Calculation Method of Time-Varying Meshing Stiffness and Transmission Error of Helical Gears Based on the Slice Method

Abstract

The present invention relates to a calculation method for the time-varying meshing stiffness and transmission error of helical gears based on the slicing method, comprising the following steps: discretizing the tooth model of the helical gear into multiple straight gear slices of equal thickness along the tooth width direction; calculating the time-varying meshing stiffness of each slice from the tooth tip to the tooth root; deriving the meshing stiffness from the tooth root to the tooth tip in the y direction of the meshing plane of the helical gear; discretizing the meshing plane into (n + 1)×(m + 1) meshing points and obtaining the meshing stiffness of each point; deriving the variation law of the meshing line; by accumulating and summing the meshing stiffnesses of each meshing point on the meshing line, deriving the time-varying meshing stiffness of the helical gear under one meshing period; establishing a tooth surface load balance equation through the relationship between the gear load deformation amount and the transmission error; and iteratively obtaining the transmission error of the helical gear under one meshing period by giving the allowable deviation value of the total gear load. The present invention aims to improve the calculation accuracy of the time-varying meshing stiffness and transmission error of helical gears and reduce the calculation time cost.

Description

Technical Field

[0001] The present invention relates to the technical field of mechanical dynamics, and particularly relates to a calculation method for the time-varying meshing stiffness and transmission error of helical gears based on the slicing method. Background Art

[0002] Currently, helical gears are widely used in the mechanical industry due to their advantages such as strong meshing performance, high adaptability of contact ratio, and smooth transmission of load-carrying capacity. In practical applications, noise is an important indicator reflecting the meshing state, vibration characteristics, and operating effects of gears.

[0003] As one of the main excitation sources of noise, the calculation accuracy of both the transmission error and the time-varying meshing stiffness is an important criterion for gear grading and standards.

[0004] In traditional transmission error calculation methods, the method of calculating the transmission error through the complete meshing line length has exposed many defects in terms of calculation accuracy and reliability, and problems such as cumbersome calculation and poor applicability also need to be solved urgently. Summary of the Invention

[0005] Aiming at the above problems existing in the prior art, the technical problem to be solved by the present invention is: how to reduce the calculation amount of the time-varying meshing stiffness and transmission error of helical gears and improve the calculation accuracy of both.

[0006] To solve the above technical problem, the present invention adopts the following technical solution: A calculation method for the time-varying meshing stiffness and transmission error of helical gears based on the slicing method, including the following steps:

[0007] S100: Select the tooth model of any helical gear and discretize it into N equally thick spur gear slices along the tooth width direction;

[0008] S200: Calculate the time-varying meshing stiffness of each spur gear slice from the tooth tip to the tooth root. The tooth tip to the tooth root is in the vertical direction and perpendicular to the tooth width direction. One spur gear slice corresponds to one α 1 , and the calculation expression for the time-varying meshing stiffness of a single spur gear slice from the tooth tip to the tooth root is as follows:

[0009]

[0010]

[0011]

[0012]

[0013]

[0014]

[0015] Among them, is the bending stiffness of the driving gear slice; is the bending stiffness of the driven gear slice; is the shear stiffness of the driving gear slice; is the shear stiffness of the driven gear slice; is the radial compression stiffness of the driving gear slice; is the radial compression stiffness of the driven gear slice; k h is the Hertz contact stiffness of the master-slave gear slice; is the body stiffness of the driving gear slice; is the body stiffness of the driven gear slice; when i is p, it represents the driving gear, and when i is g, it represents the driven gear; α 1 is the angle between the meshing load F of the spur gear slice and the Y-axis; α 5 is the angle corresponding to the root circle on the X-axis; α 2 is the half angle corresponding to the base circle; α is the angle corresponding to different positions on the tooth surface on the X-axis; r b is the base circle radius of the tooth; E is the elastic modulus of the gear material; b 0 is the effective tooth width of the master-slave spur gear slice; v is the Poisson's ratio k h is the Hertz contact stiffness of the sliced gear; are respectively the distances between the points on the root circle of the driving gear slice and the driven gear slice on the tooth symmetry line to the intersection point of the meshing force action line and the tooth symmetry line; are respectively the arc lengths corresponding to a single tooth on the root circle of the driving gear slice and the driven gear slice; L*, M*, P*, Q* are coefficients; k total (α 1 ) is the meshing stiffness of the sliced gear with respect to the variable α 1 ;

