A tooth profile generation method for a combined involute and cycloid tooth profile

Abstract

The invention discloses a tooth profile generation method of an involute and cycloid combined tooth profile: using Fourier function to fit the transmission ratio curve, there are no two curve segments with a curvature less than the minimum curvature without root cutting, and using a polynomial curve to connect the two curve segments end to end to obtain the transmission ratio curve; after calculating the pitch curve coordinates of the driving and driven non-circular gears, the tooth top part outside the pitch curve of the driving non-circular gear and the tooth root part within the pitch curve of the driven non-circular gear adopts a cycloid tooth profile, and the tooth root part within the pitch curve of the driving non-circular gear and the tooth top part outside the pitch curve of the driven non-circular gear adopts an involute tooth profile; after the tooth top profile of the driving non-circular gear is connected with the tooth root profile, and the tooth top profile of the driven non-circular gear is connected with the tooth root profile, the connecting segments at both ends of the involute tooth profile and the cycloid tooth profile are replaced by a line segment tangent to both the involute tooth profile and the cycloid tooth profile. The invention can generate a tooth profile without root cutting for a transmission ratio curve with too small a curvature.

Description

Technical Field

[0001] The invention belongs to the technical field of gear tooth profile design, and in particular relates to a tooth profile generation method of a combined involute and cycloid tooth profile. Background Art

[0002] Gear transmission is the most common transmission mode in mechanical transmission. Generally, gears are circular. However, some special occasions require the mechanism to perform variable speed ratio transmission, and traditional circular gears cannot meet the requirements. Non-circular gears can realize non-uniform speed ratio transmission between two shafts. It combines the advantages of cams and circular gears and can accurately achieve variable transmission ratios. Non-circular gears have been used in many fields such as agriculture, light industry, metallurgy, hydraulics, medical treatment, aviation, etc., such as rice transplanters, textile machines, continuous casting machines and hydraulic motors.

[0003] In order to meet the requirements of secondary unequal amplitude and non-uniform speed ratio transmission of the seedling removal mechanism for vegetable seedlings in plug trays, some scholars have proposed a combined non-circular gear transmission mechanism in which the driving wheel is composed of an incomplete non-circular gear and a rack, and the driven wheel is composed of a partial non-circular gear and an elliptical gear. The mechanism realizes secondary unequal amplitude fluctuation of the transmission ratio within a period and has been applied to the seedling removal mechanism for vegetable seedlings in plug trays. However, the secondary unequal amplitude transmission ratio curve has a sharp point with too small curvature, and a suitable involute tooth profile cannot be formed at the sharp point. When the curvature of the involute tooth profile is too small, root cutting will occur, which will reduce the strength of the gear and even directly affect the meshing.

[0004] In order to solve the problem that the curvature of some positions in the above-mentioned quadratic unequal-amplitude transmission ratio curve is too small, a new tooth profile generation method is needed to ensure that the transmission ratio curve can be continuous at the cusp point and the generated tooth profile does not undercut. Summary of the invention

[0005] The purpose of the present invention is to solve the problems existing in the prior art and to propose a tooth profile generation method of a combined involute and cycloid tooth profile, which is a conjugate tooth profile design method for a given small curvature gear pitch curve.

[0006] The technical solution adopted by the present invention is as follows:

[0007] The tooth profile generation method of the involute and cycloid combined tooth profile of the present invention comprises the following steps:

[0008] Step 1: According to the critical relationship of non-circular gear undercutting, the minimum curvature ρ of the non-circular gear ratio curve without undercutting is obtained min ; Then, calculate the minimum curvature ρ where there are two curve segments with a curvature less than that without undercutting minThe secondary unequal amplitude transmission ratio curve of the case (the driving wheel is composed of an incomplete non-circular gear and a rack, and the driven wheel is composed of a partial non-circular gear and an elliptical gear. In the secondary unequal amplitude transmission ratio curve of the combined non-circular gear transmission mechanism, there must be two sections with a curvature less than the minimum curvature ρ where no undercutting occurs. min The curvature radius of each position is smaller than the minimum curvature ρ where no undercutting occurs. min The two positions of the quadratic unequal amplitude transmission ratio curve are divided into four curve segments I1I2, I2I3, I3I4 and I4I5. The boundary points of the four curve segments are recorded as I1, I2, I3, I4 and I5 respectively. The curvature is less than the minimum curvature ρ where no undercutting occurs. min The two positions of the curve segments are I2I3 and I4I5, the angle corresponding to I2I3 is [θ1, θ2-θ1], and the angle corresponding to I4I5 is [θ3, θ4-θ3]. The Fourier function is used to refit the I1I2 and I3I4 segments, and the polynomial curve is used to re-transition and connect them to form I2I3 and I4I5 segments. In order to ensure the smooth connection of the entire quadratic unequal amplitude transmission ratio curve, it is necessary to satisfy that the slopes of the I1I2 transmission ratio curve and the I2I3 transmission ratio curve are equal at the intersection I2, and the slopes of the I2I3 and I3I4 segments are equal at the intersection I3. The expression of the entire quadratic unequal amplitude transmission ratio curve is:

