CN109751386B

CN109751386B - Design calculation method for transmission engagement of reverse involute gear

Info

Publication number: CN109751386B
Application number: CN201910113911.8A
Authority: CN
Inventors: 谭富春
Original assignee: Chongqing Moyuan Gear Co ltd
Current assignee: Chongqing Moyuan Gear Co ltd
Priority date: 2019-02-14
Filing date: 2019-02-14
Publication date: 2021-09-21
Anticipated expiration: 2039-02-14
Also published as: CN109751386A

Abstract

The invention discloses a design calculation method for transmission engagement of a reverse involute gear, which comprises the following steps: modeling by taking a gear base circle as a reference, and establishing a functional relation of an inverse involute; and deriving a correlation meshing equation; confirming the value of the deflection coefficient of the reverse involute gear; calculating the back involute transmission parameter; checking the calculated transmission parameters; and (5) checking the machining quality of the gear. The meshing track of the reverse involute tooth profile is designed on the involute gear, so that the bearing capacity of a main working face can be improved, the elastic deformation is considered during design, the stress deformation of the gear is corrected by utilizing the elastic deformation, the requirement of an elastic conjugate tooth profile curve is met, the error of gear manufacturing can be properly relaxed, and the difficulty of meeting the precision requirement of gear manufacturing is reduced.

Description

Design calculation method for transmission engagement of reverse involute gear

Technical Field

The invention relates to the field of gear manufacturing, in particular to a design calculation method for transmission meshing of a reverse involute gear.

Background

The involute gear is a convex tooth profile consisting of two sections of symmetrical involute teeth, the tooth profile is convex-convex contact in the transmission process, according to the theory of contact stress of the herz (h.hertz) spherical surface, the convex-convex contact is elastic deformation, the stress is concentrated at a contact point and is distributed from large to small along with a deformation area, the contact stress of the contact point is large, and the gear is failed due to pitting corrosion in large-power and high-power transmission, so the service life of the gear is short, the reliability is low, and the maintenance cost of a transmission mechanism is high. The gear transmission has two forms of space intersection and plane, the involute cylindrical gear is generally plane transmission, and space transmission is rarely adopted.

Disclosure of Invention

The invention aims to provide a design and calculation method for transmission engagement of a reverse involute gear, which can enable the reverse involute gear to be in correct engagement transmission.

The technical scheme of the invention is as follows:

a design calculation method for transmission engagement of a reverse involute gear, comprising the steps of:

1): modeling by taking a gear base circle as a reference, establishing a function relation of a reverse involute, and deriving a reverse involute tooth profile meshing transmission backlash-free meshing equation by using an involute tooth profile meshing transmission backlash-free meshing equation;

2): and confirming the calculation value of the modification coefficient of the reverse involute gear according to the condition of the reverse involute tooth profile.

3): calculating the radius or diameter of the boundary circle;

4): selecting the full tooth height and the addendum circle radius, and calculating the involute tooth height;

5): checking meshing transmission interference;

6): calculating and checking a relevant sliding coefficient;

7): inspecting the machining quality of the gear;

in the step 1), a rectangular coordinate function expression of any tangent point P (x, y) trajectory curve on the gear base circle is converted into a polar coordinate function equation to obtain

invα_k＝tanα_k-α_k

Wherein r is the reference circle radius, r_kRadius of a point symmetrical to the point P on the back involute profile, alpha_kIs the pressure angle, beta_kThe involute curve is a spiral angle, which is an involute function, and the curve from a starting point on a base circle to the highest cutting point of the hob nose in the base circle is also an involute, and is called a reverse involute because the involute is symmetrical to the involute outside the base circle about the starting point in the base circle;

in the step 1), the backlash-free meshing equation is transmitted based on the involute tooth profile meshing, namely, the backlash-free meshing equation of the involute gear, wherein the equation is

x_∑n＝x_n1+x_n2 (1-5)

Or

Recombination of alpha_t、α_wt、α_nThe following relationship is established with the pitch angle β:

the tooth profile curves of main and auxiliary meshing transmission are in smooth transition connection, the displacement coefficients are the same, only the total displacement coefficients are different, so the backlash-free meshing equation is also an involute gear backlash-free meshing equation, and the equation is that

x_∑sn＝x_n1-x_n2 (1-8)

x_∑snIs the total normal deflection coefficient of the auxiliary meshing transmission because of x_n2Is a negative value, apparently alpha_wtn＞α_wtThe system is a back involute tooth profile meshing transmission backlash-free meshing equation; wherein alpha is_wtIs the end face angle of engagement, α_nFor reference circle normal pressure angle, alpha_tTo reference the end face pressure angle, x_n1For main transformation of the gear coefficient, x_n2Is a double-involute symmetrical tooth profile modification coefficient, alpha_wtnTo assist the angle of engagement, z₁Number of teeth of pinion gear, z₂The number of teeth of the big gear is shown.

Has the advantages that: the meshing track of the reverse involute tooth profile is designed on the involute gear, so that the bearing capacity of a main working face can be improved, the elastic deformation is considered during design, the stress deformation of the gear is corrected by utilizing the elastic deformation, the requirement of an elastic conjugate tooth profile curve is met, the error of gear manufacturing can be properly relaxed, and the difficulty of meeting the precision requirement of gear manufacturing is reduced.

Drawings

FIG. 1 illustrates a tooth root transition profile formation process.

FIG. 2 is a schematic view of the starting point of the involute profile in the base circle.

FIG. 3 is the meshing view of involute tooth profiles at the root of the pinion and the top of the bull gear.

FIG. 4 is the meshing diagram of the involute and the reverse involute of the top of the pinion and the root of the bull gear.

Fig. 5 is an enlarged view of fig. 4 at D.

Fig. 6 is a schematic diagram of the calculation of the back involute arc tooth thickness.

FIG. 7 is a schematic view of a gear under conditions where no undercut occurs.

FIG. 8 is a schematic view of the start point of involute at the root of a gear.

FIG. 9 is a schematic view of the engagement point between the root of the small gear and the top circle of the large gear.

FIG. 10 is a schematic view of the process for machining the root of the large gear.

FIG. 11 is a schematic diagram showing the contact ratio of the meshing of the inner and outer tooth profiles of the base circle.

Fig. 12 is a schematic view of the sliding during engagement.

Fig. 13 is an enlarged view of fig. 12 at E.

FIG. 14 is a schematic view of the common normal across the number of teeth.

Fig. 15 is a normal sectional view of the pinion hob.

Fig. 16 is a normal sectional view of the bull gear hob.

Fig. 17 shows a section through a hob normal with a transmission ratio i equal to 1.

Detailed Description

The invention will be further explained with reference to the drawings.

As shown in fig. 1 to 7, the design method of the present invention includes the steps of:

firstly, modeling analysis is carried out on the existence of reverse involute meshing transmission of the gear.

