value.

Constraint group 4 The teeth on gears 1 through 4 must

not be undercut.

Constraint group 5 The top land of the teeth on gears 1

through 4 must not vanish.

Constraint group 6 The contact ratio on stages I and II

must be greater than a specif ied value.

Constraint group 7 The normal addendum coef f icients on

both stages should be in the range [−0.5, 1].

Constraint group 8 Measurability constraint for all four

gears.

Constraint group 9 The numbers of teeth on the gears of

both stages must be relative primes.

Automated optimal design of a two-stage helical gear reducer 433

Constraint group 10 Gear 2 must not interfere with the

output shaft.

Constraint group 11 Lubrication constraint—the margin

between the minimum and maximum allowable lubricant

levels should be no less than 10 mm.

Constraint group 12 The input and output shaft ends must

have a suf f icient diameter step to allow the mounting of a

belt wheel.

Constraint group 13 The inside diameter of the tapered

roller bearings on the input and output shafts must be less

than the mounting diameter of the seal.

Constraint group 14 Geometrical constraint relating to

the space required by the outside ring of the tapered roller

bearings on the input and output shafts.

Constraint group 15 Set of manufacturability constraints

on all four gears.

Constraint group 16 Input, output and intermediary shaft

stress constraints (radial and axial loads originating from

the gearings).

Constraint group 17 The fatigue life safety factors on the

three shafts must not fall below a specif ied value.

Constraint group 18 The bending strains on the three

shafts in key locations must be below certain threshold values

to enable the correct functioning of the gearings and the

bearings.

Constraint group 19 The torsional strains in the three

shafts must be below a threshold value.

Constraint group 20 The service life of the tapered roller

bearings must exceed a specif ied value.

Constraint group 21 The shearing and crushing stresses

must not exceed a specif ied value on the keys and keyways

of the input and output shafts.

Constraint group 22 The minimum distance between adjacent