Method for designing parallel axis helical gear tooth profile having separated tooth profile

Abstract

To solve a first challenge of developing a method to use numerical integration to determine a rack tooth profile with a desired relative curvature of a tooth surface along a contact line for the tooth surfaces of a parallel axis "helical" gear, and a second challenge that although a tooth profile with uneven tooth surface contact and a small relative curvature of the tooth surface has a "tooth profile portion that connects" a tooth base side tooth profile and a tooth tip side tooth profile, there is a problem that contact stress under load is large in the connecting tooth profile portion, and it is necessary to clarify whether the connecting tooth profile portion can be eliminated.SOLUTION: Using results of research that determined a differential relational expression between a rack tooth surface curvature and a relative tooth surface curvature of a gear pair to be cut, a method was developed to obtain a rack tooth profile that gives a constant value of the relative tooth surface curvature using a numerical integration technique. Also, presence or absence of a connection tooth profile portion was examined by clarifying a relation between a trajectory of a contact point of a tooth profile and a zero pressure angle line.SELECTED DRAWING: Figure 4

Description

[Technical field]

[0001] The present invention relates to a method for designing the tooth flank form of a helical gear with a parallel axis. [Background technology]

[0002] Involute tooth profiles have many advantages, such as no effect from changes in the center distance, easy tooling, and the existence of wisdom and knowledge based on a great deal of research and proven results. However, problems arise such as tooth flank interference near the base circle of tooth profiles with a small number of teeth, and a large relative curvature of the tooth flank. In recent years, it has become possible to create complex shapes with high precision, and attempts have been made to design tooth profiles that incorporate the meshing of convex and concave tooth surfaces, whereas involute tooth profiles have convex tooth surfaces that come into contact with each other, in order to improve the surface pressure capacity, which is one of the evaluations of gear load capacity. Here, the contact stress (Hertzian stress) at the contact point of the tooth surfaces is also called "surface pressure." Also, "line pressure" is the load per unit length of the contact line.

[0003] The basic shape of a WN gear (Symmark gear; Patent Document 1) is one in which the tip of one gear tooth profile is a convex arc tooth profile and the base of the tooth profile is a concave arc tooth profile, the connecting tooth profile portion of both tooth profiles is a straight line, and the centers of these arc tooth profiles are on the rolling pitch circle. Because the contact of the arc tooth profiles causes the teeth profiles to mesh with each other instantaneously, it cannot be used as a spur gear, but is used as a helical gear. All tooth surface contact is by convex and concave tooth surfaces, the relative curvature of the tooth surfaces is extremely small, and the surface pressure load capacity is significantly greater than that of an involute tooth profile. The WN gear (Symmark gear tooth profile) uses a contact point path that does not pass through the rolling pitch point, resulting in a tooth profile with separate tooth tip and root sides, and a significantly smaller relative tooth surface curvature than the involute tooth profile. However, there is a problem in that the teeth start to mesh instantaneously at one end of the tooth width and disengage instantly at the other end of the tooth width, which causes a large excitation force. Also, the Symmark gear tooth profile is a tooth profile in which the tip side tooth profile and the base side tooth profile are separated, and the straight line part of the joint is designed not to come into contact.

[0004] If both the tooth tip and dedendum of one gear are cycloidal curves, the tooth flanks at the tip and dedendum will mesh unevenly, but the relative curvature of the tooth flanks near the pitch point will be infinite.Therefore, there are gears that connect the tooth tip and dedendum portions by removing the tooth profile near the pitch point and replacing the removed portions with an involute tooth profile (Non-Patent Document 1).In this gear, the contact stress (surface pressure) of the involute portion is still large compared to the cycloid portion, so there are also gears that recess the tooth profile portion of this connecting portion to prevent contact (Patent Document 2).

[0005] As a tooth profile that smoothly integrates a concave-convex meshing tip side tooth profile and a root side tooth profile with a joining curve like an involute that meshes convexly and concavely, there is a tooth profile that adopts a clothoid curve as a rack tooth profile, in which the curvature of the tooth profile increases in proportion to the tooth profile line length (Patent Document 3), and there is also a tooth profile in which the rack tooth profile is a sine curve, in which the curvature is small near the pitch point and the curvature increases as it approaches π / 2.When these tooth profiles are attempted to fit within the tooth gap width, if the relative curvature of the tooth flanks is attempted to be reduced as a concave-convex meshing near the tip or root, the relative curvature of the convex-convex meshing tooth flanks near the pitch point becomes large, and the relative curvature becomes larger than that of an involute tooth profile near the pitch point.

