Mathematical modeling method and error evaluation method of EPS worm gear profile

Abstract

This invention relates to the field of electric power steering systems, and provides a mathematical modeling method and error evaluation method for EPS worm gear tooth profiles. The mathematical modeling method includes: establishing O- xyz Coordinate system and O H - x H y H z H The invention employs a coordinate system to determine the zero-point position and scanning start point of characteristic points on the generatrix of the EPS worm gear or EPS worm wheel hob; it also determines the coordinate values ​​of the rotating moving points and the generatrix equation of the characteristic points on the generatrix of the EPS worm gear or EPS worm wheel hob; calculates the tooth profile curve points of the EPS worm gear or EPS worm wheel hob; determines the potential meshing points between the tooth profile curves of the EPS worm gear or EPS worm wheel hob and the EPS worm wheel; determines the transmission meshing contact line and contact range between the EPS worm gear or EPS worm wheel hob and the EPS worm wheel; and forms the G4 tooth profile curve of the EPS worm wheel. This invention enables simulation and error assessment of the EPS worm wheel tooth profile, providing an improved direction for corrective measures to reduce tooth profile errors.

Description

Technical Field

[0001] This invention relates to the field of electric power steering system technology, and in particular to a mathematical modeling method and error evaluation method for EPS worm gear tooth profile. Background Technology

[0002] Currently, EPS (Electric Power Steering) systems are widely used in automobiles, playing a decisive role in the performance and reliability of vehicle steering, as well as driving comfort and safety. Among these components, the worm gear and worm reduction mechanism is the core mechanical transmission component of the EPS system, significantly impacting the system's efficiency, reliability, and steering feel. Therefore, achieving optimal meshing between the worm gear and worm becomes a crucial technology, requiring careful design and machining of their tooth profiles.

[0003] Worms are classified into several standard types based on their tooth profile: involute worms (ZI worms) and straight-profile worms (ZN worms), among others. These, along with their corresponding worm wheels, form different types of worm drives (also known as worm wheel and worm drive systems). Theoretically, the hob parameters and shape for cutting the worm wheel must be completely identical to those of the working worm, and the milling center distance should also be the same as the transmission center distance to generate an ideal worm wheel tooth profile. The working worm (EPS worm) used in EPS systems typically uses the ZI type. Because of its small outer diameter, using this as the hob parameter for the working worm wheel (EPS worm wheel) in an EPS system results in poor hob strength and very low machining efficiency. This makes EPS worm parameters unsuitable as EPS worm wheel hob parameters. However, if the EPS worm wheel hob parameters and shape are not completely identical to those of the EPS worm, the resulting EPS worm wheel tooth profile will have design errors compared to the ideal EPS worm wheel tooth profile, making precise meshing with the EPS worm tooth profile impossible.

[0004] Currently, in the EPS system industry, the design of EPS worm gear tooth profile adopts a design method similar to that of a worm gear, which is to imitate and approximate the design by adjusting parameters. The error of EPS worm gear tooth profile is usually evaluated by processing prototypes and then testing and observing them. The evaluation process is time-consuming and expensive, and cannot provide pre-analysis conditions and evaluation benchmarks for the design. It is difficult to find the direction for improvement by relying solely on industry experience and trial and error. Summary of the Invention

[0005] This invention primarily addresses the technical problems currently faced in the EPS system industry. The design of EPS worm gear teeth often employs a similar approach to worm gears, using parameter adjustments to mimic and approximate the design. Furthermore, the evaluation of EPS worm gear tooth profile errors typically involves machining prototypes and conducting fitting tests, a process that is time-consuming, costly, and difficult to improve. This invention proposes a mathematical modeling method and error evaluation method for EPS worm gear teeth to display the distribution and movement of the meshing contact line during transmission. This method simulates and evaluates the error of the EPS worm gear teeth, providing a direction for improving correction schemes to reduce tooth profile errors.

[0006] This invention provides a mathematical modeling method for EPS worm gear tooth profile, comprising the following steps 1 to 8: Step 1, establish O- x.y.z Coordinate system and O H - x H y H z H Coordinate system; Step 2: Based on the tooth profile generation mechanism of EPS worm and EPS worm wheel hob, determine the coordinate values ​​of the initial position feature points of the generatrix and the distance from the initial position feature points to the worm gear. x O y The angle of rotation of the plane; Step 3: Determine the zero point position and scanning start point of the feature points on the EPS worm gear or EPS worm wheel hob generatrix. Step 4: Determine the coordinates of the rotating points of the characteristic points on the generatrix of the EPS worm or EPS worm wheel hob and the equation of the generatrix. Step 5: Calculate the tooth profile curve points of the EPS worm or EPS worm gear hob; Step 6: Calculate the meshing transmission relationship between the EPS worm or EPS worm wheel hob and the EPS worm wheel, and determine the potential meshing point between the tooth profile curve of the EPS worm or EPS worm wheel hob and the tooth profile curve of the EPS worm wheel. Step 7: Determine the transmission meshing contact line and contact range between the EPS worm or EPS worm wheel hob and the EPS worm wheel; Step 8, as the EPS worm gear hob rotates... i Drive EPS worm gear rotation angle f Rotation transforms the point W of the EPS worm gear G4 tooth profile curve to a fixed coordinate system. x H O H y H EPS worm gear G4 tooth profile curve point W at the zero position H This forms the G4 tooth profile curve of the EPS worm gear.

[0007] Correspondingly, the present invention also provides a method for evaluating the tooth profile error of an EPS worm gear, including the following steps 9 to 10: Step 9: Determine the reference curve for the EPS worm gear tooth profile; Step 10: Use the EPS worm gear tooth profile curve as a reference to evaluate the EPS worm gear tooth profile error.

[0008] The present invention provides a mathematical modeling method and error evaluation method for EPS worm gear tooth profile, which has the following advantages compared with the prior art: 1. According to the mathematical modeling method provided by this invention, the meshing transmission between the EPS worm or EPS worm wheel hob and the EPS worm wheel is regarded as a combination transmission of rack and pinion on the Z-section. Then, the tooth profile curve of the EPS worm wheel is obtained by using the analytical method of meshing principle. A mathematical model of the meshing transmission process of the EPS worm and EPS worm wheel can be built to show the distribution and movement state of the meshing contact line in the transmission process. The EPS worm wheel tooth profile is simulated, and the meshing state of the worm and worm wheel can be analyzed under ideal conditions. By observing the distribution and movement changes of the meshing contact line, the minimum number of meshing teeth and the tooth surface contact bearing range can be confirmed, the effectiveness of the EPS worm design parameters can be evaluated, and the workload of subsequent modifications can be reduced.

[0009] 2. According to the mathematical modeling method provided by this invention, a mathematical model of the meshing transmission process of EPS worm and EPS worm wheel can be built. Based on this, a transmission process simulation can be designed, which helps to understand the working principle of its meshing transmission, facilitates the analysis of the causes of faults, provides guidance for improvement activities, and reduces the number of test improvements and the cycle.

[0010] 3. According to the mathematical modeling method provided by this invention, a simulation model of the EPS worm gear tooth profile generation process can be built, thereby showing the influence of various parameters in the rolling process on the generated tooth profile, so as to adjust the relevant parameters appropriately to improve the process and reduce the probability of trial and error based solely on experience.

[0011] 4. The mathematical modeling method provided by this invention can be used to calculate and analyze the tooth profile error of EPS worm gears, generate ideal tooth profile curves under each section of the EPS worm gear, obtain the tooth profile error evaluation benchmark, and determine the precise meshing state.

[0012] 5. The mathematical modeling method provided by this invention can be used to calculate and analyze the tooth profile error of EPS worm gears. It can generate tooth profile curves under various sections of the EPS worm gear machined by the hob in the design, and use them as the evaluation object to perform tooth profile error analysis. This can help to improve the hob correction scheme to reduce tooth profile error and reduce the number of times and cycle of making prototypes for verification. Attached Figure Description

[0013] Figure 1 A flowchart illustrating the mathematical modeling method for the EPS worm gear tooth profile of the present invention; Figure 2 This is a schematic diagram of the EPS worm gear or EPS worm wheel hob and EPS worm wheel transmission mechanism of the present invention; Figure 3 This is a schematic diagram of the coordinate system of the EPS worm gear or EPS worm wheel hob and the EPS worm wheel transmission mechanism of the present invention; Figure 4 This is a schematic diagram illustrating the ZI tooth profile generation mechanism and the relationship between feature points on the initial position generatrix of the present invention; Figure 5 This is a schematic diagram showing the relationship between the zero point position and the scanning start point of the EPS worm gear or EPS worm wheel hob feed line L1 of the present invention. Figure 6 This is a schematic diagram of the tooth profile curve and transmission relationship between the EPS worm or EPS worm wheel hob and the EPS worm wheel of the present invention. Figure 7 is a demonstration diagram of the process of point D of the T4 tooth profile curve becoming a potential or actual engagement point on the same Z-section plane of the present invention; Figure 7(a) shows no potential engagement point; Figure 7(b) shows a potential engagement point but not engaged; Figure 7(c) shows a potential engagement point and engaged at the engagement point position; Figure 7(d) shows a potential engagement point and engaged at the zero point position; Figure 7(e) shows a potential engagement point and engaged at the node position; Figure 7(f) shows a potential engagement point and engaged at the disengagement point position; Figure 7(g) shows no potential engagement point (no potential engagement points after the disengagement point); Figure 8 For the present invention when the angle i Schematic diagram of tooth profile differences of T4 tooth profile curves on different Z-sections when =0º; Figure 9 This is a schematic diagram of the contact line and contact range between the EPS worm and the EPS worm wheel of the present invention; Figure 10 For the present invention z A demonstration diagram showing the generation process of the G4 tooth profile curve and tooth root transition curve on the Z section when 1=0; Figure 11 This is a schematic diagram of the reference and evaluated tooth profile curves of the EPS worm gear on different Z-sections according to the present invention. Figure 12 This is a schematic diagram showing the distribution of the EPS worm gear tooth profile error under evaluation in different Z-sections according to the present invention. Detailed Implementation

[0014] To make the technical problems solved by this invention, the technical solutions adopted, and the technical effects achieved clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and not intended to limit it. Furthermore, it should be noted that, for ease of description, only the parts relevant to the invention are shown in the accompanying drawings, not all of them.