[0016] S300: Combine the time-varying meshing stiffness from the tooth tip to the tooth root of all spur gear slices to obtain the meshing stiffness from the tooth tip to the tooth root in the Y direction of the helical gear meshing plane. The expression of this meshing stiffness is as follows:

[0017]

[0018] Among them, k total (y) is the meshing stiffness of the sliced gear with respect to the coordinate y; y is the tooth profile direction coordinate in the meshing plane;

[0019] S400: Preset to divide the meshing plane into n equal parts along the Y direction, and then discretize the meshing plane into (n + 1) × (m + 1) meshing points. Use formula (7) to calculate the meshing stiffness of each meshing point. The calculation expression of the meshing stiffness of the meshing point is as follows:

[0020] h = -1 / 2ε α P bt + j(ε α P bt / n), (1 ≤ j ≤ n + 1)

[0021] k ij = k total (h), (1 ≤ i ≤ m + 1); (8)

[0022] Among them, h is the coordinate value of any meshing point in the y direction of the meshing plane; i is the i-th equal division in the X-axis direction; j is the j-th equal division in the Y-axis direction; k ij is the meshing stiffness of any one of the (n + 1) × (m + 1) meshing points; n is the n equal divisions of the Y-axis direction of the meshing plane; m is the m equal divisions of the X-axis direction of the meshing plane; ε α is the face contact ratio; P bt is the base pitch;

[0023] S500: According to the change law of the meshing line on the meshing plane of the helical gear, the change law of the meshing stiffness of the meshing line with the length of the meshing line is obtained, and the expression of the change law is as follows:

[0024]

[0025] Among them, K gl is the expression of the change law of the meshing line in three states during the meshing process of the helical gear; ε β represents the axial contact ratio; y 0 represents the intersection point of the meshing line and the Y-axis;

[0026] S600: Accumulate and sum the meshing stiffness of each meshing point on the meshing line to calculate the time-varying meshing stiffness of the helical gear in one meshing period:

[0027] According to formula (9), the expression of the time-varying meshing stiffness of a single tooth of the helical gear at a certain moment is as follows:

[0028]

[0029] Among them, K dc (j) is the expression after equivalent substitution of the expression of the change law of the meshing line in three states during the meshing process of the helical gear, where 1 ≤ i ≤ m + 1;

[0030] The total meshing stiffness of the helical gear at a certain moment is calculated according to formula (10) and can be expressed by the following formula:

[0031]

[0032] Among them, K(j) is the total meshing stiffness of the helical gear at a certain moment, that is, the time-varying meshing stiffness of the helical gear;

[0033] S700: Calculate the transmission error δ, and the specific steps are as follows:

[0034] S710: Calculate the load balance equation of the helical gear tooth surface, and the expression is as follows:

[0035]

[0036] Among them, F is the load on the helical gear tooth surface, and u ij is the loaded deformation amount at each meshing point, and u ij has the following expression:

[0037] u ij = δ - ε ij ; (13)

[0038] Among them, δ is the transmission error; ε ij is the initial clearance;

[0039] S720: In formula (9), for the contact point ij, if δ > ε ij is satisfied, this point is in contact, and u ij takes a positive value, otherwise u ij takes 0; the specific steps to calculate the transmission error δ are as follows:

[0040] (1) Let k = 1, and give the initial value of the transmission error δ (1) ;

[0041] (2) Judge the deformation δ (k) - ε ij at each meshing point ij of the helical gear. If δ (k) - ε ij is less than 0, then make it equal to 0;

[0042] (3) Obtain the total load F (k) of the helical gear according to formula (9);

[0043] (4) Preset the deviation value threshold γ, and judge whether the deviation value between F (k) and the actual helical gear load P is less than the allowable γ, that is, judge whether |F (k) - P| < γ holds; if |F (k) - P| < γ does not hold, then let δ (k+1) = δ (k) - (F (k) - P) / C k , and k = k + 1, and return to step (2); if |F (k) - P| < γ holds, then the iteration stops, and output δ = δ (k), the transmission error δ is obtained;

[0044] where k is the number of iterations; δ (k+1) is the transmission error at the (k + 1)-th step; δ (k) is the transmission error at the k-th step; F (k) is the calculated meshing force at the k-th step; C k is the average meshing stiffness.

