JP4401674B2 - Helical tooth pair meshing rigidity calculation device - Google Patents

Description

[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a meshing rigidity calculation device for a helical tooth pair that calculates a meshing rigidity value from the start to the end of meshing of a helical tooth pair.
[0002]
[Prior art]
Conventionally, in helical gear pairs widely used in vehicles such as automobiles, trucks, railway vehicles, and construction machines, etc., the natural vibration of meshing is aimed at improving quietness and optimizing strength. Various analyzes such as number and vibration in the rotational direction are performed.
[0003]
For such analysis, it is required to accurately grasp the variation in the meshing rigidity from the start of meshing of the helical tooth pair to the end of meshing. It is performed by solving an integral equation using an influence function of the bending and contact deflection of the helical tooth pair using a computer.
[0004]
However, the calculation of the meshing rigidity value using the above-described integral equation is very difficult, and when a certain degree of calculation accuracy is required, the calculation time becomes enormous, and the input calculation conditions (load, Depending on the tooth surface shape etc., there is a problem that the calculated value does not converge.
[0005]
In response to this, for example, in Non-Patent Document 1, involute that can display the spring stiffness variation from the start of meshing to the end of meshing in a parallel tooth with a pressure angle of 20 ° uses an approximate equation of meshing rigidity of a pair of helical teeth, A technique for calculating a variation in meshing rigidity of a helical tooth pair is disclosed.
[0006]
[Non-Patent Document 1]
Umezawa and 2 others “Vibration characteristics of helical gears for power transmission (spring stiffness approximation)”, Transactions of the Japan Society of Mechanical Engineers (C), Vol. 51, No. 469 (Showa 60-9), P2316 to P2322
[0007]
[Problems to be solved by the invention]
However, the meshing stiffness value obtained from the approximate expression disclosed in Non-Patent Document 1 described above does not necessarily correspond to the actual meshing stiffness value sufficiently, and furthermore, it is an average tooth with a pressure angle of 20 °. Since it is specialized in length, there is a possibility that it cannot be sufficiently applied to a practical gear having various pressure angles and tooth heights.
[0008]
The present invention has been made in view of the above circumstances, and for a wide range of gear specifications, a helical tooth pair capable of accurately grasping the meshing rigidity value of a helical tooth pair with simple arithmetic processing. An object of the present invention is to provide a meshing rigidity calculation device.
[0009]
[Means for Solving the Problems]
  In order to solve the above-mentioned problem, the invention according to claim 1 sets an approximate expression of the meshing rigidity curve based on the specifications of the helical tooth pair meshing with each other, and the meshing end point from the meshing start point based on the approximate expression A helical gear pair stiffness calculating device for calculating each mesh stiffness value up to a point, wherein the helical gear pair stiffness curveThe first1, a branch point calculating means for calculating a second branch point, and a first approximate expression setting means for setting an approximate expression of the meshing rigidity curve in a section from the first branch point to the second branch point And second approximation formula setting means for setting an approximation formula of the meshing rigidity curve in a section from the meshing start point to the first branch point and from the second branch point to the meshing end point;A meshing rate comparison means for comparing the magnitude relationship between the front meshing rate and the overlapping meshing rate of the helical tooth pair;WithThe first and second approximation formula setting means set the approximation formulas of the meshing rigidity curves in the sections to different functions according to the comparison result between the front meshing rate and the overlapping meshing rate.It is characterized by that.
[0013]
  Claims2An apparatus for calculating the meshing rigidity of a helical tooth pair according to the invention described in claim1In the described invention, the first approximate expression setting means is a function in which the approximate expression is a function having coordinates on the equivalent action line of the helical tooth pair as variables, and includes quadratic and quartic terms. It is set to a function.
[0014]
  Claims3According to the described invention, the meshing configuration computing device for helical tooth pairs1 or 2In the invention described in the above, when the meshing rate comparison unit determines that the front meshing rate is smaller than the overlapping meshing rate, the first approximate expression setting unit performs meshing rigidity with a predetermined tooth width / height ratio. The approximate expression is set by modifying an approximate function created based on a curve based on the coordinates of the branch point.
[0015]
  Claims4An apparatus for calculating the meshing rigidity of a helical tooth pair according to the invention described in claimAny one of 1 to 3In the invention described in the above, when the meshing rate comparison means determines that the front meshing rate is larger than the overlapping meshing rate, the magnitude relationship between the helical angle of the helical tooth pair and a preset twisting angle set in advance And the second approximate expression setting means sets the approximate expression to a different function according to the comparison result of the twist angle comparison means.
[0016]
  Claims5An apparatus for calculating the meshing rigidity of a helical tooth pair according to the invention described in claim4In the described invention, the torsion angle comparing means compares the magnitude relationship with the set torsion angle by using a torsion angle equivalently converted at a predetermined tooth width / height ratio.
[0017]
  Claims6An apparatus for calculating the meshing rigidity of a helical tooth pair according to the invention described in claim4 or claim 5When the torsion angle comparing means determines that the torsion angle is smaller than the set torsion angle, the second approximate expression setting means determines the approximate expression and the helical tooth pair. Is set to a linear function with the coordinates on the equivalent action line as a variable.
[0018]
  Claims7An apparatus for calculating the meshing rigidity of a helical tooth pair according to the invention described in claim4 or 5When the torsion angle comparing means determines that the torsion angle is larger than the set torsion angle, the second approximate expression setting means determines the approximate expression and the helical tooth pair. A function having coordinates on the equivalent action line as a variable, an exponential function of a predetermined order defined based on the twist angle, and a function of a predetermined order defined based on the twist angle Features.
[0019]
  Claims8An apparatus for calculating the meshing rigidity of a helical tooth pair according to the invention described in claim7In the described invention, the second approximate expression setting means defines the order using a torsion angle equivalently converted with a predetermined tooth width / height ratio.
[0020]
  Claims9An apparatus for calculating the meshing rigidity of a helical tooth pair according to the invention described in claimAny one of claims 1 to 8In the invention described in claim 2, when the meshing rate comparison unit determines that the front meshing rate is smaller than the overlapping meshing rate, the second approximate formula setting unit sets the approximate formula to the helical tooth. A function having coordinates on a pair of equivalent action lines as variables, and an exponential function of a predetermined order defined based on the twist angle and a function of a predetermined order defined based on the twist angle It is characterized by.
[0021]
  Claims10An apparatus for calculating the meshing rigidity of a helical tooth pair according to the invention described in claim9In the described invention, the second approximate expression setting means defines the order using a torsion angle equivalently converted with a predetermined tooth width / height ratio.