[0015] A basic overview of the EPS worm, EPS worm wheel, and EPS worm wheel hob of this invention: The working worms (hereinafter referred to as EPS worms) used in EPS systems all have involute (ZI) tooth profiles. The EPS worm wheels that mesh with the EPS worms are usually machined using EPS worm wheel hobs. The tooth profile of the EPS worm wheel is determined by the type of EPS worm wheel hob used. Various types of EPS worm wheel hobs, such as involute type hobs (hereinafter referred to as ZI hobs) and normal straight profile type hobs (hereinafter referred to as ZN hobs), can all be considered as a type of worm. The EPS worm wheel hob and the EPS worm wheel cut the EPS worm wheel tooth profile during the meshing transmission process. Therefore, the EPS worm wheel hob and the EPS worm wheel can be considered as an ideal transmission combination. However, the transmission combination of the EPS worm wheel and the EPS worm is not ideal.

[0016] Assuming that the relevant parameters of the EPS worm gear hob are the same as those of the EPS worm (including the same mounting center distance), but the EPS worm gear hob tooth tip height coefficient ha0 is different... * The height of the EPS worm gear hob needs to be increased, typically by ha0. * =1.15), referred to in this invention as: ideal EPS worm gear hob, the EPS worm gear it hobs, is then paired with the EPS worm, which is also an ideal transmission combination. At this time, the tooth profile curve of the ideal EPS worm gear can be used as a benchmark to evaluate the tooth profile error of the EPS worm gear processed by various EPS worm gear hobs.

[0017] In practice, the parameters of EPS worm gear hobs differ from those of EPS worms. Typically, EPS worm gear hobs have a larger outer diameter. This invention provides an EPS worm gear hob of the ZI type, similar to the EPS worm, and the EPS worm gear machined from this hob will be the object of evaluation. This EPS worm gear hob is arranged in the same way as the EPS worm, with its installation dimensions and structural parameters increased accordingly to ensure that its strength and other properties meet the process requirements.

[0018] The EPS worm, EPS worm wheel, and EPS worm wheel hob involved in this invention all have a right-hand helix; the transmission or rolling arrangement of the EPS worm or EPS worm wheel hob and the EPS worm wheel is at a 90º angle.

[0019] Example 1 like Figure 1 As shown in the figure, the mathematical modeling method for EPS worm gear tooth profile provided by the present invention includes the following steps 1 to 8: Step 1, establish O- x.y.z Coordinate system and O H - x H y H z H Coordinate system.

[0020] The O- x.y.z The coordinate system is based on the axis of the EPS worm gear or EPS worm wheel hob. x The axis is defined by the bisector of the tooth groove at the midpoint of the effective section of the EPS worm gear or EPS worm wheel hob. y Axis, with x shaft and y The intersection point O of the axis is the origin, and the line passing through the origin O is perpendicular to the axis. x O y The coordinate axes of the surface are z axis.

[0021] The O H - x H y H z H The coordinate system is based on the midpoint O of the EPS worm wheel's inner hole on the cross-section of the EPS worm wheel. H As the origin, passing through the origin O H Parallel to x The coordinate axes of the axis are x H Axis, parallel to z The coordinate axes of the axis are z H Axis, then with y Axis as y H Axis, establish O H - x H y H z H Coordinate system.

[0022] Because the meshing part between the EPS worm or EPS worm wheel hob and the EPS worm wheel is always located in the upper half of the axis of the EPS worm or EPS worm wheel hob and the lower half of the axis of the EPS worm wheel, in this invention, the tooth profile curves generated by various cross-sections on the EPS worm or EPS worm wheel hob only need to be analyzed in terms of the upper half.

[0023] On the cross-section passing through the centerline of the EPS worm or the centerline of the EPS worm gear hob, and on the tooth groove at the middle position of the effective section, there exists a center line; on any tooth on the cross-section of the EPS worm gear, there also exists a center line. When the center line of the EPS worm gear coincides with the center line of the EPS worm or the EPS worm gear hob, it is taken as the zero point position of the transmission mechanism or the rolling mechanism.

[0024] Figure 2 The illustrated transmission mechanism or rolling mechanism is a transmission combination of EPS worm or EPS worm wheel hob and EPS worm wheel at the zero point position. The transmission mechanism or rolling mechanism is sectioned using the cross-section of the EPS worm wheel along the axis of the EPS worm or EPS worm wheel hob. The sectioned result is as follows: Figure 3 As shown. The axis of the EPS worm gear or EPS worm wheel hob is taken as... x The axis is defined by the bisector of the tooth groove at the midpoint of the effective section of the EPS worm gear or EPS worm wheel hob. y Axis, with x shaft and y The intersection point O of the axis is the origin, and the line passing through the origin O is perpendicular to the axis. x O y The coordinate axes of the surface are z Axis, establish O- x.y.z Coordinate system. The mid-section of the EPS worm gear and... x O y Planes coincide, with the midpoint O of the EPS worm wheel's inner hole on the cross-section of the EPS worm wheel. H As the origin, origin O H exist y On the axis, passing through the origin O H Parallel to x The coordinate axes of the axis are x H Axis, parallel to z The coordinate axes of the axis are z H Axis, then with y Axis as y H An axis can be established to create an O H - x H y H z H Coordinate system. Figure 3 It includes two coordinate systems, when the EPS worm or EPS worm wheel hob rotates... x The shaft rotates, which can drive the EPS worm gear to rotate. z H The axis rotates, at which point... x O yAn EPS worm gear or EPS worm wheel hob on a flat surface can be considered as a rack, and an EPS worm wheel can be considered as a gear. Therefore, their combined transmission can be considered as a rack in... x Moving on the shaft drives the gear to continue rotating. z H The meshing transmission of the shaft rotation can be used to demonstrate the meshing transmission process between the EPS worm or EPS worm wheel hob and the EPS worm wheel.

[0025] Z-section definition: based on offset z 1 parallel to x O y The cross-section of the plane is the Z-section. The Z-section displays the complete tooth profile curves of the EPS worm or EPS worm wheel hob and the EPS worm wheel, allowing observation of their meshing and transmission. Offset z 1 can be understood as the Z-section in z The offset position on the axis, with a positive or negative sign, when z = z When 1=0, the Z-section and x O y The planes coincide.

[0026] Definition of rotation direction and angle sign: (The rest of the text appears to be a fragmented list of characters and symbols, possibly related to rotation direction and angle.) x The axis rotates, along x Rotation in a counter-clockwise direction when viewed along the axis is considered positive, and rotation in a clockwise direction is considered negative; all rotations around the axis... z H Rotation of the axis, in the opposite direction z H Rotation along the axis is considered positive if it's counter-clockwise, and negative if it's clockwise. If the rotation starts from the zero point, the angle in the positive direction is positive, and the angle in the negative direction is negative. For example... Figure 3 As shown, the rotation angle designation for EPS worm gear or EPS worm wheel hob is... i The EPS worm gear rotation angle code is f .

[0027] Step 2: Based on the tooth profile generation mechanism of EPS worm and EPS worm wheel hob, determine the coordinate values ​​of the initial position feature points of the generatrix and the distance from the initial position feature points to the worm gear. x O y The angle of rotation of the plane.

[0028] The EPS worm gear tooth profile is a standard type ZI tooth profile; while the EPS worm wheel hob tooth profile, in this invention, is only given as a standard type ZI tooth profile. Both the EPS worm gear and EPS worm wheel hob tooth profiles can be abstractly understood as being cut by a grooving tool. The cutting edge of the grooving tool is referred to as the generatrix in this invention. x The axis rotates and in x Moving along the axis allows the toothed helical surface to be scanned. The generatrix winds around... xThe rotation angle of the axis is the scanning angle, and the scanning angle is used as... i * Indicates the scanning angle. i * In the positive direction (following) x When rotating counterclockwise (viewed from the axis), the generatrix moves towards... x Move along the negative axis, scanning angle i * In the negative direction (following) x When rotating clockwise (viewed from the axis), the generatrix moves towards... x The axis moves in the positive direction. The intersection of the generatrix with the root circle, pitch circle, and addendum circle of the corresponding EPS worm or EPS worm wheel hob is called the characteristic point.