[0045] Preferably, the content of obtaining the meshing stiffness from the tooth tip to the tooth root in the Y direction of the helical gear meshing plane in S300 is as follows:

[0046] Define the functional relationship expression of α 1 and y as follows:

[0047] c = α 6 -α 5 ; (14)

[0048] y = ε α P bt sin(-π / 2 + (c - α 1 )π / c), (-1 / 2ε α P bt ≤ y ≤ 1 / 2ε α P bt ); (15)

[0049] where c is the meshing angle change range; y is the coordinate in the Y-axis direction of the tooth profile in the meshing plane; ε α is the face contact ratio; P bt is the base pitch; α 1 is the angle between the meshing load F and the Y-axis; α 5 is the angle corresponding to the tooth root circle on the X-axis; α 6 is the angle between the meshing load at the tooth tip and the Y-axis; where, α 5 ≤ α 1 ≤ α 6 ;

[0050] Combined with formula (6), the meshing stiffness k total (y) from the tooth tip to the tooth root in the Y direction of the helical gear meshing plane is obtained.

[0051] Preferably, in S400, the meshing plane is discretized into (n + 1) × (m + 1) meshing points, specifically:

[0052] The meshing plane is cut into n equal parts along the Y direction, and the distance of each equal part is:

[0053] Δy = ε α P bt / n; (16)

[0054] Among them, Δy is the average splitting distance of the meshing plane along the y direction;

[0055] Along the X-axis direction of the meshing plane, the tooth width is divided into m equal parts, and the width of each equal part is b / m. The calculation expression of m is as follows:

[0056]

[0057] Among them, β represents the helix angle of the helical gear, and b is the tooth width of the helical gear;

[0058] According to the obtained n and m, the meshing plane is discretized into (n + 1) × (m + 1) meshing points.

[0059] Compared with the prior art, the present invention has at least the following advantages:

[0060] The present invention provides a calculation algorithm for the time-varying meshing stiffness and transmission error of helical gears based on the slicing method. When solving the time-varying meshing stiffness of helical gears, the calculation method of accumulating and summing the meshing stiffness of each meshing point on the meshing line is used instead of the calculation method of calculating the integral of the meshing line length, which greatly saves the time cost when solving the time-varying meshing stiffness of helical gears, and has very high accuracy. At the same time, it can also output the transmission error that meets the accuracy requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 It is a schematic flowchart of the implementation of the present invention.

[0062] Figure 2 It is a schematic diagram of the discretization of the helical gear of the present invention along the tooth width direction.

[0063] Figure 3 It is a schematic diagram of the calculation parameters of the meshing stiffness of a single spur gear slice.

[0064] Figure 4 It is a schematic diagram of the meshing plane of the helical gear.

[0065] Figure 5 It is a schematic diagram of the change law of the single-tooth meshing line of the helical gear.

[0066] Figure 6 It is a comparison diagram of the ROMAX finite element analysis and the calculation results of the method of the present invention when the working torque T = 10 N·m.

[0067] Figure 7 It is a comparison diagram of the ROMAX finite element analysis and the calculation results of the algorithm of the present invention when the working torque T = 100 N·m.

[0068] Figure 8 It is a comparison diagram of the ROMAX finite element analysis and the calculation results of the algorithm of the present invention when the working torque T = 200 N·m. DETAILED DESCRIPTION OF THE INVENTION

[0069] The present invention will be further described in detail below.

[0070] The present invention discloses a calculation method for the time-varying meshing stiffness and transmission error of helical gears based on the slicing method. This method improves the calculation methods for the meshing line length, time-varying meshing stiffness, and transmission error of helical gears in three-dimensional space. This method not only has a low computational time cost but also achieves high-precision calculations of the time-varying meshing stiffness and transmission error of helical gears.

