JP5458063B2 - Gear and gear tooth shape design method - Google Patents

Description

  The present invention relates to a gear, and more particularly to its tooth profile curve.

  Gears having an involute tooth profile (involute gears) are widely used because of their ease of manufacture and high versatility. In the involute gear, it is known that the meshing loss is reduced by reducing the basic circle radius, that is, by increasing the pressure angle. Further, by reducing the radius of the basic circle, the tooth thickness at the tooth base increases, so that the bending strength of the tooth is also improved. On the other hand, when the basic circle radius is reduced, the meshing length is decreased and the meshing rate is decreased. Due to the reduction of the meshing rate, vibration and noise caused by the gears may be deteriorated.

  Thus, in the conventional involute gear, the meshing loss, the bending strength of the teeth, and the meshing rate are contradictory events. In the gear of the following patent document 1, the technique which makes a pressure angle different in a tooth root side and a tooth end side is described. The pressure angle on the tooth root side of the small gear is made larger than the tooth end side to increase the tooth root strength, and the tooth depth on the tooth end side is lengthened to increase the meshing rate.

JP 2001-271889 A

  In Patent Document 1 described above, the tooth depth is increased in order to increase the engagement rate. However, an increase in toothpaste leads to an increase in gear size and weight.

  An object of this invention is to provide the gear which has a softness | flexibility regarding the change of characteristics, such as a meshing loss, the bending strength of a tooth | gear, and a meshing rate.

  The gear of the present invention is a gear applying an involute tooth profile. In the involute tooth profile, the basic circle radius, which is one of the parameters for determining the tooth profile, is a fixed value. On the other hand, in the gear of the present invention, the basic circle radius changes corresponding to the position in the tooth profile direction on the tooth surface. This change is continuous corresponding to the continuous change along the tooth profile direction of the position on the tooth surface. In other words, the tooth profile of the present invention is a tooth profile formed by splicing a minute section that is a part of an involute tooth profile having a different basic circle radius.

  The basic circle radius can be large at the end portion in the tooth profile direction, that is, at the tip and the root, and small at the central portion. On the other hand, it can be made smaller at the end in the tooth profile direction and larger at the center.

  A spur gear or a helical gear can be formed by the above tooth profile.

  Furthermore, the changing base circle radius can be represented by a cosine function or a quadratic function with respect to the rotation angle of the gear.

  The characteristics required for gears can be balanced at a high level.

It is explanatory drawing of the difference of the basic circle radius in an involute tooth profile. It is a figure which shows the difference in a contact point locus | trajectory when a base circle radius is changed in an involute tooth profile. It is a figure which shows the example of a change of a basic circle radius. It is a figure which shows the relationship between a contact point and an involute tooth profile. It is a figure which shows an example of the tooth profile which changed the basic circle radius. It is a figure which shows the contact point locus | trajectory of the tooth profile shown in FIG. It is a figure which shows the other example of the tooth profile which changed the base circle radius. It is a figure which shows the contact point locus | trajectory of the tooth profile shown in FIG. It is a figure which shows the change of the basic circle radius of the tooth profile shown to FIG. It is a figure which shows the change of the pressure angle of the tooth profile shown to FIG. It is the figure which compared the characteristic of the gearwheel which has the tooth profile shown by FIG.5, 7, especially the meshing rate and the meshing loss with the involute tooth profile. It is explanatory drawing of a parameter. It is the figure which compared the characteristic of the gearwheel which has a tooth profile shown by FIG. It is the figure which compared the characteristic of the gearwheel which the tooth profile shown in FIG.5, 7 has, especially contact stress with the involute tooth profile. It is a figure which shows the change of the twist angle when a helical gear is comprised using the tooth profile shown in FIG. It is the figure which compared the meshing rate of the helical gear comprised using the tooth profile shown by FIG. 5, 7 with the involute tooth profile. It is the figure which compared the characteristic of the helical gear comprised using the tooth profile shown by FIG.5, 7, especially the meshing rate and the meshing loss with the involute helical gear.

