CYLINDRICAL MILLING
GEARS
§ 54. BASIC INFORMATION ABOUT GEARING
Gear elements
To cut a gear, you need to know the elements gearing, i.e. number of teeth, tooth pitch, tooth height and thickness, pitch circle diameter and outer diameter. These elements are shown in Fig. 240.
Let's consider them sequentially.
In each gear there are three circles and, therefore, three corresponding diameters:
Firstly, lug circumference, which is the outer circumference of the gear blank; the diameter of the circle of the lugs, or outer diameter, is designated D e;
Secondly, pitch circle, which is a conditional circle dividing the height of each tooth into two unequal parts - the upper one, called tooth head, and the lower one, called stem of the tooth; the height of the tooth head is indicated h", height of the tooth stem - h"; The diameter of the pitch circle is designated d;
Thirdly, depression circumference, which runs along the base of the tooth cavities; the diameter of the circle of the depressions is indicated D i.
The distance between the same (i.e. facing the same direction, for example two right or two left) side surfaces (profiles) of two adjacent wheel teeth, taken along the arc of the pitch circle, is called the pitch and is designated t. Therefore, we can write:
Where t- step in mm;
d- diameter of the pitch circle;
z- number of teeth.
Module m the length corresponding to the diameter of the pitch circle per one tooth of the wheel is called; Numerically, the module is equal to the ratio of the diameter of the pitch circle to the number of teeth. Therefore, we can write:
From formula (10) it follows that the step
t = π m = 3,14m mm.(9b)
To find out the pitch of a gear, you need to multiply its module by π.
In the practice of cutting gears, the most important thing is the module, since all the elements of the tooth are related to the size of the module.
Tooth head height h" equal to modulus m, i.e.
h" = m.(11)
Tooth stem height h" equal to 1.2 modules, or
h" = 1,2m.(12)
The height of the tooth, or the depth of the cavity,
h = h" + h" = m + 1,2m = 2,2m.(13)
By number of teeth z gear, you can determine the diameter of its pitch circle.
d = z · m.(14)
The outer diameter of the gear is equal to the diameter of the pitch circle plus the height of the two tooth heads, i.e.
D e = d + 2h" = zm + 2m = (z + 2)m.(15)
Consequently, to determine the diameter of the gear blank, the number of its teeth must be increased by two and the resulting number multiplied by the module.
In table 16 shows the main dependencies between the gear elements for a cylindrical wheel.
Table 16
Example 13. Determine all dimensions required for the manufacture of a gear having z= 35 teeth and m = 3.
We determine the outer diameter, or diameter of the workpiece, using formula (15):
D e = (z + 2)m= (35 + 2) 3 = 37 3 = 111 mm.
Using formula (13), we determine the height of the tooth, or the depth of the cavity:
h = 2,2m= 2.2 3 = 6.6 mm.
We determine the height of the tooth head using formula (11):
h" = m = 3 mm.
Gear cutters
To mill gears on horizontal milling machines, shaped disk cutters with a profile corresponding to the cavity between the teeth of the wheel are used. Such cutters are called gear-cutting disk (modular) cutters (Fig. 241).
Gear cutters are selected depending on the module and number of teeth of the wheel being milled, since the shape of the cavity of two wheels of the same module, but with different numbers teeth are not the same. Therefore, when cutting gears, each number of teeth and each module should have its own gear cutter. In production conditions, several cutters for each module can be used with a sufficient degree of accuracy. To cut more precise gears, it is necessary to have a set of 15 gear-cutting disk cutters; for less precise ones, a set of 8 gear-cutting disk cutters is sufficient (Table 17).
Table 17
15 Piece Gear Cutting Disc Mill Set
8 Piece Gear Cutting Disc Mill Set
In order to reduce the number of sizes of gear cutters in the Soviet Union, gear modules are standardized, i.e., limited to the following modules: 0.3; 0.4; 0.5; 0.6; 0.75; 0.8; 1.0; 1.25; 1.5; 1.75; 2.0; 2.25; 2.50; 3.0; 3.5; 4.0; 4.5; 5.0; 5.5; 6.0; 6.5; 7.0; 8.0; 9.0; 10.0; eleven; 12; 13; 14; 15; 16; 18; 20; 22; 24; 26; 28; thirty; 33; 36; 39; 42; 45; 50.
On each gear-cutting disc cutter, all the data characterizing it is stamped, allowing you to correctly select the required cutter.
Gear cutters are made with backed teeth. This is an expensive tool, so when working with it it is necessary to strictly observe cutting conditions.
Measuring tooth elements
The thickness and height of the tooth head is measured with a tooth gauge or a caliper gauge (Fig. 242); the design of its measuring jaws and the vernier reading method are similar to a precision caliper with an accuracy of 0.02 mm.
Magnitude A on which the leg should be installed 2 dental gauge will be:
A = h" a = m a mm,(16)
Where m
Coefficient A is always greater than one, since the height of the tooth head h" is measured along the arc of the initial circle, and the value A measured along the chord of the initial circle.
Magnitude IN, on which the jaws should be installed 1
And 3
dental gauge will be:
IN = m b mm,(17)
Where m- module of the measured wheel.
Coefficient b takes into account that the size IN is the size of the chord along the initial circle, while the width of the tooth is equal to the arc length of the initial circle.
Values A And b are given in table. 18.
Since the reading accuracy of the caliper is 0.02 mm, then we discard the third decimal place for the values obtained by formulas (16) and (17) and round them to even values.