[0071] See Figures 1 - 5 , a calculation method for the time-varying meshing stiffness and transmission error of helical gears based on the slicing method, includes the following steps:

[0072] S100: Select the tooth model of any helical gear and discretize it into N straight gear slices of equal thickness along the tooth width direction; when discretizing the tooth model of the helical gear into slices, the slices to be discretized are small enough, and each slice can be regarded as a straight gear, see Figure 2 ;

[0073] S200: Calculate the time-varying meshing stiffness of each straight gear slice from the tooth tip to the tooth root. The tooth tip to the tooth root is in the vertical direction and perpendicular to the tooth width direction. One straight gear slice corresponds to an α 1 , and based on the energy method, a function regarding the variable α 1 can be obtained. The calculation expression for the time-varying meshing stiffness of a single straight gear slice from the tooth tip to the tooth root is as follows:

[0074]

[0075]

[0076]

[0077]

[0078]

[0079]

[0080] Among them, is the bending stiffness of the driving gear slice; is the bending stiffness of the driven gear slice; is the shear stiffness of the driving gear slice; is the shear stiffness of the driven gear slice; is the radial compression stiffness of the driving gear slice; is the radial compression stiffness of the driven gear slice; k h is the Hertz contact stiffness of the driving and driven gear slices; is the body stiffness of the driving gear slice; is the body stiffness of the driven gear slice; when i is p, it represents the driving gear, and when i is g, it represents the driven gear; α 1 is the angle between the meshing load F of the spur gear slice and the Y-axis; α 5 is the angle corresponding to the root circle on the X-axis; α 2 is the half angle corresponding to the base circle; α is the angle corresponding to different positions on the tooth surface on the X-axis; r b is the base circle radius of the tooth; E is the elastic modulus of the gear material; b 0 is the effective tooth width of the driving and driven spur gear slices; v is the Poisson's ratio k h is the Hertz contact stiffness of the said sliced gear; are respectively the distances between the points on the root circle of the driving gear slice and the driven gear slice on the tooth symmetry line to the intersection point of the meshing force action line and the tooth symmetry line; are respectively the arc lengths corresponding to a single tooth on the root circle of the driving gear slice and the driven gear slice; L*, M*, P*, Q* are coefficients; k total (α 1 ) is the meshing stiffness of the said sliced gear with respect to the variable α 1 ;

[0081] S300: Combine the time-varying meshing stiffness from the tooth tip to the tooth root of all spur gear slices to obtain the meshing stiffness from the tooth tip to the tooth root in the Y direction of the helical gear meshing plane. This meshing stiffness is a function of the variable y, and the expression of this meshing stiffness is as follows:

[0082]

[0083] where k total (y) is the meshing stiffness of the said sliced gear with respect to the coordinate y; y is the tooth profile direction coordinate in the meshing plane;

[0084] The content of obtaining the meshing stiffness from the tooth tip to the tooth root in the Y direction of the helical gear meshing plane in the said S300 is specifically:

[0085] Define the functional relationship expression between α 1 and y as follows:

[0086] c = α 6 -α 5 ; (14)

[0087] y = ε α P bt sin(-π / 2 + (c - α 1 )π / c), (-1 / 2ε α P bt ≤ y ≤ 1 / 2ε αP bt ); (15)

[0088] where c is the variation range of the meshing angle; y is the coordinate in the Y-axis direction of the tooth profile in the meshing plane; ε α is the face contact ratio; P bt is the base pitch; α 1 is the angle between the meshing load F and the Y-axis; α 5 is the angle corresponding to the root circle at the X-axis; α 6 is the angle between the meshing load at the tooth tip and the Y-axis; where, α 5 ≤α 1 ≤α 6 ;

[0089] Combined with formula (6), the meshing stiffness k total (y) from the tooth tip to the tooth root in the Y direction of the helical gear meshing plane is obtained.

[0090] Draw the distribution diagram of the meshing line on the helical gear meshing plane. During the meshing process of the helical gear, when the contact ratio is between 2.0 and 3.0, the schematic diagram of the meshing line on the meshing plane is as Figure 4 shown. The solid rectangle A 1 A 2 C 2 C 1 in the figure is the meshing plane. Point C 2 and point A 1 are respectively the starting point and the ending point of a meshing cycle. The straight line A 1 A 2 and C 2 C 1 are respectively the front end face and the rear end face of the helical gear meshing plane. The straight lines 1, 2, and 3 are the meshing lines on three pairs of teeth at a certain moment;

[0091] Represent the helical gear meshing plane on the coordinate axes o-xy. The change of the meshing stiffness along the y-axis direction on the meshing plane is the change process from the meshing line A 2 C 2 at the tooth root of the driving gear to the meshing line A 1 C 1 at the tooth tip. The meshing stiffness of the spur gear slice also changes with the different positions from the tooth root to the tooth tip (it is a function of the variable α 1 ). Therefore, by expressing y with an expression containing the variable α 1 , the time-varying meshing stiffness along the Y-axis direction on the helical gear meshing plane can be deduced.