Hereinafter, embodiments of the present invention will be described with reference to the drawings. 1 and 2 are explanatory diagrams relating to involute tooth profiles 10A and 10B having different basic circle radii. In FIG. 1, the tooth profile 10A, the basic circle 12A, and the contact point locus 14A when the radius is large (R _b1 ) are shown by solid lines, and the tooth profile 10B, the basic circle 12B, and the contact point locus when the radius is small (R _b1 ′). 14B is represented by a broken line. In the involute gear, the contact point trajectories 14A and 14B are straight lines. The relationship between the basic circle radius R _b1 and the pressure angle α is expressed by the following equation where the number of teeth is z ₁ and the module is m.
R _b1 = (z ₁ mcosα) / 2

FIG. 2 is a diagram showing how the meshing length changes when the base circle radius is changed. In FIG. 2, the locus of the tooth tip of the first gear is denoted by reference numeral 16 ₁ , and the locus of the tooth tip of the second gear on the other side is denoted by reference numeral 16 ₂ . When the basic circle radius is large, the tooth profile 10A passes from the position indicated by reference numeral 10A-1 in the figure to the position indicated by reference numeral 10A-2 through the position indicated by reference numeral 10A-0 where the contact point is the pitch point P. Engagement continues until it reaches, and the length of the contact point locus 14A during this period becomes the engagement length L. When the basic circle radius is small, the tooth profile 10B passes from the position indicated by reference numeral 10B-1 in the figure to the position indicated by reference numeral 10B-2 through the position indicated by reference numeral 10B-0 where the contact point is the pitch point P. Engagement continues until it reaches, and the length of the contact point locus 14B during this period becomes the engagement length L ′. As can be understood from the figure, in the involute tooth profile, it is understood that the meshing length is shortened when the basic circle radius is small. When the meshing length is shortened, the front meshing rate is decreased, and vibration and noise associated with the meshing cycle of the gear may be deteriorated.

On the other hand, when the basic circle radius is reduced, the tooth thickness of the tooth root is increased, so that the bending rigidity of the tooth root is increased. In the involute gear, the slip rate increases as the distance from the pitch point P to the tooth tips 16 ₁ and 16 ₂ on the contact point trajectories 14A and 14B increases. Therefore, the slip rate is larger when the base circle radius is larger, and therefore the loss due to slip is also greater.

  As described above, in the involute tooth profile, when the base circle radius is reduced, the bending rigidity and slip rate of the tooth base are improved, but the front meshing rate is deteriorated. In the gear according to the present embodiment, by changing the basic circle radius without changing it to a fixed value, it becomes possible to balance the bending rigidity, slip rate, front meshing rate, and the like of the tooth root in a high dimension.

  Table 1 shows specifications of a general involute gear to be compared with the gear according to the present embodiment. In the following, the characteristics of the gear of the present embodiment will be examined with reference to this gear, and this gear will be described as a reference gear. First, a spur gear (with a torsion angle of 0) will be examined, and then a helical gear, particularly its overlapping mesh rate will be examined.

FIG. 3 is a diagram showing how the basic circle radius changes. A solid line 18 indicates a basic circle radius R _b10 of the gear I (reference gear) in Table 1. A broken line 20 indicates a basic circle radius R _b11 (fixed value) which is smaller than that of the solid line 18. The involute gear having this small basic circle radius is hereinafter referred to as a small basic circular gear. A chain line 22 indicates the basic circle radius of the gear according to the present embodiment. In the gear according to the present embodiment, the basic circle radius R _b1 changes with respect to the rotation angle θ _{1 of the} gear. The base circle radius changes continuously with respect to the continuous change of the rotation angle. Furthermore, the change of the base circle radius can be changed smoothly. The basic circle radius can be determined by a cosine function, for example, as in the following equation (1).

In equation (1), R _b10 is the basic circle radius of the reference gear. Δ is the rate of change of the base circle radius, and is expressed as Δ = (R _b10 −R _b11 ) / R _b10 . θ _1h is a rotation angle at the time of tooth contact. The basic circle radius of the mating gear is determined by the following equation (2).