Table 18
Values a And b for installing a caliper
| Number of teeth measured wheels | Coefficient values | Number of teeth measured wheels | Coefficient values | ||
| a | b | a | b | ||
| 12 | 1,0513 | 1,5663 | 27 | 1,0228 | 1,5698 |
| 13 | 1,0473 | 1,5669 | 28 | 1,0221 | 1,5699 |
| 14 | 1,0441 | 1,5674 | 29 | 1,0212 | 1,5700 |
| 15 | 1,0411 | 1,5679 | 30 | 1,0206 | 1,5700 |
| 16 | 1,0385 | 1,5682 | 31-32 | 1,0192 | 1,5701 |
| 17 | 1,0363 | 1,5685 | 33-34 | 1,0182 | 1,5702 |
| 18 | 1,0342 | 1,5688 | 35 | 1,0176 | 1,5702 |
| 19 | 1,0324 | 1,5690 | 36 | 1,0171 | 1,5703 |
| 20 | 1,0308 | 1,5692 | 37-38 | 1,0162 | 1,5703 |
| 21 | 1,0293 | 1,5693 | 39-40 | 1,0154 | 1,5704 |
| 22 | 1,0281 | 1,5694 | 41-42 | 1,0146 | 1,5704 |
| 23 | 1,0268 | 1,5695 | 43-44 | 1,0141 | 1,5704 |
| 24 | 1,0257 | 1,5696 | 45 | 1,0137 | 1,5704 |
| 25 | 1,0246 | 1,5697 | 46 | 1,0134 | 1,5705 |
| 26 | 1,0237 | 1,5697 | 47-48 | 1,0128 | 1,5706 |
| 49-50 | 1,023 | 1,5707 | 71-80 | 1,0077 | 1,5708 |
| 51-55 | 1,0112 | 1,5707 | 81-127 | 1,0063 | 1,5708 |
| 56-60 | 1,0103 | 1,5708 | 128-135 | 1,0046 | 1,5708 |
| 61-70 | 1,0088 | 1,5708 | Rail | 1,0000 | 1,5708 |
Example 14. Install a gear gauge to check the tooth dimensions of a wheel with a module of 5 and a number of teeth of 20.
According to formulas (16) and (17) and table. 18 we have:
A = m a= 5 · 1.0308 = 5.154 or, rounded, 5.16 mm;
IN = m b= 5 · 1.5692 = 7.846 or, rounded, 7.84 mm.
If the size of this arc is taken as many times as there are teeth on the wheel, i.e. z times, then we also obtain the length of the initial circle; hence,
Π d = t z
from here
d = (t/Π)z
Step ratio t of a link to a number Π is called the module of the link, which is denoted by the letter m, i.e.
t / Π = m
The module is expressed in millimeters. Substituting this notation into the formula for d, we get.
d = mz
where
m = d/z
Therefore, the module can be called the length corresponding to the diameter of the initial circle per one tooth of the wheel. The diameter of the protrusions is equal to the diameter of the initial circle plus two heights of the tooth head (Fig. 517, b) i.e.
D e = d + 2h"
The height h" of the tooth head is taken equal to the module, i.e. h" = m.
Let's express it in terms of module right side formulas:
D e = mz + 2m = m (z + 2)
hence
m = D e: (z +2)
From fig. 517, b it is also clear that the diameter of the circle of the depressions is equal to the diameter of the initial circle minus two heights of the tooth stem, i.e.
D i= d - 2h"
The height h" of the tooth leg for cylindrical gears is taken equal to 1.25 modules: h" = 1.25m. Expressing the right-hand side of the formula for D in terms of the modulus i we get
D i= mz - 2 × 1.25m = mz - 2.5m
or
Di = m (z - 2.5m)
The entire tooth height h = h" + h" i.e.
h = 1m + 1.25m = 2.25m
Consequently, the height of the tooth head is related to the height of the tooth stem as 1: 1.25 or as 4: 5.
The tooth thickness s for unprocessed cast teeth is taken to be approximately equal to 1.53m, and for machined teeth (for example, milled) - equal to approximately half the pitch t engagement, i.e. 1.57m. Knowing that step t engagement is equal to the thickness s of the tooth plus the width s in the cavity (t = s + s in ) (step size t determined by the formula t/ Π = m or t = Πm), we conclude that the width of the cavity for wheels with cast raw teeth.
s in = 3.14m - 1.53m = 1.61m
A for wheels with machined teeth.
s in = 3.14m - 1.57m = 1.57m
The design of the rest of the wheel depends on the forces that the wheel experiences during operation, on the shape of the parts in contact with this wheel, etc. Detailed calculations of the dimensions of all elements of the gear wheel are given in the course “Machine Parts”. To perform a graphic representation of gears, the following approximate relationships between their elements can be accepted:
Rim thicknesse = t/2
Shaft hole diameter D in ≈ 1 / in D e
Hub diameter D cm = 2D in
Tooth length (i.e. thickness of the wheel ring gear) b = (2 ÷ 3) t
Disk thickness K = 1/3b
Hub length L=1.5D in: 2.5D in
The dimensions t 1 and b of the keyway are taken from table No. 26. After determining the numerical values of the engagement module and the diameter of the hole for the shaft, it is necessary to coordinate the resulting dimensions with GOST 9563-60 (see table No. 42) for modules and for normal linear dimensions in accordance with GOST 6636-60 (table No. 43).
ORDER OF USE OF TABLES / PROGRAM
To select replacement wheels, the required gear ratio is expressed as a decimal fraction with the number of decimal places corresponding to the required accuracy. In the “Basic tables” for selecting gears (page 16-400) we find a column with a heading containing the first three digits of the gear ratio; Using the remaining numbers, we find the line on which the number of teeth of the driving and driven wheels is indicated.
You need to select replacement guitar wheels for a gear ratio of 0.2475586. First we find the column with the heading 0.247-0000, and below it the closest value to the subsequent decimal places of the desired gear ratio (5586). In the table we find the number 5595, corresponding to a set of replacement wheels (23*43) : (47*85). Finally we get:
i = (23*43)/(47*85) = 0.2475595. (1)
Relative error compared to a given gear ratio:
δ = (0.2475595 - 0.2475586) : 0.247 = 0.0000037.
We strictly emphasize: in order to avoid the influence of a possible typo, it is necessary to check the resulting relationship (1) on a calculator. In cases where the gear ratio is greater than one, it is necessary to express its reciprocal value as a decimal fraction, using the value found in the tables to find the number of teeth of the driving and driven replacement wheels and swap the driving and driven wheels.
It is required to select replacement guitar wheels for the gear ratio i = 1.602225. We find the reciprocal value 1:i = 0.6241327. In the tables for the nearest value 0.6241218 we find a set of replacement wheels: (41*65) : (61*70). Considering that the solution has been found for the inverse of the gear ratio, we swap the driving and driven wheels:
i = (61*70)/(41*65) = 1.602251
Relative selection error
δ = (1.602251 - 1.602225) : 1.602 = 0.000016.