1): modeling by taking a gear base circle as a reference, and obtaining a functional relation of an inverse involute through geometric analysis;

it is known that when a hob or a slotting tool performs outer cutting on a base circle, an involute generating line is formed along the base circle, when the hob or the slotting tool extends into the base circle, the direction of the involute generating line changes at the starting point of the base circle, the starting point is taken as a tangent of the base circle, the starting point is connected with the center of the base circle and extends, a line connecting the line and the tangent are perpendicular to each other to form a plane rectangular coordinate system, the tangent is taken as an x line, the line connecting the line is a y line, the origin of the coordinate system is taken as a starting point O, the involute in the base circle is symmetrical with the involute outside the base circle about the starting point, a base circle and a reverse base circle are formed, and the generating line and the reverse generating line are mirror images completely corresponding to each other, namely, the forward involute and the reverse involute are two mirror images symmetrical about the origin in two quadrants and four quadrants respectively in the plane rectangular coordinate system.

The specific geometric analysis and modeling is derived as follows:

as shown in FIG. 1, the hob is actually a rack, the gear to be cut rotates at an angular velocity ω, the teeth on the hob move forward at a linear velocity vk, the arc center of the hob nose (center) always moves along a straight line, v_k＝ω_bsThe center point of the circular arc of the tool nose is tangent with a circle which has the radius equal to the radius r of the root circle_fRadius rho of arc of the hob nose_fSum of additions, i.e. r_bs＝r_f+ρ_fBecause the tool nose is an arc, the cutting point of the hob is on the arc of the tool nose, and the radius of the circle will change if the cutting points are different. The vertical line passing through the center of the circular arc of the hob tip as the cutting edge of the hob upper teeth intersects with the side line at a point P (x, y), and also intersects with the circular arc at a point P (x, y), which is the highest point of the cutting edge of the hob teeth of the hob, because the hob tip always moves away from the center of the gear when the gear rolls, the point P (x, y) always contacts with the inner tooth profile of the gear base circle, which is the actual cutting point of the hob in the inner curve of the base circle, which is the envelope curve of the highest cutting point of the side line of the hob teeth, therefore, the included angle of the tangent line of the circular arc cutting point of the hob tip is equal to the tooth profile angle alpha of the hob end face_tThe radius of the tangent circle of the highest cutting point is reduced by rho_fsinα_tThis is the radius of the tangent circle of the actual tooth profile cut, r_sIt is shown that,

r_s＝r_f+ρ_f(1-sinαt) (1-1)

because the tip of the hob is a section of circular arc, the involute in the base circle also has a starting point, and because the hob teeth have a certain thickness, the tooth root circle to the starting point of the involute should be composed of two sections of curves, and the highest cutting point of the tooth edge of the hob is the dividing point of the two sections of curves.

Establishing a rectangular coordinate system with the center O of the gear as the origin, wherein P (x, y) is any tangent point, and a straight line O₂Q＝r_f+ρ_f，△POO₂And Δ OO₂Q is a right triangle. Then there are

Obtained according to the above relationship

In the formula r_sk＝r_f+ρ_f(1-sinα_k) Is the radius of the tangent circle, r, of any tangent point P (x, y)_skAngle alpha with pressure_kAnd changes accordingly. Helix angle beta to helical gear tangent point P (x, y)_k

Substituting the formula (1-2) into the above formula to convert into a function formula of the pressure angle alpha k

Rectangular coordinate function of any tangent point P (x, y)

X＝O₂Psinα_k cosβ_k

Y＝O₂Pcosα_k

Due to O₂P and relatedSubstituted into the above relation, converted into alpha_kIs a function of

According to the trigonometric function relationship:

(sinα_k)²+(cosα_k)²＝1

therefore, any point coordinate P (x, y) of the initial segment satisfies

In the initial stage

Is a constant and is thus a rectangular coordinate function of an ellipse with a major axis of

Short axis of

It is proved that the transition curve of the tooth root portion is still an elliptic curve. It is now demonstrated that the curve from the point demarcated by the highest cutting point of the hob nose to the starting point on the base circle is an involute curve.

Changing the cutting point P (x, y) to the pressure angle alpha according to the formula (1-2)_kThe rectangular coordinate function of (1):

X＝r_sktanα_kcosβ_k

Y＝r_skcosβ_k

this is a function of the rectangular coordinates of the trace of any tangent point P (x, y), and also indicates that the position of the tangent point is closely related to the pressure angle. Converting the rectangular coordinate function expression into a polar coordinate function equation to obtain

invα_k＝tanα_k-α_k

This is the involute function, and according to the above research and analysis, it is proved from the mathematical model that the curve in the base circle from the starting point on the base circle to the highest cutting point of the hob nose is also the involute, and the involute is called as the inverse involute because the involute is symmetrical with the involute outside the base circle about the starting point in the base circle.

The starting points of the transition curve and the involute in the base circle can also be expressed as mathematical formulas.

As shown in FIG. 2, the point P is the starting point of the involute in the base circle, α_dIs the pressure angle of the starting point, the extension straight line of the point P and the circumcircle of the base circle intersect at the point N,

from the illustration, the following relationship can be obtained:

PN＝P_JN-PP_J

P_JN＝r_ssinα_t

obtaining:

in the formula, alpha_tIs the tooth form angle of the hob, h_aIs the coefficient of tooth crest height, x_nIs the coefficient of variation of the gear, r_sIs the radius of the tangent circle.

If the coordinate system is changed into a plane rectangular coordinate system which is established by taking the starting point of the involute outside the base circle as the origin and taking the tangent line of the starting point and the base circle and the connecting line of the starting point and the circle center of the base circle, the two involutes are symmetrical about the origin.

2): deducing an auxiliary meshing transmission backlash-free meshing equation from the main meshing transmission backlash-free meshing equation;

as shown in FIG. 3, the main meshing transmission is the meshing transmission of the tooth profile of the top of the large gear and the root of the small gear, and r is set_bFor the base radius of the large gear, r is set_aSetting a point P as a node for the radius of the top circle of the gearwheel, setting a section of pitch line of the pitch circle of the pinion as 1-1, setting a section of pitch line of the pitch circle of the gearwheel as 2-2, and taking a corresponding point B2, B on the pitch line 2-2₂Arc B to point P₂P，B₂P is the common normal line, C₂The curve is also the envelope curve of the hob or slotting cutter. Hob or slotting cutter tooth profile C₁The curve is an involute curve, and the tooth profile C of the cut gear can be proved according to the characteristics of 'basic law of tooth profile meshing' and the involute curve₂Is also an involute, whereby the backlash free engagement equation is also an involute gear backlash free engagement equation, which equation is

x_∑n＝x_n1+x_n2 (1-5)

Or

In the formula, z₁Is the number of pinion teeth, z₂Is the number of large gear teeth, α_wtIs the end face angle of engagement, alpha_tIs the reference circle end face pressure angle, alpha_nIs the reference circle normal pressure angle, x_∑nIs the main meshing transmission total normal deflection coefficient, x_n1Is pinion normal index, x_n2Is the normal deflection coefficient of the gearwheel. Alpha is alpha_t、α_wt、α_nThe following relationship is established with the pitch angle β:

where a is the actual center distance between two gears and a' is the theoretical center distance between two gears.