[0006] If the tooth surface has some areas where the contact stress is high and some where it is low due to the meshing portion of the tooth profile, damage will occur in the areas where the contact stress is high, so a tooth profile with a constant relative curvature has been proposed over the entire tooth profile that follows the tooth profile of a "straight-tooth" gear (Patent Document 4). Involute tooth profiles have problems with tooth profile interference and large relative curvature in the vicinity of the base circle, but this method helps to solve these problems. It has also been proposed to make the vicinity of the pitch point an involute tooth profile, rather than the entire tooth profile, and to make only the tip and root portions of the tooth profile a constant curvature tooth profile (Patent Document 5).

[0007] A tooth profile with a constant relative curvature along the tooth profile of a "straight" gear is available in which a tooth profile with a value of the relative curvature of the tooth profile over the entire tooth profile is determined to obtain a load capacity greater than that of an involute tooth profile (Patent Document 6), and this tooth profile for a "straight" gear is converted to the tooth profile of a "helical" gear. Similar to the WN tooth profile, this tooth profile with a constant relative curvature is a tooth profile with a tooth profile separated into a "tooth tip side tooth profile portion" and a "tooth base side tooth profile portion", and a "connecting tooth profile portion" to connect these separated tooth profiles, and is composed of three tooth profile portions. The tooth contact stress at the tooth tip and root is small, but the tooth contact stress at the connecting tooth part is large. However, the connecting tooth part is narrow and is not considered to be a big problem in practice, and some efforts have been made to prevent the connecting tooth part near the pitch point from contacting (Patent Document 7). In this tooth profile, the relative curvature of the tooth profile is constant for the "straight" gear, and the tooth profile for the "straight" gear is converted to a "helical" gear, so the relative curvature of the tooth profile is not strictly constant for the tooth surface of the "helical" gear. It is unclear whether this is related, but (Patent Document 8) shows a design method for determining the value of the relative curvature of the tooth profile based on the face width and the tooth ratio.

[0008] The trend of improving the surface pressure load capacity of the tooth flank has been described above, but at present, the trend is to impose a "constant relative curvature constraint" that keeps the tooth flank relative curvature constant over the entire tooth flank in order to improve the load capacity. Also, the trend is to make the value of the constant relative curvature smaller than the value of the tooth profile relative curvature at the pitch point of the involute tooth profile, and closer to the relative curvature of the WN tooth profile. The tooth profiles introduced in the previous section (Patent Documents 6, 7, and 8) meet these requirements, but still have two issues. First, the angle between the contact line and the tooth trace line of a "helical" gear tooth surface is not constant in the case of a constant relative curvature tooth profile, so the relative curvature for a "straight" tooth surface cannot be used to calculate the accurate relative curvature of a "helical" tooth surface. It is necessary to develop a method for calculating a constant relative curvature tooth surface that can be applied to a "helical" gear tooth surface. The second point is that gears with small relative curvatures on the tooth tip and base sides, aiming for uneven tooth surface contact, can be said to be a common issue for several gear tooth profiles, including gears that combine cycloids and involutes, but the relative curvature of the tooth surface of the "connecting tooth surface portion" between the two tooth surface portions with small relative curvatures is large, and how to deal with this tooth surface portion is a common issue for several gears. While a method of making the "connecting tooth surface portion" recessed to prevent contact is an effective method, it is also a meaningful issue to clarify whether it is possible to reduce this tooth surface portion to zero. [Prior art documents] [Patent documents]

[0009] [Patent Document 1] Published patent, 1980-132459, Application date: 1979.4.2, Matsunaga, Nogami, Shifted circular-arc tooth gear [Patent Document 2] US0134184A, Filed: Jan.31,2002, RM Hawkins, Title: Non-involute gears with conformal contact. [Patent Document 3] JP109838, Application dated April 6, 2006, Miyaoku, Cornu Spiral Tooth Gear [Patent Document 4] US3631736A, Filed: Dec 12.29,1969, Patented Jan. 1.4,1972, OE Saari, Title: Gear tooth Form [Patent Document 5] US4640149A, Filed: Mar.4,1983, R. Drago, Title: High profile contact ratio, Non-involute gear tooth form and method [Patent Document 6] US6101892A, Filed:Apr.10,1998, BE Berlinger, JR Colbourne, Title: Gear Form Constructions [Patent Document 7] JP523405A, Application (April 21, 2003), Applicant: Genesis Partners, Gear Tooth Profile [Patent Document 8] JP514911A, Application (2004.12.16), Applicant: Genesis Partners, Gear tooth profile curvature [Non-patent literature]