To calculate the tooth profile and contact point trajectory when the base circle radius changes, the formula for the involute gear based on tangential polar coordinates introduced in, for example, Journal of the Japan Society of Mechanical Engineers, Volume C, Volume 62, 603, pp.235-240 Available. The rotation axes of the gears 1 and 2 indicating the tangential polar coordinate system in FIG. 4 are the z ₁ axis and z ₂ axis, and the direction in which the screw advances when the right screw is rotated in the rotation direction of the gear is z ₁ axis and z ₂ , respectively. Determined as the positive orientation of the axis. In FIG. 4, the z ₁ axis is positive on the paper surface and the z ₂ axis is positive on the paper surface. In the following, “ ₁ ” is attached to the coordinate system and variables relating to the gear 1, and “ ₂ ” is attached to the gear 2. A common axis orthogonal to each of the z ₁ axis and the z ₂ axis is defined as a v ₁ axis and a v ₂ axis. The directions of the v ₁ axis and the v ₂ axis are positive from the z ₁ axis toward the z ₂ axis. z ₁ orthogonal to the axis and v each _one axis, v _uniaxial defining an u ₁ axis to the direction of travel of the right screw and positive when rotating toward the z ₁ axis. The u ₂ axis is defined with the positive direction of the right-hand thread when the v ₂ axis is rotated toward the z ₂ axis. The tooth profile of the gear 1 is expressed by the following equation.

In the equations (3) and (4), when the basic circle radius R _b1 is expressed by an arbitrary function of θ ₁ and ξ ₁ , a tooth profile shape in which the basic circle radius changes and a contact point locus having the tooth profile are obtained. As the tooth profile, when θ ₁ = 0 and ξ ₁ is changed, the tooth profile contacting at the pitch point can be calculated. Here, the point of ξ ₁ = 0 is a point on the tooth profile that intersects with the pitch point. The contact point trajectory can be calculated by changing θ ₁ with ξ ₁ = 0. In the case of a spur gear, since the tooth profile is constant with respect to the z ₁ axis, it is only necessary to calculate the tooth profile in the tooth width central section (z ₁₀ = 0), and the basic cylindrical torsion angle ψ _b1 is ψ _b1 = 0. . By setting θ ₁ = 0, the gear 1 is in contact with the pitch point. In the above formulas (3) and (4), if z ₁₀ = θ ₁ = ψ _b1 = 0, the following formula is obtained.

Next, the basic circle radius is expressed as a cosine function of θ ₁ and ξ ₁ . When calculating the tooth profile uses base radius R _b1 (ξ _1), expressed as a function of xi] _1.

If the formula (5) is substituted into the formulas (3) ′ and (4) ′ and ξ ₁ is changed in the range of (−θ _1h , θ _1h ), the tooth profile is coordinated (u ₁ , v ₁ ) Can be expressed as

In FIG. 4, the tooth profile represented by the solid line is an involute tooth profile at the tooth tip Ph and the tooth root Pt. That is, the involute tooth profile of the basic circle radius R _b11 at θ ₁ = −θ _1h and θ _1h in FIG. 3 is shown. The tooth profile represented by the broken line is the involute tooth profile at the pitch point P0 (θ ₁ = θ ₁₀ ), and the tooth profile represented by the chain line is the arbitrary point Pi (θ ₁ = θ _1i between the pitch point and the tooth tip. The involute tooth profile in FIG. As the gear 1 rotates, the base circle radius changes continuously, and the contact point trajectory is obtained by connecting the contact points. Moreover, when the tooth profile at each rotation angle is connected, a tooth profile shape whose base circle radius is changed is obtained.

  FIG. 5 is a diagram illustrating an example of a tooth profile of a spur gear whose basic circle radius changes. In formula (1) or formula (5), tooth profile 30A is shown when Δ = 0.1. As shown in the figure, a convex shape is formed on the tooth root side from the pitch point P, and a concave shape is formed on the tooth tip side.

