The dimensions of the wheels, as well as the entire engagement, depend on the numbers Z1 and Z2 of the teeth of the wheels, on the modulus m of the engagement (determined from the calculation of the wheel tooth for strength), common to both wheels, and also on the method of their processing.

Let's assume that the wheels are made according to the running-in method with a rack-type tool (tool rack, worm cutter), which is profiled on the basis of the original contour in accordance with GOST 13755-81 (Fig. 10).

The process of manufacturing a gear wheel (Fig. 10) with a tool rack according to the running-in method is that the rack in motion with respect to the wheel being machined rolls without sliding one of its pitch lines (DP) or the middle line (SP) along the pitch circle of the wheel (motion running-in) and at the same time performs fast reciprocating movements along the wheel axis, while removing chips (working movement).

The distance between the middle straight rack (SP) and that pitch line (DP), which during the running-in process rolls over the pitch circle of the wheel, is called the offset X of the rack (see clause 2.6). It is obvious that the offset X is equal to the distance by which the middle straight rail is moved away from the pitch circle of the wheel. The offset is considered positive if the middle straight line is moved away from the center of the cut wheel.

The offset value X is determined by the formula:

where x is the bias factor, which has a positive or negative value (see clause 2.6).

Figure 10. Machine engagement.

Gears made without tool rack displacement are called zero gears; rails made with a positive bias are positive, with a negative bias - negative.

Depending on the values ​​x Σ, gears are classified as follows:

a) if x Σ \u003d 0, and x1 \u003d x2 \u003d 0, then the link is called normal (zero);

b) if x Σ \u003d 0, with x1 \u003d -x2, then the engagement is called equidisplaced;

c) if x Σ ≠ 0, then the link is called non-equidisplaced, and at x Σ > 0, the link is called positive non-equidisplaced, and when x Σ < 0 – отрицательным неравносмещенным.

The use of normal gears with a constant tooth head height and a constant engagement angle is caused by the desire to obtain a system of interchangeable gears with a constant distance between centers for the same sum of the number of teeth, on the one hand, and on the other hand, to reduce the number of gear cutting tool sets in the form of modular cutters, which are supplied with tool shops. However, the condition of changing gears with a constant distance between the centers can be satisfied when using helical gears, as well as wheels cut with a tool offset. Normal gears are most widely used in gears with a significant number of teeth on both wheels (with Z 1 > 30), when the efficiency of using tool displacement is much less.

With equidisplaced engagement (x Σ \u003d x 1 + x 2 \u003d 0), the thickness of the tooth (S 1) along the pitch circle of the gear increases by reducing the thickness of the tooth (S 2) of the wheel, but the sum of the thicknesses along the pitch circle of the mating teeth remains constant and equal to the pitch . Thus, there is no need to move apart the axles of the wheels; the initial circles, as well as for normal wheels, coincide with the dividing ones; the engagement angle does not change, but the ratio of the heights of the heads and legs of the teeth changes. Due to the fact that the strength of the wheel teeth decreases, such engagement can only be used with a small number of gear teeth and significant gear ratios.

With unequal engagement (х Σ \u003d x 1 + x 2 ≠ 0), the sum of the tooth thicknesses along the pitch circles is usually greater than that of zero wheels. Therefore, the axles of the wheels have to be moved apart, the initial circles do not coincide with the pitch ones, and the engagement angle is increased. Non-equidisplaced gearing has more possibilities than equidisplaced gearing, and therefore has a wider distribution.

By applying a tool offset when cutting gears, you can improve the quality of the gearing:

a) eliminate the cutting of the gear teeth with a small number of teeth;

b) increase the bending strength of the teeth (up to 100%);

c) increase the contact strength of the teeth (up to 20%);

d) increase the wear resistance of teeth, etc.

But it should be borne in mind that the improvement of some indicators leads to the deterioration of others.