[0009]

[0010] In formula (1), θ is the angular displacement of the active non-circular gear, a 11 、b 11 、c 11 、w 11 、a 22 、b 22 、c 22 and w 22 are the coefficients of the Fourier function, which are known quantities after the I1I2 and I3I4 segments are fitted; a1, b1, c1, d1, a2, b2, c2 and d2 are the coefficients of the polynomial curve, which are the quantities to be determined.

[0011] Step 2: Since the slope of the I1I2 segment at point I2 and the slope of the I3I4 segment at point I3 of the quadratic unequal amplitude transmission ratio curve are equal to the slope of the I2I3 segment at point I2 and the slope at point I3 respectively, the slope of the I1I2 segment at point I2 is the left derivative i′ at point I2 - (θ1), the slope of the I2I3 segment at point I3 is the left derivative i′ at point I3 - (θ2), the slope of the I2I3 segment at point I2 is the right derivative i′ at point I2 + (θ1), the slope of segment I3I4 at point I3 is the right derivative i′ at point I3 +(θ2), we get i′ + (θ1)=i′ - (θ1) and i′ - (θ2) = i′ + (θ2), and then we get:

[0012]

[0013] Because the I2I3 transition curve passes through points I2 and I3, combined with equation (2), the coefficients a1, b1, c1 and d1 in the equation of the I2I3 segment curve are obtained.

[0014] Similarly, since the slope of segment I3I4 at point I4 and the slope of segment I4I5 at point I5 of the quadratic unequal-amplitude transmission ratio curve are respectively equal to the slope of segment I4I5 at point I4 and the slope of segment I1I2 at point I1, and since the I4I5 transition curve passes through points I4 and I5, the coefficients a2, b2, c2 and d2 in the equation of segment I4I5 curve are obtained by solving.

[0015] Then, the pitch curves of the driving non-circular gear and the driven non-circular gear are determined by the expression of the quadratic unequal amplitude transmission ratio curve and the given center distance, as follows:

[0016] The angular displacement of the driven non-circular gear have to:

[0017]

[0018] For a given center distance l, according to the transmission ratio i, the angular displacement θ of the driving non-circular gear and the angular displacement θ of the driven non-circular gear c , find the pitch curve radius r of the driving non-circular gear and the pitch curve radius r of the driven non-circular gear c :

[0019]

[0020] Then find the coordinates (x a ,y a ) and the pitch curve coordinates (x c ,y c ):

[0021]

[0022]

[0023] Step 3: For the non-circular gear transmission with a secondary unequal amplitude transmission ratio curve, its pitch curve is prone to have a portion with a too small radius of curvature, and the involute tooth profile of this portion is severely undercut. The present invention adopts a combination of involute and cycloid to design the tooth profile of the pitch curve with a small radius of curvature, and adopts a cycloid tooth profile for the tooth top portion outside the pitch curve of the active non-circular gear and the tooth root portion within the pitch curve of the driven non-circular gear, and adopts an involute tooth profile for the tooth root portion within the pitch curve of the active non-circular gear and the tooth top portion outside the pitch curve of the driven non-circular gear; then, the active non The tooth top profile of the circular gear is connected with the tooth root profile, and the tooth top profile of the driven non-circular gear is connected with the tooth root profile. Since the connection section is not smooth when the involute tooth profile is connected with the cycloid tooth profile, that is, there is a tiny sharp point at the intersection of the involute tooth profile and the cycloid tooth profile, based on this, in order to ensure a completely smooth transition connection between the involute and the cycloid, the connection sections at both ends of the involute tooth profile and the cycloid tooth profile are replaced by a line segment tangent to both the involute tooth profile and the cycloid tooth profile, thereby forming a combined involute and cycloid tooth profile of the active non-circular gear and the driven non-circular gear.

[0024] Preferably, the critical relationship of non-circular gear undercutting is as follows:

[0025]

[0026] Where: m0 is the module of the involute profile circular gear tool for solving the non-circular gear profile; h a * is the tooth top height coefficient of the involute gear cutter; α0 is the pressure angle of the involute gear cutter; z0 is the number of teeth of the involute gear cutter; m0, α0 and h a * Standard values ​​are taken for both. z0 is determined according to the transmission ratio at a certain position of the non-circular gear transmission ratio curve and the pitch curve of the active non-circular gear or the radial direction of the driven non-circular gear at that position of the tooth profile to be solved.

