Gear cylindrical gears GOST. Basic information about involute gearing

When designing a gear transmission, it may be necessary to change the tooth profile by changing the parameters of the original contour (using the running-in method). The use of non-standard initial contours is limited by the need to manufacture special cutting and measuring tools. This need may arise, for example, in the manufacture of wheels with the number of teeth. In this case, it may turn out that the heads of the tool teeth cut into the legs of the teeth of the wheel being manufactured. This phenomenon is accompanied by cutting off part of the tooth in the area of ​​the stem and weakening of the section, where the greatest stresses act. This phenomenon is called cutting a tooth. It occurs when the line or circle of the tool tips intersects the engagement line at point (A) outside the active section (point M) (Figure 5.11). To cut such wheels with a standard tool, an offset of the cutting tool relative to the workpiece is used. The cutting tool is positioned relative to the workpiece so that the pitch surface of the tool does not touch the pitch circle of the wheel being cut at a certain distance –x, called displacement the original contour (Figure 5.13). When manufacturing offset wheels, the tooth profile is changed by using a different section of the involute of the same basic circle. Let us determine the required displacement of the rack when cutting a wheel with a standard tool at . In Fig. 5.13, the pitch line of the rack is shifted relative to the pitch circle of the wheel by a displacement amount x, which ensures the limiting position of the point of intersection of the active section of the engagement line (N-N) with the line of the heads of the rack teeth (point M).

The segment, as can be seen from Fig. 5.13, is equal to:

Where - displacement coefficient equal to the ratio of the displacement x to the engagement module m.

Line segment equal to the radius pitch circle of the wheel being cut

From the triangles we have:

Reducing by the modulus value from the resulting expression we can obtain

What, taking into account (5.20), will be

A gear that includes at least one wheel cut with an offset is called offset transmission .

Displacement of the rack from the wheel axis – positive bias transmission (), to the axis – negative bias transmission ( ).

The use of offset gears allows:

Eliminate undercutting of gear teeth at , which makes it possible to reduce the dimensions of the gear.

Enter the gear into the specified center distance while maintaining the specified gear ratio.

Increase smoothness of engagement, contact and bending strength of teeth, and reduce slip and wear.

Combination of different gear wheels can give transmission no offset - (), equally displaced – (),

With positive offset –() And with negative offset – ().

The method for determining the size of gears cut with offset depends on the type of gearing and the total offset. For spur involute gears with external gearing at known and .

1. Calculate the total displacement coefficient

2. Determine the equivalent foreign exchange angle corresponding to the engagement angle

Where: , , - profile angle of the original contour.

We determine foreign currencies using the tables.

3. Center distance

4. Diameters of initial circles:

Where is the gear ratio.

When the wheels are engaged with an offset, the smallest distance between the pitch circles is called perceived displacement. The difference between the total and perceived displacements – equalization bias. Ratio of perceived displacement to modulus – perceived displacement coefficient

Ratio of equalization bias to modulus – equalization coefficient

5) Diameters of circles of peaks and valleys

6) Dividing circumferential thickness of the tooth

By analyzing these formulas, the following features of various gears can be established.

In transmission without offset (

Center distance

Engagement angle.

Dividing circumferential thickness of the tooth.

Tooth head height.

Tooth height.

Equally offset gear

Center distance

Engagement angle.

Bias coefficients are assigned for the purpose of:

– increasing the bending strength of the tooth by increasing its dangerous cross-section near the base;

– increasing the contact strength of the tooth by using involute sections more distant from the main circle;

– alignment of maximum specific slips;

– preventing undercutting of a small wheel in gear;

– increasing the smoothness of the transmission by lengthening the active engagement line;

– ensuring a given center distance;

– ensuring double-pair engagement in the pole and other purposes.

3.10. Calculation of geometric dimensions of gears

The initial data for calculating dimensions are: number of wheel teeth z 1 and

z 2, wheel module m, profile angle of the original contour, displacement coefficients

As can be seen from the figure, the interaxle distance is equal to the sum of the radii of the initial circles, i.e. a w r w 1 r w 2, therefore

The product of the first two terms in this formula is called dividing center distance. It occurs when the gear is made zero, that is, when the total displacement coefficient is zero. In this case, w and cosines cancel.

Radius of circles of depressions

When a zero wheel is formed, its centroid, as always, is the dividing circle (Fig. 3.18), and the centroid of the tool is its dividing line (in the figure, the profile of the tool and its dividing line and straight vertical line)

tires are shown as thin lines). Therefore, the radius r of the circle of the depressions is zero-

r r h c m f 0

th wheel is equal to the difference a. When the tool is shifted by

magnitude xm, the radius of the circle of the depressions increases by the same amount and acquires the value

rf r ha c m x ​​m.