There are simple systems that allow you to determine the offset using the simplest empirical formulas. These systems improve gear performance over zero, but they don't take advantage of the full potential of shifting.

a) with the number of gear teeth Z 1 ≥ 30, normal wheels are used;

b) with the number of gear teeth Z 1< 30 и the total number of teeth Z 1 + Z 2 > 60 apply equidisplacement gearing with displacement coefficients x 1 \u003d 0.03 (30 - Z 1) and x 2 \u003d -x 1;

x Σ = x 1 + x 2 ≤ 0.9 if (Z 1 + Z 2)< 30,

c) with the number of gear teeth Z 1< 30 и total number of teeth Z 1 + Z 2< 60 применяют неравносмещенное зацепление с коэффициентами:

x 1 \u003d 0.03 (30 - Z 1);

x 2 \u003d 0.03 (30 - Z 2).

The total offset is limited to:

x Σ ≤ 1.8 - 0.03 (Z 1 + Z 2), if 30< (Z 1 + Z 2) < 60.

For critical transmissions, the offset factors should be selected in accordance with the main performance criteria.

This manual also contains tables 1 ... 3 for unequal gearing, compiled by Professor V. N. Kudryavtsev, and Table. 4 for equidisplaced gearing, compiled by the Central Design Bureau of Gear Building. The tables contain the values ​​of the coefficients x1 and x2, the sum of which x Σ is the maximum possible when the following requirements are met:

a) there should be no cutting of teeth when processing them with a tool rack;

b) the maximum allowable thickness of the tooth along the circumference of the protrusions is 0.3m;

c) the smallest value of the overlap coefficient ε α = 1.1;

d) ensuring the greatest contact strength;

e) ensuring the greatest bending strength and equal strength (equality of bending stresses) of gear and wheel teeth made of the same material, taking into account the different direction of friction forces on the teeth;

f) the greatest wear resistance and the greatest resistance given (equality of specific slips at the extreme points of engagement).

These tables should be used as follows:

a) for uneven external gearing, the displacement coefficients x1 and x2 are determined depending on the gear ratio

i 1.2: at 2 ≥ i 1.2 ≥ 1 according to the table. one; at 5 ≥ i 1.2 > 2 according to the table. 2, 3 for given Z 1 and Z 2 .

b) for equidisplaced external gearing, the displacement coefficients x 1 and x 2 = -x 1 are determined in table. 4. When selecting these coefficients, it must be remembered that the condition x Σ ≥ 34 must be met.

After determining the displacement coefficients, all engagement dimensions are calculated using the formulas given in Table. 5.

Controlled dimensions of involute gears

In the process of cutting a gear involute wheel, it becomes necessary to control its dimensions. The workpiece diameter is usually known. When cutting teeth, it is necessary to control 2 dimensions: tooth thickness and tooth pitch. There are 2 controlled sizes that indirectly determine these parameters:

1) tooth thickness along a constant chord (measured with a tooth gauge),

2) the length of the common normal (measured with a bracket).

Let's imagine that we cut an involute gear wheel, and then a rack was brought into engagement with it (put a rack on it). The contact points of the rail with the tooth will be located symmetrically on both sides of the tooth. The distance between the points of contact is the thickness of the tooth along a constant chord.

Let's draw a tooth of an involute wheel. To do this, we draw a vertical axis of symmetry (Fig. 4) and with the center at the point O we draw the radius of the circle of the protrusions r a and the radius of the dividing circle r. Let us place the wheel tooth and the rack cavity symmetrically with respect to the pole of the machine gear P c , which is located at the intersection of the vertical axis of symmetry and the dividing circle. The dividing line of the rail passes through the pole of the machine gear P c. The angle between the pitch line and the tangent to the base circle is the engagement angle during the cutting process, which is equal to the profile angle of the lath a.

Let us denote the points of contact of the rack with the tooth of the wheel A and B, and the point of intersection of the line connecting these points with the vertical axis - D.

Segment AB is a constant chord. The constant chord is denoted by the index. Let us determine the thickness of the wheel tooth by a constant chord. Figure 4 shows that

From triangle ADP c we define

Let us denote the segment EC on the dividing line - the width of the rack cavity along the dividing line, which is equal to the arc thickness of the wheel tooth along the dividing circle

Segment AP c is perpendicular to the rack profile and is tangent to the main circumference of the wheel. Let us define a segment AP c from a right triangle EAP c

Figure 4 - Tooth thickness along a constant chord

Substitute the resulting expression into the previous formula

But the segment is therefore

Thus, the thickness of the tooth along a constant chord

As can be seen from the formula obtained, the thickness of the tooth along the constant chord does not depend on the number of cut teeth of the wheel z, which is why it is called constant.