[0027] More preferably, the analytical method for involute tooth profile generation is as follows:

[0028] Coordinates of the center of curvature of the driving noncircular gear pitch curve or the driven noncircular gear pitch curve:

[0029]

[0030] in, (x,y) is the active non-circular gear pitch curve coordinate (x a ,y a ) or the driven non-circular gear pitch curve coordinate (x c ,y c ).

[0031] Pitch radius of involute profile gear cutter:

[0032]

[0033] As the pitch circle of the involute gear cutter rolls along the active noncircular gear pitch curve or the driven noncircular gear pitch curve without sliding, the rotation center of the involute gear cutter generates a path accordingly, and the offset of this path relative to the active noncircular gear pitch curve or the driven noncircular gear pitch curve is equal to the pitch circle radius r of the involute gear cutter. d , and at every moment, the curvature center of the active noncircular gear pitch curve or the driven noncircular gear pitch curve, the meshing point between the active noncircular gear pitch curve or the driven noncircular gear pitch curve and the pitch circle of the involute profile circular gear cutter, and the rotation center I of the involute profile circular gear cutter are collinear. When the active noncircular gear pitch curve or the driven noncircular gear pitch curve is convex, the coordinates of the rotation center I (x I ,y I ):

[0034]

[0035] Where ρ is the radius of curvature of the active noncircular gear pitch curve or the driven noncircular gear pitch curve at the point of meshing with the pitch circle of the involute profile circular gear cutter.

[0036] When the pitch curve of the active non-circular gear or the pitch curve of the driven non-circular gear is concave, the coordinate of the rotation center I of the involute gear profile circular gear cutter is obtained according to the three-point collinearity property:

[0037]

[0038] Involute angle θ of involute profile circular gear cutter k :

[0039] θ k =tan(α0)-α0 (11)

[0040] Base circle radius r of involute tooth profile circular gear cutter b :

[0041] r b =r d cos(α0) (12)

[0042] Assume that the arc length of the involute gear cutter pitch circle rolling from the initial position M1 along the active noncircular gear pitch curve or the driven noncircular gear pitch curve to the position M2 is ds, then ∠M1IM2 is the rotation angle of the involute profile circular gear cutter

[0043]

[0044] Suppose that after the pitch circle of the involute gear tool rolls for an arc length of ds, the meshing point between the involute of the involute gear tool and the pitch curve of the active non-circular gear or the tooth profile of the driven non-circular gear is S, and the intersection point between the tangent of the base circle of the involute gear tool passing through the tooth profile meshing point S and the base circle of the involute gear tool is recorded as N, then ∠M2IN=α0, and since after the pitch circle of the involute gear tool rolls for an arc length of ds, Where M is the involute starting point of the involute profile circular gear cutter. According to equations (11), (12), and (13), the length of the SN segment is obtained as follows:

[0045]

[0046] In ΔSIN, according to the Pythagorean theorem and equations (12) and (14), the length of the SI line segment is obtained as:

[0047]

[0048] The pressure angle of the tooth profile meshing point S is obtained in ΔSIN:

[0049]

[0050] According to formula (16), ∠SIM2 is obtained as:

[0051] ∠SIM2=α s -α0 (17)

[0052] In ΔSIM2, the coordinates of the pitch curve meshing point M2 and the coordinates of the rotation center I of the involute profile circular gear tool are known, and Therefore, according to equations (8), (10), (15) and (17), the coordinates of the tooth profile meshing point S relative to the tool rotation center I are obtained: The matrix equation is:

[0053]

[0054] According to formula (18), the coordinates of the tooth profile meshing point S are obtained:

[0055]

[0056] After the coordinates of each tooth profile meshing point are solved, all tooth profile meshing points are connected in sequence to obtain the root part tooth profile within the pitch curve of the active non-circular gear or the top part tooth profile outside the pitch curve of the driven non-circular gear.

[0057] Preferably, the analytical method for generating the cycloid tooth profile is as follows:

[0058] The pitch circle of the cycloid tooth profile circular gear tool rolls purely around the pitch curve of the active non-circular gear or the driven non-circular gear. The center coordinates of the pitch circle of the cycloid tooth profile circular gear tool are shown in formula (20):

[0059]

[0060] In the formula, r g is the pitch circle radius of the cycloid gear cutter, f is the convexity determination coefficient of the active non-circular gear or driven non-circular gear pitch curve (convex f = 1, concave f = -1), (x q ,y q ) is the curvature center coordinate of the active non-circular gear pitch curve or the driven non-circular gear pitch curve, (x, y) is the coordinate of the active non-circular gear pitch curve (x a ,y a ) or the driven non-circular gear pitch curve coordinate (x c ,y c ).