[0022]
DETAILED DESCRIPTION OF THE INVENTION
Embodiments of the present invention will be described below with reference to the drawings. The drawings relate to an embodiment of the present invention, FIG. 1 is a flowchart showing a helical gear pair engagement rigidity calculation routine, FIG. 2 is a schematic configuration diagram of a helical tooth pair engagement rigidity calculation device, and FIG. FIG. 4 is a schematic diagram showing an example of a computer for realizing the meshing rigidity calculating device for a helical tooth pair, FIG. 4 is an explanatory diagram showing an action plane of the helical gear pair, and FIG. 5 is a helical tooth in each case. FIG. 6 is a diagram showing a pair of meshing rigidity curves, FIG. 6 is an explanatory diagram showing a meshing branch point in a case (case 1) where the front meshing rate is larger than the overlapping meshing rate, and FIG. 7 is a case where the front meshing rate is equal to the overlapping meshing rate. FIG. 8 is an explanatory diagram showing a meshing branch point in a case (case 3) where the front meshing rate is smaller than the overlapping meshing rate, and FIG. FIG. 10 is a chart showing a meshing rigidity curve between points, and FIG. Fig. 11 is a diagram showing the stiffness curve, Fig. 11 is a diagram showing the meshing stiffness curves of Fig. 10 after coordinate conversion, and Fig. 12 is outside the branch point in case 1 when the branch point is near the mesh start point and mesh end point. FIG. 13 is a chart showing a meshing rigidity curve, FIG. 13 is a chart showing a meshing rigidity curve outside the branching point in case 1 when the branching point is close to the meshing center point, and FIG. FIG. 15 is a chart showing a comparison result between the theoretical value and the approximate value of the meshing rigidity curve in case 1, and FIG. 16 is a comparison result between the theoretical value and the approximate value of the meshing rigidity curve in case 2. The chart and FIG. 17 are charts showing comparison results between theoretical values and approximate values of the meshing rigidity curve in case 3.
[0023]
Here, prior to describing the configuration of the helical stiffness pair computing device according to the present embodiment, the results of analysis of the mesh stiffness values at each specification of the helical tooth pair by the present applicants. Will be described.
[0024]
In FIG. 4, reference numeral 100 denotes an action plane of the driving side gear 101 and the driven side gear 102 constituting the helical gear pair. As shown in FIG. 4, since the meshing contact line CC of the helical tooth pair is twisted with respect to the gear shaft, the meshing starts from point contact unlike the spur gear. That is, the meshing of the helical tooth pair starts from the point S, the oblique meshing contact line CC advances in parallel on the working plane 100 while changing the length, and finally ends at the point E.
[0025]
First, the present applicants solve each meshing from the start of meshing to the end of meshing by solving the following integral equation of bending deflection and tooth surface contact deflection of helical gear pairs of various gear specifications. Deflection amount δ in contact line CC_x And the amount of deflection δ_xFrom the above, the meshing rigidity value K (X) was obtained.
[0026]
δ_x= ∫K_b(X, ξ) · P (ξ) dξ + ∫K_c(X = ξ) · P (ξ) dξ (1)
P_s= ∫P (ξ) dξ (2)
K (X) = P_s/ Δ_x  ... (3)
[0027]
Here, in equation (1), K is an influence function of surface contact deflection._cIs the free end load proposed by Suzuki et al. (See Suzuki and Umezawa, “Nearly approaching the tooth contact of the gears that come into contact with each other”, The Japan Society of Mechanical Engineers, Vol. 52, No. 481 (1986), P2449). The theoretical formula between the rollers considering the influence of distribution was used.
[0028]
K_c[X = ξ, y = η = fuc (x)]
= 25 · (1-γ²) ・ ∫ (1-x^Four)^1/4dx
/ Π · E · ΔB · (1-x^Four)^1/4(However, the integration range is 0 to 1) (4)
[0029]
In addition, K is a bending bending influence function of teeth._bIs the tooth width direction proposed by Kano et al. (See Kano / Saiki, “New Bending and Deflection Influence Function of Gear Rack”, Annual Meeting of the Japan Society of Mechanical Engineers 2002 (V) 2314, P27). A high-precision formula that takes into account the bending characteristics of different tooth height directions was used.
[0030]
K_b(X, y, ξ, η)
= U · G (η) · [ν (r) / ν (η)]
・ [[[F (x) · F (ξ)] / F (| x−ξ |)]^1/2  ... (5)
r = η + [[λ · (x−ξ)]²+ (Y−η)²]^1/2  (6)
[0031]
The variables in the formulas (1) to (6) are as follows.
[0032]
(X, y): Coordinate value of the deflection observation point
(Ξ, η): Coordinate value of unit concentrated load point
P (ξ): Load distribution on the meshing contact line
E: Young's modulus [2.068 × 10¹¹Nm²]
γ: Poisson's ratio [0.3]
ΔB: Calculated division width on each meshing contact line
X: Coordinate value on the equivalent action line (see Fig. 4) of the helical tooth pair with the contact tooth width center being 0
U: Absolute value of deflection at the origin when the origin is concentrated
λ: Coordinate conversion coefficient to the concentric distribution of the bent elliptical distribution
r: Radius of bending concentric distribution with tooth tip as origin
ν (r): Deflection characteristic function of equivalent concentric distribution
G (η): Deflection characteristic function just below the concentrated load point in the tooth height direction
F (ξ): Deflection characteristic function just below the concentrated load point in the tooth width direction
[0033]
Using the above calculation formula, as a result of calculating and analyzing the theoretical value of each meshing rigidity from the start to the end of meshing of the helical tooth pair according to various gear specifications, the present applicants There is a branching point (hereinafter referred to as the first branching point) where the behavior of the meshing rigidity curve changes between the meshing start point and the meshing center point, and the same between the meshing center point and the meshing end point. It was found that there is a branch point (hereinafter referred to as a second branch point). And, as a result of examining this branch point in more detail, the position is the front meshing rate ε_αAnd overlap mesh rate ε_βIn addition to being determined by the magnitude relationship, for example, it has been found that it can be classified into three cases (case 1 to case 3) as shown in each example of FIG. In FIG. 5, the example shown in case 1 is BH [ratio of tooth width to tooth height (tooth width / tooth height)] = 2, H_k(Tooth height ratio) = 2.35, α_n(Pressure angle) = 20.5 °, β₀(Twist angle on pitch cylinder) = Each meshing stiffness value K (X) at 4 ° is meshing stiffness value K at the meshing center point_p(K (0)) is a meshing rigidity curve standardized by (K (0)). The example shown in Case 2 is BH = 3, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 36 ° is the meshing stiffness value K at the meshing center point_pIs an engagement rigidity curve standardized by the following, and an example shown in case 3 is BH = 6, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 36 ° is the meshing stiffness value K at the meshing center point_pIt is the meshing rigidity curve standardized by.