[0010] [Non-Patent Document 1] Terauchi, Nagamura, and Saijo, Design and Performance of Involute-Cycloid Composite Tooth Gears, Transactions of the Japan Society of Mechanical Engineers, C, (1981) 47-417, pp.663-674. [Non-Patent Document 2] Sakai, Study on the Tooth Profile of Hypoid Gears, Transactions of the Japan Society of Mechanical Engineers, (1955) 21-102, pp.164-170 [Non-Patent Document 3] Shimaji and Fujii, Study on the Design Reference Point of Non-Intersecting Gears (2nd Report, Regarding Interference), Transactions of the Japan Society of Mechanical Engineers, (1977-4) 43-368 Summary of the Invention [Problem to be solved by the invention]

[0011] At the center of the tooth width of a "helical" gear, there is a full-length contact area where the contact line extends from the base of the tooth to the tip of the tooth. If you try to incorporate contact between concave and convex tooth flanks at the contact line of the full-length contact area of ​​the tooth flank of a helical gear, if you make the relative curvature of the tooth flank smaller in a part, the relative curvature of the tooth flank in the remaining part will become larger. Therefore, in order to maximize the surface pressure load capacity within a limited tooth height, it is effective to flatten the contact stress distribution along the contact line, in other words, to make the relative curvature of the tooth flank a constant value along the contact line, or to impose a "constant value relative curvature constraint" that controls that value. In Patent Document 4 and Patent Document 6, which apply the "constant relative curvature constraint," a relational equation for the path of contact points that is applicable to "straight" gears (spur gears) and that keeps the relative curvature of the tooth profile constant is used. In the case of the tooth flank of a "helical" gear, the angle between the tooth trace line and the contact line changes along the tooth profile, so the relational equation for a "straight" gear cannot be used. To address this issue, the first objective of the present invention is to develop a method for setting the relative curvature of the tooth flank of a "helical" gear to a desired value along the contact line.

[0012] As a result of aiming for concave-convex tooth surface contact, the tooth surface parts with small relative curvature of the concave-convex tooth surface are formed on the tooth tip side and the tooth root side, but conversely, the tooth surface parts with large relative curvature of the convex-convex tooth surface are formed near the pitch point. The tooth surface parts with large relative curvature of the tooth surface are "tooth profile parts connecting" the tooth tip and root tooth surfaces. Such gears are a common issue for several gear tooth profiles, including WN gears and gears that combine cycloids and involutes. It is a meaningful issue to clarify whether or not the "connecting tooth profile parts", which are the tooth surface parts with large relative curvature of the tooth surface, can be made zero. In this invention, the second issue is to clarify whether or not the "connecting tooth profile parts" can be made zero under the "constraint of constant relative curvature". [Means for solving the problem]

[0013] The theory of general gear tooth profiles shows that the relative curvature of the tooth flanks of the work gear pair can be calculated by giving the curvature of the tool gear tooth flank that is in line contact with the work gear pair. Using this, the curvature of the rack tooth profile can be determined by giving a desired value as the value of the relative curvature of the tooth flanks. Therefore, at the Nth contact point, the tooth profile curvature K is calculated on the normal line of the rack tooth profile. N The center point of the tooth profile arc is determined, and the (N+1)th tooth profile point is determined on the tooth profile arc in the vicinity of the Nth point, thereby obtaining the rack tooth profile by numerical integration.

[0014] The problem is to investigate the shape of the "connected tooth profile portion" under the "constraint of constant relative curvature." Patent Document 6 specifies a point near the pitch point as the starting point of the tooth profile, but does not mention how to determine this nearby point. The tooth profile between the pitch point and the nearby point is the "connected tooth profile portion," and a means of addressing this problem is to investigate the tooth profile near the pitch point based on the shape of the path of contact points of the tooth flanks where the value of the tooth flank relative curvature is constant in parallel axis gears. Effect of the Invention

[0015] According to the present invention, it is possible to accurately determine a tooth flank such that the relative curvature of the tooth flank along the tooth profile of a parallel axis "helical" gear is a desired value, which simplifies the evaluation of the surface pressure load capacity of the tooth flank and simplifies the tooth flank design system. It is also possible to show a calculation method that can eliminate the "tooth profile portion connecting" the tooth tip side tooth flank and the tooth base side tooth flank, which have a small relative curvature of the tooth flank. [Brief description of the drawings]