[0029] Figure 4 This is a schematic diagram of the ZI tooth profile generation mechanism and the relationship between characteristic points on the initial position generatrix of an EPS worm gear or EPS worm wheel hob, where line segments L1 and L2 are generatrixes; Figure 4 (a) is to combine the EPS worm gear or EPS worm wheel hob with O- x.y.z A simplified diagram of the coordinate system; Figure 4 (b) is the opposite. z A schematic diagram showing the initial positional relationship of feature points on the generatrix when viewed along the axis. Figure 4 (c) is following x A schematic diagram showing the initial positional relationship of feature points on the generatrix when viewed along the axis. Figure 4 (d) is in reverse y A schematic diagram showing the initial positional relationship of feature points on the generatrix when viewed along the axis.

[0030] The tooth profile generation mechanism is determined by the busbar placement position and angle, and the busbar scanning process. x The position of the axis movement determines the generated tooth profile surface (helical tooth surface). ZI type tooth profile generation mechanism: The generatrix L1 is tangent to the lower side of the base circle of the EPS worm or EPS worm wheel hob, and is at the base circle lead angle γ. b Tiltd placement; busbar L2 is tangent to the upper side of the base circle, and at the base circle lead angle γ b When placed at an angle, busbars L1 and L2 are wrapped around... x The axis rotates, and each revolution is followed by a lead P. z Length along x If the shaft moves axially, the generatrix L1 and generatrix L2 can be scanned (i.e., the cutting edge of the grooving tool cuts out) the left and right helical tooth surfaces of the EPS worm or EPS worm wheel hob groove. After the left or right helical tooth surface is cross-sectioned (perpendicular to the center line of the EPS worm shaft), the edge curve of the cross-section is the involute.

[0031] The left and right helical surfaces scanned by generatrix L1 and L2 follow the same pattern. In this invention, only the tooth profile information related to the left helical surface scanned by generatrix L1 is analyzed (the same applies below). Feature points on generatrix L1 include: feature point P where generatrix L1 intersects with the root circle. 11 ( x P11 , y P11 , z P11 The characteristic point P where the generatrix L1 intersects the pitch circle. 12 ( x P12 , y P12 , z P12 The characteristic point P where the generatrix L1 intersects the tooth tip circle. 13 ( x P13 , y P13 , z P13 Note: Other types of EPS worm gears or EPS worm wheel hobs described below should also be treated in the same way, only for feature point P. 11 Feature point P 12 Feature point P 13 The tooth profile information and related information were analyzed.

[0032] Given conditions for EPS worm gear or EPS worm wheel hob with ZI tooth profile: number of starts Z1, normal module m n Tooth profile angle α, pitch circle lead angle γ, EPS worm gear addendum coefficient ha1 * Or EPS worm gear hob tooth tip height coefficient ha0 * Pitch coefficient c1 * axial tooth thickness distribution ratio λ, characteristic point P on generatrix L1 11 of z Shaft setting value z P11 Based on the known conditions of the EPS worm gear or EPS worm wheel hob with ZI tooth profile, the axial tooth pitch P can be calculated. x , pitch circle axial chord tooth thickness S n Pitch circle radius r1, base circle radius r b1 Base circle lead angle γ b , tooth tip circle radius r a1 Root radius r f1 .

[0033] according to Figure 4 The relationship shown can be used to solve for the feature point P at the initial position on bus L1. 11 Feature point P 12Feature point P 13 The coordinate value expression is as follows: (1); (2); (3); like Figure 4 As shown, to make bus L1 reach the zero position, bus L1 must be rotated around... x The axis rotates, along x Rotating counterclockwise when viewed along the axis i * 12 Corner i * 12 A positive value makes feature point P on bus L1... 12 arrive x O y On a plane. Press Figure 4 The relationship shown can be used to solve for the feature point P at the initial position. 11 arrive x O y Planar rotation angle i * 11 Feature point P 12 arrive x O y Planar rotation angle i * 12 and feature point P 13 arrive x O y Planar rotation angle i * 13 The expression is as follows: (4); Step 3: Determine the zero point position and scanning start point of the feature points on the EPS worm gear or EPS worm wheel hob mother line.

[0034] like Figure 4 As shown, in O- x.y.z In the coordinate system, when the generatrix L1 at the initial position is rotated... x Axis rotation i * 12 After that, feature point P 12 Reachable x O y On the surface, and make feature point P 12 arrive yWhen the axial distance is half the axial tooth groove width of the EPS worm or EPS worm wheel hob, the generatrix L1 is at the zero point position. Then, the characteristic point P at the initial position of the generatrix L1... 11 Feature point P 12 Feature point P 13 Feature point M that can be converted into zero position 11-0 Feature point M 12-0 Feature point M 13-0 That is: P 11 ( x P11 , y P11 , z P11 → M 11-0 ( x M11-0 , y M11-0 , z M11-0 ), P 12 ( x P12 , y P12 , z P12 →M 12-0 ( x M12-0 , y M12-0 , z M12-0 ), P 13 ( x P13 , y P13 , z P13 → M 13-0 ( x M13-0 , y M13-0 , z M13-0 After transformation, the feature points on the busbar L1 at the zero point are as follows: Figure 5 As shown, Figure 5 (a) refers to the EPS worm gear or EPS worm wheel hob and its rotation angle. i and O- x.y.z A simplified diagram of the coordinate system. Figure 5 (b) is the opposite. z A schematic diagram showing the relationship between the zero point position and the scanning start point position of the feature points on the generatrix when viewed along the axis. Figure 5 (c) is following x A schematic diagram showing the angular relationship between the zero point position and the scanning start point position of the feature points on the generatrix when viewed along the axis. Figure 5(d) is in reverse y This diagram illustrates the relationship between the zero-point position and the scanning start point position of the feature point on the generatrix when viewed along the axis. At this point, feature point M... 12-0 Scan angle point position at zero position i * =0, let the characteristic point M be the zero point position. 13-0 The scanning angle point position is i * A0 ,but: i * A0 = i * 12 - i * 13 Let M be the characteristic point at the zero point position. 11-0 The scanning angle point position is i * B0 ,but i * B0 = i * 12 - i * 11 .

[0035] (a) The coordinates of the characteristic point at the zero point position on bus L1.

[0036] ; ; ; (ii) Coordinates of the scanning start point on bus L1.

[0037] To ensure that the tooth profile curve scanned by the busbar L1 on the specified cross-section changes with the rotation (angle) of the EPS worm or EPS worm gear hob. i If it can mimic rack and pinion movement, then the feature point at the zero point position needs to be converted into the scanning starting point M. 11 ( x M11 , y M11 , z M11 ), Scan start point M 12 ( x M12 , y M12 , z M12 ), Scan start point M 13 (x M13 , y M13 , z M13 ).like Figure 5 As shown, these scanning starting points vary with the rotation angle. i exist x The new position after the axis has been moved.

[0038] The expression for the scan start point coordinates on bus L1 is as follows: ; In the formula, This represents the rate of change of axial displacement, and is a constant. , It is the lead of the EPS worm gear or EPS worm wheel hob.

[0039] ; ; Step 4: Determine the coordinates of the rotating moving point of the characteristic point on the generatrix of the EPS worm or EPS worm gear hob and the equation of the generatrix.

[0040] In O- x.y.z In the coordinate system, the generatrix L1 starts from the zero point and increases with the rotation angle of the EPS worm gear or EPS worm wheel hob. i Entering the scanning start position, at this time when bus L1 re-circles x Axis rotation i * Angle, then the scanning start point M on bus L1 11 Scan start point M 12 Scan start point M 13 It can be converted into a rotating moving point D 11 Rotating point D 12 Rotating point D 13 That is: M 11 ( x M11 , y M11 , z M11 → D 11 ( x D11 , y D11 , z D11 M 12 ( x M12 , y M12 , z M12 → D12 ( x D12 , y D12 , z D12 M 13 ( x M13 , y M13 , z M13 → D 13 ( x D13 , y D13 , z D13 The expression for the rotating point on bus L1 is as follows: (11); (12); (13); In the formula, Indicates that busbar L1 is wound x The scanning angle of the axis rotation.

[0041] In O- x.y.z Take a rotating point D in the coordinate system 11 Rotating point D 12 The line connecting two points represents the spatial position of busbar L1. When busbar L1 rotates... x Axis rotation i * After the angle, the equation of its generatrix L1 is as follows: (14); Step 5: Calculate the tooth profile curve points of the EPS worm gear or EPS worm wheel hob. Step 5 includes the following steps 501 to 504: Step 501: Generate the tooth profile curve of the EPS worm or EPS worm wheel hob, and number the tooth profile curve of the EPS worm or EPS worm wheel hob.