[0061] Since the pitch circle of the cycloid gear tool rolls purely on the pitch curve of the active non-circular gear or the driven non-circular gear, the cycloid gear tool pitch circle revolves and rotates at the same time. The rotation angle displacement θ0 has a quadrant transformation. The calculation formula of θ0 is as follows:

[0062]

[0063] The cycloid tooth profile coordinates of the pitch curve of the active non-circular gear or the driven non-circular gear are shown in formula (22):

[0064]

[0065] After the coordinates of the meshing points of the cycloid tooth profile are solved, all the meshing points of the cycloid tooth profile are connected in sequence to obtain the tooth profile of the tooth top part outside the pitch curve of the active non-circular gear and the tooth root part inside the pitch curve of the driven non-circular gear.

[0066] The present invention designs a new tooth profile generation method for transmission ratio curves with too small curvature at certain positions such as quadratic unequal amplitude transmission ratio curves, so that the transmission ratio curve can be continuous at the cusp point and the generated tooth profile does not have undercutting. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 is a schematic diagram of a secondary unequal amplitude transmission ratio curve adopted in an embodiment of the present invention;

[0068] Figure 2 is a schematic diagram of the pitch curve meshing of the driving non-circular gear and the driven non-circular gear in the present invention;

[0069] Figure 3It is an analytical diagram for generating a combined involute and cycloid tooth profile of the driving non-circular gear or the driven non-circular gear in the present invention;

[0070] Figure 4 It is an enlarged view of the involute and cycloid combined tooth profile of the active non-circular gear in the present invention and the connection position of the involute and cycloid;

[0071] Figure 5 It is a schematic diagram of the meshing of the involute and cycloid combined tooth profiles of the driving non-circular gear and the driven non-circular gear in the present invention. DETAILED DESCRIPTION

[0072] The present invention will be further described below in conjunction with the accompanying drawings.

[0073] The tooth profile generation method of the involute and cycloid combined tooth profile comprises the following steps:

[0074] Step 1: Quote the critical relationship of undercutting in the literature of China Mechanical Engineering "Concavity and Convexity of High-order Variable Eccentric Conjugate Non-circular Gears and Undercutting Discrimination":

[0075]

[0076] Where: m0 is the module of the involute profile circular gear tool for solving the non-circular gear profile; h a * is the tooth top height coefficient of the involute tooth profile circular gear cutter; α0 is the pressure angle of the involute tooth profile circular gear cutter; z0 is the number of teeth of the involute tooth profile circular gear cutter; z0 is determined according to the transmission ratio at a certain position of the non-circular gear transmission ratio curve and the pitch curve of the active non-circular gear of the tooth profile to be solved or the radial direction of the driven non-circular gear at this position.

[0077] Obtain the minimum curvature ρ of the non-circular gear ratio curve without undercutting min Then, calculate whether there are two curve segments with curvature less than the minimum curvature ρ min The curvature radius of each position of the quadratic unequal amplitude transmission ratio curve is smaller than the minimum curvature ρ. min The quadratic unequal amplitude transmission ratio curve is divided into four curve segments: I1I2, I2I3, I3I4 and I4I5. The boundary points of the four curve segments are recorded as I1, I2, I3, I4 and I5 respectively. The curvature is less than the minimum curvature ρ min The curve segments where the two positions are located are I2I3 and I4I5. The angle corresponding to I2I3 is [θ1, θ2-θ1], and the angle corresponding to I4I5 is [θ3, θ4-θ3]. Figure 1As shown; the I1I2 segment and the I3I4 segment are refitted using the Fourier function, and the I2I3 segment and the I4I5 segment are reconnected using the polynomial curve to ensure that the slopes of the I1I2 segment transmission ratio curve and the I2I3 segment transmission ratio curve are equal at the intersection I2, and the slopes of the I2I3 segment and the I3I4 segment are equal at the intersection I3; the expression of the entire quadratic unequal amplitude transmission ratio curve is:

[0078]

[0079] In formula (1), θ is the angular displacement of the active non-circular gear, a 11 、b 11 、c 11 、w 11 、a 22 、b 22 、c 22 and w 22 are the coefficients of the Fourier function, which are known quantities after the I1I2 segment and the I3I4 segment are fitted; a1, b1, c1, d1, a2, b2, c2 and d2 are the coefficients of the polynomial curve, which are the quantities to be determined;

