. Reducing for cable or belt carrying
. Increasing for screw carrying (or direct drive)

Example of application on Otocoup machine :

X axis(horizontal) gear transmission, carrying with belts.
stepper gear 12 teeth, output gear 90 dents, so for one motor rotation, 12/90 = 0.1333 output rotation

Y axis (vertical) gear transmission, carrying with belts.
stepper gear 12 teeth, output gear 100 teeth, so for one motor rotation, 12/100 = 0.12 output rotation

Z axis (cutting depth) gear transmission, carrying with simple screw.
stepper gear 58 teeth, output gear 38 teeth, so for one motor rotation, 58/38 = 1.528 output rotation

gear module chosen is 1 (see module definition herebelow), but this does not influence calculations (but it does influence size !)

If first stage is based upon belts, calculation is exactly the same, it is only based upon teeth number.

It must be noted that for screw transmission, it is simpler to use gears that to try to align motor and screw.

If we take a belt sprocket of 14 teeth, on a belt HTD 5M, pitch 5mm
Development of the belt on one rotation is 5 mm x 14 teeth, so 70 mm

So on X axis, for one stepper rotation we have 12/90*70 = 9.33 mm
So for a 200 step motor 9.333/200= 0.0466 mm/step, so 0.02333 mm/half-step

So on Y axis, for one stepper rotation we have 12/100*70 = 8.4 mm
So for a 200 step motor 8.4/200= 0.042 mm/step, so 0.021 mm/half-step

For Z axis, we use screw/nut M8, pitch being 1.25 mm
So on Z axis, for 1 stepper rotation we have 58/38*1.25 = 1.908 mm
So for a 200 step motor 1.908/200= 0.00954 mm/step, so 0.00477 mm/half-step

Nominal torque of given steppers is 6.2kg.cm, so 620mN.m
For a sprocket of 70 mm development, radius is 70/2/pi = 11.14 mm
When we reduce speed, torque increase

So, on X axis, output torque is 6.2 * 90/12 = 46.5 kg.cm
Radius being 1.114 cm (beware of units),
Blocking force is 46.5/1.114 = 41.74 kg

So, for Y axis, output torque is 6.2 * 100/12 = 51.67 kg.cm
Same sprocket as X axis
Blocking force is 46.5/1.114 = 46.37 kg

ISO thread pitch :

M5	M6	M8	M10	M12	M16
0.8 mm	1 mm	1.25 mm	1.5 mm	1.75 mm	2 mm

Torque indicated for stepper is maximum torque, stopped, with two coils energised.
Practically, we will work in half-step, that means that alternatively, 1 or 2 coils will be energised.
When only one coil is energised, torque is reduced of around 30 %.
When we run at some speed, torque reduces significantly. We can esteem 50% drop.
Moreover, there are mechanical plays, resonances, irregularitys of computer step sends, accelerations to manage.
At the end, we can estimate to have only 25% of the theoretical force.
For Y carriage, we can account on a force of 46.4 kg * 0.25 = 11.6 kg
As this carriage is hanged of its all weight (6 to 9 kg) to the belt, the weight must be deduced of the force.
If we have a carriage of 6 kg, the force which can be applied on the tool is around 11.6 kg - 6kg, so 5.6 kg.
This is sufficient, but we can see there is some loss !

If you want to use a heavy router (3.5 -4 kg), instead of Kress router (1.6 kg), or if you want to install a second head, you must either choose a stonger stepper, or modify the reduction.
The problem is that a gear of 12 teeth, module 1, is practically the minimum gear possible (the stepper shaft is 6.35 mm diameter).
Gear 100 teeth, module 1 is the larger gear we can install in the designed carriage.
Only remain belt spocket. It may be possible to find a sprocket 12 teeth instead of 14. 10 teeth sprocket probably don't exist. So, the room for modification is not wide !

For X carriage, we have a force of 41.7 * 0.24 = 10.4 kg
Beam weight around 13 kg, you have to add Y carriage weight, so a total around 20 kg. A force of half the weight is fairly suffcient.

In case of a cable carrying, it is cpomparable to belt carrying, but development is calculated differently. In fact, the cable is inserted in the screw thread. At first approach, we can estimate the cable roll diameter at external screw diameter less half of the screw pitch. Practically, it depends a lot of cable diameter vs the screw pitch.

Practical application, my own machine :
Carrying on a screw M8. Screw pitch is 1 mm, so an estimated roll diameter of 8-1/2 = 7.75 mm, so a development of 7.75 * pi = 24.35 mm
Stepper gear 20 teeth, flywheel gear 58 teeth.
So for one stepper rotation, a displacment of 20/58*24.35 = 8.40 mm/rotation
That give, per step (200 steps motor): 0.042 mm/step, so 0.021 mm/half-step
Stepper give a nominal torque of 6.2 kg.cm so on output shaft 58/20 * 6.2 = 17.98 kg.cm
Rolling radius being 7.75/2 = 3.875 mm = 0.3875 cm, we find a force of :
17.98 / 0.3875 = 46.4 kg

Operation in half-step is required to limit resonances at small speed. Though, it is not possible to totally avoid them.