[0042] Arbitrary sectioning of an EPS worm or EPS worm wheel hob can generate its tooth profile curve. However, this invention requires the study and analysis of meaningful tooth profile curves that can both reflect the meshing transmission characteristics of the EPS worm or EPS worm wheel hob and the EPS worm wheel, and generate a standard EPS worm wheel tooth profile. In this invention, the tooth profile curves to be studied and analyzed are obtained by Z-sectioning. When the transmission mechanism of the EPS worm or EPS worm wheel hob and the EPS worm wheel is sectioned with a Z-section, multiple tooth profile curves are generated. Since the transmission mechanism has a maximum of three teeth participating in meshing simultaneously, selecting four sets of tooth profile curves can completely reflect the meshing transmission relationship of the transmission mechanism, such as... Figure 6 As shown, Figure 6 In the diagram, A1 is the potential meshing point of T2 and G2, A2 is the engagement point, A3 is the meshing line of T4 and G4, A4 is the meshing point of T4 and G4, A5 is the base circle of the EPS worm gear when T4 and G4 mesh, A6 is the meshing point of T6 and G6, A7 is the meshing line of T6 and G6, A8 is the base circle of the EPS worm gear when T6 and G6 mesh, A9 is the disengagement point, A10 is the pitch circle of the EPS worm gear, A11 is the horizontal line of the pitch circle of the EPS worm or EPS worm gear hob, C is the node, and P... x It refers to the axial tooth pitch and S of the EPS worm gear or EPS worm wheel hob. n This refers to the axial chord tooth thickness of the EPS worm gear or EPS worm wheel hob; Setting: The EPS worm gear with corresponding head number Z1=2 uses... i =±360º rotation drive operation, corresponding to EPS worm gear hobs with Z1=4 (or 6 or 8) heads. i =±180º (or ±120º or ±90º) angular drive operation, converting rotational motion into linear motion to mimic the meshing transmission of a rack and pinion. In O- x.y.z In a three-dimensional coordinate system, let D be the point on the tooth profile curve of the EPS worm or EPS worm wheel hob on the Z-section. x D , y D , z D ),in z D = z 1 = constant, which always holds true, making the tooth profile curve point D ( x D , y D , z D )and z D = z 1 o'clock atx O y Point D of the tooth profile curve in a two-dimensional coordinate system x D , y D They are equivalent to each other, and their tooth profile curves are completely identical.

[0043] Figure 6 The tooth profile curve of the EPS worm gear or EPS worm wheel hob shown is similar to that of the rack tooth profile curve. The tooth profile curve of the EPS worm gear or EPS worm wheel hob is numbered T. k (Where k=1, 2, 3, 4, 5, 6, 7, 8), the tooth profile curve of the EPS worm gear is similar to that of a gear tooth profile curve, and the EPS worm gear tooth profile curve is numbered G. k (where k = 1, 2, 3, 4, 5, 6, 7, 8), at the corner i Within a limited range, these tooth profile curves T k With EPS worm gear tooth profile curve G k These are paired meshing configurations, such as: T1-G1, T2-G2, T3-G3, T4-G4, T5-G5, T6-G6, T7-G7, T8-G8. The tooth profile curve T4 is formed by the generatrix L1 varying with the scanning angle. i * Generated by rotational scanning, i.e., varying with the scanning angle. i * After rotation, the generatrix L1 and the Z-section will produce countless consecutive intersection points. Connecting these intersection points forms the tooth profile curve T4. However, not all segments of the tooth profile curve T4 are valid; only those in D... 11 - D 13 Only the portion of the tooth profile curve T4 generated by scanning within the line segment can be considered a valid tooth profile curve T4; the portion exceeding D... 11 -D 13 Curves generated outside the line segment are invalid.

[0044] When the number of hob heads for the EPS worm gear or ideal EPS worm wheel is Z1=2, the tooth profile curve T8 is also generated by scanning the generatrix L1. Similarly, tooth profile curves T2 and T6, T1 and T5, and T3 and T7 are each generated by scanning the same generatrix. When the number of hob heads for the EPS worm wheel is Z1=4, each code curve is generated by scanning different generatrixes. Regardless of whether the tooth profiles are generated by the same generatrix, each even-numbered code tooth profile (such as...) Figure 6 The tooth profiles on the left side of the tooth socket shown are spatially identical, and similarly, all odd-numbered tooth profiles (such as...) Figure 6The tooth profiles on the right side of the tooth groove shown are also identical in space. Therefore, they are cut into tooth profile curves by the same Z-section. When simulating the linear motion of the rack, as long as they are at the same position point, the tooth profile curves shown are completely consistent. However, the tooth profile curves cut into different Z-sections are different.

[0045] Step 502: Determine the general expression for point D (a point on tooth profile curve T4) of T4 tooth profile curve.

[0046] From the Z-section equation: z = z Combining 1 = constant with equation (14), we can obtain that in O- x.y.z Point D of the T4 tooth profile curve of the EPS worm or EPS worm wheel hob in the coordinate system ( x D , y D , z D The expression is as follows: (15); Step 503: Generate the limit scanning angle of the effective tooth profile curve of the EPS worm or EPS worm gear hob.

[0047] To ensure that the tooth profile curve point T4 in equation (15) can be a valid tooth profile curve point, it should be determined that at D 11 - D 13 Within the line segment, bus L1 follows i * Assign the minimum scan angle of rotation i * min and maximum scanning angle i * max To ensure that bus L1 is i * min ~ i * max All tooth profile curve points generated by scanning within the range are valid tooth profile curve points.

[0048] Z-section offset | z 1| is within half the tooth width B / 2 of the EPS worm gear, there is | z 1|≤B / 2, while the tooth tip circle radius r of the EPS worm gear or EPS worm wheel hob a1 B / 2 always holds true, that is, r a1 >| z 1| Valid; for rotating point D 13 Then the bus L1 is scanned by the angle i * Rotation always results in the rotating point D being able to rotate. 13 Falling onto the Z-section, the corresponding scanning angle at this time i * A It is a limit scanning angle ( i * min or i * max ), scanning angle i * A The expression is as follows: (16); Because the root circle radius r of the EPS worm gear or EPS worm wheel hob f1 When the Z-section is used to section the EPS worm gear, two situations will occur: r f1 ≥| z 1| Situation and r f1 <| z 1|Situation.

[0049] ① When r f1 ≥| z In case 1, the bus L1 is scanned at an angle i * Rotation always results in the rotating point D being able to rotate. 11 Falling onto the Z-section, the corresponding scanning angle at this time i * B It is an extreme scanning angle.

[0050] When r f1 ≥| z Limiting scanning angle in case 1 i * B expression: (17); ②When r f1 <| z In case 1, the busbar L1 is scanned and rotated, and the rotating point D... 11 It cannot fall on the Z-section, let D be. 11 -D 13 A rotating point D within a line segment w Rotating point D w distance x Let the axis radius be r w =| z 1|, then D w -D 11 Line segments will not generate tooth profile curves after scanning, Dw -D 13 Line segments can generate effective tooth profile curves after scanning.

[0051] When r f1 <| z Limiting scanning angle in case 1 i * w Rotating point D w Set the starting position of the scan to M. w ( x Mw , y Mw , z Mw ), Scan starting position M w The angle of the scanning starting point is set to i * w0 When this is taken as the starting point and the scanning angle i * = i * w At that time, the rotating point D can be made to rotate. w If it falls tangentially onto the Z-section, then i * w It is an extreme scanning angle.

[0052] Scan start point M w In M 11 -M 12 On a straight line, the following relationship should be satisfied: (18); In O- x.y.z In the coordinate system, the scanning start point M w In x With the axis as the center line, and r w =| z On a cylindrical surface with radius 1, the following relationship is satisfied: Combining it with equation (18), we can obtain the following relation: (19) In the formula, ; Equation (19) contains only one unknown quantity. y Mw ,set up ,but Using Newton's iterative formula: Solve f ( x When )=0, the accuracy requirements are met in engineering. xIf the value is found, then we can obtain... y Mw The specific values ​​will then be obtained. y Mw The parameter value can be obtained by substituting it into equation (18). x Mw , z Mw The parameter value indicates the scan start position M. w ( x Mw , y Mw , z Mw In coordinate system O- x.y.z This allows for accurate positioning and the acquisition of the scanning starting point angle. i * w0 and limit scanning angle i * w The expression is as follows: (20); Step 504: Generate the minimum scanning angle i * min and maximum scanning angle i * max .

[0053] From the obtained limit scanning angle i * A Limiting scanning angle i * B or extreme scanning angle i * w The minimum scanning angle can be obtained. i * min and maximum scanning angle i * max as follows: (twenty one); Step 6: Calculate the meshing transmission relationship between the EPS worm or EPS worm wheel hob and the EPS worm wheel, and determine the potential meshing point between the tooth profile curve of the EPS worm or EPS worm wheel hob and the tooth profile curve of the EPS worm wheel.