Deducing an auxiliary meshing transmission backlash-free meshing equation:

as shown in FIGS. 4 and 5, let J₂Is the base circle of the bull gear J₂' is the reverse base circle of the reverse involute profile at the root of the bull gear, and the point A is the meshing point, alpha_t1Is the end face pressure angle, alpha, of the pinion at point A_t2Is the end face pressure angle, alpha, of the gearwheel at point A_wtnIs the engagement angle at point a, i.e., the angle between the pinion and pinion forces and the horizontal (X-axis). The root of the bull gear is a reverse involute which is the same as the transmission of an internal gear, so the change rule of the tooth thickness is also opposite, the tooth profile curves of the main and auxiliary meshing transmissions are in smooth transition connection, the modification coefficients are the same, but the total modification coefficients are different, so the backlash-free meshing equation is also an involute gear backlash-free meshing equation which is the same as the involute gear backlash-free meshing equation

x_∑sn＝x_n1-x_n2 (1-8)

x_∑snIs the total normal deflection coefficient of the auxiliary meshing transmission because of x_n2Is a negative value, apparently α_wtn＞α_wtThe calculation result is completely consistent with the situation of the drawing, and the stress directions of all points of the gear tooth profile are inconsistent when the double-involute gear is in meshing transmission.

Under the condition that the back involute exists in the gear, the corresponding tooth profile parameters need to be calculated, wherein,

1. calculating the arc tooth thickness of the back involute tooth profile;

as shown in FIG. 6, let O be the center of the bull gear, O 'be the center of the reverse base circle, let any tangent point P (x, y) be selected at point A, A' be a point on the reverse involute profile symmetrical to A, r_iIs the radius of any point A, r_kIs the radius of the point of symmetry A', theta_iIs the spread angle, θ, of any point A_kIs the spread angle of the point of symmetry a'. Two triangles DeltaAOO & ^' OO ^ θ ^ can be proved_k＝θ_i. Because OA is r_i，OA′＝r_k，OO′＝2r_bIn the AOO' triangle, there are the following corner relationships

According to the nature of the involute

According to the property of the involute, the included angle of the starting point of the involute at the base circle

According to the involute property

θ_k＝θ_i＝invα_k＝invα_i

Simultaneous solution to obtain an equation

Because θ i is small, θ i ═ inv α_kIs approximately equal to 0 and can be approximately obtained

Namely have

r_b-r_k＝r_i-r_b

Or

r_k＝2r_b-r_i

r_i＝2r_b-r_k

According to an involute function formula

This is a property with respect to the back involute: the included angle between any symmetrical point of the involute and the reverse involute and the central point of the gear is bisected by the connecting line of the starting point and the center of the gear, or the included angle between the connecting line of any symmetrical point of the involute and the reverse involute and the central point of the gear and the connecting line of the starting point and the center of the gear is equal. According to the calculation formula of the arc tooth thickness, the calculation formula of the arc tooth thickness at any point A' in the base circle of the large gear is

Thus, once the A' point, i.e., r, is determined_kThe tooth thickness can be found by numerical values.

2. Calculating a top clearance and a tooth crest height shortening coefficient;

the correct meshing of a pair of gears is ensured, and a certain gap is reserved between the root part and the top part of the gear, and the gap is called a tooth top gap, which is called a top gap for short. The modified gear selects the installation center distance according to the requirement of ensuring no-backlash meshing, so that the top backlash cannot be ensured, and the installation center distance is selected according to the requirement of ensuring the top backlash, so that the top backlash cannot be ensured

δ_y＝x_∑n-y_n (1–13)

In the formula y_nIs the center separation coefficient, the calculation formula

Substituting the formulas (1-6) and (1-14) into the formulas (1-13) to obtain the delta y calculation formula

The limit of a pair of involute gear mesh transmissions is the angle of engagement alpha_wtIs more than or equal to 0 and is obtained according to the formula (1-15)

This is also the condition for obtaining the back involute profile. The conditions for obtaining the back involute tooth profile by hobbing and gear shaping cutting can also be determined according to the relationship between the tooth height and the base circle. I.e. the root circle must be smaller than the base circle.

r-2h_f＜r_b

According to the calculation formula of involute gear

h_f＝(h_a ^*-δ_y+c^*)m_n

Substituting into a correlation calculation formula to obtain

The range of the crest height reduction factor is

The range of the reduction of the tooth height of the reverse involute gear is also a condition for forming the reverse involute tooth profile, and the range can be used for determining parameters for designing the reverse involute gear and checking whether the tooth height of the reverse involute gear is reasonable or not.

3. Comprises a conditional expression that an involute gear is not undercut;

as shown in fig. 7, the reverse involute gear is also a large negative displacement involute gear, so that the tooth number and the displacement coefficient of the large gear and the helix angle should satisfy a certain relationship or should be selected to ensure that the gear does not undercut and the meshing transmission does not interfere. According to the principle of modified gear, the modified gear is that the hob is moved x along radial direction from standard position_nm_nAway from the cut gear. Wherein x_nIs the normal deflection coefficient, m_nIs the normal modulus.

Taking a tangent point position N corresponding to the intersection point of the plane of the top end of the reference circle tooth and the base circle₁The condition of not undercutting the interference is

N₁Q＞h_a-x_nm_n

Or

x_nm_n＞h_a-N₁Q

In the formula h_aIs the pitch circle tooth crest height, alpha_tIs the gear end face reference circle pressure angle, r is the gear reference circle radius, which are respectively calculated by the following formula

h_a＝(h_a ^*-δ_y)m_n

N₁Q＝P N₁sinα_t

PN₁＝rsinα_t

Substituting the above formula into a conditional formula without generating undercut interference, and obtaining a check formula of the displacement coefficient by mathematical operation

Or checking the number of teeth according to the displacement coefficient, helix angle, pressure angle

Or checking the helix angle according to the displacement coefficient, tooth number and pressure angle

Formulas (1-17), (1-18) and (1-19) are used for checking that the displacement coefficient, the tooth number and the helix angle meet the conditions mutually, wherein ha is the tooth crest coefficient, and delta is the tooth crest coefficient_yFor decreasing the crest coefficient of tooth, m_nIs the modulus.