FIG. 6 is a diagram illustrating a contact point locus. A contact point locus 32A in the case of the tooth profile 30A of the present embodiment and a contact point locus 32S based on the tooth profile of the reference gear are shown. The position of the tooth surface at the start of engagement is indicated by reference numeral 30A-1, and the position of the tooth surface at the end of engagement is indicated by a tooth profile 30A-2. The position of the tooth surface when the contact point becomes the pitch point P0 is indicated by a tooth profile 30A-0. Further, arcs indicated by reference numerals 34 ₁ and 34 ₂ are tooth tip loci, and arcs indicated by reference numeral 36 are pitch circles.

An arbitrary point on the tooth surface comes into contact with the gear on the other side once during the meshing period from the meshing start to the meshing end. That is, an arbitrary point on the tooth surface becomes a contact point once during the meshing period, and the rotation angle θ _{1 at} that time is specified. That is, there is a one-to-one relationship between an arbitrary point on the tooth surface and the rotation angle of the gear when this point becomes a contact point. Further, the basic circle radius R _b1 is determined to continuously change with respect to the rotation angle θ ₁ as described above. When a point on the tooth surface continuously changes its position along the tooth profile, the basic circle radius corresponding to the position of the point changes continuously.

In FIG. 6, the position of the tooth surface when an arbitrary point CN on the tooth surface becomes a contact point is indicated by reference numerals 30A-N. The rotation angle θ ₁ N at this time is an angle at which the arc P 0 PN looks at the center, where the pitch point on the tooth surface 30A-N is the point PN. When the point CN moves on the tooth surface, the tooth surface 30A-N for making the point CN a contact point also moves, and the rotation angle θ ₁ N also changes. The basic circle radius R _b1 also changes according to a predetermined relationship, for example, equation (1). When the position of the point CN changes continuously along the tooth profile, since the contact point locus 32A is continuous, the rotation angle θ ₁ N also changes continuously. Since the basic circle radius is determined to change continuously with respect to the rotation angle, if the position of the point CN changes continuously along the tooth profile, the corresponding basic circle radius also changes continuously. To do.

Further, since the relationship between the rotation angle θ ₁ and the basic circle radius R _b1 is determined, the involute tooth profile at an arbitrary rotation angle can be specified. The contact point at this rotation angle is a point on the involute tooth profile.

The change of the basic circle radius R _b1 can be determined by the following equation (6), for example.

  Equation (6) is obtained by shifting the phase of the cosine function of Equation (1) by π. FIG. 7 is a diagram showing the tooth profile 30B when Δ = 0.1 in Equation (6). As shown in the drawing, a concave shape is formed on the tooth root side from the pitch point P, and a convex shape is formed on the tooth tip side, which is opposite to the tooth profile 30A shown in FIG.

FIG. 8 is a diagram showing a contact point locus. A contact point locus 32B in the case of the tooth profile 30B of the present embodiment and a contact point locus 32S based on the tooth profile of the reference gear are shown. The position of the tooth surface at the start of engagement is indicated by reference numeral 30B-1, and the position of the tooth surface at the end of engagement is indicated by a tooth profile 30B-2. The position of the tooth surface when the contact point becomes the pitch point P0 is indicated by a tooth profile 30B-0. Arcs indicated by reference numerals 34 ₁ and 34 ₂ are tooth tip loci similar to those in FIG.

FIG. 9 is a diagram illustrating a change in the basic circle radius of the tooth profile 30B. A solid line 18, a broken line 20, and a curve 22 indicate the basic circle radius R _b10 of the reference gear, the basic circle radius R _b11 of the small basic circle gear, and the basic circle radius of the tooth profile 30A, as in FIG. Curve 23 is the basic circle radius of tooth profile 30B. FIG. 10 is a diagram illustrating a change in the pressure angle of the tooth profiles 30A and 30B. Pressure angle of the solid line 40 is the reference wheel, showing the pressure angle of the broken line 42 is a small base circle gear pressure angle of the curve 44 is the tooth-shaped 30A, curve 46 a pressure angle of the teeth form 30B.