In order to be able to control the thickness of the tooth by a constant chord with a tooth gauge, we need to determine one more dimension - the distance from the circumference of the protrusions to the constant chord. This size is called the height of the tooth to a constant chord and is indicated by an index (Fig. 4).

As can be seen from Fig.4

From a right triangle we determine

But, therefore

Thus, we obtain the height of the tooth of the involute wheel to a constant chord

The obtained dimensions make it possible to control the dimensions of the tooth of the involute wheel during the cutting process.

Cylindrical gears.

Calculation of geometric parameters

Terms and designations are given in Table. 1, see definitions of terms in GOST 16530-83 and 16531-83.

1. Terms and designations of spur gears

Dividing center distance - a

Center distance - a w

Spur gear ring width - b

Gear ring working width - b w

Radial clearance of a pair of initial contours - c

Radial clearance coefficient of normal initial contour - c*

Tooth height of spur gear - h

The height of the dividing head of the tooth of a spur gear - h a

Height coefficient of the head of the original contour - h a *

Height to the chord of the wheel tooth -

Height to permanent tooth chord -

Height to the chord of the circular arc -

The depth of the teeth of the wheel, as well as the depth of the teeth of the original racks -

The height of the tooth indexing leg of the wheel - h f

The limiting height of the wheel tooth - h l

Gear pitch diameter - d

The diameter of the tops of the teeth of the wheel - d a

Main gear diameter - d b

Gear cavity diameter - d f

Diameter of the circle of the boundary points of the gear - d l

Initial gear diameter - d w

Gear radius - r

Estimated module of spur gear - m

Normal tooth module - m n

Circumferential tooth module (face) - m t

Involute gear pitch - p b

Normal rack tooth pitch - p n

Face pitch of rack teeth - p t

Axial pitch of rack teeth - p x

Basic normal tooth pitch - p bn

Primary circumferential tooth pitch - p bt

Basic normal tooth thickness - s bn

Permanent tooth chord -

Normal rack tooth thickness - s n

Rack tooth axial thickness - s x

Rack tooth end thickness - s t

Tooth chord thickness -

Circumferential thickness at a given diameter d y - s ty

Thickness along the chord -

Gear wheel normal length - W

Shift coefficient of the original contour - x

The coefficient of the least displacement of the original contour - x min

The coefficient of the sum of displacements x Σ

Perceived displacement coefficient - y

Equalization bias coefficient - Δу

Number of gear teeth (number of sector gear teeth) - z

The smallest number of teeth free from undercutting - z min

Number of teeth in the length of the common normal - z w

Normal backlash of an involute spur gear - j n

involutetooth profile angle - inv a

involuteangle corresponding to the profile point on the circle d y – inv a y

Gear wheel speed per minute - n

Gear ratio (z 2 / z 1; d 2 / d 1; n 1 / n 2) - u

Tooth profile angle of the original contour in normal section - a

Tooth profile angle in end section - a t

Angle of engagement - a tw

Profile angle at a point on a concentric circle of a given diameter d y - a y

The angle of inclination of the tooth line of a coaxial cylindrical surface of diameter d y - β y

The angle of inclination of the tooth line - β

The main angle of inclination of the tooth line (helical gear on its main cylinder) - β b

Tooth involute angle - v

Half the angular thickness of the tooth - ψ

Half of the angular thickness of a tooth of an equivalent gear corresponding to a concentric circle of diameter d y /cos 2 β y - ψ yv

Angular speed - ω

A gear is a transmission gear with a smaller number of teeth, a wheel with a large number of teeth. With the same number of teeth of gear wheels, the gear is called the driving gear, and the driven gear is called the wheel. Index 1 - for the values ​​related to the gear, index 2 - related to the wheel.