3): finally, the forming conditions of the back involute tooth profile are obtained through the calculation, the calculation value of the back involute gear modification coefficient is required to be confirmed, and the confirmation steps are as follows:

the conditions for forming the reverse involute are that the smallest circle of the working tooth profile is smaller than the radius of the basic circle, and the working tooth profile has the following characteristics regardless of the top clearance

r-(h_a ^*-x_n)m_n＜r_b

Where r is the reference circle radius, r_bIs the radius of the base circle, and is obtained by substituting the radius into a related calculation formula

Taking values according to absence of undercuttingObtaining the gear wheel deflection coefficient x by the range condition, namely the formula (1-17) and the condition for forming the back involute, namely the formula (1-20)_nIn the selection range of

In selecting the shift coefficient x_nIn time, the influence of the crest shortening factor is not considered, so the displacement factor x_nIs in the value range of

The two-section tooth profile of the double-involute gear is uniformly distributed, and the working tooth profile takes the base circle as a symmetrical line and has

r is the reference circle radius, r_bIs the radius of the base circle, and is obtained by substituting the radius into a calculation formula

The expression (1-22) is the condition for obtaining the back involute tooth profile and the value range of the modification coefficient, and the expression (1-23) is the optimum modification coefficient for designing the back involute gear.

The back involute tooth profile meshing transmission is similar to the involute tooth profile internal tooth meshing transmission, but the back involute tooth profile meshing transmission is different from the involute tooth profile meshing transmission, so the involute tooth profile internal tooth meshing transmission and the back involute tooth profile meshing transmission have the following same points and different points:

same point

The back involute tooth profile is meshed with the internal tooth profile and is the same as the internal tooth profile.

The difference.

The tooth profile of the internal gear is distributed along the inner circumference, and the back involute tooth profile is distributed along the outer circumference; the motion directions of the internal gear transmissions are the same, and the motion directions of the back involute tooth profiles are opposite; the back involute gear transmission is the coexistence of main and auxiliary meshing transmission, i.e. the convex tooth profile and the convex-concave tooth profile are in contact coexistence, and the internal gear transmission is only in contact with the convex-concave tooth profile.

Based on the above derivation calculation process, it can be seen that the centers of the two gears meshed with each other are relatively fixed, i.e. the center distance is not changed, when the pinion is at ω₁When rotating at an angular speed, the large gear is driven to rotate at omega₂Angular velocity rotation, shown in the diagrams of fig. 1 to 4, J₁Is the base circle of the pinion, J₂The tooth profile of the large gear and the tooth profile of the small gear are outside the base circle of the large gear, when the contact tooth profiles of the small gear and the large gear are outside the base circle of the large gear, the tooth profiles of the large gear and the small gear must pass through a node, so that the transmission ratio and the rotation angular velocity are constant, and the meshing transmission of the involute tooth profiles, namely the main meshing transmission, is realized when the meshing angle is consistent with the stress direction of the tooth surfaces of the two gears.

When the contact point of the small gear and the big gear is in the base circle of the big gear, the section is auxiliary meshing transmission, and the meshing angle is set as alpha_wtnPressure angle of pinion gear is alpha_t1The pressure angle of the gearwheel is alpha_t2，α_t1And alpha_wtnAnd alpha_t2The triangles have the following relations:

α_t1＞α_wtn＞α_t2

this relationship states that when the contact point of the pinion and the gearwheel is within the base circle of the gearwheel, the profile of the gearwheel is an inverse involute, which is an inner envelope profile opposite to the involute, the normal of which contact point is tangent to the inverse base circle, the normal of which contact point is tangent to the base circle of the pinion, and the profile of the pinion and gearwheel has no common normal but follows a minor meshing angle α_wtnForming resultant force acting on the tooth surfaces of the two gears; but the sizes of the three angles are not equal, a contact point always has a tangential component force to generate sliding, which is equivalent to grinding processing, and tests show that after the back involute tooth profile runs for a period of time through large-load transmission, the root part of the bull gear and the top part of the pinion are smooth, like grinding processing,as a result of the interaction of a tangential force component. This slip phenomenon also exists in involute gear drives where the pressure angle and the mesh angle are unequal when a modified involute profile is used and alpha is positive modification_wt＞α_tWhen negative deflection is used, α_wt＜α_tThat is, the involute profile modified tooth meshing transmission is not pure rolling, and has sliding, and the sliding coefficient is usually evaluated by the sliding coefficient, and the sliding coefficient is required not to exceed a certain value. Similarly, it is allowable to control the relative sliding coefficient of the back involute tooth profile not to exceed a certain value, and because the tooth profile is a secondary meshing transmission, the motion track of the tooth profile is controlled by the primary meshing transmission, and only the secondary supporting function is realized.

According to the analysis, the involute and back involute tooth profiles are completely applicable to meshing transmission, but the condition is that an optimal use range needs to be found, and the sliding coefficient is ensured to be within a condition allowable range so as to design and calculate the double involute gear.

When the number of teeth of the big gear and the small gear is the same, the transmission ratio is 1, namely the input rotating speed is the same as the output rotating speed, the speed is not changed, if the reverse involute tooth profile meshing transmission is still adopted, if the x is adopted_n1＝x_n2Then α is_wtn＝α_tThis is not allowed, and it can be confirmed that if x is not allowed according to the back involute forming conditions (1 to 21) or (1 to 22) and the backlash free meshing equations (1 to 4) and (1 to 5)_n1＝x_n2Then the back involute condition cannot be satisfied; therefore, when the number of teeth of the small gear is the same as that of the large gear, there is a relation x_n1＝x_∑n-x_n2This is a special case of a double involute gear drive. Generally, this method can be used when the transmission ratio is small.

In addition, according to the characteristics of the curve in the base circle, the back involute and the involute are symmetrical about the starting point of the involute, so that the boundary circle of the double involute is the base circle. For a pair of gears in meshing transmission, the base circle is also the pitch circle, and the pitch circle of the corresponding pinion, i.e. the demarcation circle of the pinion, is thus

r_J2＝r_b2 (1–24)

r_J1＝a–r_b2 (1–25)

In the formula, r_b2Is the base radius of the gearwheel, r_J2Is the dividing circle radius of the big gear, a is the actual center distance of the gear pair, r_J1Is the pinion dividing circle radius.

Obtaining a correlation formula of the back involute of the gear, adopting values of full tooth height and involute tooth height to confirm the addendum coefficient, the tip clearance coefficient and the chamfer coefficient, and specifically comprising the following steps:

calculating the addendum coefficient, the tip clearance coefficient and the chamfering coefficient, setting the addendum coefficient ha and the tip clearance coefficient c of the double involute gear, adopting the involute gear standard, generally taking the addendum coefficient ha as 1, taking the tip clearance coefficient c as 0.25, adopting the ha to take the long gear more than 1 for the double involute gear, generally requiring the addendum to remove sharp corners, namely the addendum to chamfer, generally using the chamfer c for removing the sharp corner burrs, and adopting the involute gear with the tip clearance coefficient ha as the center of the double involute gear_aX 45 deg., where c_aCalculated by the following formula

c_a＝c_a ^*m_n (2–1)