[0016] [Figure 1] Given the Nth tooth profile point, normal, and curvature radius of a tooth profile, the procedure for finding the N+1th tooth profile point and normal is as follows: [Diagram 2] The change in the contact locus depending on the value of the relative tooth flank curvature is shown. [Diagram 3] The contact lines on the rack teeth are shown separating. [Figure 4] Illustrates an example of a rack tooth profile and the progression of contact between the teeth as the gears rotate. [Diagram 5] This shows how the gap between the tooth profiles of both racks changes when the amount of reduction in the center distance is changed. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0017] The first issue is to develop a method for designing the tooth surface of a helical gear so that the relative curvature of the tooth surface along the contact line is the desired value. Patent Document 4 adopts a method for determining the shape of the locus of contact points so that the straight tooth profile has a constant relative curvature from the differential relationship of the locus of contact points in the cross section perpendicular to the axis. Patent Document 6 uses a method for determining the straight gear tooth profile from the Euler-Savary equation, which shows the relationship between a pair of tooth profile curvatures of the large and small gears at points on the locus of contact points. In contrast to these, this paper adopts a method for determining the shape of the rack tooth profile for a helical gear so that the value of the relative curvature of the tooth surface is the desired value.

[0018] There is a study that discusses the tooth profile of general gears and obtains the relational expression between the curvature of the tooth surface of a tool gear and the relative curvature of the tooth surface at the contact point of the tooth surface of a line contact gear. In parallel axis gears, the condition that the rack is a tool gear is added to the relational expression, and the relative curvature of the tooth surface at the cross section perpendicular to the contact line, k S can be obtained by the formula (1-a). The formula (1-a) describes the tooth flank shape relationship that handles the contact and envelopment relationship between the tooth flanks of general gears as a vector operation formula, and the differential amount of the tooth flank normal is expressed in affinol. In order to prevent misuse, it is best to read Non-Patent Document 2 when using the formula.

[0019]

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[0020] Hope k S There are several methods for calculating k conversely from k, but since the curvature k of the rack tooth profile can be determined by a trial and error iterative approximation method, for example, there is no problem. In Fig. 1, the Nth rack tooth profile point r N , the tooth surface normal at that point n N and curvature k N Here is an image of the numerical integration to find the N+1th point when the Nth point r N Curvature k N The reciprocal of the radius of curvature is R N Then, point r N From normal n N Distance R on the extension line ofN Point O N Then, point N is set to this point O N Around a small angle δ N Rotated by r N+1 Let be the N+1th point. The surface normal of the N+1th point is the normal n of point N. N Similarly, the small angle δ N Rotated normal n N+1 Now, the position r of the N+1th point is given as N+1 and normal n N+1 Next, change the rack position to find the position where the N+1th point contacts, and calculate the curvature k of the rack tooth profile at the contact point. N+1 The same process is then repeated.

[0021] When designing a rack tooth profile, the teeth and tooth spaces are placed within the width of one pitch. For example, if there is no addendum in a standard rack tooth profile, the tooth thickness width and tooth space width on the rack rolling line are set to be equal, and the position of the intersection O of the rack rolling line and the tooth profile is determined. In Patent Document 6, the tooth profile is found by starting integration of the tooth profile with point A near intersection point O, and proceeding with numerical integration in a direction tangent to the tooth root curve set at the tooth groove bottom. There is no description of how to determine point A, and it is simply stated that point A is indicated by polar coordinates (s, φ) in a polar coordinate system that represents the path of contact points, and that the radius s and the angle φ between the rack rolling line and the radius line are given. Here, three methods will be shown to clarify whether it is possible to eliminate the "interlocking tooth portion", which is the subject of the present invention. In the first method, the integration start point B is placed on the curve (root circle) at the bottom of the tooth groove, and integration is carried out so that the rack tooth profile with a constant tooth flank relative curvature reaches intersection point O. The position of integration start point B and the rack tooth profile length are adjusted to determine the tooth profile. With this method, a tooth profile with the desired relative curvature can be obtained from point B to point O. Conversely, in the second method, end point A of a straight tooth profile with a short length e extending from intersection point O and having a tooth flank pressure angle φ is set as the integration starting point, and the length of the tooth profile from point A to the tangent point B between the rack tooth profile and the tooth root circle and the length e of the straight tooth profile are adjusted. Patent Document 6 is similar to this method. In this method, the tooth flank relative curvature of the straight tooth profile portion up to point A is quite large, but this large value gradually changes until it becomes the desired small relative curvature, and the tooth profile in the vicinity of point A does not have the desired relative curvature. The third method, which will be described later, is a method in which integration proceeds from the intersection point O in the direction of the tooth root curve. The first method is a basic method for determining a rack tooth profile having a desired relative tooth flank curvature in the present invention. In the second method, if the value of the relative tooth flank curvature of the short straight tooth profile portion (the value of the relative tooth flank curvature at the pitch point) is set equal to the value of the relative tooth flank curvature given as a constant value, a tooth profile similar to an involute tooth profile is obtained, and for convenience, this tooth profile is called the IS tooth profile. The second method is an effective method for determining the IS tooth profile with high accuracy. There is a method called Method 2-1. In this method, after the rack tooth profile is obtained using Method 2, numerical integration is performed in the reverse direction from point E, which is 2 to 3 times the distance OA, toward point O from point A. This method can cause a problem in which the tooth profile cannot reach intersection point O, requiring a different measure, but this will not be discussed here. EXAMPLES