[0034]
Case 1 has a front meshing ratio ε_αIs the overlap meshing ratio ε_βIs a larger case. In this case, as shown in FIG. 6, the length of the meshing contact line CC that translates on the working plane 100 from the meshing start point S to the meshing end point E is such that the meshing contact line CC is toothed on the working plane 100. It is the longest in a state that crosses the width direction. In this case, the first branch point exists at the moment when the length of the meshing contact line CC is the longest, and the second branch point exists at the moment when the length is not the longest.
[0035]
Case 2 has a front meshing ratio ε_αIs the overlap meshing ratio ε_βIs the same case. In this case, as shown in FIG. 7, the length of the meshing contact line CC that translates from the meshing start point S to the meshing end point E on the working plane 100 is the same as the meshing contact line CC on the working plane 100. It becomes the longest at the moment of crossing in the width direction and the tooth height direction. In this case, the first and second branch points simultaneously exist at the moment when the length of the meshing contact line CC becomes the longest.
[0036]
Case 3 has a front meshing ratio ε_αIs the overlap meshing ratio ε_βIs a smaller case. In this case, as shown in FIG. 8, the length of the meshing contact line CC that translates from the meshing start point S to the meshing end point E on the working plane 100 is the same as the meshing contact line CC on the working plane 100. It is the longest in a state that crosses the length direction. In this case, the first branch point exists at the moment when the length of the meshing contact line CC is the longest, and the second branch point exists at the moment when the length is not the longest.
[0037]
Thus, each meshing rigidity curve is greatly influenced by the difference in the branch point position. The position of each branch point is one of the above three cases depending on only the gear specifications. Here, the coordinates on the action line of the meshing start point, the meshing end point, and the first and second branch points, that is, X_s, X_e, X_d1, X_d2Is calculated as follows.
[0038]
X_s=-(Ε_α+ Ε_β) / [(Ε_α/ M_n・ H] ... (7)
X_e= (Ε_α+ Ε_β) / [(Ε_α/ M_n・ H] ... (8)
X_d1=-| (Ε_α−ε_β) / [(Ε_α/ M_n) ・ H] |… (9)
X_d2= | (Ε_α−ε_β) / [(Ε_α/ M_n) ・ H] |… (10)
[0039]
Here, the variables in the equations (7) to (10) are as follows.
[0040]
m_n: Tooth right angle module
H: Tooth length
As is clear from the equations (9) and (10), the X coordinate of the first and second branch points is the meshing center point X._pOn behalf of these, X_d2Simply X_dAlso called.
[0041]
Based on the above points, first, between the first and second branch points (X_d1≦ X ≦ X_d2)) Approximate function of each meshing rigidity value.
[0042]
FIG. 9 shows the meshing rigidity curves of the helical tooth pairs of various specifications classified as case 1. In FIG. 9, BH = 2, H_k= 2.35, α_n= 20.5 °, β₀= Each meshing rigidity value K (X) at 20 ° is the meshing rigidity value K at the meshing center point_pMeshing rigidity curve standardized by BH = 4, H_k= 2.35, α_n= 20.5 °, β₀= Each meshing rigidity value K (X) at 20 ° is the meshing rigidity value K at the meshing center point_pMeshing rigidity curve standardized with BH = 6, H_k= 2.35, α_n= 20.5 °, β₀= Each meshing rigidity value K (X) at 20 ° is the meshing rigidity value K at the meshing center point_pThe meshing rigidity curves standardized in FIG. As can be seen from FIG. 9, in case 1, each meshing rigidity curve is on substantially the same parabola, regardless of the tooth width / height ratio BH.
[0043]
FIG. 10 shows the meshing rigidity curves of the helical tooth pairs of various specifications classified as case 3. In FIG. 10, BH = 4, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 36 ° is the meshing stiffness value K at the meshing center point_pMeshing rigidity curve standardized with BH = 5, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 36 ° is the meshing stiffness value K at the meshing center point_pMeshing rigidity curve standardized with BH = 6, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 36 ° is the meshing stiffness value K at the meshing center point_pThe meshing rigidity curves standardized in FIG. In this case, as shown in FIG. 10, the meshing rigidity value K (X) of each helical tooth pair is affected by the tooth width B, and therefore the behavior varies depending on the tooth width / tooth height ratio BH. On the other hand, it can be seen that the meshing rigidity value curves of each helical tooth pair substantially coincide on the same parabola when coordinate transformation for matching each branch point position is performed, as shown in FIG. FIG. 11 shows an example in which the first and second branch point positions of BH = 4 and 5 are made to coincide with the first and second branch point positions of BH = 6 by coordinate transformation.
[0044]
As can be seen from the above, each meshing rigidity curve between the branch points can be approximated by a bilaterally symmetric parabolic function having a quadratic term and a quadratic term and having a meshing center point as a vertex. In this case, the parabola shape of the case 1 is constant regardless of the tooth width tooth height ratio BH, but the parabola shape of the case 3 changes depending on the tooth width tooth height ratio BH. Therefore, taking these into consideration, the present applicants have decided that between the branch points (−X_d1≦ X ≦ X_d2) Was created as follows.
[0045]
Case 1 (ε_α> Ε_β)in the case of
K (X) / K_p= 1 + a₂・ X²+ A_Four・ X^Four  ... (11)
a₂= F₁(Β₀)
= -1.40884 + 0.00726. (Β₀-4)
-0.00111 ・ (β₀-4)²  (12)
a_Four= F₂(Β₀)
= -2.35761-0.12962 · (β₀-4)
+0.00470 ・ (β₀-4)²  ... (13)
[0046]
Case 3 (ε_α<Ε_β)in the case of
K (X) / K_p= 2 + a₂・ X_N ²+ A_Four・ X_N ^Four-Y_N  ... (14)
X_N= X + X_{d (BH = 6)}-X_d  ... (15)
Y_N= 1 + a₂・ (X_{d (BH = 6)}-X_d)²+ A_Four・ (X_{d (BH = 6)}-X_d)^Four  ... (16)
a₂= F₆(Β₀)
= −0.51970 · exp [−0.36190 · (β₀-28)] ... (17)
a_Four= F₇(Β₀)
= −16.9460 · exp [−0.34700 · (β₀-28)] ... (18)
Here, the approximate function defined by the equations (14) to (18) is based on the meshing rigidity curve with the tooth width ratio BH = 6, and the branch point X_dThe meshing rigidity curve at each tooth width / height ratio BH is approximated by performing coordinate conversion based on the above. In other words, the helical tooth pair having the tooth width ratio BH = 6 is one of the most frequently used helical tooth pairs, and the helical tooth pair is a tooth width tooth height. Since the ratio BH is large, it is possible to set a large number of calculations for the meshing rigidity value. From these facts, in the formulas (14) to (18), each tooth width tooth height ratio is based on the meshing rigidity curve of the helical tooth pair at a practical and highly accurate tooth width tooth height ratio BH = 6. Approximate the meshing stiffness curve at BH. In this case, the X coordinate (X_d) Can be equivalently converted with a tooth width / height ratio BH = 6 as in the following equation.