In the formula, a sharp corner burr removing and chamfering coefficient is called as an addendum chamfering coefficient for short, ca is 0.05-0.1, the larger the modulus is, the smaller the value is, and the chamfer is ensured to be between 0.1-0.5;

and secondly, calculating the normal modulus and the normal pressure angle, wherein the involute gear is provided with a reference circle, the normal modulus, the normal pressure angle and the spiral angle are calculated by reference circles which are called the reference circle normal modulus, the reference circle normal pressure angle and the reference circle spiral angle, and the reference circle is a parameter which is set as required by design and calculation. The pressure angle is also called the profile angle, the number of teeth z, the pitch diameter d, and the normal modulus m_nThe helix angle β has the following relationship:

reference circle normal pressure angle alpha_nAnd indexingThe relationship of the pressure angle of the rounded end face,

base circle diameter d_bReference circle diameter d, reference circle end face pressure angle alpha_tIn relation to (2)

d_b＝dcosα_t

Gear end face module m_tModulus of normal m_nIn relation to (2)

Calculating the involute tooth height, the full tooth height and the addendum circle radius, mainly calculating and selecting the involute tooth height, calculating the working tooth height and the pinion addendum circle radius and calculating other parameters, specifically,

(1) the tooth profiles of the double involute gears are respectively outside a base circle and inside the base circle, the tooth height outside the base circle can be calculated, the tooth height outside the base circle is hb, and the calculation formula is

h_b＝r_a–r_b

Calculating the radius r of reference circle and the radius r of addendum circle according to the involute gear_aRadius of base circle r_bFormula (2)

r_a＝r+(h_a ^*+x_n–δ_y)m_n

Is substituted into the above formula to obtain

The tooth height of the tooth profile of the outer part of the base circle is calculated according to the involute gear, the double-involute gear can be normally meshed for transmission, parameters such as interference, sliding coefficient and the like are also checked, and therefore the involute tooth height of the double-involute gear is smaller than h_bI.e. involute tooth height h_JSelection range

h_J≤h_b

The formula (2-2) is an involute tooth height calculation formula, and the formula is a selected numerical range;

(2) calculating the full tooth height and the addendum circle radius of the large gear;

firstly, the working tooth height and the radius of the top circle of the pinion are calculated, and according to the design calculation of the involute gear, the working tooth height (not including the tooth top clearance and the tooth root clearance) calculation formula of the gear pair

h_w＝(2h_a ^*-δ_y)m_n

In the transmission of double-involute gear engagement, the working tooth height of the pinion is less than or equal to the working tooth height calculated according to the involute gear, i.e.

h_w1≤(2h_a ^*-δ_y)m_n (2–3)

According to the installation requirement of the gear pair, the root circle radius r of the pinion_f1To satisfy the conditions

r_f1≤a–r_a2＝a–r_b2–h_b

In the formula

Root radius r of pinion_f1Is satisfied with

Addendum circle with pinionRadius r_a1Radius of addendum circle of pinion

r_a1＝r_f1+h_w1

The tooth tip clearance is c m_nAdding the tooth top clearance to the actual addendum circle radius of the pinion, and substituting the actual addendum circle radius into the formulas (2-2) and (2-3) to obtain the addendum circle radius

The formula (2-3) is to determine the working tooth height of the pinion, and the formula (2-4) is to select the radius range of the top circle of the pinion

Fourthly, calculating the full tooth height and the addendum circle radius of the large gear, wherein the full tooth height is one clearance more than the working tooth height, namely, a full tooth height calculation formula

h₁＝h_w1+c^*m_n (2–5)

Calculation formula for obtaining addendum circle radius and full tooth height of large gear by same theory

r_a2＝a–r_f1+h₁ (2–6)

h₂＝h₁+c_f ^*m_n (2–7)

Wherein c is the coefficient of tooth crest clearance, and is 0.25 according to the involute gear standard_fThe root clearance coefficient of the bull gear is generally 0.05-0.1.

After the related parameters are calculated, checking is needed, and the checking comprises the following steps:

the meshing transmission check of the external involute tooth profile of the base circle of the bull gear comprises the following specific steps,

the involute tooth profile of the bull gear is in meshing transmission check, and actually, the tooth top of the bull gear is in meshing transmission check with the root of the pinion gear. The tooth top of the big gear must be engaged with the involute of the root of the small gear to avoid the interference of transition curves, because the big gear is a double-involute tooth profile, and whether the small gear can pass through the base circle of the big gear should be checked.

(1) Pressure angle alpha of processing starting point of involute at root of pinion_d

As shown in fig. 8, point B is assumed to be the starting point of the involute. The pressure angle of the involute processing starting point is set as alpha_dHas the following relational expression

PN₁＝r sinα_t

r_b＝rcosα_t

Where r is the pinion pitch circle radius, α_tIs the end face reference circle pressure angle, H is the small tooth, the actual tooth height of the root of the wheel

H＝(h^* _a–x_n1)m_n

Thus, it is possible to provide

(2) Pressure angle alpha of tooth crest meshing starting point of gearwheel_c

As shown in FIG. 9, let B be the starting point of the engagement of the tooth profile at the root of the pinion and the starting point of the top circle of the bull gear, α_cIs the tooth profile working mesh initiation point pressure angle, N₁N₂Is tangent line and has the following formula

BN₁＝N₁N₂–N₂B

N₁N₂＝(r_b1+r_b2)tanα_wt

N₂B＝r_b2tanα_at2

In the formula, r_b1Is a pinionRadius of base circle, r_b2Is the base radius of the gearwheel, alpha_wtIs the angle of engagement, calculated according to the formula (1-4), alpha_at2The pressure angle of the top end face of the gear of the big gear is calculated by the following formula

Thus is provided with

tanα_c＝tanα_wt–i(tanα_at2–tanα_wt)

According to the condition that no transition curve interference occurs, the method comprises

tanα_c≥tanα_d

That is, the checking formula that the addendum circle of the big gear does not generate transition curve interference at the root of the small gear is

Similarly, the formula for checking that the addendum circle of the pinion does not generate transition curve interference at the root of the bull gear is obtained

The formulas (3-2) and (3-3) are general checking formulas of gear meshing transmission, and because the double involute gear transmission is in homodromous involute tooth profile contact, the calculated value is a negative number, and the absolute value should be used for checking, or the pressure angle alpha of the contact starting point of the gear root part is directly calculated_fcIs checked, i.e.

tanα_fc≥tanα_d

With respect to alpha_fcCan be calculated according to the contact position of the starting point

In the formula O₁B＝r_fc1Is the pinion mesh starting point radius, as shown in FIG. 9, at Δ O₁BO₂In

O₁O₂＝a

O₂B＝r_a2

∠O₁O₂B＝α_at2–α_wt

According to a triangle corner formula

The radius rfc2 of the contact starting point of the root of the large gear can be obtained by the same method

The root of the bull gear is a back involute tooth profile, and an involute calculation formula cannot be used, so that the calculation is different from the calculation of the formula (3-4), and the formula is required to be based on

Is calculated, i.e.

Obtaining a checking formula according to the formulas (3-2) and (3-3) as follows:

the formulas (3-8) and (3-9) are suitable for checking the meshing transmission interference of the double-involute gear.