[0054] along with i * Rotational scanning, the tooth profile curve T4 obtained from equations (15) and (21) in O- xyz z On the Z-section of the coordinate system, the T4 tooth profile curve point D ( x D , y D , z D )of z D = z The coordinate value is eliminated, causing the tooth profile curve T4 to be shifted to... x O y In the plane coordinate system, the meshing transmission relationship between the tooth profile curve T4 and the EPS worm gear tooth profile curve G4 on the Z-section is as follows: Figure 6 As shown. If the normal of point D on the T4 tooth profile curve passes through node C (0, r1'), then point D on the T4 tooth profile curve becomes a potential meshing point between tooth profile curve T4 and EPS worm gear tooth profile curve G4. This potential meshing point has three characteristics: ① As the rotation angle of the EPS worm or EPS worm gear hob increases... i Rotation, tooth profile curve T4 will be x Moving on the axis, each point on it can only move when corresponding to... i Only at the proper location of the corner point can it become a potential engagement point; ② Corner i Only within a certain rotation angle range can all points on the tooth profile curve T4 become potential meshing points; ③ The T4 tooth profile curve point D that becomes a potential meshing point and the rotation angle i Scanning angle i * The normal to point D of the T4 tooth profile curve (its slope is set to...) k D They correspond to each other and are unique.

[0055] exist x O y In a planar coordinate system, find the slope of the normal line at point D on the T4 tooth profile curve. k D The process is as follows: (twenty two); The derivative can be obtained from equation (15). The expression is as follows: (twenty three); The derivative can be obtained from equation (11). The expression is as follows: (twenty four); The derivative can be obtained from equation (12). The expression is as follows: (25); From equations (22) to (25) of the above process, we can obtain that...x O y The slope of the normal line at point D on the T4 tooth profile curve in the planar coordinate system k D As the EPS worm gear or EPS worm wheel hob rotates to... i Angle point, the normal of point D on the T4 tooth profile curve and y The intersection point of the axes is set as C1(0, y C1 When intersection point C1 coincides with node C (i.e.) y C1 Only when =r1' can point D on the T4 tooth profile curve become a potential meshing point.

[0056] exist x O y In the plane coordinate system, point D of the T4 tooth profile curve ( x D , y D The normal and y Intersection value y C1 The expression is as follows: (26); like y C1 When r1' = r1', the intersection point C1 coincides with the node C, then point D on the T4 tooth profile curve becomes the potential meshing point. r1' represents the pitch circle radius of the EPS worm or EPS worm wheel hob, which is a known parameter.

[0057] Therefore, one problem needs to be solved: what happens when the Z-section offset of the EPS worm gear or EPS worm wheel hob... z 1 and corner i Given the busbar scanning angle i * What value is needed to make the offset... z 1. Corner i Scanning angle i * The corresponding T4 tooth profile curve point D ( x D , y D This becomes the only potential engagement point. If the precise scanning angle value is obtained by reverse derivation of the relational formula... i * It is very difficult, but it can be obtained through programming calculations using relevant software. This invention recommends using Excel software to write all the above parameter expressions into cells, as long as the Z-section offset is given. z 1. Angle value i and scanning angle value i *This allows for the forward calculation of point D on the T4 tooth profile curve. x D , y D The normal to point D of the T4 tooth profile curve and y Intersection value y C1 etc. When the Z-section offset z 1 and corner i Given the information, VBA programming is used in an Excel environment to iteratively assign scanning angles. i * Assigning values ​​in a positive calculation, gradually reducing the size. i * By assigning a range of values, a scanning angle value that meets the engineering accuracy requirements can be found. i * , so that the difference |r1'- y C1 | Approaching zero (making intersection point C1 approach coincidence with node C), then at this time, the T4 tooth profile curve point D ( x D , y D This becomes the only potential engagement point.

[0058] Note: When there are no potential meshing points on the T4 tooth profile curve, the scan angle calculated by VBA programming is... i * = i * min or i * = i * max , so that |r1'- y C1 The error is minimized. This indicates that any [object] on the Z-section [is at its minimum]. i At an angular point, the existence of a potential meshing point within the tooth profile curve segment cannot be guaranteed, such as... Figure 6 As shown, at the location of the T8 tooth profile curve, there are no potential meshing points within its tooth profile curve segment. Similarly, the T4 tooth profile curve varies with the rotation angle. i When moved to position T8, there is also no potential engagement point.

[0059] Based on the above analysis and calculations, when the angle iFigure 7 illustrates the process of point D on the T4 tooth profile curve becoming a potential engagement point on the same Z-section, at different corner positions. Figure 7(a): no potential engagement point; Figure 7(b): potential engagement point but not engaged; Figure 7(c): potential engagement point and engaged at the engagement point; Figure 7(d): potential engagement point and engaged at the zero point; Figure 7(e): potential engagement point and engaged at the node; Figure 7(f): potential engagement point and engaged at the disengagement point; Figure 7(g): no potential engagement point (no potential engagement points beyond the disengagement point). When the Z-section offset z1 is changed, the tooth profile of the T4 tooth profile curve differs significantly on different Z-sections, such as... Figure 8 As shown.

[0060] Based on the above, similarly, we can obtain tooth profile curves T2, T6, and T8 on the same Z-section, with identical tooth shapes. When the rotation angle... i At different corner positions, they also exhibit the same evolutionary process as point D of the T4 tooth profile curve shown in Figure 7, becoming a potential meshing point. Therefore, tooth profile curves T2, T6, and T8 can be derived by replicating the T4 tooth profile curve, according to a P... x The spacing between tooth pitches moves together with the tooth profile curve T4.

[0061] Step 7: Determine the transmission meshing contact line and contact range between the EPS worm or EPS worm wheel hob and the EPS worm wheel.

[0062] As the EPS worm gear or EPS worm wheel hob rotates... i Rotation, the tooth profile curve T4 on the Z-section will be x Moving on the axis, its T4 tooth profile curve point D ( x D , y D Only after becoming a potential engagement point and entering the engagement and disengagement points on the line of engagement (which is also the normal to point D, which is dynamic) can point D on the T4 tooth profile curve become a true engagement point (as shown in Figures 7(c), 7(d), 7(e), and 7(f), point D has become a true engagement point). This engagement point W ( x w , y w ( ) is the contact point where the EPS worm or EPS worm wheel hob meshes with the EPS worm wheel drive, and it also belongs to the G4 tooth profile curve point of the EPS worm wheel. On the Z-section, the line of meshing intersects with the section arc of the EPS worm wheel tooth tip circle (the radius of the section arc of the EPS worm wheel tooth tip circle is set as...). The intersection point is the engagement point, denoted as W1. x w1 , yw1 The line of meshing is the horizontal line of the section of the EPS worm or EPS worm wheel hob tooth tip circle (the height of the horizontal line of the section of the EPS worm or EPS worm wheel hob tooth tip circle is set to...). The intersection is the disengagement point, denoted as W2. x w2 , y w2 ).

[0063] The horizontal height of the section line of the tip circle of the EPS worm or EPS worm wheel hob on the Z-section. The radius of the cross-sectional arc of the EPS worm gear tooth tip circle The expression is as follows: (27); In the formula, This indicates the tip circle radius of the EPS worm gear or EPS worm wheel hob. This indicates the offset of the Z-section. This indicates the radius of the tip circle (throat circle) of the EPS worm gear. This indicates the radius of the tip circle (throat circle / generator circle) of the EPS worm gear. Indicates the radius of the outer circle (maximum circle) of the EPS worm gear; Entry point W1 ( x w1 , y w1 The expression is as follows: (28); In the formula, , , .

[0064] Disengagement point W2 ( x w2 , y w2 The expression is as follows: (29); Combining equation (15), on the Z-section, point D of the T4 tooth profile curve ( x D , y D ) can become the meshing point (or G4 tooth profile curve point) W ( x w , y w The conditional expression for ) is as follows: (30); The tooth profile of the tooth profile curve T4 is different on different Z-sections, even with the hob rotation angle of the EPS worm or EPS worm wheel. i At the same corner point, their current potential engagement points or actual engagement points are also different. Based on the above analysis and calculations, when the Z-section offset is continuously applied... z 1. Different assignments (assignment range: -B / 2≤) z 1≤B / 2), and at the same time, the rotation angle of the EPS worm or EPS worm wheel hob is continuously increased. i Different assignments (assignment range: -2π≤) i After ≤2л), different T4 tooth profile curves can be obtained on different Z-sections, and the current... i Different potential engagement points or actual engagement points at the corner points.

[0065] In this invention, Excel is used for VBA programming to calculate and give the T4 tooth profile curve point D ( x D , y D By listing the data of the actual meshing points, these situations can be displayed, as well as the changes in the transmission meshing contact line and the contact range.

[0066] Based on the above calculations, the data reflecting the changes in the meshing point position of the T4 tooth profile curve on different Z-sections can be plotted as follows. Figure 9 The image shows the contact trajectory of the EPS worm gear meshing with the EPS worm wheel drive, and how it changes with the rotation angle. i Rotation can display the dynamic changes of the contact line.

[0067] Step 8, as the EPS worm gear hob rotates... i Drive EPS worm gear rotation angle f Rotation transforms the point W of the EPS worm gear G4 tooth profile curve to a fixed coordinate system. x H O H y H EPS worm gear G4 tooth profile curve point W at the zero position H This forms the G4 tooth profile curve of the EPS worm gear.

