1、 Basic requirements

(1) The basic law of tooth profile meshing, the properties of involutes, the basic parameters of gears and the basic concepts related to the meshing characteristics are clarified. ★★★★

(2) Master the basic parameters and geometric dimension calculation of involute standard spur gear and its gear transmission. ★★★★

(3) Understand the principle of involute gear processing by generating method, undercutting phenomenon of gear and the concept of modified gear.

(4) Understand the geometric dimension calculation of modified gear transmission, the type of involute gear transmission and the application of modified gear.

(5) Understand the characteristics of other types of gear transmission and geometric dimension calculation.

Summary of basic knowledge

The basic law of meshing and involute profile

(1) Basic law of tooth profile meshing

The tooth profile of driving gear is meshed with that of driven gear at point K, and passing through point K is taken as the common normal line N1N2 of two tooth profile. According to the three center theorem, the intersection point P (P is called meshing node) of common normal line N1N2 and second gear connecting center line is instantaneous center of relative speed of second gear, and linear speed of second gear at point P shall be equal. That is to say, the transmission ratio of a pair of gears in any position is inversely proportional to the length of the two segments divided by the common normal line of the meshing tooth profile at the contact point.

(2) Characteristics of involute tooth profile and involute tooth profile meshing transmission

① The length of the generating line rolling along the base circle kn is equal to the arc length AK

② The normal of any point on the involute must be the tangent of the base circle, and there is no involute in the base circle.

③ The shape of the involute depends only on the size of the base circle. When the radius of base circle is infinite, the involute will become a straight line, and the straight line is a special involute.

The characteristics of involute tooth profile meshing transmission are as follows:

① The instantaneous transmission ratio is constant.

② The change of center distance does not affect the transmission ratio.

③ The line of engagement is a straight line. When friction is ignored, the force between tooth surfaces acts along the common normal line of the contact point, that is, along the direction of the meshing line. Therefore, no matter where the gear teeth are meshed, the direction of the acting line of the force between the tooth surfaces remains unchanged.

④ Center distance a 'of involute gear meshing transmission

Calculation of basic parameters and geometric dimensions of involute spur gear

(1) Basic parameters of involute spur gear

The five basic parameters of involute gear are as follows: the number of teeth Z, the number of index circle modulus m, the pressure angle of indexing circle a, the coefficient of addendum height ha * and the coefficient of backlash c *.

The module on the dividing circle of gear is called modulus for short, which has been standardized and must be selected according to the national standard. The modulus reflects the size of the gear teeth and each part. The larger the modulus, the greater the gear pitch, thickness, height and diameter of the dividing circle (when the number of teeth is unchanged); the pressure angle a of the dividing circle has been standardized in China, and the standard value of a = 20 ° is adopted in China, and sometimes a = 15 ° is taken; The addendum height coefficient ha * and clearance coefficient C * are standard values. Normal teeth ha * = 1, c * = 0.25; short teeth ha * = 0.8, c * = 0.3.

(2) Geometric dimension calculation of involute standard spur gear

The dividing circle is the circle with standard modulus and standard pressure angle in the gear. It is the basis for calculating the size of each part of the gear. Each gear has a completely determined dividing circle, and there is only one dividing circle. The gear with tooth thickness equal to the width between teeth (s = e) on the dividing circle is the standard gear. The geometric dimension calculation of each part of standard gear is one of the key contents of this chapter

Meshing transmission of involute spur gear

(1) Correct engagement conditions (also known as pairing conditions)

The correct meshing condition of involute supported cylindrical gear is that the modulus and pressure angle of two gears are equal respectively and are standard values.

(2) Continuous transmission conditions

① Gear meshing process

② Continuous transmission conditions. In order to make gear continuous transmission, it is necessary to ensure that before the first pair of teeth is out of engagement, the second pair of teeth has entered into meshing.

③ Coincidence degree: in order to meet the requirements of continuous transmission conditions, the actual meshing line segment should be greater than the normal pitch Pb of the gear.

Calculation formula: ε α = [Z1 (Tan α A1 Tan α′) + Z2 (Tan α A2 Tan α′)] / (2 π)

The influencing factors are as follows: a. the coincidence degree is independent of modulus M;

b. It increases with the increase of the number of teeth Z;

c. It increases with the decrease of meshing angle α′ and the increase of addendum coefficient ha *;

d. When the number of teeth tends to infinity, the limit coincidence degree is 1.981.

Cutting principle and undercutting phenomenon of involute tooth profile

(1) Cutting method can be divided into profiling method and generating method

The principle of profiling method: on the milling machine, the milling cutter with the same blade shape as the tooth profile on both sides of the tooth slot of the gear to be cut is used to cut the tooth slots one by one.

Principle of generating method: cutting tooth profile by using the basic law of tooth profile meshing.

(2) Root cutting phenomenon and its causes

Definition: when cutting a gear by generating method, sometimes the top of the cutter will cut into the root of the tooth too much, cutting part of the involute profile of the tooth root. This phenomenon is called undercutting of the tooth.

Causes: a. the cutting edge of the cutter cuts from point B1 on the meshing line to the intersection point B2 of the meshing line and the tool tooth top line, forming the involute part of the tooth profile;

b. If point B2 is below the meshing limit point N1, the tooth profile of the gear to be cut is involute from point B2 to tooth top, and the transition curve from point B2 to tooth root circle is non involute;

c. If the number of teeth of the gear to be cut is small and the meshing limit point N1 falls below the tool's tooth top line, the tool cuts from position II to position III, thus cutting part of the tooth root involute profile near N1 'point, resulting in undercutting of teeth.