[0022] In the first method of numerical integration, the relative curvature of the tooth surface was set to a constant value along the contact line, and the path of the contact point was calculated by changing the value of the constant. The specifications of the trial calculation example are as follows. Number of teeth: 15 / 45, Determine the ratio of rack tooth groove width to tooth thickness width based on design requirements and determine the intersection point O of the rolling pitch line and tooth profile. The center of the tooth root circle with a radius of 0.38 m is placed at the (1 / 4) pitch position. Rack tooth profile bottom depth = 1.25m (m: module), A "helical" gear with a twist angle of 30°.

[0023] We will explain the coordinate system used to explain the path of the contact point. The line connecting both gear axis points in a plane perpendicular to the gear axis is called the "center line," the "pitch point" on the center line, and the line that passes through the pitch point and is perpendicular to the center line is called the "zero pressure angle line." The movement of the contact point within the coordinate system created by these is called the "path of the contact point." The shape of the locus of contact points has a one-to-one correspondence with the tooth flank shapes of helical and straight bevel gears and racks, and is useful for grasping the characteristics of the performance of gear tooth profiles. Furthermore, no matter where in the axial direction the plane perpendicular to the gear axis is located, the shape of the locus of contact points of helical and straight bevel gears does not change.

[0024] In order to focus on the state of the contact locus near the pitch point and the zero pressure angle line, the numerical integration step was made extremely fine to obtain the contact locus, as shown in Figure 2a. Figure 2a shows only the contact locus on the root side of the pinion. Relative tooth surface curvature k S When = 0.12, the path of the contact point is almost a straight line, and the rack tooth profile is close to the straight line of the involute tooth profile. For convenience, this type of tooth profile will be called the IS tooth profile. Relative tooth surface curvature k S k greater than =0.12 S The locus of contact points when = 0.24 intersects with the center line connecting the two gear shafts and is away from the zero pressure angle line. This type of tooth profile is called the FS tooth profile. Relative tooth surface curvature k S A tooth profile having a locus of contact points for a relative curvature smaller than .times.0.12 is referred to as an SS tooth profile. What should be noted is whether the trajectory of the contact points of the SS type intersects with the center line connecting the gear shafts and with the zero pressure angle line. This could not be determined from the trajectory curve calculated numerically.

[0025] In tooth profile theory, as written in technical books, a tooth profile with a "path of contact points that crosses the zero pressure angle line" cannot be used because the sign of the relative curvature of the tooth surface changes at the crossing point. Also, "path of contact points close to the zero pressure angle line" means that the tooth surface pressure angle is close to zero, and it is known that tooth surface interference occurs just by doing this. These points apply to the WN tooth profile, and the contact line (which is also the path of the contact point in the WN tooth profile) cannot be close to the zero pressure angle line. For the SS tooth profile to be able to avoid these points, common sense would dictate that the path of the contact point of the SS tooth profile must extend along the zero pressure angle line. However, as will be explained below, the path of the contact point of the SS tooth profile crosses the zero pressure angle line, which is unreasonable.