Rice. 1. Initial contour of involute gears in accordance with GOST 13755-81 and bevel gears with straight teeth in accordance with GOST 13754-81

Index n - for quantities related to the normal section, t - to the circumferential (end) section. In those cases where there can be no discrepancy and ambiguity, the indices n and t can be excluded.

The terms of the parameters of the normal source circuit and the normal source generating circuit, expressed in fractions of the modulus of the normal source circuit, are formed by adding the word "coefficient" before the term of the corresponding parameter.

The designations of the coefficients correspond to the designations of the parameters with the addition of the “*” sign, for example, the radial clearance coefficient of a pair of initial contours with *.

Modules (according to GOST9563-60). This standard applies to involute spur gears and bevel gears with straight teeth and specifies:

for cylindrical wheels - the values ​​of normal modules;

for bevel gears - the values ​​of the outer circumferential dividing modules.

Numeric values ​​of modules:

Row 1	Row 2	Row 1	Row 2	Row 1	Row 2	Row 1	Row 2
	1,125
1,25	1,375		2,75
	1..75
	2,25

Notes:

1. When choosing modules, row 1 should be preferred to row 2.

2. For cylindrical gears, it is allowed:

a) in the tractor industry, the use of modules 3.75; 4.25 and 6.5mm;

b) in the automotive industry, the use of modules other than those specified in this standard;

c) in gearbox building application of modules 1.6; 3.15; 6.3; 12.5m.

3. For bevel gears it is allowed:

a) determine the module at the average cone distance;

b) in technically justified cases, the use of modules that differ from those indicated in the table.

4. The standard provides for the use of modules in the range of values ​​from 0.05 to 100mm.

Initial contour of spur gears.The initial contour of the wheels (Fig. 1) means the contour of the teeth of the rack in a section normal to the direction of the teeth. Radial clearance c = 0.25m, radius of curvature of the transition curve of the tooth p f = 0.4m. It is allowed to increase the radius p f if this does not violate the correct engagement, and an increase in up to 0.35m when processing wheels with cutters and shavers and up to 0.4m when grinding teeth.

For cylindrical wheels with external gearing at peripheral speeds more than those indicated in Table. 2 apply the original contour with the modification of the profile of the tooth head (Fig. 2). In this case, the modification line is straight, the modification coefficient h g * should be no more than 0.45, and the modification depth coefficient Δ* should not be more than 0.02.

Main elementsgearing are shown in fig. 3 and 4 in accordance with the designation according to the table. one.

Displacement of gear wheels with external gearing.To increase the strength of the teeth in bending, reduce contact stresses on their surface and reduce wear due to the relative sliding of the profiles, it is recommended to mix the tool for cylindrical (and bevel) gears, in which z 1 ≠ z 2 . The greatest result is achieved in the following cases:

Rice. 2. Original contour with profile modification

2. The circumferential speed of the wheels depending on their accuracy

Wheel type	Peripheral speed in m / s with the degree of accuracy of the wheel in accordance with GOST 1643-81
Spurs
Helical

3. Coefficient of depth of modification Δ* depending on the modulus and the degree of accuracy

Module m, mm	The degree of accuracy according to the standards of smooth operation in accordance with GOST 1643-81
Up to 2	0,010	0,015	0,020
St. 2 to 3.5	0,009	0,012	0,018
» 3.5 » 6.3	0,008	0,010	0,035
» 6.3 » 10	0,006	0,008	0,012
» 10 » 16	0,005	0,007	0,010
» 16 » 25		0,006	0,009
» 25 » 40			0,008

1) when shifting gears in which the gear has a small number of teeth (z 1< 17), так как при этом устраняется под­рез у корня зуба;

2) with large gear ratios, since in this case the relative slip of the profiles is significantly reduced.

Rice. 3

Rice. 4

The position of the original generating circuit relative to the wheel being cut, at which the dividing straight rail touches the dividing circle of the wheel, is called the nominal position (Fig. 5, a). A wheel whose teeth are formed at the nominal position of the original producing rail is called a wheel cut without mixing the original contour (according to the old terminology - uncorrected wheel).