Typically, it is necessary to select wheels for gear ratios expressed to the sixth, fifth, and in some cases to the fourth decimal place. Then the seven-digit numbers given in the tables can be rounded to the appropriate decimal place. If the existing set of wheels is different from the normal one (see page 15), then, for example, when adjusting the differential or break-in chains, you can select a suitable combination from a number of adjacent values with an error that satisfies the conditions set out on pages 7-9. In this case, some numbers of teeth can be replaced. So, if the number of teeth in a set is not more than 80, then
(58*65)/(59*95) = (58*13)/(59*19) = (58*52)/(59*76)
The “heel” combination is preliminarily transformed as follows:
(25*90)/(70*85) = (5*9)/(7*17)
and then, using the obtained factors, the number of teeth is selected.
DETERMINING THE PERMISSIBLE SETUP ERROR
It is very important to distinguish between absolute and relative tuning errors. The absolute error is the difference between the obtained and required gear ratios. For example, it is required to have a gear ratio i = 0.62546, but the result is i = 0.62542; the absolute error will be 0.00004. Relative error is the ratio of the absolute error to the required gear ratio. In our case, the relative error
δ = 0.00004/0.62546 = 0.000065
It should be emphasized that the accuracy of the adjustment must be judged by the relative error.
General rule.
If any value A obtained by tuning through a given kinematic chain is proportional to the gear ratio i, then with a relative tuning error δ, the absolute error will be Aδ.
For example, if the relative error of the gear ratio is δ = 0.0001, then when cutting a screw with a pitch t, the deviation in the pitch, depending on the setting, will be 0.0001 * t. The same relative error when adjusting the differential of a gear hobbing machine will result in additional rotation of the workpiece not to the required arc L, but to an arc with a deviation of 0.0001 * L.
If a product tolerance is specified, the absolute size deviation due to adjustment inaccuracy should be only a certain fraction of this tolerance. In the case of a more complex dependence of any value on the gear ratio, it is useful to resort to replacing actual deviations with their differentials.
Adjusting the differential chain when processing screw products.
The following formula is typical:
i = c*sinβ/(m*n)
where c is the chain constant;
β - angle of inclination of the helix;
m - module;
n is the number of cuts of the cutter.
Having differentiated both sides of the equality, we obtain the absolute error di of the gear ratio
di = (c*cosβ/m*n)dβ
then the permissible relative adjustment error is
δ = di/i = dβ/tgβ
If tolerance express the helix angle dβ not in radians, but in minutes, we get
δ = dβ/3440*tgβ (3)
For example, if the angle of inclination of the helix of the product is β = 18°, and the permissible deviation in the direction of the tooth is dβ = 4" = 0",067, then the permissible relative adjustment error
δ = 0.067/3440*tg18 = 0.00006
On the contrary, knowing the relative error of the given gear ratio, we can use formula (3) to determine the permissible error in the helix angle in minutes. When establishing the permissible relative error, you can use trigonometric tables in such cases. Thus, in formula (2) the gear ratio is proportional to sin β. According to trigonometric tables for the taken numerical example it can be seen that sin 18° = 0.30902, and the difference in sines per 1" is 0.00028. Therefore, the relative error per 1" is 0.00028: 0.30902 = 0.0009. The permissible deviation of the helix is 0.067, therefore the permissible error of the gear ratio is 0.0009 * 0.067 = 0.00006, the same as when calculating using formula (3). When both mating wheels are cut on the same machine and using the same differential chain setting, significantly larger errors in the direction of the tooth lines are allowed, since both wheels have the same deviations and only slightly affect the lateral clearance when the mating wheels engage.
Setting up the running chain when machining bevel wheels.
In this case, the setting formulas look like this:
i = p*sinφ/z*cosу or i = z/p*sinφ
where z is the number of teeth of the workpiece;
p is the running-in chain constant;
φ is the angle of the initial cone;
y is the angle of the tooth stem.
The radius of the main circle is proportional to the gear ratio. Based on this, you can set the permissible relative adjustment error
δ = (Δα)*tgα/3440
where α is the engagement angle;
Δα is the permissible deviation of the engagement angle in minutes.
Settings for processing screw products.
Setting formula
δ = Δt/t or δ = ΔL/1000
where Δt is the deviation in the propeller pitch due to tuning;
ΔL is the accumulated error in mm per 1000 mm of thread length.
The value Δt gives absolute mistake step, and the value ΔL essentially characterizes the relative error.
Adjustment taking into account screw deformation after processing.
When cutting taps taking into account the shrinkage of steel after subsequent heat treatment or taking into account the deformation of the screw due to heat during machining, the percentage of shrinkage or expansion directly indicates the required relative deviation in gear ratio compared to what would have happened without taking these factors into account. In this case, the relative deviation of the gear ratio, plus or minus, is no longer an error, but a deliberate deviation.
Setting up dividing circuits. Typical tuning formula
where p is a constant;
z is the number of teeth or other divisions per revolution of the workpiece.
A normal set of 35 wheels provides absolutely accurate tuning up to 100 divisions, since the numbers of wheel teeth contain all the prime factors up to 100. In such tuning, the error is generally unacceptable, since it is equal to:
where Δl is the deviation of the tooth line at the workpiece width B in mm;
pD is the length of the initial circle or the corresponding other circumference of the product in mm;
s - feed along the axis of the workpiece per revolution in mm.
Only in rough cases this error may not play a role.
Setting up gear hobbing machines in the absence of the required multipliers in the number of teeth of replacement wheels.
In such cases (for example, at z = 127) you can tune the division guitar to approximately a fractional number teeth, and make the necessary correction using a differential. Usually the formulas for tuning guitars for division, feed and differential look like this:
x = pa/z ; y = ks ; φ = c*sinβ/ma
Here p, k, c are, respectively, the constant coefficients of these circuits; a is the number of cuts of the cutter (usually a = 1).
We tune the specified guitars according to the formulas
x = paA/Az+-1 ; y = ks ; φ" = pc/asA
where z is the number of teeth of the wheel being processed;
A is an arbitrary integer chosen so that the numerator and denominator of the gear ratio are factorized into factors suitable for selecting replacement wheels.
The sign (+) or (-) is also chosen arbitrarily, which makes factorization easier. When working with a right-handed cutter, if the (+) sign is selected, the intermediate wheels on the guitars are placed as they are done according to the manual for working on this machine for a right-handed workpiece; if the (-) sign is selected, the intermediate wheels are installed as for a left-handed workpiece; when working with the left cutter, it’s the other way around.
It is advisable to choose A within
then the differential chain ratio will be from 0.25 to 2.