[0068] The EPS worm gear tooth profile curve is formed by hobbing with a corresponding EPS worm gear hob, as shown in the reference. Figure 6 Or, as shown in Figure 7, taking the EPS worm gear tooth profile curve G4 as an example, it is generated analytically from the paired tooth profile curve T4 according to the meshing principle. On the Z-section, as the EPS worm gear hob rotation angle... i Rotation, T4 tooth profile curve point D ( x D , y D Once the point becomes the meshing point, this point becomes the W point of the EPS worm gear G4 tooth profile curve. xw , y w ).

[0069] On the same Z-section, when the EPS worm gear hob rotation angle is continuously applied... i After assignment, the EPS worm gear rotates at an angle. f Continuous rotation yields the EPS worm gear tooth profile curve G4. Figure 10 What is shown is z This diagram demonstrates the generation process of the EPS worm gear tooth profile curve G4 and tooth root transition curve on the Z-section when 1=0. The thick solid lines in the diagram represent the generated tooth profile curve and tooth root transition curve. The diagram also shows the rotation angle. i = During the range of -163º to 259º, the hobbing cutter edge cuts out the G4 tooth profile curve, at the corner. i The hobbing cutter tip cuts out the tooth root transition curve between 85º and 259º, where... i = -163º when the hob cutting edge just enters the cutting tooth profile curve position (i.e., the meshing point of T4 and G4). i = At 85º, the hob tip just enters the position of the cutting tooth root transition curve (i.e., the hob tip just falls on the...). y (on the axis) i = At 259º, the hob cutting edge and tip are just disengaged from the cutting position (i.e., the disengagement point of T4 and G4).

[0070] As the EPS worm gear hob rotates... i Drive EPS worm gear rotation angle f Rotation has become the point W of the EPS worm gear G4 tooth profile curve. x w , y w () is a moving point and needs to be transformed to a fixed coordinate system. x H O H y H EPS worm gear G4 tooth profile curve point W at the zero position H ( x H_w , y H_w This allows for a comprehensive observation of the EPS worm gear tooth profile curve G4. EPS worm or EPS worm gear hob rotation angle. i With EPS worm gear rotation angle f The relationship is as follows: (31); In the formula, Z1 represents the number of heads of the EPS worm gear or EPS worm wheel hob, and Z2 represents the number of teeth of the EPS worm wheel.

[0071] According to the EPS worm gear rotation angle f Using the coordinate system transformation formula, the meshing point (or the point on the tooth profile curve of the EPS worm gear G4) W (30) is transformed into the coordinate system transformation formula. x w , y w ) transformed into x H O H y H Point W of the EPS worm gear G4 tooth profile curve at the zero position in the coordinate system H ( x H_w , y H_w Its expression is as follows: (32); In the formula, H represents the center distance between the EPS worm or EPS worm wheel hob and the EPS worm wheel, and in this formula, it specifically refers to the center distance between the EPS worm wheel hob and the EPS worm wheel.

[0072] The corresponding EPS worm gear G4 tooth profile curve can also be obtained on different Z-sections. Example 2

[0073] This invention provides a method for evaluating the tooth profile error of an EPS worm gear, comprising the following steps 9 to 10: The EPS worm gear tooth profile curve reference refers to the ideal tooth profile curve produced by hobbing with an ideal EPS worm gear hob. The EPS worm gear tooth profile curve reference is different on different Z-sections.

[0074] EPS worm gear tooth profile curve point error refers to the minimum distance between a point on the evaluated tooth profile curve and the tooth profile curve reference, denoted by Δ. j The tooth profile curve points being evaluated may be distributed on the same side or both sides of the reference, but the error of the EPS worm gear tooth profile curve points is always positive in numerical value.

[0075] EPS worm gear tooth profile error refers to the average value of the errors at all evaluated EPS worm gear tooth profile curve points on a specified cross-section and at a specified position, denoted by Δ. In this invention, the worm gear mid-section (i.e., when...) is at the zero-point position. z The evaluation of the EPS worm gear tooth profile curve on the Z-section (when 1=0) coincides with its reference pitch circle (pitch circle point) to analyze different Z-sections (when 1=0). z The distribution of EPS worm gear tooth profile error on different Z-sections.

[0076] Step 9: Determine the reference curve of the EPS worm gear tooth profile.

[0077] To obtain the EPS worm gear tooth profile curve reference as the evaluation benchmark for tooth profile error, this invention employs an ideal EPS worm gear hob. Using the mathematical modeling method for the EPS worm gear tooth profile provided in any embodiment of this invention, a hob rotation angle can be generated on the Z-section. i The unique corresponding ideal tooth profile curve point of the EPS worm gear, the angle of continuous rotation of the hob. i It can generate countless ideal tooth profile curve points, and connecting these points forms the reference EPS worm gear tooth profile curve on the current Z-section.

[0078] In this invention, Excel is used to list the EPS worm gear tooth profile curve point data, which serve as the reference for the EPS worm gear tooth profile curve. Based on this, a tooth profile curve reference graphic can be generated. Similarly, tooth profile curve reference graphics on different Z-sections can be generated, such as... Figure 11 As shown.

[0079] This invention uses an EPS worm gear hob of the same ZI type and with the same installation method (90º angle installation) as the EPS worm gear. The main parameters include: Z1=8, m n =2, α=14.5º, γ=20º, h a0 * =1.15, using this hob to machine the EPS worm gear tooth profile as the evaluation object. This hob, when machining the EPS worm gear, still follows the mathematical modeling method for the EPS worm gear tooth profile provided in any embodiment of this invention, and can generate the tooth profile curve point data to be evaluated, and such as... Figure 11 The tooth profile curve shown is illustrated.

[0080] Step 10: Use the EPS worm gear tooth profile curve as a reference to evaluate the EPS worm gear tooth profile error.

[0081] Depend on Figure 11 It can be seen that there is a significant difference between the reference EPS worm gear tooth profile curve and the evaluated EPS worm gear tooth profile curve. This is mainly due to the large difference in size between the EPS worm gear hob and the EPS worm. Therefore, it is necessary to limit the effective range of the evaluated EPS worm gear tooth profile curve. This range is determined with reference to the actual meshing range of the EPS worm, that is, the worm gear tooth profile curve segment between the EPS worm gear tooth tip circle and the EPS worm tooth tip circle on the Z-section is the effective evaluation range. x H O H y H In the coordinate system, point W is the tooth profile curve of the EPS worm gear being evaluated. H ( x H_w , y H_w )of y H_w The value is used for limitation judgment. From equation (27), the expression for the effective evaluation range is as follows: (33); Because there is a dimensional difference between the EPS worm gear tooth tip circle and the EPS worm wheel hob tooth tip circle, while the EPS worm wheel tooth tip circle is the same, and the worm wheel tooth tip circle constraint condition (G4 tooth profile curve point W expression (30) related to the EPS worm wheel tooth tip circle engagement point condition) is different... The EPS worm gear tooth profile curve points have been used in the process of obtaining the EPS worm gear tooth profile curve points. All EPS worm gear tooth profile curve points have met the limiting condition. Therefore, the effective evaluation range is limited only to the EPS worm tooth tip circle. Thus, equation (33) can be simplified to the following conditional expression for the effective evaluation range of EPS worm gear tooth profile error: (34); In the formula, The formula represents the tip circle radius of the EPS worm or EPS worm wheel hob, specifically referring to the tip circle radius of the EPS worm in this formula. H represents the center distance between the EPS worm or EPS worm wheel hob and the EPS worm wheel, specifically referring to the center distance between the EPS worm and the EPS worm wheel in this formula.

[0082] According to equation (34), the following can be drawn: Figure 11 The curve shows the limitation of the effective evaluation range.

[0083] exist x H O H y H In the coordinate system, let the reference point of the EPS worm gear tooth profile curve on the Z-section be designated as W. H0 ( x H0_w , y H0_w The reference point locations are numbered 0, 1, 2, ..., i-1, i, i+1, ..., 719, 720. The code for the EPS worm gear profile curve point being evaluated is still set to W. H ( x H_w , y H_w Let their position numbers be 0, 1, 2, ..., j-1, j, j+1, ..., 719, 720. Let the point W of the EPS worm gear tooth profile curve being evaluated be... H Reference point W of EPS worm gear tooth profile curve H0 The distance between them is L j,i (Note that: if point W does not exist at the position corresponding to the number) H0 ( x H0_w , y H0_w ) or point W H ( xH_w , y H_w If the distance value L does not exist, then there is no distance value L. j,i Let all distances L j,i The minimum distance is L j_min1 The corresponding reference point is W. H01 ( x H0_w1 , y H0_w1 The second minimum distance is L. j_min2 The corresponding reference point is W. H02 ( x H0_w2 , y H0_w2 At this point, the tooth profile curve point W of the EPS worm gear number j being evaluated... H to the reference point W corresponding to these two minimum distances H01 and W H02 The distance between the lines is the error Δ of the EPS worm gear tooth profile curve point. j Its expression is as follows: (35); In the formula, .