[0047]
X_{d (BH = 6)}= [Ε_β(6 / BH) -ε_α] / [(Ε_α/ M_n・ H] ... (19)
Needless to say, an approximation function equivalent to the equations (14) to (18) may be created on the basis of the meshing rigidity curve other than the tooth width / height ratio BH = 6.
[0048]
Next, outside the first and second branch points (X_s≦ X ≦ X_d1, X_d2≦ X ≦ X_e)) Approximate function of each meshing rigidity value.
[0049]
FIG. 12 shows the case where the branch point is close to the meshing start point and the meshing end point among the meshing rigidity curves of the helical tooth pairs of various specifications classified as case 1 (that is, the twist angle β₀Is relatively small). In FIG. 12, BH = 3, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 4 ° is the meshing stiffness value K at the meshing center point_pMeshing rigidity curve standardized by B, BH = 3, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 8 ° for each meshing stiffness value K (X)_pMeshing rigidity curve standardized by B, BH = 3, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 12 ° is the meshing stiffness value K at the meshing center point_pMeshing rigidity curve standardized by B, BH = 3, H_k= 2.35, α_n= 20.5 °, β₀= Each meshing stiffness value K (X) at 16 ° is the meshing stiffness value K at the meshing center point_pMeshing rigidity curve standardized with BH = 3, H_k= 2.35, α_n= 20.5 °, β₀= Each meshing rigidity value K (X) at 20 ° is the meshing rigidity value K at the meshing center point_pThe meshing rigidity curves standardized in FIG.
[0050]
FIG. 13 shows the case where the branch point is close to the meshing center point among the meshing rigidity curves of the helical tooth pairs of various specifications classified as case 1 (that is, the twist angle β₀Is relatively large). In FIG. 13, BH = 3, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 24 ° is the meshing stiffness value K at the meshing center point_pMeshing rigidity curve standardized by B, BH = 3, H_k= 2.35, α_n= 20.5 °, β₀= Each meshing stiffness value K (X) at 28 ° is the meshing stiffness value K at the meshing center point_pMeshing rigidity curve standardized with BH = 3, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 32 ° is the meshing stiffness value K at the meshing center point_pThe meshing rigidity curves standardized in FIG.
[0051]
FIG. 14 shows the meshing rigidity curves of the helical tooth pairs of various specifications classified as case 2. In FIG. 14, BH = 3, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 36 ° is the meshing stiffness value K at the meshing center point_pMeshing rigidity curve standardized by BH = 4, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 30 ° is the meshing stiffness value K at the meshing center point_pMeshing rigidity curve standardized with BH = 5, H_k= 2.35, α_n= 20.5 °, β₀= Each meshing stiffness value K (X) at 26 ° is the meshing stiffness value K at the meshing center point_pMeshing rigidity curve standardized with BH = 6, H_k= 2.35, α_n= 20.5 °, β₀= Meshing stiffness value K (X) at 22.5 ° is the meshing stiffness value K at the meshing center point_pThe meshing rigidity curves standardized in FIG. Here, the meshing rigidity curves of the helical tooth pairs of various specifications classified as case 3 are substantially the same as those of case 2 outside the branch point.
[0052]
As is apparent from FIG. 12, the twist angle β₀Is relatively small, the meshing stiffness curve outside the branch point can be approximated by a straight line. As is apparent from FIGS. 13 and 14, the twist angle β₀Is relatively large, it is difficult to approximate the meshing stiffness curve outside the branch point with a straight line. Based on these, the applicants conducted further studies on each meshing rigidity curve outside the branch point, and as a result, the twist angle β₀With tooth width / height ratio BH = 6 (β_{0 (BH = 6)}) When equivalent conversion to_{0 (BH = 6)}<12 ° meshing stiffness curve can be linearly approximated, β_{0 (BH = 6)}It has been found that a meshing rigidity curve of ≧ 12 ° can be approximated by an exponential function having an A-order term and a function having a B-order term. Therefore, in consideration of these, the applicants are outside the branch point (X_s≦ X ≦ X_d1, X_d2≦ X ≦ X_e) Was created as follows.
[0053]
In addition, equivalent transformation helix angle β_{0 (BH = 6)}Is defined as:
[0054]
β_{0 (BH = 6)}= Tan^-1[(BH / 6) · tanβ₀] (20)
[0055]
β_{0 (BH = 6)}<12 °
K (X) / K_p= A₀+ A₁・ | X | (21)
a₀= (| X_d｜ ・ Y_s-Y_d・ | X_s|) / (| X_d|-| X_s|) (22)
a₁= (Y_d-Y_s) / (| X_d|-| X_s|) (23)
[0056]
β_{0 (BH = 6)}When ≧ 12 °
K (X) / K_p= [Ka (X) + Kb (X)] / 2 (24)
Ka (X) = a₀・ Exp (a₁・ | X |^A... (25)
Kb (X) = b₀+ B₁・ | X |^B  ... (26)
a₀= Exp [(| X_d｜^A・ InY_s− | X_s｜^A・ InY_d)
/ (| X_d｜^A− | X_s｜^A]] ... (27)
a₁= (LnY_d-LnY_s) / (| X_d｜^A− | X_s｜^A(28)
b₀= (| X_d｜^B・ Y_s-Y_d・ | X_s｜^B) / (| X_d｜^B− | X_s｜^B... (29)
b₁= (Y_d-Y_s) / (| X_d｜^B− | X_s｜^B... (30)
[0057]
Here, in the formulas (24) to (30), the orders A and B in each case are the twist angles β₀Is defined as follows.
[0058]
Case 1 (ε_α> Ε_β)in the case of
A = f_Four(Β_{0 (BH = 6)})
= 3.79864-0.09678.beta_{0 (BH = 6)}  ... (31)
B = f_Five(Β_{0 (BH = 6)})
= 4.42725-0.19427.beta_{0 (BH = 6)}
+ 0.00451 · β_{0 (BH = 6)} ²  ... (32)
Cases 2 and 3 (ε_α≤ε_β)in the case of
A = f₈(Β_{0 (BH = 6)})
= -0.13500 + 0.08200 · β_{0 (BH = 6)}  ... (33)
B = f_Five(Β_{0 (BH = 6)})
= 4.42725-0.19427.beta_{0 (BH = 6)}
+ 0.00451 · β_{0 (BH = 6)} ²  ... (34)
In the formulas (24) to (30), Y_sIndicates the ratio between the meshing stiffness value at the meshing start point and the meshing stiffness value at the meshing center point, and is obtained, for example, as follows.
