It is possible to parametrize only two gear wheels, number 293270 and 293310, two annular gear wheels with straight teeth which directly mate, because of the following reasons:

- the measures of the box cannot be changed;

- the parts constituting the whole reduction gear have been designed and assembled to minimize the overall dimension.

The first gear wheel is characterized by these data:

- module = 3

- number of teeth = 18

- reference rack = UNI 6587-69

- pitch diameter = 54

- pressure angle = 20°

The torque passes from the first to the second gear wheel, featured by:

- module = 3

- number of teeth = 63

- reference rack = UNI 6587-69

- pitch diameter = 189

- pressure angle = 20°

On the same shaft there are two other helical gear wheels, one right handed, the other left handed, so that the forces are counterbalanced and so that there is a constraint for the shaft on which these two gear wheels are fit - otherwise the shaft would move towards the bottom of the box-. Each of these last two gear wheels mates with another gear wheel fit on the two output shafts.

The parametrization must leave the wheel base unchanged = 121,5mm.

In other words, the sum of the teeth belonging to the two gear wheels must be constant = 81 teeth.

Analysing the free spaces and the dimensions of the other near objects in the assembly, we can observe the velocity ratio and the output torque change while changing the number of teeth 4 times:

z = teeth number of the gear wheel

m = module

d = pitch diameter of the gear wheel    d = z*m

t= velocity ratio         t 1-2 = z1/z2      

                                                          t tot = Pt _i    = 1/50                                                          

C = output torque                              C = C input / ttot

C input = 12 Nm

Increasing the number of teeth of the first gear wheel and reducing that of the driven one, the velocity ratio increases.

Consequently the output torque available at the output shafts decreases. Together with the torque, the preloading axial force decreases, and this is due to the tangential force on the output pigno n, which lessens.

C = output torque

F _tg = nominal axial force on output pignon

F _max = maximum tangential force on pignon, found multiplying the nominal axial force by a safety factor 

F _{tg p} = preloading tangential force on pignon, equal to 40% of maximum tangential force

C _p = preloading torque on output pignon shaft

F _{tg prel} = tangential force on double preloading output helical pignon

F _ax = preloading axial force

Necessary formula for the computations:

F _tg = C / (d/2)

F _max = 1,25* F _tg

F _{tg p} = 0,4* F _max

C _p = F _{tg p} *d/2

F _{tg prel} = [C _p *(z2/z1)]/(d_p/2)

F _ax = 2* F _{tg prel} tgb

Symbols:

d = pitch diameter of output pignon = 100,798mm

d_p = pitch diameter of helical preloading pignon = 77,28mm

z1 = number of teeth of annual gear 293340 = 56

z2 = number of teeth of preloading helical pignon 293330 = 21

b = rake of preloading helical pignon = 18°

Forces are expressed in N and torques in Nm.

On the one hand, a further increase of the first gear wheel number of teeth could be interesting for the further reduction of the velocity ratio, and the dimensions of the two parts would be compatible with the other parts in the assembly. On the other hand, this would mean a further reduction of the output torque, while a machine tool requires a certain value of available torque, which allows better finish turning, higher chip thickness,...

Also the output helical gears should be parametrized, but this would mean increasing their number of teeth (decreasing is not compatible with the geometry of the box), which would mean a further reduction of the output torque.

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