Rice. 5. The position of the producing rack contour relative to the workpiece:

a - nominal; b - with a negative bias; c - with positive bias

Rice. 6. Graph for determining the lower limit value z 1 depending on z 2 at which ε a \u003d 1.2 (x 1 \u003d x 2 \u003d 0.5)

Rice. 7. Graph to determine x min depending on z and β or z min - x and β

(rounded up to the nearest whole number)

Examples.

1. Given: z = 15; β = 0. According to the schedule, we determine x min= 0.12 (see dashed line).

2. Given: x = 0; β = 30°. According to the schedule, we determine the smallest number of teeth(c m. dashed line)

Rice. 8. Influence of displacement of the original contour on the geometry of the teeth

If the original producing rail in the machine engagement is displaced from the nominal position and set so that its dividing line does not touch the dividing circle of the cut wheel, then as a result of processing, a wheel cut with an offset of the original contour will be obtained (according to the old terminology, a corrected wheel).

Rice. 9. Engagement (in a section parallel to the front) of a gear wheel with an offset with the original producing rail

4. Displacement coefficients for spur gears

Displacement factor		Application area
y gear x 1	wheel x 2
		0.5(z1 + z2)m or not set	Kinematic transmission	z1 ≥ 17
				12 ≤ z1< 16 и z 2 ≥ 22
		Center distance a w is set equal to 0.5(z1 + z2)m	Power transmission	z1 ≥ 21
				14 ≤ z 1 ≤ 20 and u ≥ 3.5
		Center distance a w not specified		z1 > 30
				10 ≤ z 1 ≤ 30. Within 10 ≤ z 1 ≤ 16 lower limit the value of z 1 is determined from the graph (Fig. 6)

5. Displacement coefficient for helical and herringbone gears

Displacement factor		Application area
y gear x 1	wheel x 2
		Center distance a w is set equal to (z 1 +z 2)m/(2cosβ) or not set	Kinematic transmission
			Power transmission

Rice. 10. Tooth thickness along a constant chord and height to a constant chord in normal section

The distance from the pitch line of the original generating rail (or original circuit) to the pitch circle of the wheel is the offset value.

The ratio of the displacement of the original contour to the calculated modulus is called the displacement coefficient (x).

If the dividing line of the original contour intersects the dividing circle of the gear (Fig. 5, b), the offset is called negative (x<0), если не пере­секает и не соприкасается (рис. 5, в) - по­ложительным (х > 0). At the nominal position of the original contour, the offset is zero (x = 0).

The displacement factor x is provided by setting the tool relative to the gear workpiece in the machine gear.

The displacement coefficients for gear wheels are recommended to be selected according to Table. 4 for spur gear and according to table. 5 - for helical and chevron gears.

The main elements of gearing with offset are shown in fig. 8, 9, 10.

6. Breakdown of the coefficient of the sum of the displacement x Σ y of the spur gear into components x 1 and x 2

Shift sum coefficient x Σ	Displacement factor		Application area
	y gear x 1	wheel x 2
0 < x Σ ≤ 0.5	x Σ		Kinematic gears

Figure 3. Involute gear parameters.

The main geometric parameters of an involute gear include: module m, pitch p, profile angle α, number of teeth z, and relative displacement coefficient x.

Types of modules: divisive, basic, initial.

For helical gears, they additionally distinguish: normal, end and axial.

To limit the number of modules, GOST has established a standard range of its values, which are determined by the dividing circle.

Module- this is the number of millimeters of the diameter of the pitch circle of the gear per tooth.

pitch circle is the theoretical circle of the gear, on which the modulus and pitch take on standard values

The dividing circle divides the tooth into a head and a stem.

is the theoretical circle of the gear belonging to its initial surface.

tooth head- this is the part of the tooth located between the pitch circle of the gear and its circle of vertices.

Tooth stalk- this is the part of the tooth located between the pitch circle of the gear and its circle of depressions.

The sum of the heights of the head ha and the stem hf corresponds to the height of the teeth h:

Top circle- this is the theoretical circle of the gear, connecting the tops of its teeth.

d a =d+2(h * a + x - Δy)m

Trough circumference- this is the theoretical circle of the gear, connecting all its cavities.

d f = d - 2(h * a - C * - x) m

According to GOST 13755-81 α = 20°, C* = 0.25.