[0059]
Y_s= K (X_s) / K_p= F_Three(Β₀, BH)
= [C_C0+ C_C1・ (Β₀-4) + C_C2・ (Β₀-4)²] / Β₀  ... (35)
C_C0= 0.07139-0.01574 (BH-2)
+0.00206 ・ (BH-2)²  ... (36)
C_C1= 0.01714-0.00725 (BH-2)
+ 0.00071 · (BH-2)²  ... (37)
C_C2= 0.00005 + 0.00009 · (BH-2)
+0.000002 ・ (BH-2)²  ... (38)
In the formulas (24) to (30), Y_dIs the ratio of the meshing stiffness value at the bifurcation point to the meshing stiffness value at the meshing center point [K (X_d) / K_p, And an approximate expression set based on the expressions (11) to (13) or an approximate expression set based on the expressions (14) to (18).
[0060]
Next, an arithmetic unit that calculates the meshing rigidity of the helical tooth pair using each of the above approximate functions will be described.
[0061]
In FIG. 2, reference numeral 1 denotes a helical gear pair meshing rigidity calculating device. The meshing rigidity calculating device 1 includes an input unit 5 for inputting the specifications of the helical tooth pair, and an input unit 5. The calculation unit 6 for calculating the meshing rigidity value of the helical tooth pair on the basis of the input specifications and the meshing rigidity calculation routine executed by the calculation unit 6 are stored, and the input specifications from the input unit 5 and The storage unit 7 that appropriately stores the calculation result and the like in the calculation unit 6 and the output unit 8 that outputs the calculation result and the like in the calculation unit 6 are configured.
[0062]
Here, the meshing rigidity computing device 1 is realized by a computer system 10 shown in FIG. 3, for example. In the computer system 10, for example, a keyboard 12, a display device 13, and a printer 14 are connected to a computer main body 11 via a connection cable 15 to constitute a main part. In the computer system 10, for example, various drive devices, a keyboard 12, and the like arranged in the computer main body 11 function as the input unit 5, and a CPU, ROM, RAM, and the like built in the computer main body 11 are arithmetic units. 6 functions. Further, a hard disk or the like built in the computer main body 11 functions as the storage unit 7, and the display device 13, the printer 14, and the like function as the output unit 8.
[0063]
In the present embodiment, the meshing rigidity calculation routine stored in the storage unit 7 sets and sets the meshing rigidity approximation formula corresponding to the input specifications using the above formulas (7) to (38) as appropriate. This is for calculating the meshing rigidity value of each helical tooth pair in each meshing state based on the meshing rigidity approximation formula.
[0064]
Then, the calculation unit 6 loads and executes the meshing rigidity calculation routine stored in the storage unit 7, thereby executing a branch point calculation unit, a meshing rate comparison unit, and a first approximate expression setting unit (approximation formula setting unit). Each function as a second approximation formula setting unit (approximation formula setting unit) and a twist angle comparison unit is realized.
[0065]
Next, the meshing rigidity calculation routine executed by the calculation unit 6 in the meshing rigidity calculating apparatus 1 having the above-described configuration will be described.
[0066]
This routine is executed after the specifications of the helical tooth pair are input through the input unit 5. Here, in the present embodiment, the meshing rigidity computing device 1 includes, as specifications, for example, a tooth right angle module m._n, Tooth height coefficient K_s, Crevice coefficient C_k, Pressure angle α_n, Tooth width B, and twist angle β₀Is entered.
[0067]
When the routine starts, first, in step S101, the calculation unit 6 uses a well-known calculation method, based on the input specifications, the tooth height H, the tooth width / tooth height ratio BH, and the front meshing ratio ε._α, And overlap meshing ratio ε_βCalculate the specifications.
[0068]
In subsequent step S102, the calculation unit 6 uses the equations (7) and (8) to calculate the X coordinate (X of the engagement start point and the engagement end point of the helical tooth pair._sAnd X_e) Is calculated.
[0069]
In subsequent step S103, the calculation unit 6 uses the equations (9) and (10) to calculate the X coordinates (X of the first and second branch points of the helical tooth pair._d1And X_d2) Is calculated.
[0070]
In step S104, the calculation unit 6 determines that the front meshing ratio ε obtained in step S101._αAnd overlap mesh rate ε_βAnd the magnitude relationship with ε_α> Ε_βIf it is determined that the helical tooth pair based on the current input specifications is a helical tooth pair classified as case 1, the process proceeds to step S105.
[0071]
When the process proceeds from step S104 to step S105, the calculation unit 6 uses the equations (11) to (13) to calculate the distance between the first and second branch points (X_d1≦ X ≦ X_d2) Is set as an approximate expression for the meshing rigidity curve.
[0072]
In subsequent step S106, the calculation unit 6 uses the approximate expression set in step S105 to calculate the ratio Y between the meshing rigidity value at the branch point and the meshing rigidity value at the meshing center point._d[= K (X_d) / K_p] Is calculated.
[0073]
In subsequent step S107, the calculation unit 6 uses the equations (35) to (38) to calculate the ratio Y between the meshing rigidity value at the meshing start point and the meshing rigidity value at the meshing center point._sIs calculated.
[0074]
In subsequent step S108, the calculation unit 6 uses the equation (20) to calculate the equivalent conversion torsion angle β when the tooth width / height ratio BH = 6._{0 (BH = 6)}Is calculated.
[0075]
In step S109, the calculation unit 6 determines that the equivalent conversion torsion angle β_{0 (BH = 6)}Is compared with the preset twist angle (for example, 12 °), and β_{0 (BH = 6)}If it is determined that it is <12 °, the helical tooth pair based on the current input specifications is outside the branch point (X_s≦ X ≦ X_d1, X_d2≦ X ≦ X_e), It is determined that the meshing rigidity curve in the section can be linearly approximated, and the process proceeds to step S110.
[0076]
When the process proceeds from step S109 to step S110, the calculation unit 6 sets an approximate expression of the meshing rigidity curve in the section outside the branch point using Expressions (21) to (23).
[0077]
Then, when proceeding from step S110 to step S117, the calculation unit 6 uses the approximate expression of the meshing rigidity curve in each section set in step S105 and step S110, and the meshing start point X of the helical tooth pair._sFrom meshing end point X_eEach meshing rigidity value is calculated until the calculation result is output through the output unit 8, and then the routine is terminated.
[0078]
On the other hand, in step S109, β_{0 (BH = 6)}If it is determined that ≧ 12 °, it is determined that the helical tooth pair based on the present input specifications is difficult to linearly approximate the meshing rigidity curve in the section outside the branch point. Then, the process proceeds to step S111.