  The characteristics of the tooth profiles 30A and 30B shown in FIGS. 5 and 7 will be examined in comparison with the reference gear and the small basic circular gear.

  FIG. 11 is a diagram illustrating the relationship between the meshing rate and the meshing loss. The meshing loss and the front meshing rate can be calculated by the following equations (7) and (8).

μは摩擦係数、Ｐは荷重、Ｖは滑り速度、Ｔは伝達トルク、Ｒ_b は基礎円半径、ω₁ ，ω₂ は歯車の角速度、Ｄは接触線長さ、α_t は正面圧力角を表す。接触線長さＤは、ピッチ点から接触点までの距離を表し、図６で説明すれば、ピッチ点Ｐ0と接触点ＣNを結ぶ線分の長さに相当する。

μ is the friction coefficient, P is the load, V is the sliding speed, T is the transmission torque, R _b is the basic circle radius, ω ₁ and ω ₂ are the angular velocities of the gears, D is the contact line length, and α _t is the front pressure angle. Represent. The contact line length D represents the distance from the pitch point to the contact point and corresponds to the length of the line segment connecting the pitch point P0 and the contact point CN as described with reference to FIG.

In FIG. 11, the characteristics (meshing ratio, meshing loss) of each gear are shown as a ratio to the reference gear. As described above, in the involute tooth profile, when the meshing loss is reduced, the meshing rate is correspondingly lowered. In the case of the tooth profile 30 </ b> A shown in FIG. 5, it is possible to suppress a decrease in the meshing rate as compared with the case where an involute gear is employed. As shown in FIG. 6, in the tooth profile 30A, the contact point of the tooth tip approaches the pitch point, the sliding speed is lowered, and the meshing loss is reduced. Base radius R _b1 in the addendum, since equal to the small base circle gear as shown in FIG. 3, the addendum of the contact points of the teeth 30A becomes the same position as the small base circle gears, meshing loss is substantially equal. On the other hand, the length (meshing length) of the contact point locus is longer in the tooth profile 30A having a curved locus than the small basic circular gear having a straight locus, and this effect improves the meshing rate. In the case of the tooth profile 30B, the meshing loss is equivalent to that of the small basic circular gear, but the meshing rate is not improved but rather deteriorates. This is because the normal pitch is a function of the pressure angle as shown in the equation (8). Therefore, in the tooth profile 30B, the influence of the increase in the normal pitch is larger than the increase in the length of the contact point locus. Because.

Next, the root bending strength will be examined. The bending stress σ _b is calculated by the following equation (9).

Each parameter in the equation is shown in FIG. 12A shows the reference gear, FIG. 12B shows the small basic circular gear, and the gears of the tooth profiles 30A and 30B described in FIGS. s _f is the critical section tooth thickness, the tooth thickness at the position where the tooth tip of the counterpart gear contacts, s is the tooth tip thickness, P _n is the tooth tip load, λ is the tooth tip load direction, and h is the tooth tip The height from the critical section at the point where the extension line of the load intersects the tooth thickness center line, b is the tooth width.

FIG. 13 is a diagram showing the root bending stress σ _b of each gear. Also in this figure, the characteristic (tooth root bending stress) of each gear is expressed as a ratio to the reference gear. The small basic circular gear, the gear of the tooth profile 30A, and the gear of the tooth profile 30B all have a base circle radius smaller than that of the reference gear, so that the tooth thickness of the tooth root increases and the tooth root bending stress is improved. .

Next, the Hertz contact stress is examined. Hertz stress is calculated by the following equation (10).

In the formula, P _U is a load per unit tooth width, E is Young's modulus, and R is a relative radius of curvature. FIG. 14 is a diagram showing the average value of the Hertz stress of each gear on the tooth profile. Also in this figure, the characteristic (Hertz stress) of each gear is expressed as a ratio to the reference gear. In the gear of the tooth profile 30B, since the contact between the tooth surfaces becomes the contact between the concave surface and the convex surface, the relative curvature radius R is increased, and the tooth surface strength can be improved.