Secondly, the meshing transmission checking of the tooth profile of the basic circle of the gearwheel comprises the specific steps of,

when the tooth profile of the pinion passes through the base circle of the bull gear, alpha_at2Since the engagement angle is constant, the engagement angle α can be obtained from the formula (1-4) as 0_wt

Therefore, the formula for checking the interference when the tooth top of the small gear passes through the base circle of the large gear is

If the above formula cannot be satisfied, the parameters such as the displacement coefficient, the helical angle, and the number of teeth must be selected again.

Thirdly, the meshing transmission check of the back involute tooth profile in the gear wheel base circle, the curve in the gear wheel base circle is a back involute and is similar to the internal gear transmission, the tooth top of the pinion and the root of the gear wheel are also required to avoid the interference of transition curves, the concrete steps are that,

the highest point in the process of machining the root of the large gear

As shown in FIG. 10, the hob tooth effective depth h_dI.e. height of pinion tooth profile, radius of nose fillet of hob is rho_fThe tooth form angle of the hob is alpha_nThe highest point of the back involute profile is

r_c＝r _a2–h_d+(1–sinα_n)ρf

② highest point of pinion tooth crest and meshing passing condition

If the central distance of the gear pair is a, the highest point of the tooth top of the pinion is a-r_a1Thus, the condition for avoiding interference is

a–r_a1≥r_c

According to the condition that the hob processes the highest point of the back involute tooth profile, the pinion passes through in a meshing way

h_d≥r_a1+r_a2–a+(1–sinα_n)ρ_f (3–10)

h_d＝h₁

Tooth thickness s of hob at effective tooth height_dSatisfies the conditions

In the formula s_a2The tooth thickness of the circular arc tooth top of the large gear can be obtained according to a calculation formula of an involute gear, and s is set₂Is the indexing arc tooth thickness of the big gear, alpha_a2Is the addendum circle pressure angle, alpha, of the gearwheel_tIs the reference circle end face pressure angle, x_n2Is the coefficient of deflection, beta is the helix angle, then

In order to improve the strength of the tooth profile root, the corner radius of the hob tip should be increased, and in order to ensure that the meshing does not interfere, the corner radius of the hob tip should be reduced as much as possible, so that the corner radius of the hob tip should be reasonably selected, which is related to the modulus, and is usually calculated according to the following formula

ρ_f＝ρ^*m_n (3–12)

Wherein ρ is 0.15 to 0.38, usually 0.25.

And (IV) calculating an overlapping coefficient, wherein the overlapping coefficient is the ratio of the length of the actual meshing line of the two gears to the normal pitch of the gears, and respectively calculating an outer involute overlapping coefficient epsilon of the base circle_wEnd, endCoefficient of area overlap epsilon_dOverlap coefficient epsilon of involute in base circle_nAs shown in fig. 11, in this example,

J₁is the base circle of the pinion, J₂Is the base circle of the gearwheel, N₁N₂Is a common normal line, 1-1 and 2-2 are pitch circles of the pinion and the bull gear, respectively, P is a node, B is a pitch circle₁B is the line of engagement of the base circle outer tooth profile, B₁C is the meshing line of the base circle internal back involute tooth profile and the pinion tooth profile, B₁B is the length of the base circle outer involute meshing line, B₁C is the length of the meshing line of the back involute tooth profile in the base circle of the gearwheel. Alpha is less than 1_Jt1Is the end face pressure angle of the pinion at the dividing circle of the big gear wheel, and the angle 6 is alpha_at1Is the pressure angle of addendum circle end face of the pinion, and the angle 4 is alpha_at2Is the pressure angle of the end face of the addendum circle of the bull gear, and the angle 3 and the angle 5 are equal to the meshing angle alpha_wt。

(1) Base circle outer involute overlap coefficient epsilon_w

According to FIG. 11, B₁B is the length of the base circle outer involute meshing line,

B₁B＝P B₁+PB

PB₁＝r_b1(tanα_Jt1–tanα_wt)

PB＝r_b2(tanα_at2–tanα_wt)

in the formula, alpha_at2The pressure angle of the tip circle end surface of the gearwheel is calculated by the following formula

α_Jt1The pressure angle of the end face of the pinion at the dividing circle of the bull gear is calculated by the formula

In the formula, r_J1Is the pinion dividing circle radius, calculated according to equation (1-25). To obtain

B₁The ratio of B to the normal, i.e. the contact ratio outside the base circle

(2) End face overlap coefficient ε d

The calculation of the end overlap ratio is the same as the calculation of the involute, i.e.

Where b is the effective width of the gear teeth.

(3) The overlap coefficient of involute in base circle is epsilon n

According to FIG. 11, B₁C is the length of involute meshing line in base circle

B₁C＝r_b1(tanα_at1–tanα_Jt1)

Where rb1 is the pinion base radius, α Jt1 is the pinion tip face pressure angle,

α_at1is the end face pressure angle of the pinion at the base circle of the big gear, and is calculated by the formula (3-14), namely the calculation formula is

The normal pitch of the gear is equal to the circumference pitch of the end surface of the base circle

Thus, the involute overlap coefficient ε n in the base circle

(4) Total degree of lamination (overlap factor)

ε_Σ＝ε_w+ε_d+ε_n (3–18)

Because the engagement mode of the back involute tooth profile is different from that of the involute tooth profile, in order to ensure the reliability of the engagement transmission, the contact ratio of main engagement is more than 1 in general conditions, k is a number more than 1 in order to evaluate the engagement condition in universality and adaptability, and the condition for checking the contact ratio is that

ε_w+ε_d＞k (3–19)

The value of k is determined based on the basic conditions of the engaged transmission and the transmission usage requirements.

(V) the calculation and check of the related sliding coefficients, as shown in fig. 12 to 13, includes the steps of calculating the tooth crest sliding coefficients of the large gear according to the sliding coefficient formula:

when the tooth profile of the pinion is in contact with the base circle of the bull gear, and the relative sliding coefficient of the tooth profile of the pinion at the base circle of the bull gear is eta J1,

from η_J1＝tanα_Jt1 (3–22)

The formula for deducing the sliding coefficient at the base circle of the large gear is as follows:

the formula of the sliding coefficient of the top of the pinion is as follows:

the three position points with the maximum sliding coefficient in the meshing transmission of the reverse involute gear have different positions and different required sliding coefficients, and need to be checked respectively.

Sixthly, checking the total tooth number and the displacement coefficient, which comprises the following steps,

the negative involute gear is a negative profile gear, and the total profile modification factor is negative and also has a limit, which is called the minimum total profile modification factor. According to basic law of tooth profile meshing, pitch circle of a pair of involute gear meshing transmission must be out of base circle, namely the sum of radii of base circles of two gears is smaller than installation center distance, and a conditional expression is obtained according to a base circle calculation formula

Therefore, the sum of the numbers of teeth of the back involute gear pair satisfies the condition

The above formula shows that as long as the center distance and the modulus are determined, the maximum tooth number of the gear pair is determined, and the formula is substituted into the backlash-free meshing equation (1-6) to obtain the gear pair

Limit case, α _wt0, obtained poleLimit value

This is the condition for the minimum shift coefficient.