(3) Minimum number of teeth of standard gear without undercutting

In order to avoid undercutting, the meshing limit point N1 must be above the tool tooth top line, that is, the minimum number of teeth without undercutting is Zmin = 2ha * / sin2 α.

Involute modified gear

Modification method: this method of cutting gear by changing the relative position of cutter and wheel blank is called modification method.

The concept of modified gear: the dividing line of the cutter is not tangent to the dividing circle of the gear blank, so the gear processed is called modified gear (positive modification / negative modification).

Modification coefficient: the moving distance XM of rack tool is called radial displacement, where m is modulus and X is radial modification coefficient.

(1) Minimum modification coefficient to avoid undercutting

The minimum modification coefficient to avoid undercutting is xmin = ha * (zmin-z) / Zmin.

(2) Geometric dimension of modified gear (taking positive modification as an example)

The alveolar width is e = (π / 2-2xtan α) M;

The tooth root height is HF = ha * m + C * m - XM = (ha * + C * - x) M;

The apical height was ha = ha * m + XM = (ha * + x) M;

The radius of the addendum circle is ra = R + (ha * + x) M;

(for gears with negative modification, the above formula is also applicable, just note that the modification coefficient X is negative.)

(3) Modified gear transmission

(1) The correct meshing condition and continuous transmission condition of the modified gear transmission are the same as those of the standard gear transmission. (2) the center distance of the modified gear transmission should meet the requirements of meshing without backlash and taking the top clearance as the standard value

The meshing equation inv α′ = 2tan α (x1 + x2) / (z1 + Z2) + inv α

The reduction coefficient of addendum height Δ y is called the reduction coefficient of addendum height, and its value is Δ y = (x1 + x2) - Y. b. The addendum height of the gear is ha = ha * m + XM - Δym = (ha * + X - Δ y) M.

(4) Types and characteristics of modified gear transmission

According to the difference of modification coefficient and (x1 + x2) value of two meshing gears, the modified gear transmission can be divided into three basic types.

① Standard gear transmission X1 + x2 = 0, and X1 = x2 = 0.

② The gear transmission with equal modification (gear transmission with high modification) X1 + x2 = 0, and X1 = - x2 ≠ 0.

③ Unequal modification gear transmission (angle modification gear transmission) X1 + x2 ≠ 0

a. Positive transmission properties X1 + x2 > 0 α′ > α, a ′ > A, Y > 0, Δ Y > 0

b. Negative transmission properties X1 + x2 < 0 α′ < α, a ′ < A, y < 0, Δ Y > 0

(5) Design steps of modified gear transmission

The known conditions are Z1, Z2, m, α, a ′, and the design steps are as follows: ★

a. Determine the engagement angle, i.e. α′ = arccos [(ACOS α) / a ′].

b. The sum of the modification coefficients is determined, i.e. X1 + x2 = (INV α′ - inv α) (z1 + Z2) / (2tan α)

c. The coefficient of variation of center distance is determined, i.e. y = (a '- a) / m.

d. The reduction coefficient of addendum height is determined as Δ y = (x1 + x2) - Y.

e. The modification coefficients X1 and X2 are distributed, and the geometric dimensions of gears are calculated according to table 10-4 of the textbook.

The known conditions are Z1, Z2, m, α, x1, X2, and the design steps are as follows: ★

a. The engagement angle is determined as inv α′ = 2tan α· (x1 + x2) / (z1 + Z2) + inv α.

b. Determine the center distance, i.e. a '= ACOS α / cos α'.

c. The center distance variation coefficient y and the addendum height reduction coefficient Δ y are determined.

d. According to table 10-4 of the textbook, calculate the geometric dimension of the modified gear.

Helical cylindrical gear

(1) Calculation of basic parameters and geometric dimensions of helical gears

The conversion relationship between normal surface parameters (Mn, α n, Han *, CN *) and end face parameters (MT, α T, XT, St, etc.) is as follows: ★

(1) The modulus conversion relationship is Mn = mtcos β

(2) The conversion relation of meshing angle is tan α n = Tan α TCOS β

(3) The diameter of the dividing circle on the end face is d = zmt = zmn / cos β

(4) The standard center distance is a = (D1 + D2) / 2 = Mn (z1 + Z2) / (2cos β)

(5) The relationship between the face modification coefficient XT and the normal surface modification coefficient xn is XT = xncos β

(2) Meshing transmission of a pair of helical gears

Conditions for correct engagement of a pair of helical gears

① The modulus and pressure angle are equal, that is, Mn1 = Mn2, α N1 = α N2

② The conditions of helix angle are as follows: A. external meshing β 1 = - β 2; B. internal meshing β 1 = β 2.

(3) Coincidence degree of helical gear transmission

① Coincidence degree of helical gear transmission

The coincidence degree of helical gear transmission is ε γ = (L + Δ L) / PBT = ε α + ε β

② End face coincidence

The calculation formula of ε α is ε α = [Z1 (Tan α at1-tan α t ′) + Z2 (Tan α at2 Tan α t ′)] / (2 π)

③ Coincidence degree of axial plane

The calculation formula of ε β is ε β = bsin β / (π Mn)

Calculation of geometric parameters and dimensions of spur bevel gear transmission

① The diameter of dividing circle is D1 = 2rsin δ 1, D2 = 2rsin δ 2

② Transmission ratio

a. In general, the transmission ratio of two wheels is i12 = ω 1 / ω 2 = Z2 / Z1 = D2 / D1 = sin δ 2 / sin δ 1.

b. When the shaft intersection angle ∑ = 90 ° between the two bevel gears, the transmission ratio of the two wheels is i12 = ω 1 / ω 2 = Z2 / Z1 = D2 / D1 = cot δ 1 = Tan δ 2