In Fig. 3.18, the location of the tool in relation to the wheel being cut is shown in bold lines.

Vertex circle radii

The calculation of the radii of the circles of the vertices is clear from Fig. 3.19, which presents those gearing elements that are associated with this calculation. It can be seen directly from the figure that the radius of the circle of the vertices of the first wheel is equal to

ra 1 aw rf 2 c m ,

the radius of the circle of the vertices of the second wheel is equal to

ra 2 aw rf 1 c m .

Tooth thickness along pitch circle

The thickness of the wheel tooth along the pitch circle is determined by the width of the tool rack cavity along the machine-initial straight line (Fig. 3.20), which during the manufacture of the wheel rolls along its pitch circle

the depressions of the tool rack along its pitch circle and two legs of right-angled triangles, shaded in Figure 3.20, which are located on the machine-initial straight rack. The vertical legs of these triangles are equal to xm, since they represent the amount of displacement of the tool from the center

wheels when cutting it, which is essentially equal to the distance between the pitch and machine-initial straight lines. Each horizontal leg of a right triangle is equal to xm tg. Taking these considerations into account, the tooth thickness S can be

express it like this

S m 2 xm tg,

or in its final form, after a simple transformation

2 x tg.

In all calculation formulas geometric dimensions gears, the displacement coefficients must be substituted with their own signs.

Self-test questions

1. What is the essence of the basic law of gearing?

2. Which wheel tooth profiles are called conjugate?

3. What is the involute of a circle producing a straight line?

4. What properties does the involute of a circle have?

5. What is an involute function?

6. Name the elements of a gear wheel, what lines outline the tooth profile?

7. What is called the pitch of the wheel, module, head, leg of the tooth?

8. Where is the tooth thickness and wheel cavity width measured?

Cylindrical gears.

Calculation of geometric parameters

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

1. Terms and designations of spur gears

Pitch distance - a

Center distance - a w

Spur gear crown width - b

Working width of the gear ring - b w

Radial clearance of a pair of original contours - c

Radial clearance coefficient of normal initial contour – c*

Spur gear tooth height - h

Height of pitch head of cylindrical gear tooth - h a

Head height coefficient of the original contour – h a *

Height to chord of wheel tooth -

Height to constant tooth chord -

Height to the chord of the arc of a circle -

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

Height of the gear tooth pitch - h f

Wheel tooth limit height - h l

Gear pitch diameter - d

Wheel tooth tip diameter - d a

Main diameter of gear - d b

Gear wheel diameter - d f

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

Initial gear diameter - d w

Gear radius - r

Spur gear calculation module - m

Normal tooth module - m n

Circumferential module of teeth (end) - m t

Involute gear pitch - p b

Normal rack tooth pitch - p n

Rack tooth pitch - p t

Axial pitch of rack teeth - p x

Basic normal tooth pitch - p bn

Main circumferential pitch of teeth - p bt

Basic normal tooth thickness - s bn

Constant chord of the tooth -

Normal rack tooth thickness - s n

Axial thickness of the rack tooth - s x

Rack tooth end thickness - s t

Thickness along the chord of the tooth -

Circumferential thickness at a given diameter d y - s ty

Chord thickness -

Length of the common normal of the gear - W

Initial contour displacement coefficient - x

Coefficient of the smallest displacement of the original contour - x min

Displacement sum coefficient x Σ

Perceived displacement coefficient - y

Equalization bias coefficient - Δу

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

The smallest number of teeth free from undercutting - z min

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

Normal side clearance 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 rotation speed per minute - n

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

The angle of the tooth profile of the original contour in the normal section is a

Tooth profile angle in the end section - a t

Engagement angle - a tw

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

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

Tooth line angle - β

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

Angle of tooth involute - v

Half angular thickness of tooth - ψ

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

Angular velocity - ω

Gear is a transmission gear with a smaller number of teeth, a wheel with a larger number of teeth. When the number of teeth of the gear wheels is the same, the drive gear is called the gear, and the driven gear is called the wheel. Index 1 - for quantities related to the gear, index 2 - related to the wheel.

Rice. 1. Initial contour of cylindrical gears of involute gearing according to GOST 13755-81 and bevel gears with straight teeth according to GOST 13754-81

Index n - for values ​​related to the normal section, t - to the circumferential (end) section. In cases where there can be no discrepancy or ambiguity, the subscripts n and t can be omitted.