The only mean to avoid resonances is to use 'micro-step' mode, which need special costly boards. Generally, on amateur routers, we 'do with' the resonances, which don't affect too much quality of the work, even if that drive to some noise (noise you will not hear as soon as you start the router...).

In full-step mode, it arrives that the machine simply cannot work at some speeds (this is the case of Otocoup machine)

Useful diameter on a gear is the average diameter where there is contact between teeth. This diameter is called 'primitive' diameter. Gear dimension is defined by the unit called 'module'. A module of 1 give a primitive diameter of 1 mm per teeth.

As example :
. a gear 58 teeth module 1 will have a primitive diameter of 58 mm
. a gear 38 teeth module 1.5 will have a primitive diameter of 38x1.5 = 57 mm

Teeth height is around twice the module, and distance between two teeth is evidently pi*module. So one tooth module 1.5 will have a pitch of 1.5 x pi = 4.71 mm

External diameter is equal to primitive diameter + 2*module, our 58 teeth gear module 1 have an external diameter of 60 mm.

Timing belts have evolved in recent years, and the extent of the proposals can be troublesome.
US manufacturer 'Gates' is world leader on that market, and have defined modern belts standards.

Long length belts for moving carriages are called 'LL' belts

1) Classical tooth belts (CTD)

Pitch in inches, elastomeric material.
Arming cables are in fiberglass, and can be in steel on demand.
Strength improvment given by steel cables is 30%. I don't know the stiffness imlprovment, which is the important thing for us.

Sizes : (L for Light, and H : Heavy)
MXL : pitch 2.032 mm. width 6.35, 9.53 mm
In printer and small machines
XL : pitch 5.08 mm. width 6.35, 9.53 mm
This is the size used in CNC router 'Coquery' type
L : pitch 9.52 mm width 12.7, 19.05, 25.4 mm
H : pitch 12.7 mm width 19.05, 25.4, 38.10, 50.80, 76.20 mm
XH : pitch 22.23 mm
XXH : pitch 31.75 mm
Minimum teeth number for sizes MXL, XL et L is 10

2) HTD belt type

Elastomeric material, teeth are larger and higher (with same pitch) that CTD belts. Teeth shape is rounded.
Strength at same width and pitch is around four times than the strength of CTD belts. So, price is somewhat higher.

Sizes : (M for metric)
3M : pitch 3 mm
5M : pitch 5 mm, widths 6, 10, 15, 25 mm
8M : pitch 8 mm, width 10, 15, 20, 30, 50, 85 mm
14M : pitch 14 mm
20M : pitch 20 mm
Minimum teeth for sizes :
3M : 10 teeth, 5M : 12 teeth, 8M: 22 teeth.

3) T and AT belts

Theses belts (metric) are stronger than CTD belts. They are similar (trapezoidal shape), but teeth are higher and in case of AT belts, very large.
Nevertheless, these belts are not as strong as HTD belts.
As minimum roll diameter of T belts is much lower than AT belts, we will choose T belts.

4) High performances belts

Since ten years exist high power belts in polyurethane, with sometimes an arming material being kevlar (aramide fiber). These are Polychain belts (Gates brand). These belts are as strong as metallic chains

These belts have for us a big default, their minimum roll diameter is relatively high, which render them difficult to use for a carriage move, because it will require another reduction stage.

For a moving usage, there is an increase over HTD belts, but it is not as high as the gain between HTD and CTD belt. I don't know the stiffness increase, but it can be important for aramide fiber (Kevlar) armored belts.

5) Final choice

For 'Otocoup' machine, regarding it's large size, i recommend to use :
. HTD 5M, width 25 mm
and , if not possible :
. T5, width 25

While 'Coquery' type machines are generally equipped with CTD XL belts, i feel it might be preferable to use HTD 5M belts, but in width 10 or 15mm, to improve precision and stiffness. It depends of machine use, but if you wish to machine aluminium, higher cost can be balanced.

6) Alternative

Considering that it is done on professional machines, and also that the cost is lower than good quality belts, it is now proposed an alternative for transmission with rack and pinion.

2 meters size gantry is a rectangular beam done with 4 mason rulers 100 x 18 x 1.

Medium panels are too heavy compared to aluminium at equal stiffness, but i have no numbers. Also, medium protection against humidity is a problem, even if some suggests to use scavenged oil mixed with petrol as protective coating... You have right to prefer linseed oil.

Mason rulers are in nearly pure aluminium (to ease extrusion), and they are easy to be marked or bent. This fragility is a problem, because you must change the rulers in case of any incident which will drive to an impact.

On main beam, present drawings shows angles in anodised aluminium glued (epoxy) on the beam. That will reinforce the stiffness, and also increase the resistance to marking.

The horizontal aluminium rulers can be replaced by steel beams, which is recommended, as proposed as an option in the drawings. As they are not moving, the higher weight will not be a concern.