[0084] On any Z-section, the reference point W of the tooth profile curve H0 ( x H0_w , y H0_w ) and the evaluated tooth profile point W H ( x H-w , y H-w List them separately in Excel Sheet1 and Sheet2, with their row numbers starting from a corner. i Distributed across ROW(10:730), with each row representing an angle, and column numbers based on Z-section offset. z 1 is distributed in column (2:203) with each pair of columns representing an offset, and the corner. i With offset z 1. Fill in W at the intersection position H0 ( x H0_w , y H0_w ) or W H ( x H_w , y H_w Coordinate values.

[0085] The calculated tooth profile error data can be plotted as follows: Figure 12 The diagram shows the distribution of the EPS worm gear tooth profile error under evaluation at the zero position on different Z-sections. Figure 12 It is evident that the EPS worm gear tooth profile error is significant in the design. This indicates that even with an EPS worm gear hob of the same ZI type as the EPS worm, the EPS worm gear produced by increasing the structural dimensions of the EPS worm gear hob cannot be used in the product.

[0086] In summary, this invention provides a mathematical modeling method and error assessment method for EPS worm gear tooth profiles. It treats the meshing transmission between the EPS worm or EPS worm gear hob and the EPS worm gear as a combination of rack and pinion transmission on the Z-section. Then, it uses the analytical method based on the meshing principle to obtain the EPS worm gear tooth profile curve. Based on this, a mathematical model of the meshing transmission process of the EPS worm and worm gear and the EPS worm gear tooth profile can be constructed to display the distribution and movement state of the meshing contact line during the transmission process. This allows for simulation of the EPS worm gear tooth profile and errors, providing an evaluation benchmark for the design process and enabling improvements to the correction schemes for reducing tooth profile errors.

[0087] 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 foregoing embodiments, those skilled in the art should understand that modifications to the technical solutions described in the foregoing embodiments, or equivalent substitutions for some or all of the technical features, do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A mathematical modeling method for EPS worm gear tooth profile, characterized in that, Includes the following steps 1 through 8: Step 1, establish O- xyz Coordinate system and O H - x H y H z H Coordinate system; Step 2: Based on the tooth profile generation mechanism of EPS worm and EPS worm wheel hob, determine the coordinate values ​​of the initial position feature points of the generatrix and the distance from the initial position feature points to the worm gear. x O y The angle of rotation of the plane; Step 3: Determine the zero point position and scanning start point of the feature points on the EPS worm gear or EPS worm wheel hob generatrix. Step 4: Determine the coordinates of the rotating points of the characteristic points on the generatrix of the EPS worm or EPS worm wheel hob and the equation of the generatrix. Step 5: Calculate the tooth profile curve points of the EPS worm or EPS worm gear hob; Step 6: Calculate the meshing transmission relationship between the EPS worm or EPS worm wheel hob and the EPS worm wheel, and determine the potential meshing point between the tooth profile curve of the EPS worm or EPS worm wheel hob and the tooth profile curve of the EPS worm wheel. Step 7: Determine the transmission meshing contact line and contact range between the EPS worm or EPS worm wheel hob and the EPS worm wheel; Step 8, as the EPS worm gear hob rotates... θ Drive EPS worm gear rotation angle φ Rotation transforms the point W of the EPS worm gear G4 tooth profile curve to a fixed coordinate system. x H O H y H EPS worm gear G4 tooth profile curve point W at the zero position H This forms the G4 tooth profile curve of the EPS worm gear.

2. The mathematical modeling method for EPS worm gear tooth profile according to claim 1, characterized in that, The O- xyz The coordinate system is based on the axis of the EPS worm gear or EPS worm wheel hob. x The axis is defined by the center line of the tooth groove at the middle position of the effective section of the EPS worm gear or EPS worm wheel hob. y Axis, with x shaft and y The intersection point O of the axis is the origin, and the line passing through the origin O is perpendicular to the axis. x O y The coordinate axes of the surface are z axis; The O H - x H y H z H The coordinate system is based on the midpoint O of the EPS worm wheel's inner hole on the cross-section of the EPS worm wheel. H As the origin, passing through the origin O H Parallel to x The coordinate axes of the axis are x H Axis, parallel to z The coordinate axes of the axis are z H Axis, then with y Axis as y H axis.

3. The mathematical modeling method for EPS worm gear tooth profile according to claim 1, characterized in that, In step 2, the tooth profile generation mechanism includes: The busbar L1 is tangent to the lower side of the base circle of the EPS worm or EPS worm gear hob, and is at the base circle lead angle γ. b Tiltd placement; busbar L2 is tangent to the upper side of the base circle, and at the base circle lead angle γ b When placed at an angle, busbars L1 and L2 are wrapped around... x The axis rotates, and each revolution is followed by a lead P. z Length along x If the shaft moves axially, the left and right helical tooth surfaces of the EPS worm or EPS worm wheel hob groove can be scanned by the generatrix L1 and L2 respectively. After the left or right helical tooth surface is cross-sectioned at the end, the edge curve of the cross-section is an involute. Feature points on busbar L1 include feature point P where busbar L1 intersects with the tooth root circle. 11 ( x P11 , y P11 , z P11 The characteristic point P where the generatrix L1 intersects the pitch circle. 12 ( x P12 , y P12 , z P12 The characteristic point P where the generatrix L1 intersects the tooth tip circle. 13 ( x P13 , y P13 , z P13 ); Feature point P at the initial position on bus L1 11 Feature point P 12 Feature point P 13 The coordinate value expression is as follows: （1）； （2）； （3）； Among them, P x S represents the axial tooth pitch. n r1 represents the pitch circle axial chord tooth thickness, and r represents the pitch circle radius. b1 Indicates the base circle radius. Represents feature point P on busbar L1 11 of z Axis setting value, γ b The base circle lead angle, r a1 Indicates the radius of the tooth tip circle; Feature point P at the initial position 11 arrive x O y Planar rotation angle θ * 11 Feature point P 12 arrive x O y Planar rotation angle θ * 12 and feature point P 13 arrive x O y Planar rotation angle θ * 13 The expression is as follows: （4）； Where, r f1 This represents the radius of the tooth root circle, and has... .