The terms of the parameters of the normal initial contour and the normal initial generating contour, expressed in fractions of the module of the normal initial contour, are formed by adding the word “coefficient” before the term of the corresponding parameter.

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

Modules (according to GOST9563-60). The standard applies to involute cylindrical gears and bevel gears with straight teeth and establishes:

for cylindrical wheels - values ​​of normal modules;

for bevel wheels - the values ​​of the external circumferential dividing modules.

Numerical values ​​of modules:

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 that differ from those specified in this standard;

c) in gearbox manufacturing 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 conical distance;

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

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

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

For cylindrical wheels of external gearing at a peripheral speed greater than that indicated in the table. 2, the original contour is used with a modification of the tooth head profile (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 be no more than 0.02.

Essential elementsgearing are shown in Fig. 3 and 4 in accordance with the designation according to table. 1.

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

Rice. 2. Original contour with profile modification

2. Peripheral speed of wheels depending on their accuracy

3. Modification depth coefficient Δ* depending on the module and degree of accuracy

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) at large gear ratios, since in this case the relative sliding of the profiles is significantly reduced.

Rice. 3

Rice. 4

The position of the original generating contour relative to the wheel being cut, at which the straight pitch rack touches the pitch 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 rack is called a wheel cut without mixing the original contour (according to the old terminology - uncorrected wheel).

Rice. 5. Position of the producing rack-and-pinion circuit 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 = 1.2 (x 1 = x 2 = 0.5)

Rice. 7. Graph for determination x min depending on z and β or z min - x and β

(rounded to the nearest higher integer)

Examples.

1. Given: z = 15; β = 0. From the graph we determine x min= 0.12 (see dashed line).

2. Given: x = 0; β = 30°. Using the graph, we determine the smallest number of teeth(see dashed line)

Rice. 8. The influence of the displacement of the original contour on the geometry of the teeth

If the initial producing rack in the machine gearing is shifted from the nominal position and installed so that its dividing line does not touch the dividing circle of the wheel being cut, then the result of processing will be a wheel cut with a displacement of the original contour (according to the old terminology - a corrected wheel).

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

4. Displacement coefficients for spur gears

5. Displacement coefficient for helical and herringbone gears

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

The distance from the pitch line of the original generating rack (or the original contour) to the pitch circle of the wheel is the displacement value.

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

If the dividing line of the original contour intersects the pitch circle of the gear wheel (Fig. 5, b), the displacement 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 displacement is zero (x = 0).

The displacement coefficient x is ensured by installing the tool relative to the gear blank in the machine gear.

It is recommended to select the displacement coefficients of gears according to the table. 4 for spur gear and according to table. 5 - for helical and chevron gears.

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

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

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 are further distinguished: normal, face and axial.

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

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

Pitch circle− this is the theoretical circle of the gear wheel on which the module and pitch take standard values

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

is the theoretical circumference 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 vertex circle.

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

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

Vertex circle- This is the theoretical circumference of a gear, connecting the tops of its teeth.

d a =d+2(h * a + x - Δy)m

Depression circumference- This is the theoretical circle of a gear that connects all its cavities.

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

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

Equalization displacement coefficient Δу:

Circular step, or step p− this is the distance along the arc of the pitch 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

Step along the main circle− 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 pitch 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 pitch circle between opposite points of the profiles of adjacent teeth

Tooth thickness Sb along the main circumference− 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 opposite points of the profiles of one tooth.

− this 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 meshing gears, or a gear and a rack. In the first case, it serves to transmit rotational motion from one shaft to another, in the second - to transform rotational motion into translational motion.

Used in mechanical engineering the following types gears: cylindrical (Fig. 1) with parallel shafts; conical (Fig. 2, A) with intersecting and intersecting shafts; screw and worm (Fig. 2, b And V) with intersecting shafts.

The gear that transmits rotation is called the driving gear, and the gear that is driven into rotation is called the driven gear. The wheel of a gear pair with a smaller number of teeth is called a gear, and the paired wheel with a larger number of teeth is called a wheel.

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

The kinematic characteristic of a gear transmission is the gear ratio i , which is the ratio of the angular speeds of the wheels, and at constant i - and the ratio of wheel angles

If at i there are no indices, then the gear ratio should be understood as the ratio angular velocity driving wheel to the angular speed 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 teeth, and the other - internal teeth (see Fig. 1, c).

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

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

The most widely used gears are 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 a mating pair of gears; 2) manufacturing is easier, since wheels can be cut with the same tool

Rice. 1.