[0092] S400: Preset to divide the meshing plane into n equal parts along the Y direction, and then discretize the meshing plane into (n + 1)×(m + 1) meshing points. Use formula (7) to calculate the meshing stiffness of each meshing point. The calculation expression of the meshing stiffness of the meshing point is as follows:

[0093] h = -1 / 2ε α P bt +j(ε α P bt / n),(1 ≤ j ≤ n + 1)

[0094] k ij =k total (h),(1 ≤ i ≤ m + 1); (8)

[0095] Among them, h is the coordinate value of any meshing point in the y direction of the meshing plane; i is the i-th equal part in the X-axis direction; j is the j-th equal part in the Y-axis direction; k ij is the meshing stiffness of any point among the (n + 1)×(m + 1) meshing points; n is to divide the meshing plane into n equal parts in the Y-axis direction; m is to divide the meshing plane into m equal parts in the X-axis direction; ε α is the face contact ratio; P bt is the base pitch;

[0096] In the S400, the meshing plane is discretized into (n + 1)×(m + 1) meshing points, specifically:

[0097] Divide the meshing plane into n equal parts along the Y direction, and the distance of each equal part is:

[0098] Δy = ε α P bt / n; (16)

[0099] Among them, Δy is the average division distance of the meshing plane along the y direction;

[0100] Along the X-axis direction of the meshing plane, divide the tooth width into m equal parts, and the width of each equal part is b / m. The calculation expression of m is as follows:

[0101]

[0102] Among them, β represents the helix angle of the helical gear, and b is the tooth width of the helical gear;

[0103] Discretize the meshing plane into (n + 1)×(m + 1) meshing points according to the obtained n and m.

[0104] S500: According to the change law of the meshing line on the meshing plane of the helical gear, obtain the change law of the meshing stiffness of the meshing line with the length of the meshing line. The change law expression is as follows:

[0105]

[0106] Among them, K gl is the expression of the variation law of the line of action in three states of the meshing process of the helical gear; ε β represents the face contact ratio; y 0 represents the intersection point of the line of action and the Y-axis;

[0107] In further describing the evolution law of the helical gear line of action, as Figure 5 shown, during the meshing process of the helical gear, the meshing process of a certain tooth is as follows: starting from the tooth root C 2 (at the tooth tip of the driven gear) on one end face of the driving gear, after entering the meshing plane, the contact line changes from short to long, which is the area A 2 B 1 C 2 C′ 2 in the figure; then it enters the area C′ 1 A 2 C′ 2 C 1 , and the contact line always remains the longest and unchanged; finally, when entering the area A 1 C′ 1 C 1 B 2 , the contact line changes from long to short again, and finally separates completely at the tooth tip A 1 (tooth root of the driven gear) on the other end face of the driving gear. The triangles A 2 B 1 C 2 and the triangles A 1 C 1 B 2 in the figure are the extended spaces for convenient calculation, and there is no actual line of action in this space. In the above three regions, the meshing stiffness of the line of action changes with the length of the line of action, and the variation law is obtained based on this change.

[0108] S600: Accumulate and sum the meshing stiffness of each meshing point on the line of action to calculate the time-varying meshing stiffness of the helical gear in one meshing cycle:

[0109] According to formula (9), the expression of the time-varying meshing stiffness of a single tooth of the helical gear at a certain moment is as follows:

[0110]

[0111] Among them, K dc (j) is the expression after equivalent substitution of the expression of the variation law of the line of action in three states of the meshing process of the helical gear, where 1 ≤ i ≤ m + 1;

[0112] The total meshing stiffness of a helical gear at a certain moment is calculated according to Equation (10) and can be expressed by the following formula:

[0113]

[0114] Among them, K(j) is the total meshing stiffness of the helical gear at a certain moment, that is, the time-varying meshing stiffness of the helical gear; for the case where the total contact ratio is between 2.0 and 3.0, during the meshing process of the helical gear teeth, multiple teeth will participate in meshing. That is to say, on the meshing plane, there will be two or three meshing lines at the same time, and the difference between adjacent teeth in the o-xy coordinate system is P bt ;