It is especially necessary to emphasize that when taking replacement wheels on a guitar, the actual feed must be determined in order to be substituted into the differential adjustment formula with great accuracy. It is better to calculate it using the kinematic diagram of the machine, since the constant coefficient k in the feed adjustment formula in the machine manual is sometimes given approximately. If this instruction is not followed, the wheel teeth may become noticeably beveled instead of straight.
Having calculated the feed, we practically obtain precise tuning using the first two formulas (4). Then the permissible relative error in tuning the guitar differential is
δ = sA*Δl/пmb (5)
de b is the width of the workpiece gear rim;
Δl is the permissible deviation of the tooth direction at the width of the crown in mm.
In the case of cutting wheels with helical teeth, it is necessary, using a differential, to provide the cutter with additional rotation to form a helix and additional rotation to compensate for the difference between the required number of divisions and the actually adjusted number of divisions. The resulting setup formulas are:
x = paA/Az+-1 ; y = ks ; φ" = c*sinβ/ma +- pc/asA
In the formula for x, the sign (+) or (-) is chosen arbitrarily. In these cases:
1) if the screw direction of the cutter and the workpiece is the same, in the formula for φ" they take the same sign as chosen in the formula for x;
2) if the direction of the screw for the cutter and the workpiece is different, then in the formula for φ" the sign is taken opposite to that chosen for x.
The intermediate wheels on guitars are placed as indicated in the instructions for this machine, according to the direction of the screw teeth. Only if it turns out that φ"
Non-differential setting.
In some cases, when processing screw products, it is possible to use more rigid non-differential machines if a secondary passage of the processed cavities is not required from the same installation and with an accurate hit into the cavity. If the machine is set up at a predetermined feed rate, due to the small number of replacement wheels or the presence of a feed box, then setting up the division chain requires great accuracy, i.e. it must be carried out as precision. Permissible relative error
δ = Δβ*s/(10800*D*cosβ*cosβ)
where Δβ is the deviation of the product helix in minutes;
D is the diameter of the initial circle (or cylinder) in mm;
β is the angle of inclination of the workpiece tooth to its axis;
s - feed per revolution of the workpiece along its axis in mm.
To avoid time-consuming precision tuning, proceed as follows. If a sufficiently large set of wheels can be used for a guitar feed (25 or more, in particular the normal set and tables in this book), then first consider the given feed s approximate. Having adjusted the division chain and considering the adjustment to be quite accurate, they determine what the axial feed s should be for this.
The usual fission chain formula is rewritten as follows:
x = (p/z)*(T/T+-z") = ab/cd (6)
where p is the constant coefficient of the fission circuit;
z - number of divisions of the product (teeth, grooves);
T = pmz/sinβ - pitch of the workpiece helix in mm (it can be determined in another way);
s" - tool feed along the axis of the workpiece per revolution in mm. The sign (+) is taken for different directions of the screw of the cutter and the workpiece; sign (-) for the same.
Having selected, in particular from the tables in this book, the drive wheels with the numbers of teeth a and b, and the driven ones - c and d, from formula (6) we determine the exact required feed
s" = T(pcd - zab)/zab (7)
Substitute the value s" into the feed adjustment formula
The relative error δ of the feed setting causes a corresponding relative error of the helix pitch T. Based on this, it is not difficult to establish that when tuning a guitar’s pitch, a relative error can be allowed
δ = Δβ/3440*tgβ (9)
From a comparison of this formula with formula (3) it is clear that the permissible error in tuning the pitch guitar in this case is the same as it is with the usual tuning of the differential circuit. It is worth emphasizing once again the need to know exact value coefficient k in the feed formula (8). If in doubt, it is better to check it by calculation using the kinematic diagram of the machine. If the coefficient k itself is determined with a relative error δ, then this causes an additional deviation of the helix by Δβ, determined for a given β from relation (9).
CONDITIONS OF ADJACTION OF REPLACEMENT WHEELS
In machine manuals, it is useful to provide graphs that make it easy to assess in advance the adhesion capabilities of a given wheel combination. In Fig. Figure 1 shows the two extreme positions of the guitar, determined by circular grooves B. In Fig. Figure 2 shows a graph in which arcs of circles are drawn from points Oc and Od, which are the centers of the first drive wheel a and the last driven wheel d (Fig. 3). The radii of these arcs on the accepted scale are equal to the distances between the centers of interlocking interchangeable wheels with the sums of the numbers of teeth 40, 50, 60, etc. These sums of the numbers of teeth for the first pair of interlocking wheels a + c and the second pair b + d are placed at the ends corresponding arcs.
Let a set of wheels be found from the tables (50*47) : (53*70). Will they mate in the order 50/70 * 47/53? The sum of the numbers of the teeth of the first pair is 50 + 70 = 120 The center of the finger should lie somewhere on the arc marked 120 drawn from the center Oa. The sum of the numbers of teeth of the wheels of the second pair is 47 + 53 = 100. The center of the pin should be on the arc marked 100 drawn from the center Od. As a result, the center of the finger will be established at point c at the intersection of the arcs. According to the diagram, wheel traction is possible.
For the combination 30/40 * 20/50, the sum of the numbers of teeth of the first pair is 70, the second is also 70. Arcs with such marks do not intersect inside the figure, therefore, wheel traction is impossible.
In addition to the graph shown in Fig. 2, it is advisable to also draw the outline of the box and other parts that may interfere with the installation of gears on the guitar. To make the best use of the tables in this book, it is advisable for the guitar designer to follow following conditions, which are not strictly required, but desirable:
1. The distance between the permanent AXLES Oa AND Od must be such that two pairs of wheels with total amount 180 teeth could still engage in mutual engagement. The most desirable distance Oa - Od is from 75 to 90 modules.
2. A wheel with a number of teeth of at least 70 should be installed on the first drive roller, and up to 100 on the last driven roller (if the dimensions allow, up to 120-127 can be provided for some cases of refined settings).
3. The length of the guitar slot at the extreme position of the finger should ensure the adhesion of the wheels located on the finger and on the axis of the guitar with a total of teeth of at least 170-180.
4. The extreme angle of deviation of the guitar groove from the straight line connecting the centers Oa and Od must be at least 75-80°.
5. The box must have sufficient dimensions. The adhesion of the most unfavorable combinations should be checked according to the graph included in the machine manual (see Fig. 2).
The machine or mechanism adjuster should use the graph given in the manual (see Fig. 2), but, in addition, take into account that the larger the gear on the first drive shaft (with at this moment forces), the less force on the teeth of the first pair; the larger the wheel on the last driven shaft, the less force on the teeth of the second pair.