Rice. 2.

With different number teeth, but the same module and engagement angle; 3) wheels of the same module are mated to 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 come into 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 when the gear pair operates, they roll over one another without slipping. The concept of an initial 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 using the rolling method, the pitch circle is like a production initial circle that arises during the manufacturing process of the wheel. In the case of transmission without displacement, the pitch circles coincide with the initial ones.

Rice. 3. :

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

When cylindrical gears operate, the point of contact of the teeth moves along a straight line MN, tangent to the main circles, passing through the meshing pole and called the meshing line, which is the 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 - diameter of the tooth tips; d - pitch diameter; df is the diameter of the depressions; h - tooth height - the distance between the circles of the peaks and valleys; ha - height of the pitch head of the tooth - the distance between the circles of the pitch and the tops of the teeth; hf - the height of the pitch leg of the tooth - the distance between the circles of the pitch and the cavities; pt - circumferential pitch of teeth - the distance between the same profiles of adjacent teeth along the arc of the concentric circle of the gear wheel;

st - circumferential thickness of the tooth - the distance between different tooth profiles along a circular arc (for example, along the pitch, initial); ra - step of involute gearing - the distance between two points of the same surfaces of adjacent teeth located on the normal MN to them (see Fig. 3).

Circumferential modulus mt-linear quantity, in P(3.1416) times less than the circumferential step. The introduction of the module simplifies the calculation and production of gears, as it allows one to express various wheel parameters (for example, wheel diameters) in whole numbers, rather than in infinite fractions associated with a number P. GOST 9563-60* established the following modulus values, mm: 0.5; (0.55); 0.6; (0.7); 0.8; (0.9); 1; (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); 8; (9); 10; (eleven); 12; (14); 16; (18); 20; (22); 25; (28); 32; (36); 40; (45); 50; (55); 60; (70); 80; (90); 100.

Rice. 4.

The values ​​of the pitch circumferential pitch pt and the engagement pitch ra for various modules are presented in Table. 1.

1. Values ​​of pitch circumferential pitch and engagement pitch for various modules (mm)

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

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

When calculating involute gearing, the concept of involute angle of the tooth profile (involute), denoted 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. Using this formula, involution tables are calculated, which are given in reference books.

Radian is equal to 180°/p = 57° 17" 45" or 1° = 0.017453 glad. The angle expressed in degrees must be multiplied by this value to convert it to radians. For example, ax = 22° = 22 X 0.017453 = 0.38397 rad.

Initial outline. When standardizing gears and gear cutting tools, the concept of an initial contour was introduced to simplify the determination of the shape and size of the cut teeth and tools. This is the outline of the teeth of the nominal original rack when sectioned by a plane perpendicular to its pitch plane. In Fig. Figure 5 shows the initial contour in accordance with 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 coefficient h*a = 1; leg height coefficient h*f = 1.25; coefficient of radius of curvature of the transition curve р*f = 0.38; coefficient of tooth engagement depth in a pair of initial contours h*w = 2; radial clearance coefficient in a pair of original contours C* = 0.25.

It is allowed to increase the radius of the transition curve рf = р*m, if this does not interfere with the correct engagement in the gear, as well as an increase in the radial clearance C = C*m before 0.35m when processing with cutters or shavers and before 0.4m when processing for gear grinding. There may be gears with a shortened tooth, where h*a = 0.8. The part of the tooth between the pitch surface and the surface of the tops of the teeth is called the pitch head of the tooth, the height of which ha = hf*m; the part of the tooth between the dividing surface and the surface of the depressions - the dividing leg of the tooth. When the teeth of one rack are inserted into the valleys of another until their profiles coincide (a pair of initial contours), a radial gap is formed between the peaks and valleys With. The approach 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 the original contour, its value p = pm, and the thickness of the rack tooth in the pitch plane is half the pitch.

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

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

The displacement of the original contour (Fig. 6) is the normal distance between the pitching surface of the gear and the pitching plane of the original gear rack at its nominal position.

When cutting gears without displacement with a rack-type tool (hobs, combs), the pitch circle of the wheel is rolled without sliding along the center line of the rack. In this case, the thickness of the wheel tooth is equal to half the pitch (if we do not take into account the normal side clearance *2, the value of which is small.

Rice. 7. Lateral and radial in gear clearances

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

half a step.

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

When processing teeth using the rolling method, gears with a displacement of the original contour are cut with the same tool and with the same machine settings as wheels without displacement. Perceived displacement is the difference between the center distance of the gear with the displacement and its pitch distance.

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

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