[0080] Step 2: Since the slope of the I1I2 segment at point I2 and the slope of the I3I4 segment at point I3 of the quadratic unequal amplitude transmission ratio curve are equal to the slope of the I2I3 segment at point I2 and the slope at point I3 respectively, the slope of the I1I2 segment at point I2 is the left derivative i′ at point I2 - (θ1), the slope of the I2I3 segment at point I3 is the left derivative i′ at point I3 - (θ2), the slope of the I2I3 segment at point I2 is the right derivative i′ at point I2 + (θ1), the slope of segment I3I4 at point I3 is the right derivative i′ at point I3 + (θ2), we get i′ + (θ1)=i′ - (θ1) and i′ - (θ2) = i′ + (θ2), and then we get:

[0081]

[0082] Because the I2I3 transition curve passes through points I2 and I3, combined with equation (2), the coefficients a1, b1, c1 and d1 in the I2I3 segment curve equation are solved;

[0083] Similarly, since the slope of the I3I4 segment at point I4 and the slope of the I4I5 segment at point I5 of the quadratic unequal amplitude transmission ratio curve are respectively equal to the slope of the I4I5 segment at point I4 and the slope of the I1I2 segment at point I1, and since the I4I5 transition curve passes through points I4 and I5, the coefficients a2, b2, c2 and d2 in the equation of the I4I5 segment curve are obtained by solving;

[0084] Then, the pitch curves of the driving non-circular gear and the driven non-circular gear are determined by the expression of the quadratic unequal amplitude transmission ratio curve and the given center distance, as follows:

[0085] The angular displacement of the driven non-circular gear have to:

[0086]

[0087] For a given center distance l, according to the transmission ratio i, the angular displacement θ of the driving non-circular gear and the angular displacement θ of the driven non-circular gear c , find the pitch curve radius r of the driving non-circular gear and the pitch curve radius r of the driven non-circular gear c :

[0088]

[0089] Then find the coordinates (x a ,y a ) and the pitch curve coordinates (x c ,y c ):

[0090]

[0091]

[0092] The meshing conditions of the pitch curves of the driving non-circular gear and the driven non-circular gear are as follows: Figure 2 shown.

[0093] Step 3: A cycloid tooth profile is used for the tooth top part outside the pitch curve of the driving non-circular gear and the tooth root part inside the pitch curve of the driven non-circular gear, and an involute tooth profile is used for the tooth root part inside the pitch curve of the driving non-circular gear and the tooth top part outside the pitch curve of the driven non-circular gear; then, the tooth top profile of the driving non-circular gear is connected with the tooth root profile, and the tooth top profile of the driven non-circular gear is connected with the tooth root profile; finally, the connecting segments at both ends of the involute tooth profile and the cycloid tooth profile are replaced by a line segment tangent to both the involute tooth profile and the cycloid tooth profile, thereby forming a combined involute and cycloid tooth profile of the driving non-circular gear and the driven non-circular gear, such as Figure 5 The involute and cycloid combined tooth profile of the active non-circular gear is shown in Figure 4 As shown in the ABCDEF curve segment, point O is the curvature center of the active non-circular gear. Figure 4 It can be seen that when the connecting segment EGD of the involute tooth profile and the cycloid tooth profile is not replaced by a line segment, there will be a tiny cusp at the intersection of the involute tooth profile and the cycloid tooth profile.

[0094] As a preferred embodiment, the analytical method for generating the involute tooth profile is as follows:

[0095] Coordinates of the center of curvature of the driving noncircular gear pitch curve or the driven noncircular gear pitch curve:

[0096]

[0097] in, (x,y) is the active non-circular gear pitch curve coordinate (x a ,y a ) or the driven non-circular gear pitch curve coordinate (x c ,y c );

[0098] Pitch radius of involute profile gear cutter:

[0099]

[0100] Involute tooth profile gear tool pitch circle ( Figure 3 The circle marked with number 2) is along the driving non-circular gear pitch curve or the driven non-circular gear pitch curve ( Figure 3 When the curve (with the number 3 in the middle) rolls without sliding, the rotation center of the involute gear cutter generates a path ( Figure 3 The curve marked with number 4 in the middle), the center of curvature of the active noncircular gear pitch curve or the driven noncircular gear pitch curve, the meshing point between the active noncircular gear pitch curve or the driven noncircular gear pitch curve and the pitch circle of the involute gear profile circular gear cutter, and the rotation center I of the involute gear profile circular gear cutter are collinear; when the active noncircular gear pitch curve or the driven noncircular gear pitch curve is convex, the coordinates of the rotation center I (x I ,y I ):

[0101]

[0102] Where ρ is the radius of curvature of the active noncircular gear pitch curve or the driven noncircular gear pitch curve at the point of meshing with the pitch circle of the involute profile circular gear cutter;