Let us consider decelerating transmissions, i.e. the case when i
z1/z3 * z2/z4 ; z2/z3 * z1/z4 (10)
The second combination is preferable. It provides a lower moment of force on the intermediate shaft and allows you to comply with the requirements additional conditions(see Fig. 3):
a+c > b+(20...25); b + d > c+(20...25) (11)
These conditions are set to prevent replacement wheels from resting on the corresponding shafts or fastening parts; the numerical term depends on the design of the guitar in question. However, the second of combinations (10) can only be adopted if the wheel Z2 is installed on the first drive shaft and if the gear z2/z3 is slow or does not contain much acceleration. It is desirable that z2/z3
For example, the combination (33*59) : (65*71) is better used in the form 59/65 * 33/71 But in a similar case, the ratio 80/92 * 40/97 is not applicable if the wheel z = 80 is not placed on the first shaft. Sometimes, to fill in the corresponding intervals of gear ratios, inconvenient combinations of wheels are given in the tables, for example 37/41 * 92/79 With this order of wheels, condition (11) is not met. The drive wheels cannot be swapped, since the z = 92 wheel is not placed on the first shaft. These combinations are indicated for cases where a more accurate gear ratio must be obtained by any means. In these cases, you can also resort to methods for refined settings (p. 401). For acceleration gears (i > 1), it is advisable to split i = i1i2 so that the factors are as close as possible to each other and the speed increase is distributed more evenly. Moreover, it is better if i1 > i2
MINIMUM REPLACEMENT WHEELS SETS
The composition of sets of replacement wheels depending on the area of application is given in table. 2. For particularly precise settings, see page 403.
table 2
To set up the dividing heads, you can use the tables provided by the factory. It’s more complicated, but you can choose the appropriate heel combinations from the “Basic tables for selecting gears” given in this book.
Chapter 2
CUTTING CYLINDRICAL WHEELS WITH WORM CUTTERS
BASIC INFORMATION ABOUT THE PROCESS
Cutting teeth with a hob cutter is carried out on gear hobbing machines using the rolling method. The profile of the cutting part of a hob cutter in its axial section is close to the profile of the rack, so cutting teeth with a hob cutter can be represented as the engagement of the rack with a gear wheel.
The working stroke (cutting movement) is carried out by rotating cutter 4 (Fig. 1). To ensure running-in, the rotation of the cutter and workpiece 3 must be coordinated in the same way as when engaging the worm 1 and wheel 2, i.e., the rotation speed of the table with the workpiece must be less than the rotation speed of the cutter as many times as the number of teeth being cut more number cutter passes (with a single-pass cutter, the table with the workpiece rotates 1/2 times slower than the cutter).
The feed movement is carried out by moving the caliper with the cutter relative to the wheel being cut (parallel to its axis). New machine designs also have radial feed (plunging). When slicing helical wheels additional
1. Main kinematic chains of gear hobbing machines
| Chain | What is provided | Extreme elements of the chain | Movements to be connected | Setting organ |
| Express | Cutting speed u, m/min (cutter rotation speed n, rpm) | Electric motor - milling spindle | Rotation of the electric motor shaft ( ne, rpm) and cutters ( n, rpm) | Guitar speeds |
| Axial (vertical) feed chain | Innings Soi mm/rev | Table - caliper feed screw | One revolution of the workpiece - axial movement of the caliper by the amount Eo | Guitar feed |
| Fission circuit | Number of teeth cut z | Table - milling spindle | One revolution of the cutter k/z table revolutions | Guitar division |
| Differential chain | The angle of inclination of the cut teeth in | Table - caliper feed screw | Moving the caliper by an axial step ta- additional rotation of the workpiece | Guitar differential |
Rice. 1. Operating principle of gear hobbing machines:
1 - worm; 2 - dividing worm wheel; 3 - workpiece; 4 - cutter; 5 - division guitar
rotation of the table with the workpiece associated with the feed movement. Therefore, the gear hobbing machine has kinematic chains and their adjustment organs (guitars) indicated in Table. 1.
GEAR MILLING MACHINES
Design and technical characteristics of machines
Depending on the position of the workpiece axis, gear hobbing machines (Table 2-4) are divided into vertical and horizontal. Vertical gear hobbing machines (Fig. 2) are made of two types: with a feed table and with a feed column (stand).
Rice. 2. General view of a vertical gear hobbing machine:
1 - table; 2 - bed; 3 - control panel; 4 - column; 5 - milling support; 6 - bracket; 7 - support stand
A machine with a feed table on which the workpiece is fixed has a fixed column with a milling support and a rear support column with or without a cross member. The approach of the cutter and the workpiece is carried out by horizontal movement of the table (along the guides).
A machine with a feed column that moves to approach the workpiece mounted on a stationary table can be made with or without a rear stand. Large machines usually do this.
Notes:
1. Machines with the letter “P” in the designation, as well as models 5363, 5365, 5371, 5373, 531OA, are machines of increased and high precision and are intended, in particular, for cutting turbine gears.
2. Large machines (mod. 5342, etc.) have a single division mechanism for working with disk and finger cutters using optional overhead heads: for cutting wheels with external teeth with a finger cutter (see Table 5), wheels with internal teeth a disk or finger cutter or a special hob cutter (see Table 1). On request, a broaching support for cutting worm wheels with tangential feed and a mechanism for cutting wheels with a cone angle of the tooth tips up to 10°, a reverse mechanism for cutting chevron wheels without a groove with a finger cutter are supplied.
3. Machines mod. 542, 543, 544, 546 and machines created on their basis are designed for cutting large high-precision worm wheels, for example index wheels of gear cutting machines.
4. Horizontal machines mod. 5370, 5373, 5375 and machines created on their basis are designed to work with a hob, finger and disk cutters; other domestically produced machines are used only for working with a hob cutter.
5. The letters indicated in brackets after the model name indicate variants of this model: for example, 5K324 (A, P) means that there are models 5K324, 5K324A and 5K324P.
3. Main table dimensions (in mm) of gear hobbing machines, number of index wheel teeth z k
Rice. 3. Horizontal gear hobbing machine:
1 - bed; 2 - tailstock; 3 - milling support; 4 - faceplate; 5 - front headstock
Horizontal hobbing machines(Fig. 3), intended primarily for cutting teeth of gear shafts (gears made integral with the shaft) and small gears with hobs, are made with a feed spindle headstock carrying the workpiece, or with a feed milling support.