Equalizing displacement coefficient Δу:

District step, or step p- this is the distance along the arc of the dividing circle between the same points of the profiles of adjacent teeth.

is the central angle enclosing the arc of the pitch circle corresponding to the circumferential pitch

Base circle step- this is the distance along the arc of the main circle between the same points of the profiles of adjacent teeth

p b = p cos α

Tooth thickness s along the pitch circle- this is the distance along the arc of the dividing circle between opposite points of the profiles of one tooth

S = 0.5 ρ + 2 x m tg α

Depression width e along the pitch circle- this is the distance along the arc of the dividing circle between opposite points of the profiles of adjacent teeth

Tooth thickness Sb on the base circle- this is the distance along the arc of the main circle between opposite points of the profiles of one tooth.

Tooth thickness Sa along the circumference of the vertices- this is the distance along the arc of the circle of the vertices between the opposite points of the profiles of one tooth.

is an acute angle between the tangent t - t to the tooth profile at a point lying on the pitch circle of the gear and the radius vector drawn to this point from its geometric center

Chapter 1GENERAL INFORMATION

BASIC CONCEPTS ABOUT GEARS

A gear train consists of a pair of meshed gears or a gear and a rack. In the first case, it serves to transfer rotational motion from one shaft to another, in the second - to convert rotational motion into translational.

In mechanical engineering, the following types of gears are used: cylindrical (Fig. 1) with a parallel arrangement of shafts; conical (Fig. 2, a) with intersecting and crossing shafts; screw and worm (Fig. 2, b and in) with cross shafts.

The gear that transmits the rotation is called the driver, which is driven into rotation - the driven. The wheel of a gear pair with a smaller number of teeth is called a gear, the paired wheel paired with it with a large number of teeth is called a wheel.

The ratio of the number of teeth of the wheel to the number of teeth of the gear is called the gear ratio:

The kinematic characteristic of the gear train is the gear ratio i , which is the ratio of the angular velocities of the wheels, and at a constant i - and the ratio of the angles of rotation of the wheels

If at i If there are no indexes, then the gear ratio should be understood as the ratio of the angular velocity of the driving wheel to the angular velocity of the driven wheel.

Gearing is called external if both gears have external teeth (see Fig. 1, a, b), and internal if one of the wheels has external and the second has internal teeth (see Fig. 1, c).

Depending on the profile of the gear teeth, there are three main types of engagement: involute, when the tooth profile is formed by two symmetrical involutes; cycloidal, when the tooth profile is formed by cycloidal curves; Novikov engagement, when the tooth profile is formed by circular arcs.

An involute, or development of a circle, is a curve that is described by a point lying on a straight line (the so-called generating line) that is tangent to the circle and rolls along the circle without slipping. A circle whose development is an involute is called the base circle. As the radius of the base circle increases, the involute curvature decreases. When the radius of the main circle is equal to infinity, the involute turns into a straight line, which corresponds to the rack tooth profile outlined in a straight line.

The most widely used are gears with involute gearing, which has the following advantages over other types of gearing: 1) a slight change in the center distance is allowed with a constant gear ratio and normal operation of the mated pair of gears; 2) manufacturing is facilitated, since the wheels can be cut with the same tool

Rice. one.

Rice. 2.

with a different number of teeth, but the same module and engagement angle; 3) the wheels of the same module are mated with each other regardless of the number of teeth.

The information below applies to involute gearing.

Scheme of involute engagement (Fig. 3, a). Two wheels with involute tooth profiles are in contact at point A, located on the line of centers O 1 O2 and called the engagement pole. The distance aw between the axles of the transmission wheels along the center line is called the center distance. The initial circles of the gear wheel pass through the engagement pole, described around the centers O1 and O2, and during the operation of the gear pair, they roll over one another without slipping. The concept of the pitch circle does not make sense for one individual wheel, and in this case, the concept of a pitch circle is used, on which the pitch and engagement angle of the wheel are respectively equal to the theoretical pitch and engagement angle of the gear cutting tool. When cutting teeth by the running-in method, the pitch circle is, as it were, a production initial circle that occurs during the manufacture of the wheel. In the case of transmission without offset, the pitch circles coincide in the initial ones.