4. The mathematical modeling method for EPS worm gear tooth profile according to claim 1, characterized in that, Step 3 includes: In O- xyz In the coordinate system, when the generatrix L1 at the initial position is rotated... x Axis rotation θ * 12 After that, feature point P 12 Reachable x O y On the surface, and make feature point P 12 arrive y When the axial distance is half the axial tooth groove width of the EPS worm or EPS worm wheel hob, the generatrix L1 is at the zero point position. Then, the characteristic point P at the initial position of the generatrix L1... 11 Feature point P 12 Feature point P 13 Feature point M that can be converted into zero position 11-0 Feature point M 12-0 Feature point M 13-0 At this time, feature point M 12-0 Scan angle point position at zero position θ * =0, let the characteristic point M be the zero point position. 13-0 The scanning angle point position is θ * A0 ,but: θ * A0 = θ * 12 - θ * 13 Let M be the characteristic point at the zero point position. 11-0 The scanning angle point position is θ * B0 ,but θ * B0 = θ * 12 - θ * 11 ; The coordinates of the characteristic point at the zero point position on bus L1: （5）； （6）； （7）； To ensure that the tooth profile curve scanned by the busbar L1 on a specified cross-section can mimic the movement of the rack as the EPS worm or EPS worm gear hob rotates, the feature point at the zero position needs to be converted into the scanning starting point M. 11 ( x M11 , y M11 , z M11 ), Scan start point M 12 ( x M12 , y M12 , z M12 ), Scan start point M 13 ( x M13 , y M13 , z M13 ); The expression for the scan start point coordinates on bus L1 is as follows: （8）； In the formula, This indicates the rotation angle of the EPS worm gear or EPS worm wheel hob. This represents the rate of change of axial displacement, and is a constant. , It is the lead of the EPS worm gear or EPS worm wheel hob; （9）； （10）。 5. The mathematical modeling method for EPS worm gear tooth profile according to claim 4, characterized in that, Step 4 includes: In O- xyz In the coordinate system, the generatrix L1 starts from the zero point and increases with the rotation angle of the EPS worm gear or EPS worm wheel hob. θ Entering the scanning start position, at this time when bus L1 re-circles x Axis rotation θ * Angle, then the scanning start point M on bus L1 11 Scan start point M 12 Scan start point M 13 It can be converted into a rotating moving point D 11 Rotating point D 12 Rotating point D 13 ; The expression for the rotating point on bus L1 is as follows: （11）； （12）； （13）； In the formula, θ * Indicates that busbar L1 is wound x The scanning angle of the axis rotation; In O- xyz Take a rotating point D in the coordinate system 11 Rotating point D 12 The line connecting two points represents the spatial position of busbar L1. When busbar L1 rotates... x Axis rotation θ * After the angle, the equation of its generatrix L1 is as follows: （14）。 6. The mathematical modeling method for EPS worm gear tooth profile according to claim 5, characterized in that, Step 5 includes the following steps 501 to 504: Step 501: Generate the tooth profile curve of the EPS worm or EPS worm wheel hob, and number the tooth profile curve of the EPS worm or EPS worm wheel hob. Step 502: Determine the general expression for point D of the T4 tooth profile curve; In O- xyz Point D of the T4 tooth profile curve of the EPS worm or EPS worm wheel hob in the coordinate system ( x D , y D , z D The expression is as follows: （15）； Step 503: Generate the limit scanning angle of the effective tooth profile curve of the EPS worm or EPS worm gear hob. For the rotating point D 13 Bus L1 scan angle θ * Rotation always results in the rotating point D being able to rotate. 13 Falling onto the Z-section, the corresponding scanning angle at this time θ * A It is a limit scanning angle, scanning angle θ * A The expression is as follows: （16）； For the rotating point D 11 There are two possibilities: r f1 ≥| z 1| Situation and r f1 <| z 1| Situation; When r f1 ≥| z In case 1, the bus L1 is scanned at an angle θ * Rotation always results in the rotating point D being able to rotate. 11 Falling onto the Z-section, the corresponding scanning angle at this time θ * B It is a limit scanning angle; When r f1 ≥| z Limiting scanning angle in case 1 θ * B expression: （17）； When r f1 <| z In case 1, the busbar L1 is scanned and rotated, and the rotating point D... 11 It cannot fall on the Z-section, let D be. 11 -D 13 A rotating point D within a line segment w Rotating point D w distance x Let the axis radius be r w =| z 1|, then D w -D 11 Line segments will not generate tooth profile curves after scanning, D w -D 13 Line segments can generate effective tooth profile curves after scanning; When r f1 <| z Limiting scanning angle in case 1 θ * w Rotating point D w Set the starting position of the scan to M. w ( x Mw , y Mw , z Mw ), Scan starting position M w The angle of the scanning starting point is set to θ * w0 When this is taken as the starting point and the scanning angle θ * = θ * w At that time, the rotating point D can be made to rotate. w If it falls tangentially onto the Z-section, then θ * w It is a limit scanning angle; Scan start point M w In M 11 -M 12 On a straight line, the following relationship should be satisfied: （18）； In O- xyz In the coordinate system, the scanning start point M w In x With the axis as the center line, and r w =| z On a cylindrical surface with radius 1, the following relationship is satisfied: Combining it with equation (18), we can obtain the following relation: （19）； In the formula, , ; Equation (19) contains only one unknown quantity. y Mw ,set up ,but Using Newton's iterative formula: Solve f ( x When )=0, the accuracy requirements are met in engineering. x If the value is found, then we can obtain... y Mw The specific values ​​will then be obtained. y Mw The parameter value can be obtained by substituting it into equation (18). x Mw , z Mw The parameter value indicates the scan start position M. w ( x Mw , y Mw , z Mw In coordinate system O- xyz This allows for accurate positioning and the acquisition of the scanning starting point angle. θ * w0 and limit scanning angle θ * w The expression is as follows: （20）； Step 504: Generate the minimum scanning angle θ * min and maximum scanning angle θ * max ; From the extreme scanning angle θ * A Limiting scanning angle θ * B or extreme scanning angle θ * w The minimum scanning angle can be obtained. θ * min and maximum scanning angle θ * max as follows: （21）。 7. The mathematical modeling method for EPS worm gear tooth profile according to claim 6, characterized in that, Step 6 includes: The three characteristics that make point D on the T4 tooth profile curve a potential meshing point between tooth profile curve T4 and EPS worm gear tooth profile curve G4 are: (1) As the rotation angle of the EPS worm or EPS worm gear hob... θ Rotation, tooth profile curve T4 will be x Moving on the axis, each point on it can only move when corresponding to... θ A corner point must be in the correct position to become a potential engagement point; a corner θ Only within a certain rotation angle range can all points on the tooth profile curve T4 become potential meshing points; the T4 tooth profile curve point D that becomes a potential meshing point is related to the rotation angle. θ Scanning angle θ * The normals to point D on the T4 tooth profile curve are mutually corresponding and unique; exist x O y In a planar coordinate system, find the slope of the normal line at point D on the T4 tooth profile curve. k D : （22）； The derivative can be obtained from equation (15). The expression is as follows: （23）； The derivative can be obtained from equation (11). The expression is as follows: （24）； The derivative can be obtained from equation (12). The expression is as follows: （25）； From equations (22) to (25) of the above process, we can obtain that... x O y The slope of the normal line at point D on the T4 tooth profile curve in the planar coordinate system k D As the EPS worm gear or EPS worm wheel hob rotates to... θ Angle point, the normal of point D on the T4 tooth profile curve and y The intersection point of the axes is set as C1(0, y C1 Only when the intersection point C1 coincides with the node C can point D on the T4 tooth profile curve become a potential meshing point; exist x O y In the plane coordinate system, point D of the T4 tooth profile curve ( x D , y D The normal and y Intersection value y C1 The expression is as follows: （26）； like y C1 When r1', the intersection point C1 coincides with the node C, then point D on the T4 tooth profile curve becomes the potential meshing point; r1' represents the pitch circle radius of the EPS worm or EPS worm wheel hob.

8. The mathematical modeling method for EPS worm gear tooth profile according to claim 7, characterized in that, Step 7 includes: As the EPS worm gear or EPS worm wheel hob rotates... θ Rotation, the tooth profile curve T4 on the Z-section will be x Moving on the axis, its T4 tooth profile curve point D ( x D , y D Only when it becomes a potential engagement point and enters the area between the engagement and disengagement points on the engagement line can point D on the T4 tooth profile curve become a true engagement point. x w , y w ) is the contact point where the EPS worm or EPS worm wheel hob meshes with the EPS worm wheel drive, and it also belongs to the G4 tooth profile curve point of the EPS worm wheel; on the Z section, the intersection of the meshing line and the section arc of the EPS worm wheel tooth tip circle is the engagement point, denoted as W1 ( x w1 , y w1 The point where the line of engagement intersects the horizontal line of the section of the tip circle of the EPS worm or EPS worm wheel hob tooth is the disengagement point, denoted as W2. x w2 , y w2 ); The horizontal height of the section line of the tip circle of the EPS worm or EPS worm wheel hob on the Z-section. The radius of the cross-sectional arc of the EPS worm gear tooth tip circle The expression is as follows: （27）； In the formula, This indicates the tip circle radius of the EPS worm gear or EPS worm wheel hob. This indicates the offset of the Z-section. This indicates the radius of the tip circle of the EPS worm gear teeth. This indicates the radius of the tip circle of the EPS worm gear teeth. Indicates the outer radius of the EPS worm gear; Entry point W1 ( x w1 , y w1 The expression is as follows: （28）； In the formula, , , ; Disengagement point W2 ( x w2 , y w2 The expression is as follows: （29）； Combining equation (15), on the Z-section, point D of the T4 tooth profile curve ( x D , y D ) can become the meshing point W ( x w , y w The conditional expression for ) is as follows: （30）。 9. The mathematical modeling method for EPS worm gear tooth profile according to claim 8, characterized in that, Step 8 includes: According to the EPS worm gear rotation angle φ The meshing point W of equation (30) x w , y w ) transformed into x H O H y H Point W of the EPS worm gear G4 tooth profile curve at the zero position in the coordinate system H ( x H_w , y H_w Its expression is as follows: （32）； In the formula, H represents the center distance between the EPS worm or EPS worm wheel hob and the EPS worm wheel.

10. A method for evaluating the tooth profile error of an EPS worm gear, characterized in that, Includes the following steps 9 to 10: Step 9: Determine the reference curve for the EPS worm gear tooth profile; Using an ideal EPS worm gear hob, an ideal tooth profile curve point of EPS worm gear that uniquely corresponds to the hob rotation angle θ can be generated on the Z-section. Continuous rotation of the hob rotation angle θ can generate countless ideal tooth profile curve points. Connecting these points is the reference for the EPS worm gear tooth profile curve on the current Z-section. Step 10: Use the EPS worm gear tooth profile curve reference to evaluate the EPS worm gear tooth profile error; The conditional expression for the effective evaluation range of EPS worm gear tooth profile error: （34）； In the formula, The radius of the addendum circle of the EPS worm or EPS worm wheel hob is indicated by H, and the center distance between the EPS worm or EPS worm wheel hob and the EPS worm wheel is indicated by H. exist x H O H y H In the coordinate system, let the reference point of the EPS worm gear tooth profile curve on the Z section be designated as W. H0 ( x H0_w , y H0_w The reference point locations are numbered 0, 1, 2, ..., i-1, i, i+1, ..., 719, 720. The code for the EPS worm gear profile curve point being evaluated is still set to W. H ( x H_w , y H_w Let its position number be 0, 1, 2, ..., j-1, j, j+1, ..., 719, 720; let the evaluated EPS worm gear tooth profile curve point W H Reference point W of EPS worm gear tooth profile curve H0 The distance between them is L j,i Let all distances L j,i The minimum distance is L j_min1 The corresponding reference point is W. H01 ( x H0_w1 , y H0_w1 The second minimum distance is L. j_min2 The corresponding reference point is W. H02 ( x H0_w2 , y H0_w2 At this point, the tooth profile curve point W of the EPS worm gear number j being evaluated... H to the reference point W corresponding to these two minimum distances H01 and W H02 The distance between the lines is the error Δ of the EPS worm gear tooth profile curve point. j Its expression is as follows: （35）； In the formula, , .

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