[0103] When the pitch curve of the active non-circular gear or the pitch curve of the driven non-circular gear is concave, the coordinate of the rotation center I of the involute gear profile circular gear cutter is obtained according to the three-point collinearity property:

[0104]

[0105] Involute profile circular gear cutter involute ( Figure 3 The curve with the label 5) has an angle θ k :

[0106] θ k =tan(α0)-α0 (11)

[0107] Base circle of involute profile circular gear cutter ( Figure 3 The radius of the circle with number 1 is r b :

[0108] r b =r d cos(α0) (12)

[0109] Assume that the pitch circle of the involute gear cutter changes from the initial position M1 (the initial position is as follows Figure 3 (a) starts to roll along the active non-circular gear pitch curve or the driven non-circular gear pitch curve to position M2 (as shown in Figure 3 The arc length of (b) is ds, then ∠M1IM2 is the rotation angle of the involute profile circular gear cutter

[0110]

[0111] Suppose that after the pitch circle of the involute gear tool rolls for an arc length of ds, the meshing point between the involute of the involute gear tool and the pitch curve of the active non-circular gear or the tooth profile of the driven non-circular gear is S, and the intersection point between the tangent of the base circle of the involute gear tool passing through the tooth profile meshing point S and the base circle of the involute gear tool is recorded as N, then ∠M2IN=α0, and since after the pitch circle of the involute gear tool rolls for an arc length of ds, Where M is the involute starting point of the involute profile circular gear cutter. According to equations (11), (12), and (13), the length of the SN segment is obtained as follows:

[0112]

[0113] In ΔSIN, according to the Pythagorean theorem and equations (12) and (14), the length of the SI line segment is obtained as:

[0114]

[0115] The pressure angle of the tooth profile meshing point S is obtained in ΔSIN:

[0116]

[0117] According to formula (16), ∠SIM2 is obtained as:

[0118] ∠SIM2=α s -α0 (17)

[0119] In ΔSIM2, the coordinates of the pitch curve meshing point M2 and the coordinates of the rotation center I of the involute profile circular gear tool are known, and Therefore, according to equations (8), (10), (15) and (17), the coordinates of the tooth profile meshing point S relative to the tool rotation center I are obtained: The matrix equation is:

[0120]

[0121] According to formula (18), the coordinates of the tooth profile meshing point S are obtained:

[0122]

[0123] After the coordinates of each tooth profile meshing point are solved, all tooth profile meshing points are connected in sequence to obtain the root part tooth profile within the pitch curve of the active non-circular gear or the top part tooth profile outside the pitch curve of the driven non-circular gear.

[0124] As a preferred embodiment, the analytical method for generating the cycloid tooth profile is as follows:

[0125] The pitch circle of the cycloid tooth profile circular gear tool rolls purely around the pitch curve of the active non-circular gear or the driven non-circular gear. The center coordinates of the pitch circle of the cycloid tooth profile circular gear tool are shown in formula (20):

[0126]

[0127] In the formula, r g is the pitch radius of the cycloid gear cutter, f is the concavity determination coefficient of the pitch curve of the active non-circular gear or the driven non-circular gear, (x q ,y q ) is the curvature center coordinate of the active non-circular gear pitch curve or the driven non-circular gear pitch curve, (x, y) is the coordinate of the active non-circular gear pitch curve (x a ,y a ) or the driven non-circular gear pitch curve coordinate (x c ,y c );

[0128] The calculation formula for the rotational angular displacement θ0 of the cycloid profile circular gear cutter pitch circle doing pure rolling on the active non-circular gear or driven non-circular gear pitch curve is as follows:

[0129]

[0130] The cycloid tooth profile coordinates of the pitch curve of the active non-circular gear or the driven non-circular gear are shown in formula (22):

[0131]

[0132] After the coordinates of the meshing points of the cycloid tooth profile are solved, all the meshing points of the cycloid tooth profile are connected in sequence to obtain the tooth profile of the tooth top part outside the pitch curve of the active non-circular gear and the tooth root part inside the pitch curve of the driven non-circular gear.