[0079]
When the process proceeds from step S109 to step S111, the calculation unit 6 sets an approximate expression of the meshing rigidity curve in the section outside the branch point using the expressions (24) to (32).
[0080]
Then, when proceeding from step S111 to step S117, the calculation unit 6 uses the approximate expression of the meshing rigidity curve in each section set in step S105 and step S111, and the meshing start point X of the helical tooth pair._sFrom meshing end point X_eEach meshing rigidity value is calculated until the calculation result is output through the output unit 8, and then the routine is terminated.
[0081]
Also, ε_α≤ε_βIf it is determined that the helical tooth pair based on the current input specifications is a helical tooth pair classified as case 2 or case 3, the process proceeds to step S112.
[0082]
When the process proceeds from step S104 to step S112, the calculation unit 6 uses the equations (14) to (19) to calculate the distance between the first and second branch points (X_d1≦ X ≦ X_d2) Is set as an approximate expression for the meshing rigidity curve.
[0083]
In subsequent step S113, the calculation unit 6 uses the approximate expression set in step S112 to calculate the ratio Y between the meshing rigidity value at the branch point and the meshing rigidity value at the meshing center point._d[= K (X_d) / K_p] Is calculated.
[0084]
In subsequent step S114, the calculation unit 6 uses the expressions (35) to (38) to calculate the ratio Y between the meshing rigidity value at the meshing start point and the meshing rigidity value at the meshing center point._sIs calculated.
[0085]
In subsequent step S115, the arithmetic unit 6 uses the equation (20) to calculate the equivalent conversion torsion angle β with the tooth width / height ratio BH = 6._{0 (BH = 6)}Is calculated.
[0086]
Then, when the process proceeds from step S115 to step S116, the arithmetic unit 6 uses the equations (24) to (30), (33), and (34) to obtain an approximate expression of the meshing rigidity curve in the section outside the branch point. Set.
[0087]
Then, when the process proceeds from step S116 to step S117, the calculation unit 6 uses the approximate expression of the meshing rigidity curve in each section set in steps S112 and S116 to start the meshing start point X of the helical tooth pair._sFrom meshing end point X_eEach meshing rigidity value is calculated until the calculation result is output through the output unit 8, and then the routine is terminated.
[0088]
Next, a specific calculation example of the meshing rigidity value using the above-described meshing rigidity calculating device 1 will be described. Note that the calculation formulas shown below and the calculation results thereof are slightly different because the calculation is performed by giving a predetermined number of calculation divisions on the meshing rigidity calculation device 1.
[0089]
(Operation example 1)
In this calculation example 1, as the specifications of the helical tooth pair, the meshing rigidity calculation device 1 has m_n= 1, K_s= 1, C_k= 0.35, α_n= 20.5 °, B = 4.7, β₀An example in which = 4 ° is input will be described.
[0090]
When the above specifications are input and the meshing rigidity calculation routine starts, first, in step S101, the calculation unit 6 determines, based on the input specifications, the tooth height H, the tooth width / height ratio BH, and the front meshing ratio ε._α, And overlap meshing ratio ε_βCalculate the specifications.
[0091]
H = 2.35
BH = 2
ε_α= 1.919
ε_β= 0.104
[0092]
In subsequent step S102, the calculation unit 6 uses the equations (7) and (8) to calculate the X coordinate (X of the engagement start point and the engagement end point of the helical tooth pair._sAnd X_e) Is calculated as follows.
[0093]
X_s=-(1.919 + 0.104)
/[(1.919/1)·2.35]
= -0.448
X_e= (1.919 + 0.104)
/[(1.919/1)·2.35]
= 0.448
[0094]
In subsequent step S103, the calculation unit 6 uses the equations (9) and (10) to calculate the X coordinates (X of the first and second branch points of the helical tooth pair._d1And X_d2) Is calculated as follows.
[0095]
X_d1=-| (1.919-0.104)
/ [(1.919 / 1) · 2.35] |
= -0.403
X_d2= | (1.919-0.104)
/ [(1.919 / 1) · 2.35] |
= 0.403
[0096]
And ε_α= 1.919, ε_β= 0.104, so in step S104, the calculation unit 6_α> Ε_βIt is determined that the helical tooth pair based on the current input specifications is a helical tooth pair classified as case 1, and the process proceeds to step S105.
[0097]
When the process proceeds from step S104 to step S105, the calculation unit 6 uses the equations (11) to (13) to calculate the distance between the first and second branch points (X_d1≦ X ≦ X_d2), The approximate expression of the meshing rigidity curve in the section is set as follows.
[0098]
a₂= F₁(Β₀)
= -1.40884 + 0.00726 ・ (4-4)
-0.00111 ・ (4-4)²
= -1.40846
a_Four= F₂(Β₀)
= -2.35761-0.12962 (4-4)
+0.00470 ・ (4-4)²
= -2.33571
K (X) / K_p= 1-1.4084.X²-2.35761X^Four... (39)
[0099]
In subsequent step S106, the calculation unit 6 uses the approximate expression (39) to calculate the ratio Y between the meshing rigidity value at the branch point and the meshing rigidity value at the meshing center point._d[= K (X_d) / K_p] Is calculated as follows.
[0100]
Y_d= K (X) / K_p
= 1.-1.4084. (0.403)²
-2.35761 (0.403)^Four
= 0.714
[0101]
In subsequent step S107, the calculation unit 6 uses the equations (35) to (38) to calculate the ratio Y between the meshing rigidity value at the meshing start point and the meshing rigidity value at the meshing center point._sIs calculated as follows.
[0102]
C_C0= 0.07139-0.01574 (2-2)
+0.00206 ・ (2-2)²
= 7.139E-2
C_C1= 0.01714-0.00725 (2-2)
+ 0.00071 · (2-2)²
= 1.714E-2
C_C2= 0.00005 + 0.00009 (2-2)
+0.000002 ・ (2-2)²
= 4.823E-5
Y_s= K (X_s) / K_p= F_Three(Β₀, BH)
= [7.139E-2 + (1.714E-2). (4-4)
+ (4.823E-5), (4-4)²] / 4
= 1.785E-2
[0103]
In subsequent step S108, the calculation unit 6 uses the equation (20) to calculate the equivalent conversion torsion angle β when the tooth width / height ratio BH = 6._{0 (BH = 6)}Is calculated.
[0104]
β_{0 (BH = 6)}= Tan^-1[(2/6) tan4 °]
= 1.335 °
[0105]
And β_{0 (BH = 6)}= 1.335 °, the calculation unit 6 determines that β is β in step S109._{0 (BH = 6)}It is determined that the angle is <12 °, and the helical tooth pair based on the present input specifications is outside the branch point (X_s≦ X ≦ X_d1, X_d2≦ X ≦ X_e), It is determined that the meshing rigidity curve in the section can be linearly approximated, and the process proceeds to step S110.