[0115] S700: Calculate the transmission error δ. By using the relationship between the gear load deformation and the transmission error, establish the tooth surface load balance equation. During the loading process of the helical gear, contact deformation will occur at each meshing point on the meshing line, and the product of the meshing stiffness at each point and its load deformation is the total load borne by the gear. The specific steps are as follows:

[0116] S710: Calculate the tooth surface load balance equation of the helical gear. The expression is as follows:

[0117]

[0118] Among them, F is the tooth surface load of the helical gear, u ij is the load deformation at each meshing point, and the expression of u ij is as follows:

[0119] u ij =δ - ε ij ; (13)

[0120] Among them, δ is the transmission error; ε ij is the initial clearance;

[0121] S720: By specifying the allowable deviation value between the calculated total load and the actual total load of the gear, iteratively find the transmission error of the helical gear under one meshing cycle; in Equation (9), for the contact point ij, if δ > ε ij is satisfied, the point is in contact, and u ij takes a positive value, otherwise u ij takes 0; the specific steps to calculate the transmission error δ are as follows:

[0122] (1) Let k = 1, and specify the initial value of the transmission error δ (1) ;

[0123] (2) Judge the magnitude of the deformation δ (k) -ε ij at each meshing point ij of the helical gear. If δ (k) -ε ij is less than 0, then make it equal to 0;

[0124] (3) Obtain the total load F of the helical gear according to formula (9). (k) ;

[0125] (4) Preset the deviation value threshold γ, and judge whether the deviation value between F (k) and the actual load P of the helical gear is less than the allowable γ, that is, judge whether |F (k) - P| < γ holds; if |F (k) - P| < γ does not hold, then let δ (k+1) = δ (k) - (F (k) - P) / C k , and k = k + 1, and return to step (2); if |F (k) - P| < γ holds, then the iteration stops, and output δ = δ (k) , that is, the transmission error δ is obtained;

[0126] where k is the number of iterations; δ (k+1) is the transmission error at the (k + 1)-th step; δ (k) is the transmission error at the k-th step; F (k) is the calculated meshing force at the k-th step; C k is the average meshing stiffness.

[0127] Embodiment 1: The parameters and material properties of the selected helical gear are shown in Table 1 as follows:

[0128] Table 1 Helical gear parameters

[0129] Gear parameters Symbol Driving gear / Driven gear Number of teeth z1 / z2 53 / 63 Module (mm) mn 2.0 Pressure angle (°) αp / αg 14.5 Helix angle (°) βp / βg 27 Mass (kg) mp / mg 1.586 / 2.158 Tooth width (mm) b 23.2

[0130] According to the established helical gear load-bearing contact model, the transmission error analysis is carried out. In this embodiment, the correctness and accuracy of the algorithm of the present invention are verified by comparing with the ROMAX finite element results. The comparison and verification are carried out under three working torques T of 10 N·m, 100 N·m, and 200 N·m respectively. The comparison results of the transmission error are respectively as Figure 6 、 Figure 7 、 Figure 8 shown.

[0131] It can be seen from the comparison that the change trend of the transmission error of the algorithm of the present invention is to increase first. After increasing to 3 / 10 of the meshing cycle, it gradually decreases, and slowly increases after 8 / 10 of the meshing cycle. The change trend of the algorithm of the present invention is almost the same as that of the ROMAX finite element within the same meshing cycle, thus verifying the correctness of the gear meshing situation of the algorithm of the present invention:

[0132] When the working torque T is 10 N·m, the peak-to-peak value of the transmission error obtained by the algorithm of the present invention is 0.019 μm, and the peak-to-peak value of the transmission error obtained by ROMAX finite element is 0.022 μm, and the maximum difference between the algorithm of the present invention and the ROMAX finite element error is 0.025 μm;

[0133] When the working torque T is 100 N·m, the peak-to-peak value of the transmission error obtained by the algorithm of the present invention is 0.184 μm, and the peak-to-peak value of the transmission error obtained by ROMAX finite element is 0.208 μm, and the maximum difference between the algorithm of the present invention and the ROMAX finite element error is 0.092 μm;

[0134] When the working torque T is 200 N·m, the peak-to-peak value of the transmission error obtained by the algorithm of the present invention is 0.361 μm, and the peak-to-peak value of the transmission error obtained by ROMAX finite element is 0.394 μm, and the maximum difference between the algorithm of the present invention and the ROMAX finite element error is 0.089 μm;

[0135] Within the working torque range, the differences in the transmission error and the peak-to-peak value of the transmission error are both below the micron level, thus verifying the accuracy of the gear meshing condition of the algorithm of the present invention.