[0026] The theoretical formula for the relative curvature K of a gear tooth profile (the basic part of Non-Patent Document 2) is expressed as a fractional formula, whose denominator is the square of the relative velocity V, and whose numerator is the product of the sine value sin(φ) of the tooth flank pressure angle φ and the slip ratios σ1 and σ2 of the tooth flank. Unless the value is special, such as zero or infinity, the slip ratio at the intersection of the contact path and the zero pressure angle line will not be a special value. In this case, the common sense result described in the previous section will be obtained. Therefore, the intersection point is considered to be a special point where the sliding ratio of one tooth surface is infinite. First, a tooth profile having a "locus of contact points that crosses the zero pressure angle line" cannot be used because the sign of the relative curvature of the tooth flank changes at the point of crossing. In the case of a special point, the product of the sliding ratios σ1 and σ2 of both tooth profiles is (infinity x 1), and since the pressure angle φ at the tooth flank contact point is zero, the relative curvature of the tooth flank becomes indefinite (zero x infinity) at the limit to the intersection point, and the relative curvature K can maintain a finite value. Here, infinite slip ratio means that the normal to the contact point path at a particular point passes through the gear axis. Look at the part (circled area) in Figure 2 where the path line with a relative curvature value of 0.04 approaches the zero pressure angle line. The figure also shows a circular arc path that is tangent to the path in question and where the slip ratio is infinite. There is no contradiction in thinking that the path line crosses the zero pressure angle line at the circled area where the slip ratio is infinite at the intersection with the zero pressure angle line. In terms of tooth profile theory, this intersection is the point where both the sliding ratios σ1 and σ2 become indeterminate. However, going back to the original definition of sliding ratio, if σ1 keeps a large value while approaching the intersection, σ 21 can be understood as approaching the intersection while remaining close to 1. The circled area in Figure 2a corresponds to this explanation. Furthermore, in another numerical calculation performed for confirmation and verification, it was confirmed that the locus of contact points with a constant relative curvature intersects with the zero pressure angle line, and even after passing this line, the positive and negative signs do not change, satisfying the requirement of a constant relative curvature, and there is no contradiction with the tooth profile theory. Next, "the locus of contact points is close to the zero pressure angle line" means that the tooth flank pressure angle is close to zero, and we will consider whether tooth flank interference occurs due to this alone. Regarding tooth flank interference, when the pressure angle of the tooth flank of a parallel axis gear is zero, NQ (crossover) interference generally occurs, but interference can be avoided at the singular point where the slip ratio is infinite (Non-Patent Document 3 explains NQ interference in the tooth flank of a general gear). There is no problem in this regard either. As shown in Figure 4, the meshing progression diagram of the tooth profile also confirms that no interference occurs. In this way, by satisfying certain special conditions, it is possible to create a "hole" in the zero pressure angle line through which the locus of contact points can pass.

[0027] If the path of the contact points used as the tooth profile is used up to the point where it intersects with the zero pressure angle line, the tooth profile will include a point where the pressure angle is zero. When creating the tooth surface by generating processing, it is expected that there will be a problem of whether the tooth profile can be created accurately. The tooth profile near the point where the pressure angle is zero is the part where the cutting edge cuts the vicinity of the zero pressure angle point many times during generating processing, compared to the tooth profile part with a large pressure angle, and it is expected that the tooth profile will be overcut. As for tooth surface modification, I will touch on it at the end, but I think that it is possible to obtain a tooth surface modification effect like a recess, which is theoretically impossible, which is desired as a response to the center distance error. In addition, even if the tooth surface is as theoretically expected, the method described below can be considered.

[0028] It shows that the position of the hole where the contact point locus can pass through the zero pressure angle line in the rack tooth profile tangent to the dedendum circle is determined by the value of the relative curvature of the tooth surface, and the first method of numerical integration automatically passes through the hole. In contrast, the position of the hole is unknown, and the second method of numerical integration, which advances the integration in the direction away from the zero pressure angle line, inevitably includes a "connected tooth profile part." Also, the contact point locus of the connected tooth profile passes through the pitch point.

[0029] In the third method, a method for finding the tooth profile by giving a relative curvature value along the tooth profile will be explained. There is a possibility of finding the path of the contact point by setting the position of the hole on the zero pressure angle line, through which the path of the contact point can pass, as the starting point of the numerical integration. In this method, it is necessary to give the tangent direction of the path of the contact point at the hole position and the integration start point so that the slip ratio of one of the gear tooth profiles becomes infinite. Specifically, the intersection point O of the rack rolling pitch line and the rack tooth profile is set as the integration start point, and the curvature of the rack tooth profile is set to satisfy formula (1-b) at the starting point and its immediate vicinity. The method for determining the gear tooth profile has the problem that the slip rate of one of the gear tooth profiles is infinite, i.e. the radius of curvature of the tooth profile is zero, but with the method for determining the rack tooth profile of the present invention, even if the slip rate of one of the gears is infinite, the curvature of the rack tooth profile will not become an extreme value. Figure 2b shows an example of the rack tooth profile calculated using the third method, the path of contact points, and the state where the path of contact points near intersection point O comes into contact with the line of infinite slip rate.

[0030] The fact that the path of contact points of the SS tooth profile found by the first numerical integration method extends to the zero pressure angle line means that if the tooth profile is divided into a tooth tip side tooth profile and a tooth root side tooth profile with the rolling pitch line of the rack as the boundary, the tooth profile has a constant relative curvature of the tooth surface over the entire tooth profile from the intersection of the rolling pitch line and the tooth profile to the tooth tip or root. In other words, the rack tooth profile does not have a "connecting tooth profile portion."