The above is the examination of the spur gear. However, in the case of a helical gear, it is necessary to examine the meshing rate in consideration of the overlapping meshing rate. In the gear of the present embodiment in which the basic circle radius (and the front pressure angle α _t ) changes, the torsion angle changes with respect to the rotation angle of the gear, and the overlap meshing rate differs from that of the involute gear. The twist angle β (θ ₁ ) is expressed by the following equation (11).

ψ _b1 is a torsion angle on the basic cylinder and is a constant value. FIG. 15 shows the relationship between the rotation angle θ ₁ and the twist angle β. As shown in FIG. 15, in the case of a conventional gear, the twist angle β is constant, but varies in the tooth profiles 30A and 30B of the present embodiment. The meshing rate is calculated by the following equation (12). In the tooth profile 30 </ b> A, as shown in FIG. 15, the torsion angle is larger than that of the reference gear, so that the overlap meshing rate is increased. In the tooth profile 30B, since the twist angle decreases, the overlap meshing rate decreases.

  FIG. 16 is a diagram comparing the meshing rates of the gears. The tooth profile 30A achieves a meshing rate substantially equal to that of the reference gear. FIG. 17 is a diagram showing the relationship between the meshing loss and the meshing rate in the helical gear. In the gear having the tooth profile 30A, the meshing rate can be made equal to the reference gear while reducing the meshing loss.

  As described above, by adopting the tooth profile in which the radius of the basic circle is changed, the contradictory characteristics can be improved in a well-balanced manner with respect to the conventional involute gear. For example, if the above-described tooth profile 30A is employed, it is possible to reduce the meshing loss while maintaining the meshing rate. If the tooth profile 30B is employed, both the meshing loss and the Hertz contact stress can be improved.

The function for determining the basic circle radius is not limited to the above-mentioned cosine function, but may be a continuous and smooth function. For example, it may be a quadratic function. In the case of a quadratic function, similarly to the above-described cosine function, it may be maximized or minimized at the rotation angle θ ₁ = 0 (pitch point). The above cosine function is a symmetric function with respect to the rotation angle θ ₁ = 0 (pitch point). However, both the cosine function and the quadratic function may be asymmetric.

30A, 30B tooth profile, 32A, 32B, 32S contact point locus, 34 ₁ , 34 ₂ tooth tip line.

Claims (10)

A gear having a tooth profile formed by connecting involute tooth profiles having different base circle radii in the tooth profile direction, the base circle radius continuously changing along the tooth profile direction, and being largely central at the end of the tooth profile direction. A small gear. It said gear is a spur gear, the gear according to claim 1. The gear according to claim 1, wherein the gear is a helical gear. The gear according to claim 2 or 3 , wherein the basic circle radius is expressed by a cosine function with respect to a rotation angle of the gear. The gear according to claim 2 or 3 , wherein the basic circle radius is expressed by a quadratic function with respect to the rotation angle of the gear. For each rotation angle of the gear, a basic circle having a radius that continuously changes with respect to the change in the rotation angle is determined,
  The tooth profile is determined by connecting the involute tooth profile based on the basic circle determined for each rotation angle in the tooth profile direction.
Gear tooth shape design method. The gear tooth profile design method according to claim 6, wherein the basic circle is determined to have a large radius at an end portion in a tooth profile direction and a small radius at a central portion. The gear tooth profile design method according to claim 6, wherein the basic circle is defined so as to have a small radius at the end portion in the tooth profile direction and a large radius at the center portion. The gear tooth profile design method according to any one of claims 6 to 8, wherein the basic circle is determined to have a radius represented by a cosine function with respect to a rotation angle of the gear. . The gear tooth profile design method according to any one of claims 6 to 8, wherein the basic circle is determined to have a radius represented by a quadratic function with respect to the rotation angle of the gear. Method.

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