On a feedstock machine, one end of the workpiece is secured in the spindlestock and the other is supported by the rear center. The hob cutter is located under the workpiece on the spindle of the milling support, the carriage of which moves horizontally along the guides of the machine bed parallel to the axis of the workpiece. Radial cutting of the cutter is carried out by vertical movement of the spindle head together with the rear center and the workpiece being processed.
On a machine with a feed support, the workpiece is secured in the spindle head and in rests. The hob cutter is located behind the workpiece, on the spindle of the milling support, the carriage of which, during working feed, moves horizontally along the guides of the bed, parallel to the axis of the workpiece.” Radial cutting of the cutter is carried out by horizontal movement of the milling support perpendicular to the axis of the workpiece.
The drive of the gear hobbing machine table is a worm gear - a worm with a worm wheel. The kinematic accuracy of the machine mainly depends on the accuracy of this transmission. Therefore, the table rotation speed should not be allowed to be too high to avoid heating and jamming of the teeth of the indexing worm gear. In the case of cutting wheels with a small number of teeth, as well as when using multi-start cutters, the actual sliding speed of the worm gear pair should be determined, which for cast iron wheels should not exceed 1-1.5 m/s, and for a worm wheel with a bronze rim 2-3 m/s. Sliding speed Uс(approximately equal to the peripheral speed of the worm) and rotational speed nh can be determined by formulas
where dch is the diameter of the initial circle of the dividing worm, mm; nh; n - rotation speed of the worm and cutter, rpm; zk; z - number of teeth of the dividing and cutting wheels; k is the number of passes of the hob cutter.
The designs of the machines provide the ability to adjust the dividing pair, table and spindle bearings, wedges and worm pair of the support.
Setting up gear hobbing machines
The main adjustment operations are setting up the kinematic chains of the machine (speeds, feeds, division, differential); installation, alignment, securing the workpiece and cutter; setting the cutter relative to the workpiece to the required milling depth; installation of stops for automatic shutdown of the machine.
It is convenient to consider the transmission of motion to various machine mechanisms on its kinematic diagram (Fig. 4), which greatly facilitates the derivation of formulas for setting up machine circuits.
The diagram shows the number of teeth of cylindrical, bevel and worm wheels and the number of worm starts in a worm gear. Electric motors for the main drive, accelerated movements, and axial movement of the cutter (along the axis of the milling mandrel) are also shown, which in some cases makes it possible to increase the durability of the cutter.
The diagram shows electromagnetic clutches, the inclusion of which in various combinations provides the required movements: MF1 or MF2 - rapid movement of the table or support; MF1 and MF4 - radial table feed; MF2 and MF4; MF2 and MFZ - vertical feed of the caliper up and down. Worm wheels are cut using radial feed of the cutter.
Gear hobbing machines have a differential mechanism designed for additional rotation of the workpiece when cutting helical wheels. When working with the differential turned on, the wheel z = 58 receives and transmits the main and additional rotations to the table. The main rotation is transmitted through bevel wheels z = 27, additional rotation is from the differential gear through a 27/27 bevel gear, 1/45 worm gear, carrier, differential wheels z = 27. In this case, the driven wheel rotates twice as fast as the worm wheel z = 45 and carrier (see below for setting up the differential chain). The main and additional rotations are added (the rotation of the workpiece is accelerated) if the inclination of the wheel teeth and the direction of the cutter turn are the same (for example, the right wheel is cut by the right cutter), and subtracted if they are different (for example, the right wheel is cut by the left cutter). The required direction of additional rotation relative to the main one is provided by the intermediate wheel in the differential gear.
When cutting spur wheels, the differential is turned off, the carrier is stationary, and only the main movement is transmitted (except for the setup of a machine for cutting a spur wheel with a simple number of teeth, discussed below).
Guitar tuning machines mod. 5K32A and 5K324A (see Fig. 4). Guitar speeds (rotation of cutter). The high-speed chain connects the specified rotation speed of the cutter nf with the rotational speed of the main drive electric motor ne = 1440 rpm, therefore the equation of the high-speed chain has the following form:
Where does the gear ratio of the guitar come from?
where a and b are the numbers of teeth of the replacement guitar speed wheels.
The machine is equipped with five pairs of replaceable wheels (23/64, 27/60; 31/56; 36/51; 41/46). The wheels of each pair can be installed in the specified and reverse order(for example, 64/23), which allows you to obtain, respectively, ten different cutter speeds (40, 50, 63, 80, 100, 125, 160, 200, 250, 315 rpm).
Guitar division. To cut wheels with a given number of teeth r during one revolution of the hob cutter with the number of passes k, the workpiece must make k/z, revolution, which is ensured by the selection of replacement wheels of the division guitar with a gear ratio i business
The dividing circuit equation has the following form:
IN general view The calculation formula for tuning a division guitar can be represented as follows:
The Transaction values for a number of machines are given in table. 5.
The machine is supplied with 45 replaceable wheels with a 2.5 mm module. guitars of division, feed and differential with the following numbers of teeth: 20 (2 pcs.), 23, 24 (2 pcs.), 30, 33, 34, 35, 37, 40 (2 pcs.), 41, 43, 45, 47, 50, 53, 55, 58, 59. 60, 61, 62, 67, 70 (2 pcs.), 71, 72, 75 (2 pcs.), 79, 80, 83, 85, 89, 90, 92, 95, 97 98, 100.
Other options for selecting replacement wheels are also possible, for example 30/55 35/70, etc.
To place two pairs of interchangeable wheels in any guitar, the following conditions must be met: a1 + b1 > c1; c1 + d1 > b1.
We check: 30 + 55 > 40; 40 + 80 > 55; 0b conditions are met.
Example 2. According to the table supplied with the machine, select replacement wheels for cutting a wheel z = 88 with a two-flute cutter on the machine specified in example 1.
Solution z = 88/2 = 44. Using the table we find
i div = 30 / 55 = a1 / b1
As you can see, one pair of replacement wheels is enough here. If the design of the guitar requires two pairs of replacement wheels, then the second pair is added with a gear ratio equal to one; For example:
idel = 30 / 55 40 / 40.
Feed guitar. For one revolution of the workpiece installed on the table, the support with the cutter must receive vertical movement by the amount of the axial (vertical) feed So (selected when assigning cutting modes), which is ensured by setting the feed rate.