And in general conditions, as long as the hob is qualified, the base circle outer involute is qualified, the base circle inner reverse involute can be judged to be qualified, or as long as one involute is judged to be qualified, two involutes can be judged to be qualified, and the main inspection contents are as follows:

1. common normal line length calculation

(1) And a cross-measurement tooth number calculation formula of the common normal length of the base circle outer involute tooth profile.

As shown in FIG. 14, the number of gear teeth is z, and the tip circle pressure angle is α_atThe number of teeth in the distance measured from the common normal is calculated by the following formula, and k is used for the number of teeth measured_aTo represent

Specifically, the following description is provided: the pinion tooth profile is an involute tooth profile, and the calculation of the cross-measurement tooth number of the common normal length and the calculation of the common normal length are completely the same as those of an involute gear, and are not researched here. Formula for calculation

(2) Common normal line length calculation formula

The calculation of the length of the common normal line is the same as that of the involute gear, the calculation formula is the same, and only the calculation of the number of teeth measured in a crossing mode is different, and the measured positions are different. The common normal length is expressed by wk, and a calculation formula

w_k＝m_ncosα_n[π(k_a–0.5)+z invα_t+2x_ntanα_n] (4–3)

Cross-measure tooth number is ka represents doubleThe involute gear has k as cross-measuring tooth number and is involute gear, and the meaning of parameter coincidence is noticed, and the corresponding numerical value is substituted, for distinguishing, the common normal length of the double-involute tooth profile gear is w_kaThe length of the common normal of the involute profile gear is denoted wk.

2. Demarcation circular tooth thickness calculation

The tooth thickness is the important dimension of gear transmission, also is the important parameter for manufacturing hob and processing gear, and is the very important dimension for double involute, boundary circular tooth thickness and calculation formula of large and small gear boundary circular tooth thickness

(1) S for involute and reverse involute tooth profile boundary arc tooth thickness_JDenotes the boundary circle radius r_JIt is shown that for a gearwheel the dividing circle radius is equal to the base circle radius, i.e.,

r_J2＝rb₂

the thickness s of the circular arc tooth of the boundary of the large gear_J2

(2) A pinion dividing circle radius rJ1, related to the gear pair mounting center distance a,

r_J1＝a–r_b2 (4–5)

the pinion demarcation circular arc tooth thickness is represented by sJ1,

in the formula, alpha_Jt1Is the pressure angle of the boundary circle end face

The dividing circular thickness of the gear is not convenient to measure, but is an important inspection parameter for the cutter, which is different from the involute gear.

Wu, in generalThe gear stress calculation and strength check comprises the design calculation of the actual installation center distance a and the modulus m_nIn addition, the stress analysis and the strength check of the tooth surfaces of the double involute tooth profiles further comprise: strength check of main meshing transmission big gear and strength check of auxiliary meshing transmission small gear

Wherein, the strength of the main meshing transmission gear wheel is checked, and the input torque is set as N_eThe pitch circle radius of the main meshing transmission of the large gear is r_w2The pitch circle radius of the pinion is r_w1Then the circumferential force F acting on the tooth flank_tAnd N_eAnd r_w2And r_w1Is in the relationship of

Circumferential force F acting on tooth surface_tWith positive pressure F on the tooth surface_nAnd axial force F_zAnd beta_wIs in the relationship of

The positive pressure acting on the tooth surface is perpendicular to the tooth surface, so that the stress of the gear tooth is decomposed into a tangential positive pressure F_qAnd a radial force F_rDue to the base radius r_b＝r_wcosα_wt＝r cosα_tThen tangential force F_qAnd a radial force F_rCan be calculated.

The gear teeth of the gear are equivalent to a cantilever beam, the bending moment born by the tooth root is maximum, the stress point is at a node, and the distance h from the stress point to the root is_wq2Calculated by the following formula

The radius of the root circle of the gear tooth root is equal to r_a2–h₂Calculating the thickness s of the root according to the formulas (1-11) and (1-12)_wq2

In the formula

Specifically, the bending strength was checked:

the section of the gear tooth is approximately rectangular, the tooth thickness is approximately arc tooth thickness, and the section modulus (bending modulus) W can be calculated according to a formula

Where b is the effective width of the gear tooth contact. Bending moment applied to root of gear tooth

M＝F_qh_wq2

During main meshing transmission, the bending stress of the root of the gear of the large gear is maximum, and the bending stress sigma of the large gear is obtained by substituting into related calculation_F2

The above calculation is based on static load, actually dynamic load and much more complicated situation, so that the load comprehensive coefficient K (called working condition coefficient) is considered, the force distribution proportion of the force bearing points of the main and auxiliary meshing transmission is included, and the actual stress is obtained

In general, the unit of torque is N.m, the unit of gear size parameter is mm, the unit is not uniform, and the unit of stress is MPa in order to directly calculate the unit. According to the relation of transmission ratio, the bending stress of the root of the gear of the large gear in the main meshing transmission can be calculated by all parameters of the large gear, namely

The gear transmission torque is changed, meanwhile, the stress of the gear teeth of the gear is also changed in a clearance type period mode, the gear teeth of the gear are stressed when in contact in the transmission process, and the stress is 0 when in no contact, so that the stress state of the tooth profile can be considered to be regular period change between 0 and sigma Fmax, and the gear tooth strength can be checked according to the allowable bending strength according to the fatigue strength theory. Namely, it is

σ_F2≤[σ]_F2 (5–10)

In the formula, the allowable bending stress of a gear material is adopted, and for an automobile gear, 20CrMnTi low-carbon high-quality alloy steel material is generally adopted for carburizing and quenching, and the allowable bending stress is generally 420 MPa.

Checking contact strength

The contact stress of the back involute tooth profile is the same as that of the involute tooth profile, the contact stress of the back involute tooth profile is calculated by adopting a hertz (H.Hertz) stress formula, gear meshing transmission is all rolling belt sliding, even if the pure rolling tooth profile tooth surface is still subjected to contact variable stress, and therefore the damage forms are pitting damage. The fatigue failure is a fatigue failure, has a large relation with materials, hardness and stress cycle times, and the limit stress of the contact fatigue failure is obtained by determining a cycle base number and a heat treatment specification through experiments. It is necessary to calculate the contact stress intensity.