[0031] An example of the contact line on the rack tooth surface of an IS tooth profile and an SS tooth profile is shown in Figure 3. Also, an example of the progression of meshing of an SS rack tooth profile and an SS gear tooth profile is shown in Figure 4.

[0032] The present invention makes it possible to calculate the relative tooth flank curvature of a "helical" gear. In reality, gear teeth are flexible and the tooth flank load direction at the contact point is not constant, so even if the tooth flank has a constant relative tooth flank curvature, the load (linear pressure) per unit length of the contact line is not strictly constant. However, the average value of the tooth flank pressure angle, which is the tooth flank load direction, over the entire tooth flank is close to the pressure angle of an involute tooth profile. Therefore, in order to roughly compare the surface pressure load capacity of different types of tooth profiles, we will calculate the surface pressure load capacity assuming that "the relative tooth flank curvature is constant along the contact line" and that "the linear pressure is also constant." When a load is applied to two cylinders in line contact, the maximum Hertzian stress P0 at the contact point is expressed by equation (2). "Maximum Hertzian stress" is simply called "Hertzian stress" or "surface pressure". E i are the Young's moduli of the two cylinders, and ν i is the Poisson ratio, R i is the radius of the cylinder, L is the length of the contact line of the cylinder,

[0033]

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[0034] According to formula (2), when the allowable surface pressure P0 and the relative curvature (1 / R), which is the desired relative curvature ks of the tooth surface in the case of an SS tooth profile, are given, the line load (P / L)0, which is the load per unit contact line length, is determined by formula (3). The total tooth surface load is calculated by integrating the line load over the entire contact line within the tooth tip limit.

[0035]

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[0036] Calculation using formula (3) may be useful for evaluation in the early stages of design, but in reality, it is not simple, as the direction of the force applied to the contact point and tooth deflection must be taken into consideration. The contact stress on the gear tooth surface with a constant relative tooth surface curvature is calculated using a simulator such as the finite element method, and the value of the given relative tooth surface curvature is adjusted depending on the position of the contact line so that the contact stress value is constant over the entire tooth surface. However, the evaluation method using formula (3) is considered to be quite useful for comparing the approximate load capacity of different types of tooth profiles. The main specifications of the gear for the trial calculation are the same as those of the first embodiment. The trial calculation is performed on the pinion tooth surface. The total load on the tooth surface is the maximum allowable surface pressure P0 = 200 kgf / mm 2 The results of trial calculations given the desired relative curvature of the tooth surface are shown in the upper part of Table 1. For comparison, the results for an involute tooth profile are also shown in the lower part of Table 1. In the case of IS and SS tooth profiles, the pinion tooth root tooth profile quantities are doubled for comparison with involutes. Also, the area within the gear tip limit of the contact line is considered valid. In the case of an involute tooth profile with no profile shift, the gear ratio is 3, the undercut interference limit is reached at the base of the pinion, and the relative curvature of the tooth surface becomes abnormally large, resulting in a small total load of 1128 kgf. Since this is not desirable for comparison, the value of the tooth surface equivalent to a 0.7 m profile shift is also shown.

[0037] [Table 1]

[0038] Patent Document 8 shows a design method in which, for a “straight” tooth profile, the value of the tooth profile relative curvature to be given as a constant value is uniquely determined by the “tooth ratio and face width” based on the tooth profile relative curvature at the pitch point of an involute tooth profile as a standard. In the design of the SS tooth profile of the present invention, the relative curvature of the "helical" tooth surface k S It is not believed that the value of is related to the gear ratio and face width, but rather depends on the requirements placed on the gear device. Constant relative tooth flank curvature k S The smaller the value, the shorter the length of the contact line on the tooth surface, and the closer it is to the WN tooth profile. Contact begins at the end of the helical tooth flank, and the contact line disappears at the other end. The load on the contact line, which changes in length and moves, generates a moment around the face width centerline that changes with gear rotation. This change is considered to be one of the exciting forces that cause gear vibration. curvature k S If the contact line length is short, the excitation force tends to be large, approaching the WN tooth profile, which is considered to have a large excitation force. In other words, in applications where the excitation force is not an issue, the tooth surface relative curvature k S If the value of is reduced to about 0.01 in Table 1, which is considered to be close to the theoretical limit, the surface pressure load capacity will be 3 to 4 times that of the IS tooth profile. When insensitivity to changes in center distance like an involute and small excitation force are required, the relative curvature of the tooth surface k S The value of should be chosen to be closest to the value of the IS tooth profile.