Rice. 3. :

a - basic parameters; b - involute; 1 - line of engagement; 2 - main circle; 3 - initial and dividing circles

During the operation of cylindrical gears, the point of contact of the teeth moves along the straight line MN, tangent to the main circles, passing through the gearing pole and called the gearing line, which is a common normal (perpendicular) to the conjugate involutes.

The angle atw between the engagement line MN and the perpendicular to the center line O1O2 (or between the center line and the perpendicular to the engagement line) is called the engagement angle.

Elements of a spur gear (Fig. 4): da is the diameter of the tops of the teeth; d - dividing diameter; df is the diameter of the depressions; h - tooth height - the distance between the circles of peaks and troughs; ha - the height of the dividing head of the tooth - the distance between the circumferences of the dividing and the tops of the teeth; hf - the height of the dividing leg of the tooth - the distance between the circumferences of the dividing and depressions; pt - circumferential pitch of the teeth - the distance between the same profiles of adjacent teeth along the arc of the concentric circle of the gear;

st is the circumferential thickness of the tooth - the distance between the different profiles of the wub along the arc of a circle (for example, along the dividing, initial); pa - involute engagement pitch - the distance between two points of the same-name surfaces of adjacent teeth located on the normal MN to them (see Fig. 3).

District modulus mt-linear value, in P(3.1416) times less than the circumferential step. The introduction of the module simplifies the calculation and manufacture of gears, as it allows you to express various wheel parameters (for example, wheel diameters) as integers, rather than infinite fractions associated with a number P. GOST 9563-60* established the following module values, mm: 0.5; (0.55); 0.6; (0.7); 0.8; (0.9); one; (1.125); 1.25; (1.375); 1.5; (1.75); 2; (2.25); 2.5; (2.75); 3; (3.5); 4; (4.5); 5; (5.5); 6; (7); eight; (nine); ten; (eleven); 12; (fourteen); sixteen; (eighteen); 20; (22); 25; (28); 32; (36); 40; (45); fifty; (55); 60; (70); 80; (90); 100.

Rice. 4.

The values ​​of the dividing circumferential pitch pt and the engagement pitch pa for various modules are presented in Table. one.

1. Pitch and engagement pitch values ​​for different modules (mm)

In a number of countries where the inch system (1 "= 25.4 mm) is still used, a pitch system has been adopted, according to which the parameters of the gear wheels are expressed in terms of pitch (pitch - step). The most common system is a diametrical pitch used for wheels with a pitch from one and higher:

where r is the number of teeth; d - pitch circle diameter, inches; p - diametral pitch.

When calculating the involute engagement, the concept of the involute angle of the tooth profile (involute), denoted by inv ax, is used. It represents the central angle 0x (see Fig. 3, b), covering part of the involute from its beginning to some point xi and is determined by the formula:

where ah is the profile angle, rad. According to this formula, the involute tables are calculated, which are given in reference books.

The radian is 180°/r = 57° 17" 45" or 1° = 0.017453 glad. By this value, you need to multiply the angle, expressed in degrees, to convert it to radians. For example, ax \u003d 22 ° \u003d 22 X 0.017453 \u003d 0.38397 rad.

Source outline. With the standardization of gears and gear-cutting tools, the concept of the initial contour was introduced to simplify the determination of the shape and dimensions of the cut teeth and tool. This is the contour of the teeth of the nominal original rack in section with a plane perpendicular to its dividing plane. On fig. 5 shows the original contour according to GOST 13755-81 (ST SEV 308-76) - a straight-sided rack contour with the following values ​​of parameters and coefficients: angle of the main profile a = 20°; head height factor h*a = 1; leg height factor h*f = 1.25; coefficient of radius of curvature of the transition curve p*f = 0.38; coefficient of tooth entry depth in a pair of initial contours h*w = 2; coefficient of radial clearance in a pair of initial contours C* = 0.25.