[0136] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the purpose and scope of the technical solutions of the present invention, and they should all be covered by the scope of the claims of the present invention.

Claims

1. A calculation method for the time-varying meshing stiffness and transmission error of helical gears based on the slicing method, characterized in that: It includes the following steps: S100: Select the tooth model of any helical gear and discretize it into N straight gear slices with equal thickness along the tooth width direction; S200: Calculate the time-varying mesh stiffness of each spur gear slice from the tooth tip to the tooth root, where the tooth tip to the tooth root is in the vertical direction and perpendicular to the tooth width direction, and one spur gear slice corresponds to one α 1 ; S300: Combine the time-varying meshing stiffness from the tooth tip to the tooth root of all straight gear slices to obtain the meshing stiffness from the tooth tip to the tooth root in the Y direction of the meshing plane of the helical gear; S400: Define that the meshing plane is cut into n equal parts along the Y direction, and then the meshing plane is discretized into (n + 1)×(m + 1) meshing points, and calculate the meshing stiffness of each meshing point; S500: According to the change law of the meshing line on the meshing plane of the helical gear, obtain the change law of the meshing stiffness of the meshing line with the length of the meshing line; S600: Accumulate and sum the meshing stiffness of each meshing point on the meshing line to calculate the time-varying meshing stiffness of the helical gear in one meshing period; S700: Calculate the transmission error δ, and the specific steps are as follows: S710: Calculate the tooth surface load balance equation of the helical gear, and the expression is as follows: Among them, F is the tooth surface load of the helical gear, β represents the helix angle of the helical gear, Δy is the average splitting distance of the meshing plane along the y direction, k ij is the meshing stiffness of any one of the (n + 1)×(m + 1) meshing points, i is the i-th equal division in the X-axis direction, j is the j-th equal division in the Y-axis direction, n is the n equal divisions of the meshing plane in the Y-axis direction, m is the m equal divisions of the meshing plane in the X-axis direction, ε α is the face contact ratio, P bt is the base pitch, ε β represents the axial contact ratio, u ij is the loaded deformation amount of each meshing point, and the expression of u ij is as follows: u ij = δ - ε ij ;(13) where δ is the transmission error; ε ij is the initial clearance; S720: For the contact point ij, if δ > ε ij is satisfied, the point is in contact, and u ij takes a positive value; otherwise, u ij takes 0. The specific steps for calculating the transmission error δ are as follows: (1) Let k = 1 and given the initial value δ of the transmission error (1) ; (2) Judge the magnitude of the deformation δ (k) -ε ij at the meshing point ij of each helical gear. If δ (k) -ε ij is less than 0, let it be equal to 0; (3) Calculate the total load F of the helical gear (k) ; (4) Preset the deviation value threshold γ, and judge F (k) Whether the deviation value from the actual helical gear load P is less than the allowable γ, that is, judge whether |F (k) -P| < γ holds; if |F (k) -P| < γ does not hold, then let δ (k+1) = δ (k) -(F (k) -P) / C k , and k = k + 1, and return to step (2); if |F (k) -P| < γ holds, then the iteration is terminated, and output δ = δ (k) , that is, the transmission error δ is obtained; where k is the number of iterations; δ (k+1) is the transmission error at the (k + 1)-th step; δ (k) is the transmission error at the k-th step; F (k) is the calculated meshing force at the k-th step; C k is the average meshing stiffness.