The equation of the vertical feed chain, if we consider this machine chain from the table to the milling support, has the following form (in-gear ratio of the feed guitar, 10 mm - pitch of the vertical feed screw):
Accordingly, the values of vertical and horizontal (radial) feeds for this machine were obtained:
where Disp. is a coefficient depending on the kinematic chain of a given machine.
To simplify the selection of replacement guitar feed wheels, also use the table included with the machine.
Guitar differential. When moving the caliper by the amount of the axial pitch Px of the helical wheel, the table with the workpiece, in addition to turning in the dividing chain, must make an additional turn by the magnitude of the circumferential pitch of the wheel being cut, i.e. by 1/z of a turn, which is ensured by adjusting the differential gear. Number of revolutions of the vertical feed screw in increments t=10 mm, corresponding to the movement of the nut with the caliper by the amount of the axial pitch of the wheel, nв = ta/t.
Considering the kinematic diagram of the machine from the milling support to the table through the differential guitar with a gear ratio i differential, we compose the equation of the differential circuit:
where mn and B are the normal module and the angle of inclination of the teeth of the cut wheel; k is the number of cuts of the cutter; Sdif is a coefficient that is constant for a given machine (see Table 5).
Attached to the machine are tables for selecting replacement differential wheels depending on the module and angle of inclination of the teeth B. But since the number of B values in the tables is limited, replacement wheels have to be selected by calculation. The calculation formula includes the values Pi = 3.14159 ... and sin B, so an absolutely accurate selection of replacement differential guitar wheels is impossible. The calculation is usually carried out accurate to the fifth or sixth decimal place. Then, using specially published tables for selecting replacement wheels, the result obtained according to the formula decimal with high accuracy converted into a simple fraction or into the product of two simple fractions, the numerator and denominator of which correspond to the numbers of teeth of the replacement wheels of the differential guitar.
Example 1. Select replacement wheels for the differential guitar for cutting a helical gear mn = 3 mm with a single-thread worm cutter; B = 20° 15" on a machine model 5K32A or 5K324A.
1st solution option. Using the work tables we find the nearest value i differential and corresponding numbers of teeth of replacement wheels
2nd solution. Using work tables, we will convert the decimal fraction into a simple fraction and factor it into factors:
0,91811 = 370/403 = 2*5*37/(13*31). By multiplying the numerator and denominator of the fraction by 10 = 5*2 we get
The results of selecting replacement wheels from different tables are the same, but the first solution is obtained faster, so it is more convenient to use the tables given in the work.
Example 2. Select replacement wheels for the conditions given in example 1, but at B = 28° 37".
Since the tables show values of fractions less than one, we determine the reciprocal i differential, and the values of the numbers of teeth according to the tables given in the work:
I/1.27045 = 0.7871122 = 40*55/(43*65),
i diff = 65*43/(40*55) = a3/b3 * c3/d3.
Accelerated movement of the caliper:
Smin = 1420*25/25*36/60*50/45*1/24*10 = 390 mm/min;
for the table
Smin = 1420*25/25*36/60*45/50*34/61*1/36 = 118 mm/min.
Cutting spur gears with prime numbers of teeth *1. In the absence of replacement guitar wheels, division wheels with prime tooth numbers above 100 can be cut with additional adjustment and inclusion of a differential chain.
The essence of this machine setting is as follows: the division guitar is set not to z teeth, but to z + a, where a is a small arbitrarily chosen value, which is recommended to be less than one. To compensate for the influence of this value, the differential guitar is additionally adjusted. When drawing up the adjustment equation, one should proceed from the relationship: one revolution of the cutter corresponds to k/z revolutions of the workpiece along the dividing and differential circuits. It looks like this (see Fig. 4):
k/z*96/1*1/idiv+k/z*96/1*2/26*ipod*39/65*50/45*48/32*idif*1/45X2*27/27*29/ 29*29/29*16/64 = 1 rev. cutters.
Substituting isub = 0.5s0, we obtain the following tuning formulas:
Tuning guitar division for machine tools mod. 5K32A; 5327, etc., where Sdel = 24 (see Table 5),
tuning the guitar differential for machine tools mod. 5K32A and 5K324A
If in the formula idel is taken with a plus sign, then idif should be taken with a minus sign, i.e. the differential should slow down the rotation of the table, and vice versa. The pitch guitar must be tuned precisely to ensure S0 pitch.
Example. On the machine mod. 5K324A cut a spur gear z = 139. Right cutter; k = l; S0 = 1 mm/rev. Solution.
Guitar division
*1 - Prime numbers cannot be factorized, for example 83, 91, 101, 107, ... 139, etc.
Helical teeth can be cut without adjusting the differential by appropriately selecting replacement pitch and pitch guitar wheels. In this case
where the signs (+) or (-) can be determined from the table. 6.
6. Conditions determining the sign in calculation formula i affairs
Due to the fact that the formula includes Pi and sin B, an accurate selection of replacement guitar division wheels is impossible. Therefore, they are selected approximately, with the smallest error (almost accurate to the fifth digit). Using the above formula, the nearest number of teeth of the division guitar wheels at a given feed is selected and the actual gear ratio of the division guitar is determined from them (the index “f” denotes the actual value). Then, using this ratio, we determine i replaceable guitar feed wheels are selected under and with the smallest error.
Calculation i under (accurate to the fifth digit) can be produced by the formula
Where i d.f - actual division guitar tuning.
Example. On the machine mod. 5K32A, with a non-differential setting, cut a helical gear; m = 10 mm; z = 60; B = 30° right tooth inclination. Hob cutter - right-handed single-thread, milling is carried out against the feed direction.
Solution. We take s0 = 1 mm/rev; Then
Then (see work)
If it is not possible to use the replacement wheel z = 37 occupied in the division guitar, we accept another set that gives a value close to the calculated value
i sub.f = 45/73*65/100 = 0.505385.
Actual Feed
Sof = 80/39*0.5054 = 1.03 mm/rev.
When processing teeth, splines, grooves, cutting helical grooves and other operations on milling machines, dividing heads are often used. Dividing heads, as devices, are used on cantilever universal milling and wide-universal machines. There are simple and universal dividing heads.
Simple dividing heads are used to directly divide the circle of rotation of the workpiece. The dividing disk of such heads is fixed on the head spindle and has divisions in the form of slots or holes (in the number of 12, 24 and 30) for the latch latch. Discs with 12 holes allow you to divide one revolution of the workpiece into 2, 3, 4, 6, 12 parts, with 24 holes - into 2, 3, 4, 6, 8, 12, 24 parts, and with 30 holes - into 2 , 3, 5, 6, 15, 30 parts. Specially made dividing disks of the head can be used for other division numbers, including division into unequal parts.