[0106]
When the process proceeds from step S109 to step S110, the calculation unit 6 sets the approximate expression of the meshing rigidity curve in the section outside the branch point as follows using the expressions (21) to (23).
[0107]
a₀= (| 0.403 | .1.785E-2-0.714. | -0.448 |)
/ (| 0.403 |-| -0.448 |)
= 6.28584
a₁= (0.714-1.785E-2)
/ (| 0.403 |-| -0.448 |)
= -13.92886
K (X) / K_p
= 6.28584-13.92886 · | X | (40)
[0108]
Then, when the process proceeds from step S110 to step S117, the calculation unit 6 uses the approximate expressions (39) and (40) of the meshing rigidity curves in the sections set in step S105 and step S110 to determine a pair of helical teeth. Meshing start point X_sFrom meshing end point X_eEach meshing rigidity value is calculated until the calculation result is output through the output unit 8, and then the routine is terminated.
[0109]
As a result, as shown in FIG. 15, a meshing rigidity value that is in good agreement with the theoretical value of the meshing rigidity value calculated using the equations (1) to (6) is obtained.
[0110]
(Calculation example 2)
In this calculation example 2, as the specifications of the helical tooth pair, the meshing rigidity calculation device 1 has m_n= 1, K_s= 1, C_k= 0.35, α_n= 20.5 °, B = 7.05, β₀An example in which = 36 ° is input will be described.
[0111]
When the above specifications are input and the meshing rigidity calculation routine starts, first, in step S101, the calculation unit 6 determines, based on the input specifications, the tooth height H, the tooth width / height ratio BH, and the front meshing ratio ε._α, And overlap meshing ratio ε_βCalculate the specifications.
[0112]
H = 2.35
BH = 3
ε_α= 1.347
ε_β= 1.319
[0113]
In subsequent step S102, the calculation unit 6 uses the equations (7) and (8) to calculate the X coordinate (X of the engagement start point and the engagement end point of the helical tooth pair._sAnd X_e) Is calculated as follows.
[0114]
X_s=-(1.347 + 1.319)
/[(1.347/1)·2.35]
= -0.842
X_e= (1.347 + 1.319)
/[(1.347/1)·2.35]
= 0.842
[0115]
In subsequent step S103, the calculation unit 6 uses the equations (9) and (10) to calculate the X coordinates (X of the first and second branch points of the helical tooth pair._d1And X_d2) Is calculated as follows.
[0116]
X_d1=-| (1.347-1.319)
/ [(1.347 / 1) · 2.35] |
= 0
X_d2= | (1.347-1.319)
/ [(1.347 / 1) · 2.35] |
= 0
X_d1, X_d2Is actually calculated by giving a predetermined number of calculation divisions, and in this calculation example, all are “0”.
[0117]
And ε_α= 1.347, ε_β= 1.319, so in step S104, the arithmetic unit 6 determines that ε_α> Ε_βIt is determined that the helical tooth pair based on the current input specifications is a helical tooth pair classified as case 1, and the process proceeds to step S105.
[0118]
When the process proceeds from step S104 to step S105, the calculation unit 6 uses the equations (11) to (13) to calculate the distance between the first and second branch points (X_d1≦ X ≦ X_d2), The approximate expression of the meshing rigidity curve in the section is set as follows.
[0119]
a₂= F₁(Β₀)
= -1.40884 + 0.00726 ・ (36-4)
-0.00111 ・ (36-4)²
=-2.30915
a_Four= F₂(Β₀)
= -2.35761-0.12962 (36-4)
+0.00470 ・ (36-4)²
= -1.69128
K (X) / K_p= 1-2.30915 · X²-1.69128 ・ X^Four... (41)
[0120]
In subsequent step S106, the calculation unit 6 uses the approximate expression (41) to calculate the ratio Y between the meshing rigidity value at the branch point and the meshing rigidity value at the meshing center point._d[= K (X_d) / K_p] Is calculated as follows.
[0121]
Y_d= K (X) / K_p
= 1-2.30915 ・ (0)²
-1.69128 ・ (0)^Four
= 1
[0122]
In subsequent step S107, the calculation unit 6 uses the equations (35) to (38) to calculate the ratio Y between the meshing rigidity value at the meshing start point and the meshing rigidity value at the meshing center point._sIs calculated as follows.
[0123]
C_C0= 0.07139-0.01574 (3-2)
+0.00206 ・ (3-2)²
= 5.772E-2
C_C1= 0.01714-0.00725 (3-2)
+ 0.00071 · (3-2)²
= 1.059E-2
C_C2= 0.00005 + 0.00009 · (3-2)
+0.000002 ・ (3-2)²
= 1.401E-4
Y_s= K (X_s) / K_p= F_Three(Β₀, BH)
= [5.772E-2 + (1.059E-2). (36-4)
+ (1.401E-4) ・ (36-4)²] / 36
= 1.500E-2
[0124]
In subsequent step S108, the calculation unit 6 uses the equation (20) to calculate the equivalent conversion torsion angle β when the tooth width / height ratio BH = 6._{0 (BH = 6)}Is calculated.
[0125]
β_{0 (BH = 6)}= Tan^-1[(3/6) tan 36 °]
= 19.965 °
[0126]
And β_{0 (BH = 6)}= 19.965 °, the calculation unit 6 determines that β_{0 (BH = 6)}It is determined that ≧ 12 °, and the helical tooth pair based on the present input specifications is out of the branch point (X_s≦ X ≦ X_d1, X_d2≦ X ≦ X_e), It is determined that it is difficult to linearly approximate the meshing rigidity curve, and the process proceeds to step S111.
[0127]
When the process proceeds from step S109 to step S111, the calculation unit 6 sets the approximate expression of the meshing rigidity curve in the section outside the branch point as follows using the expressions (24) to (32).
[0128]
A = 3.79864-0.09678 / 19.965 = 1.866
B = 4.42725-0.19427 / 19.965
+0.00451 ・ (19.965)²
= 2.345
a₀= Exp [[| 0 |^1.866・ In (1.500E-2)
− | −0.842 |^1.866・ Ln (1)]
/ [| 0 |^1.866− | −0.842 |^1.866]]
= 1
a₁= [Ln (1) -ln (1.500E-2)]
/ [| 0 |^1.866− | −0.842 |^1.866]
= -5.881609
b₀= [| 0 |^2.345・ (1.500E-2)
-1 ・ | -0.842 |^2.345]
/ [| 0 |^2.345− | −0.842 |^2.345]
= 1
b₁= [1- (1.500E-2)]
/ [| 0 |^2.345− | −0.842 |^2.345]
= -1.48226
Ka (X) = 1 · exp (−5.881609 · | X |^1.866)
Kb (X) = 1.-1.48226 · | X |^2.345
K (X) / K_p= [Exp (−5.881609 · | X |^1.866)
+ (1-1.482226 · | X |^2.345)] / 2 ... (42)
[0129]
Then, when the process proceeds from step S111 to step S117, the calculation unit 6 uses the approximate expressions (41) and (42) of the meshing rigidity curve in each section set in step S105 and step S111, to form a pair of helical teeth. Meshing start point X_sFrom meshing end point X_eEach meshing rigidity value is calculated until the calculation result is output through the output unit 8, and then the routine is terminated.