Claims

1. A method for generating a tooth profile of a combination of an involute and a cycloid tooth profile, characterized in that: The following steps are involved: Step 1: According to the critical relationship of non-circular gear undercutting, the minimum curvature ρ of the non-circular gear ratio curve without undercutting is obtained min ; Then, calculate the minimum curvature ρ where there are two curve segments with a curvature less than that without undercutting min The curvature radius of each position of the secondary unequal amplitude transmission ratio curve of the case is smaller than the minimum curvature ρ where no undercutting occurs. min The two positions of the quadratic unequal amplitude transmission ratio curve are divided into four curve segments, I1I2, I2I3, I3I4 and I4I5. The boundary points of the four curve segments are recorded as I1, I2, I3, I4 and I5 respectively. The two positions where the curvature is less than the minimum curvature ρmin without root cutting are the I2I3 and I4I5 segments. The angle corresponding to the I2I3 segment is [θ1, θ2-θ1], and the angle corresponding to the I4I5 segment is [θ3, θ4-θ3]. The Fourier function is used to refit the I1I2 segment and the I3I4 segment respectively, and the polynomial curve is used to re-transition and connect them to form the I2I3 segment and the I4I5 segment respectively, to ensure that the slopes of the I1I2 segment transmission ratio curve and the I2I3 segment transmission ratio curve are equal at the intersection I2, and the slopes of the I2I3 segment and the I3I4 segment are equal at the intersection I3. The expression of the entire quadratic unequal amplitude transmission ratio curve is: In formula (1), θ is the angular displacement of the active non-circular gear, a 11 , b 11 、c 11 、w 11 、a 22 , b 22 、c 22 and w 22 are the coefficients of the Fourier function, which are known quantities after the I1I2 segment and the I3I4 segment are fitted; a1, b1, c1, d1, a2, b2, c2 and d2 are the coefficients of the polynomial curve, which are the quantities to be determined; Step 2: Since the slope of the I1I2 segment at point I2 and the slope of the I3I4 segment at point I3 of the quadratic unequal amplitude transmission ratio curve are equal to the slope of the I2I3 segment at point I2 and the slope at point I3 respectively, the slope of the I1I2 segment at point I2 is the left derivative i′ at point I2 - (θ1), the slope of the I2I3 segment at point I3 is the left derivative i′ at point I3 - (θ2), the slope of the I2I3 segment at point I2 is the right derivative i′ at point I2 + (θ1), the slope of segment I3I4 at point I3 is the right derivative i′ at point I3 + (θ2), we get i′ + (θ1)=i′ - (θ1) and i′ - (θ2) = i′ + (θ2), and then we get: Because the I2I3 transition curve passes through points I2 and I3, combined with equation (2), the coefficients a1, b1, c1 and d1 in the I2I3 segment curve equation are solved; Similarly, since the slope of the I3I4 segment at point I4 and the slope of the I4I5 segment at point I5 of the quadratic unequal amplitude transmission ratio curve are respectively equal to the slope of the I4I5 segment at point I4 and the slope of the I1I2 segment at point I1, and since the I4I5 transition curve passes through points I4 and I5, the coefficients a2, b2, c2 and d2 in the equation of the I4I5 segment curve are obtained by solving; Then, the pitch curves of the driving non-circular gear and the driven non-circular gear are determined by the expression of the quadratic unequal amplitude transmission ratio curve and the given center distance, as follows: The angular displacement of the driven non-circular gear have to: For a given center distance l, according to the transmission ratio i, the angular displacement θ of the driving non-circular gear and the angular displacement θ of the driven non-circular gear c , find the pitch curve radius r of the driving non-circular gear and the pitch curve radius r of the driven non-circular gear c : Then find the coordinates (x a ,y a ) and the driven non-circular gear pitch curve coordinates (x c ,y c ): Step 3: A cycloid tooth profile is used for the tooth top part outside the pitch curve of the driving non-circular gear and the tooth root part inside the pitch curve of the driven non-circular gear, and an involute tooth profile is used for the tooth root part inside the pitch curve of the driving non-circular gear and the tooth top part outside the pitch curve of the driven non-circular gear; then, the tooth top profile of the driving non-circular gear is connected with the tooth root profile, and the tooth top profile of the driven non-circular gear is connected with the tooth root profile; finally, the connecting segments at both ends of the involute tooth profile and the cycloid tooth profile are replaced by a line segment tangent to both the involute tooth profile and the cycloid tooth profile, thereby forming a combined involute and cycloid tooth profile of the driving non-circular gear and the driven non-circular gear.

2. The method for generating a tooth profile of a combined involute and cycloid tooth profile according to claim 1, characterized in that: The critical relationship for undercutting of non-circular gears is as follows: Where: m0 is the module of the involute profile circular gear tool for solving the non-circular gear profile; h a * is the tooth top height coefficient of the involute tooth profile circular gear cutter; α0 is the pressure angle of the involute tooth profile circular gear cutter; z0 is the number of teeth of the involute tooth profile circular gear cutter; z0 is determined according to the transmission ratio at a certain position of the non-circular gear transmission ratio curve and the pitch curve of the active non-circular gear of the tooth profile to be solved or the radial direction of the driven non-circular gear at this position.