[0039] Representative evaluations of gear load capacity include surface pressure strength, bending strength, and scoring damage. In addition to these gear load capacities, it is also important to consider that the center distance after gear assembly differs from the design value by an error δ due to processing and assembly errors. In tooth profile modification as a countermeasure against center distance errors, the tooth contact center point is set approximately in the center of the tooth tip side tooth profile and the tooth root side tooth profile, and the tooth profile is modified in a direction that cuts the tooth flank in a quadratic curve, for example, as it moves away from the center point. In this case, the tooth flank near the pitch point is recessed. However, in the case of an SS tooth profile that does not have a "connected tooth profile portion," the tooth flank pressure angle at the pitch point is zero, and theoretically, while it may be possible with a forming method, it is not possible to recess the tooth flank near the pitch point with a generating method. One method of tooth profile modification is to modify the tooth profile only on the tip side of the tooth flank of both large and small gears from the center point of tooth contact on the tip side, as shown in Figure 5, and then make the center distance slightly shorter than the normal position when assembling the gears. Figure 5 shows the tooth profile clearance when the tooth tip of a module 2.5 tooth is modified by 0.015 and the normal center distance is 0.000, -0.050, and -0.100. If the assembled center distance is -0.050, an error in center distance of ±0.050 is allowed.

Claims

1. The gear tooth surface, pinion tooth surface, and rack tooth surface of a parallel-axis "helical" gear have a common contact line, and if the rack tooth surface or tooth profile is specified, the gear tooth surface is determined. There is a study that calculates the relative curvature of the tooth surface at the tooth surface contact point assuming that the curvature of the rack tooth surface is known. Using the results of this study, the relative curvature of the gear tooth surface at the Nth contact point can be set to a desired value K. Ｓ、Ｎ The curvature K of the rack tooth profile is given as Ｎ At the contact point, the tooth curvature K Ｎ The center point of the tooth profile arc is determined, and the N+1th tooth profile point is determined as a point on the tooth profile arc in the vicinity of the Nth contact point. Next, the rack is moved to the position where the N+1th point contacts, and the relative curvature of the gear tooth surface at the contact position after the movement is set to the desired value K. Ｓ、Ｎ＋１ The curvature of the rack tooth profile is given as Ｎ＋１ The desired value K is calculated along the contact line of the helical tooth surface by the numerical integration method. Ｓ、N A method for obtaining a rack tooth profile with a relative tooth surface curvature of

2. The line connecting both gear axis points in a plane perpendicular to the gear axis is called the "center line", the "pitch point" on the center line, and the line that passes through the pitch point and is perpendicular to the center line is called the "zero pressure angle line". Within the coordinate system created by these, the movement of the contact point is captured as the "locus of contact points". The value of the relative curvature at the pitch point of a tooth profile that is tangent to the rack bottom curve and whose locus of contact points passes through the pitch point is determined like the value at the pitch point of an involute tooth profile. When designing a tooth profile with a tooth surface relative curvature smaller than this value of relative curvature, the tooth profile is separated into a tooth tip side tooth profile and a tooth base side tooth profile. In order to eliminate the need for a "tooth profile connecting" the tooth tip side tooth profile and the tooth base side tooth profile by taking the tooth profile to be the point where the path of contact points of the rack and the rack bottom curve intersects the zero pressure angle line, or to a point very close to that intersection, when determining the rack tooth profile by numerical integration, the integration is advanced in the direction from the point of contact between the rack bottom curve and the rack tooth profile toward the intersection O of the rack rolling line and the rack tooth profile, or in the direction toward the zero pressure angle line in terms of the path of contact points, thereby determining a parallel axis gear tooth profile that does not have a "connecting tooth profile portion" between the tooth profile from point O to the tooth tip and the tooth profile from point O to the tooth base.

3. Regarding claim 2, when a tooth profile is obtained by numerical integration while giving a desired relative curvature value along the tooth profile, in order to obtain a tooth profile that does not have a "connecting tooth profile", the intersection position of the contact point path and the zero pressure angle line is set as the starting point of the numerical integration, and the requirement that the slip ratio of one of the gear tooth profiles as the limit value to the starting point becomes infinite is given, the curvature of the rack tooth profile at the intersection point O of the rack rolling line and the rack tooth profile is determined, and the numerical integration is carried out from the intersection point O in the direction of the rack tooth bottom curve, thereby obtaining a parallel axis gear tooth profile that does not have a "connecting tooth profile portion".

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