Universal dividing heads are used to set the workpiece being processed at the required angle relative to the machine table, rotate it around its axis at certain angles, and impart continuous rotation to the workpiece when milling helical grooves.
In the domestic industry, universal dividing heads of the UDG type are used on cantilever universal milling machines (Fig. 1, a). Figure 1, 6 shows auxiliary accessories for dividing heads of the UDG type.
On widely-universal tool milling machines, dividing heads are used that are structurally different from dividing heads of the UDG type (they are equipped with a trunk for installing the rear center and, in addition, have some differences in the kinematic diagram). The settings for both types of heads are identical.
As an example in Fig. 1, a shows a diagram of processing a workpiece by milling using a universal dividing head. The workpiece / is installed on a reference in the centers of the spindle 6 of the head 2. and the tailstock 8. The modular disk cutter 7 from the spindle of the milling machine receives rotation, and the machine table receives a working longitudinal feed. After each periodic rotation of the gear blank, the cavity between adjacent teeth is machined. After processing the cavity, the table quickly moves to its original position.
Rice. 1. Universal dividing head UDG: a - diagram of the installation of the workpiece in the dividing head (1 - workpiece; 2 - head; 3 - handle; 4 - disk; 5 - hole; 6 - spindle; 7 - cutter; 8 - headstock); b - accessories for the dividing head (1 - spindle roller; 2 - front center with a driver; 3 - jack; 4 - clamp; 5 - rigid center mandrel: 6 - cantilever mandrel; 7 - rotary plate). The cycle of movements is repeated until all the teeth of the wheel are completely processed. To install and fix the workpiece in the working position using the dividing head, rotate its spindle 6 with handle 3 along the dividing disk 4 with the dial. When the axis of the handle 3 enters the corresponding hole in the dividing disk, the spring device of the head fixes the handle 3. On the disk on both sides there are 11 circles concentrically located with the number of holes 25, 28, 30, 34, 37, 38, 39, 41, 42 , 43, 44, ^7, 49, 51, 53, 54, 57, 58, 59, 62, 66. Kinematic diagrams of universal dividing heads are shown in Fig. 2. In universal dial dividing heads, rotation of handle 1 (Fig. 2, a-c) relative to dial 2 is transmitted through gear wheels Zs, Z6 and worm gear Z7, Zs spindle. The heads are configured for direct, simple and differential division.
Rice. 2. Kinematic diagrams of universal dividing heads: a, b, c - limb; g - without limbs; 1 - handle; 2 - dividing dial; 3 - stationary disk. The direct division method is used to divide a circle into 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 24, 30 and 36 parts. When dividing directly, the rotation angle is counted using a 360" graduated disk with a division value V. The vernier allows this measurement to be carried out with an accuracy of up to 5". The angle a, degrees, of rotation of the spindle when dividing into z parts is determined by the formula
a=3600/z
where z is the specified number of divisions.
With each rotation of the head spindle, the value corresponding to the position of the spindle before rotation will be added, equal to the value angle a found using formula (5.1). The universal dividing head (its diagram is shown in Fig. 2, a) provides simple division into z equal parts, which is performed by rotating the handle relative to the stationary disk according to the following kinematic chain:
1/z=пp(z5/z6)(z7/z8)
Where (z5/z6)(z7/z8) = 1/N; pr - number of handle revolutions; N - head characteristic (usually N=40).
Then
1/z=пp(1/N)
Where pp=N/z=A/B
Here A is the number of holes through which you need to turn the handle, and B is the number of holes on one of the circles of the dividing disk. Sector 5 (see Fig. 5.12, a) is moved apart by an angle corresponding to the number A of holes, and the rulers are fastened. If the left ruler of the sliding sector 5 rests against the handle latch, then the right one is aligned with the hole into which the latch must be inserted during the next turn, after which the right ruler rests against the latch. For example, if you need to configure a dividing head for milling the teeth of a cylindrical wheel with Z = 100, with head characteristics N = 40, then we get
pr - N/z = A/B = 40/100 = 4/10 = 2/5 = 12/30, i.e. A = 12 and B = 30.
Consequently, the circumference of the dividing disk with the number of holes B = 30 is used, and the sliding sector is adjusted to the number of holes A = 12. In cases where it is impossible to select a dividing disk with the required number of holes, differential division is used. If for the number z there is no required number of holes on the disk, take the number zф (actual) close to s, for which there is a corresponding number of holes. The discrepancy (l/z- l/zф) is compensated by additional rotation of the head spindles to this equality, which can be positive (additional rotation of the spindle is directed in the same direction as the main one) or negative (additional rotation is in the opposite direction). This correction is carried out by additional rotation of the dividing disk relative to the handle, i.e., if during simple division the handle is rotated relative to the stationary disk, then during differential division the handle is rotated relative to the slowly rotating disk in the same (or opposite) direction. From the head spindle, rotation is transmitted to the disk through replaceable wheels a-b, c-d (see Fig. 2, b) a conical pair Z9 and Z10 and gears Z3 and Z4.
The amount of additional rotation of the handle is:
prl = N(1/z-1/zф)=1/z(a/b(c/d)(z9/z10)(z3/z4)
We accept (z9/z10)(z3/z6) = C (usually C = I).
Then (a/b)(c/d)=N/C((zф-z)/zф))
Let's say you want to set up a dividing head for milling the teeth of a cylindrical wheel with g = 99. It is known that N-40 and C = 1. The number of handle revolutions for simple division is PF-40/99. Considering that the dividing disk does not have a circle with the number of holes 99, we take t = 100 and the number of handle revolutions is PF-40/100 = 2/5 = 12/30, i.e. We take a disk with the number of holes on the circle B = 30 and turn the handle into 12 holes (A = 12) when dividing. The gear ratio of replacement wheels is determined by the equation
and = (a/b)(c/d) = N/C= (zф-z)/z) = (40/1)((100 - 99)/100) = 40/30 = (60/30) x (25/125).
Dividing heads without dials (see Fig. 2) do not have dividing disks. The handle is turned one turn and fixed on a fixed disk 3. When simply divided into equal parts, the kinematic chain has the form:
Considering that z3/z4=N,
We get (a2/b2)(c2/d2)=N/z