3. The method for generating a tooth profile of a combined involute and cycloid tooth profile according to claim 2, characterized in that: The analytical method for generating involute tooth profile is as follows: Coordinates of the center of curvature of the driving noncircular gear pitch curve or the driven noncircular gear pitch curve: in, (x, y) is the active non-circular gear pitch curve coordinate (x a ,y a ) or the driven non-circular gear pitch curve coordinate (x c ,y c ); Pitch radius of involute profile gear cutter: When the pitch circle of the involute gear profile circular cutter rolls along the active noncircular gear pitch curve or the driven noncircular gear pitch curve without sliding, the curvature center of the active noncircular gear pitch curve or the driven noncircular gear pitch curve, the meshing point of the active noncircular gear pitch curve or the driven noncircular gear pitch curve with the pitch circle of the involute gear profile circular cutter, and the rotation center I of the involute gear profile circular cutter are collinear; when the active noncircular gear pitch curve or the driven noncircular gear pitch curve is convex, the coordinates of the rotation center I of the involute gear profile circular cutter (x I ,y I ): Where ρ is the radius of curvature of the active noncircular gear pitch curve or the driven noncircular gear pitch curve at the point of meshing with the pitch circle of the involute profile circular gear cutter; When the pitch curve of the active non-circular gear or the pitch curve of the driven non-circular gear is concave, the coordinate of the rotation center I of the involute gear profile circular gear cutter is obtained according to the three-point collinearity property: Involute angle θ of involute profile circular gear cutter k : i k =tan(α0)-α0 (11) Base circle radius r of involute tooth profile circular gear cutter b : r b =r d cos(α0) (12) Assume that the arc length of the involute gear cutter pitch circle rolling from the initial position M1 along the active noncircular gear pitch curve or the driven noncircular gear pitch curve to the position M2 is ds, then ∠M1IM2 is the rotation angle of the involute profile circular gear cutter Suppose that after the pitch circle of the involute gear tool rolls for an arc length of ds, the meshing point between the involute of the involute gear tool and the pitch curve of the active non-circular gear or the tooth profile of the driven non-circular gear is S, and the intersection point between the tangent of the base circle of the involute gear tool passing through the tooth profile meshing point S and the base circle of the involute gear tool is recorded as N, then ∠M2IN=α0, and since after the pitch circle of the involute gear tool rolls for an arc length of ds, Where M is the involute starting point of the involute profile circular gear cutter. According to equations (11), (12), and (13), the length of the SN segment is obtained as follows: In ΔSIN, according to the Pythagorean theorem and equations (12) and (14), the length of the SI line segment is obtained as: The pressure angle of the tooth profile meshing point S is obtained in ΔSIN: According to formula (16), ∠SIM2 is obtained as: ∠SIM2=a s -a0 (17) In ΔSIM2, the coordinates of the pitch curve meshing point M2 and the coordinates of the rotation center I of the involute tooth profile circular gear tool are known. Therefore, according to equations (8), (10), (15) and (17), the coordinates of the tooth profile engagement point S relative to the tool rotation center I are obtained: The matrix equation is: According to formula (18), the coordinates of the tooth profile meshing point S are obtained: After the coordinates of each tooth profile meshing point are solved, all tooth profile meshing points are connected in sequence to obtain the root part tooth profile within the pitch curve of the active non-circular gear or the top part tooth profile outside the pitch curve of the driven non-circular gear.

4. The method for generating a tooth profile of a combined involute and cycloid tooth profile according to claim 1, 2 or 3, characterized in that: The analytical method for generating cycloid tooth profile is as follows: The pitch circle of the cycloid tooth profile circular gear tool rolls purely around the pitch curve of the active non-circular gear or the driven non-circular gear. The center coordinates of the pitch circle of the cycloid tooth profile circular gear tool are shown in formula (20): In the formula, r g is the pitch radius of the cycloid gear cutter, f is the determination coefficient of the pitch curve concavity of the active non-circular gear or the driven non-circular gear, (x q ,y q ) is the curvature center coordinate of the active noncircular gear pitch curve or the driven noncircular gear pitch curve, (x, y) is the coordinate of the active noncircular gear pitch curve (x a ,y a ) or the driven non-circular gear pitch curve coordinate (x c ,y c ); The calculation formula for the rotational angular displacement θ0 of the cycloid profile circular gear cutter pitch circle doing pure rolling on the active non-circular gear or driven non-circular gear pitch curve is as follows: The cycloid tooth profile coordinates of the pitch curve of the active non-circular gear or the driven non-circular gear are shown in formula (22): After the coordinates of the meshing points of the cycloid tooth profile are solved, all the meshing points of the cycloid tooth profile are connected in sequence to obtain the tooth profile of the tooth top part outside the pitch curve of the active non-circular gear and the tooth root part inside the pitch curve of the driven non-circular gear.

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