It is allowed to increase the radius of the transition curve pf = p*m, if this does not violate the correct engagement in the gear, as well as an increase in the radial clearance C \u003d C * m before 0.35m when processing with cutters or shavers and up to 0.4m when machining for gear grinding. There may be gears with a shortened tooth, where h*a = 0.8. The part of the tooth between the dividing surface and the surface of the tops of the teeth is called the dividing head of the tooth, the height of which ha \u003d hf * m; part of the tooth between the dividing surface and the surface of the cavities - the dividing leg of the tooth. When the teeth of one rack are inserted into the cavities of the other until their profiles coincide (a pair of initial contours), a radial gap is formed between the vertices and cavities with. The lead-in height or straight section height is 2m, and the tooth height m + m + 0.25m = 2.25m. The distance between the same profiles of adjacent teeth is called the pitch. R original contour, its value p = pm, and the thickness of the rack tooth in the dividing plane is half the step.

To improve the smoothness of the operation of cylindrical wheels (mainly with an increase in the circumferential speed of their rotation), a profile modification of the tooth is used, as a result of which the surface of the tooth is made with a deliberate deviation from the theoretical involute formula at the top or at the base of the tooth. For example, cut off the profile of the tooth at its top at a height hc = 0.45m from the circle of vertices to the depth of modification A = (0.005% 0.02) m(Fig. 5, b)

To improve the operation of gears (increase the strength of the teeth, smooth engagement, etc.), to obtain a given center distance, to avoid undercutting * 1 of the teeth, and for other purposes, the original contour is shifted.

Offset of the original contour (Fig. 6) - the distance along the normal between the dividing surface of the gear and the dividing plane of the original gear rack at its nominal position.

When cutting gears without displacement with a rack-and-pinion tool (worm cutters, combs), the pitch circle of the wheel is rolled in without sliding along the middle line of the rack. In this case, the thickness of the wheel tooth is equal to half the pitch (if you do not take into account the normal backlash * 2, the value of which is small.

Rice. 7. Lateral with and radial in gear gaps

When cutting gears with an offset, the original rail is displaced in the radial direction. The pitch circumference of the wheel is rolled not along the center line of the rack, but along some other straight line parallel to the center line. The mixing ratio of the original contour to the calculated modulus is the coefficient of displacement of the initial contour x. For offset wheels, the tooth thickness along the pitch circle is not equal to the theoretical one, i.e., half the step. With a positive displacement of the initial contour (from the wheel axis), the tooth thickness on the pitch circle is greater, with a negative (in the direction of the wheel axis) - less

half step.

To ensure lateral clearance in engagement (Fig. 7), the thickness of the tooth of the wheels is made somewhat less than the theoretical one. However, due to the small value of this displacement, such wheels are practically considered wheels without displacement.

When machining teeth by the running-in method, gears with an offset of the original contour are cut with the same tool and at the same machine setting as wheels without offset. Perceived displacement - the difference between the center distance of a transmission with an offset and its dividing center distance.

Definitions and formulas for the geometric calculation of the main parameters of gears are given in Table. 2.

2.Definitions and formulas for calculating some parameters of involute spur gears

Parameter	Designation	Definition	Calculation formulas and instructions	Picture
Initial data
Module: calculated involute gearing		Dividing normal tooth module. Linear value, n times smaller than the dividing circumferential step	According to GOST 9563 - 60*
Profile angle of the original contour		Acute angle between the tangent to the tooth profile of the rack and the straight line perpendicular to the dividing plane of the rack	According to GOST 13755-81 a = 20°
Number of teeth: gear wheel
Angle of inclination of the tooth line
Head height factor		The ratio of the distance ha between the circles of the tops of the teeth and dividing to the calculated modulus
Radial clearance factor		The ratio of the distance C between the surface of the tops of one transmission wheel and the surface of the troughs of the other to the calculation module		7
Displacement factor: at the gear at the wheel		The ratio of the distance between the pitch surface of the wheel and the pitch plane of the generating rail to the calculation module
Calculation of parameters
Gear wheel diameters: dividing		Diameters of concentric circles