2. A calculation method for the time-varying meshing stiffness and transmission error of helical gears based on the slicing method according to claim 1, characterized in that: The calculation expression for calculating the time-varying meshing stiffness from the tooth tip to the tooth root of each straight gear slice in S200 is as follows: Among them, is the bending stiffness of the driving gear slice; is the bending stiffness of the driven gear slice; is the shear stiffness of the driving gear slice; is the shear stiffness of the driven gear slice; is the radial compression stiffness of the driving gear slice; is the radial compression stiffness of the driven gear slice; k h is the Hertz contact stiffness of the master-slave gear slice; is the body stiffness of the driving gear slice; is the body stiffness of the driven gear slice; when i is p, it represents the driving gear, and when i is g, it represents the driven gear; α 1 is the angle between the meshing load F of the spur gear slice and the Y-axis; α 5 is the angle corresponding to the root circle on the X-axis; α 2 is the half angle corresponding to the base circle; α is the angle corresponding to different positions on the tooth surface on the X-axis; r b is the base circle radius of the tooth; E is the elastic modulus of the gear material; b 0 is the effective tooth width of the master-slave spur gear slice; v is the Poisson's ratio k h is the Hertz contact stiffness of the slice gear; are respectively the distances between the points on the root circle of the driving gear slice and the driven gear slice on the tooth symmetry line and the intersection point of the meshing force action line and the tooth symmetry line; are respectively the arc lengths corresponding to a single tooth on the root circle of the driving gear slice and the driven gear slice; L*, M*, P*, Q* are coefficients; k total (α 1 ) is the meshing stiffness of the slice gear with respect to the variable α 1 .

3. A calculation method for the time-varying meshing stiffness and transmission error of helical gears based on the slicing method according to claim 2, characterized in that: The specific content of obtaining the meshing stiffness from the tooth tip to the tooth root in the Y direction of the meshing plane of the helical gear in S300 is: Define α 1 The functional relationship expression with y is as follows: c＝α 6 -α 5 ； (14) y = ε α P bt sin(-π / 2 + (c - α 1 )π / c), (-1 / 2ε α P bt ≤ y ≤ 1 / 2ε α P bt );(15) Among them, c is the change range of the meshing angle; y is the coordinate in the Y-axis direction of the tooth profile in the meshing plane; ε α is the face contact ratio; P bt is the base pitch; α 1 is the angle between the meshing load F and the Y-axis; α 5 is the angle corresponding to the root circle at the X-axis; α 6 is the angle between the meshing load at the tooth tip and the Y-axis; among them, α 5 ≤α 1 ≤α 6 ; Combined with formula (6), the meshing stiffness \(k(y)\) from the tooth tip to the tooth root in the Y direction of the helical gear meshing plane is obtained, and the calculation expression is as follows: total (y), the calculation expression is as follows: where k total (y) is the meshing stiffness of the slice gear with respect to the coordinate y; y is the tooth profile direction coordinate in the meshing plane.

4. A calculation method for the time-varying meshing stiffness and transmission error of helical gears based on the slicing method according to claim 3, characterized in that: In S400, the meshing plane is discretized into (n + 1)×(m + 1) meshing points, and the formula (7) is used to calculate the meshing stiffness of each meshing point. The calculation process is as follows: The meshing plane is cut into n equal parts along the Y direction, and the distance of each equal part is: Δy = ε α P bt / n; (16) Along the X-axis direction of the meshing plane, the tooth width is divided into m equal parts, and the width of each equal part is b / m. The calculation expression of m is as follows: where b is the tooth width of the helical gear; According to the obtained n and m, the meshing plane is discretized into (n + 1)×(m + 1) meshing points; The calculation expression of the meshing stiffness of the meshing point is as follows: h = -1 / 2ε α P bt +j(ε α P bt / n), (1 ≤ j ≤ n + 1) k ij = k total (h), (1 ≤ i ≤ m + 1); (8) where h is the coordinate value of any meshing point in the y direction of the meshing plane.

5. A calculation method for the time-varying meshing stiffness and transmission error of helical gears based on the slicing method according to claim 4, characterized in that: The calculation expression for obtaining the change law of the meshing stiffness of the meshing line with the length of the meshing line in S500 is as follows: Among them, K gl is the expression of the variation law of the line of action in three states during the meshing process of the helical gear; y 0 represents the intersection point of the line of action and the Y-axis.

6. A calculation method for the time-varying meshing stiffness and transmission error of helical gears based on the slicing method according to claim 5, characterized in that: The calculation process of calculating the time-varying meshing stiffness of the helical gear in one meshing period in S600 is as follows: According to formula (9), the expression of the time-varying meshing stiffness of a single tooth of the helical gear at a certain moment can be obtained as follows: Among them, K dc (j) is the expression after equivalent substitution of the variation law expression of the meshing line in three states of the meshing process of the helical gear, where 1 ≤ i ≤ m + 1; According to formula (10), the total meshing stiffness of the helical gear at a certain moment is calculated, and it can be expressed by the following formula: where K(j) is the total meshing stiffness of the helical gear at a certain moment, that is, the time-varying meshing stiffness of the helical gear.

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