Let the modulus of elasticity of the pinion material be E₁Poisson's ratio of 1, and the modulus of elasticity of the material of the bull gear is E₂And the Poisson's ratio is 2, the coefficient of material property of the gear pair is Delta E

Obtaining contact stress sigma H according to the herz stress formula

Where the combined radius of curvature ρ Σ,

the curvature radius of the node is the pitch circle radius rw, and the transmission ratio coefficient Zi is set

The combined radius of curvature ρ Σ

Tooth profile contact line length l

The positive pressure of the tooth surface of the tooth profile is calculated according to the formula (7-6), and the contact stress sigma H is calculated according to the formula

The coefficient of material properties is a constant Δ E of 8.84 × 10-6 for both large and small gears made of carbon steel or alloy steel, so the contact stress can be easily calculated. Similarly, the above are all calculated with uniform load, so the case of uneven dynamic load is still considered, i.e. the load comprehensive coefficient K should be corrected to

The contact stress of the big gear and the small gear is the same, but the contact times of the big gear and the small gear are different, and the materials are possibly different, so the contact fatigue strength is different, and the strength check is respectively carried out. It is generally assumed that the contact stress of the pinion gear is divided by the contact stress of the bull gear. Namely, it is

When the material used for the bull gear and the pinion gear are the same, it is generally not necessary to check the contact strength of the bull gear. The comprehensive coefficient K is a complex coefficient and relates to the factors such as the working characteristics of a gear pair, dynamic load change, tooth direction load distribution, tooth space load distribution, contact line length change, tooth surface roughness and the like, and the selection method is the same as that of an involute gear and can be selected according to the actual use requirement, and is not discussed here. The contact strength of the tooth profile being checked in accordance with the permissible compressive stress, i.e.

σ_H≤[σ]_H (5–15)

In the formula [ sigma ]]_HThe allowable compressive stress of the gear material is generally 730MPa, and the automobile gear is usually carburized and quenched by adopting 20CrMnTi low-carbon high-quality alloy steel material.

Similarly, for intensity checking of the pinion of the auxiliary meshing transmission, the auxiliary meshing transmission is also called an internal meshing transmission, and has a similar place to the internal gear transmission, but is different from the internal gear meshing transmission of the involute gear. The auxiliary or internal engagement angle is calculated according to the equations (1-7) and (1-8)Calculating the node radius of the large gear at the inner node according to the formula (1-11), which is called the radius r of the inner node circle for short_wn2

The distance hwq1 from the pinion force point to the root is calculated by the following equation

h_wq1＝h₁-(r_a1-r_wn1) (5–17)

Pinion root pressure angle alpha wq1

Root arc tooth thickness s of pinion_wq1

Obtaining the tangential stress of the pinion at the inner node according to the formula (7-8)

Wherein the pinion root helix angle

Like the big gear, the section of the gear tooth is approximately rectangular, the tooth thickness is approximately arc tooth thickness, and the section modulus (bending modulus) can be calculated according to a formula

M＝F_qh_wq1

Like the bull gear, the bending stress sigma of the root of the pinion is obtained by considering the load comprehensive coefficient K (called the duty coefficient)_F1

The checking formula is

σ_F1≤[σ]_F1

The contact of the back involute tooth profile is different from that of the involute tooth profile, namely the contact area of the inner node is larger than that of the involute tooth profile, and the contact strength is more reliable, so that the calculation and the check of the contact strength of the inner node are not generally carried out.

With respect to the crest factor h_aCoefficient of backlash c, general involute gear h_aThe double involute gear is still determined as an involute gear, with c being 0.25. The hob nose radius coefficient ρ is defined in the hob standard, and for smooth root transition, the double involute gear hob is selected according to the hob standard, and the hob nose radius coefficient ρ is 0.25. In order to protect the working tooth surface, the tooth top is chamfered by removing sharp corners and burrs, the angle is 45 degrees, the chamfer height is changed according to modulus, and ca is c_am_nCoefficient of chamfering c_a0.1. The transmission ratio i is determined according to the requirement of transmission speed change, and the modulus m_nThe transmission power P and the rotational speed n or the torque Ne are used to determine the installation (actual) center distance a, and the transmission torque or the size of the mechanism space is used to determine the installation (actual) center distance a. Therefore, the crest coefficient h a, the crest clearance coefficient c, the circular arc radius coefficient rho of the hob nose, the chamfering coefficient c a, the transmission ratio i and the modulus m_nThe actual center-to-center distance a, etc. are known conditions. The number of teeth is first obtained from the transmission ratio i, the pitch circle normal pressure angle alpha_nAnd pitch circle helix angle beta and displacement coefficient x_nOr the gear can be selected preliminarily, checked, modified and determined according to the meshing conditions, and then the sizes of all parts of the gear are calculated to determine the machining parameters.

Example 1: large ratio gear calculation, as shown in FIGS. 15 and 16

The 6T53 automobile transmission first gear adopts a dotted line gear improved design. The transmission input torque 1516Nm (output torque 7200Nm), the installation center distance a is 123, the required transmission ratio i is 4.5-5, the material is selected from 20CrMnTi carburizing and quenching, the material characteristic coefficient delta E is 8.84 multiplied by 10 < -6 > characteristic system, and the allowable bending stress [ sigma ] is applied]_F420Mpa, allowable contact stress [ sigma ]]_H730MPa, and the other parameters are selected according to the standard. From the calculation, the pinion z is actually determined₁8, bull gear z₂38, modulus m_n5.75, pressure angle α_n22.5 deg., and the helix angle beta is 7 deg.. The specific calculations are shown in Table 1-1.

TABLE 1-1 Point-line gear drive design calculation table

TABLE 1-2 Gear parameter table (dotted line gear drive)

TABLE 1-3 Hob PARAMETER TABLE (pinion)

TABLE 1-4 Hob parameter table (big gear)

Example 2: gear drive calculation for ratio 1 (as shown in FIG. 17)

Cylindrical gear of Steyr driving axle, centre distance a 193, transmission ratio i 1, modulus m determined according to transmission output torque_n5.25, pressure angle α n 20 °, number of gear teeth z₁＝z₂The helix angle β is 15 °, 35 °. Adopting a dotted line gear improved design to calculate and determine a modulus m_n6.5, pressure angle α_n22.5 degrees, 15 degrees, and z degrees₁＝z₂The tooth width 36 is constant at 31. The calculation process is shown in Table 2-1.

TABLE 2-1 Gear drive design calculation Table with the same number of teeth (i ═ 1)

TABLE 2-2 Gear parameter Table (dotted line gear i ═ 1)

TABLE 2-3 Gear parameter table

The undescribed parts of the present invention are consistent with the prior art, and are not described herein.

The above description is only an embodiment of the present invention, and not intended to limit the scope of the present invention, and all equivalent structures made by using the contents of the present specification and the drawings can be directly or indirectly applied to other related technical fields, and are within the scope of the present invention.

Claims

1. A design calculation method for transmission engagement of a reverse involute gear is characterized by comprising the following steps: the design calculation method comprises the following steps:

3): calculating the radius or diameter of the boundary circle;

5): checking meshing transmission interference;

6): calculating and checking a relevant sliding coefficient;

7): inspecting the machining quality of the gear;

invα_k＝tanα_k-α_k

x_∑n＝x_n1+x_n2 (1-5)

Or

x_∑sn＝x_n1-x_n2 (1-8)