[0130]
As a result, as shown in FIG. 16, a meshing rigidity value that matches the theoretical value of the meshing rigidity value calculated using the equations (1) to (6) very well is obtained.
[0131]
(Calculation example 3)
In this calculation example 3, as the specifications of the helical tooth pair, the meshing rigidity calculation device 1 is set to m_n= 1, K_s= 1, C_k= 0.35, α_n= 20.5 °, B = 14.1, β₀An example in which = 36 ° is input will be described.
[0132]
When the above specifications are input and the meshing rigidity calculation routine starts, first, in step S101, the calculation unit 6 determines, based on the input specifications, the tooth height H, the tooth width / height ratio BH, and the front meshing ratio ε._α, And overlap meshing ratio ε_βCalculate the specifications.
[0133]
H = 2.35
BH = 6
ε_α= 1.347
ε_β= 2.638
[0134]
In subsequent step S102, the calculation unit 6 uses the equations (7) and (8) to calculate the X coordinate (X of the engagement start point and the engagement end point of the helical tooth pair._sAnd X_e) Is calculated as follows.
[0135]
X_s=-(1.347 + 2.638)
/[(1.347/1)·2.35]
= -1.258
X_e= (1.347 + 2.638)
/[(1.347/1)·2.35]
= 1.258
[0136]
In subsequent step S103, the calculation unit 6 uses the equations (9) and (10) to calculate the X coordinates (X of the first and second branch points of the helical tooth pair._d1And X_d2) Is calculated as follows.
[0137]
X_d1=-| (1.347-2.638)
/ [(1.347 / 1) · 2.35] |
= -0.407
X_d2= | (1.347-2.638)
/ [(1.347 / 1) · 2.35] |
= 0.407
[0138]
And ε_α= 1.347, ε_β= 2.638, in step S104, the calculation unit 6 determines that ε_α≤ε_βIt is determined that the helical tooth pair based on the current input specifications is a helical tooth pair classified as case 3, and the process proceeds to step S112.
[0139]
When the process proceeds from step S104 to step S112, the calculation unit 6 uses the equations (14) to (19) to calculate the distance between the first and second branch points (X_d1≦ X ≦ X_d2), The approximate expression of the meshing rigidity curve in the section is set as follows.
[0140]
X_{d (BH = 6)}= [2.638 · (6/6) −1.347] / [(1.347 / 1) · 2.35]
= 0.407
a₂= F₆(Β₀)
= -0.51970.exp [-0.36190. (36-28)]
= -2.873E-2
a_Four= F₇(Β₀)
= -16.9460.exp [-0.34700. (36-28)]
= -1.056
X_N= X + 0.407-0.407
Y_N= 1 + (-2.873E-2). (0.407-0.407)²
-1.056 ・ (0.407-0.407)^Four
= 1
K (X) / K_p= 1 + (− 2.873E−2) · X²-1.056 · X^Four... (43)
[0141]
In subsequent step S113, the calculation unit 6 uses the approximate expression (43) to calculate the ratio Y between the meshing rigidity value at the branch point and the meshing rigidity value at the meshing center point._d[= K (X_d) / K_p] Is calculated as follows.
[0142]
Y_d= K (X) / K_p
= 1 + (-2.873E-2). (0.407)²
-1.056 ・ (0.407)^Four
= 0.965
[0143]
In subsequent step S114, the calculation unit 6 uses the expressions (35) to (38) to calculate the ratio Y between the meshing rigidity value at the meshing start point and the meshing rigidity value at the meshing center point._sIs calculated as follows.
[0144]
C_C0= 0.07139-0.01574 (6-2)
+0.00206 ・ (6-2)²
= 4.146E-2
C_C1= 0.01714-0.00725 · (6-2)
+ 0.00071 · (6-2)²
= −5.643E-4
C_C2= 0.00005 + 0.00009 · (6-2)
+0.000002 ・ (6-2)²
= 4.371E-4
Y_s= K (X_s) / K_p= F_Three(Β₀, BH)
= [4.146E-2 + (-5.643E-4). (36-4)
+ (4.371E-4) / (36-4)²] / 36
= 1.308E-2
[0145]
In subsequent step S115, the arithmetic unit 6 uses the equation (20) to calculate the equivalent conversion torsion angle β with the tooth width / height ratio BH = 6._{0 (BH = 6)}Is calculated.
[0146]
β_{0 (BH = 6)}= Tan^-1[(6/6) tan 36 °]
= 36 °
[0147]
Then, when the process proceeds from step S115 to step S116, the calculation unit 6 uses the expressions (24) to (30), (33), and (34) to calculate an approximate expression of the meshing rigidity curve in the section outside the branch point. Set as follows.
[0148]
A = −0.13500 + 0.08200 · 36 = 2.817
B = 4.442525−0.19427 · 36
+0.00451 ・ (36)²
= 3.275
a₀= Exp [[| 0.407 |^2.817・ Ln (1.3082E-2)
− | −1.258 |^2.817・ Ln (0.965)]
/ [| 0.407 |^2.817− | −1.258 |^2.817]]
= 1.16738
a₁= [Ln (0.965) -ln (1.308E-2)]
/ [| 0.407 |^2.817− | −1.258 |^2.817]
= -2.334216
b₀= [| 0.407 |^3.275・ (1.308E-2)
-0.965 ・ | -1.258 |^3.275]
/ [| 0.407 |^3.275− | −1.258 |^3.275]
= 0.99007
b₁= [0.965- (1.308E-2)]
/ [| 0.407 |^3.275− | −1.258 |^3.275]
= -0.45835
Ka (X) = (1.16738)
・ Exp (−2.334216. | X |^2.817)
Kb (X) = 0.99007-0.45835. | X |^3.275
K (X) / K_p= [(1.16738)
・ Exp (−2.334216. | X |^2.817)
+ (0.99007−0.45835 · | X |^3.275)] / 2 ... (44)
[0149]
Then, when the process proceeds from step S116 to step S117, the calculation unit 6 uses the approximate expressions (43) and (44) of the meshing rigidity curve in each section set in step S112 and step S116, to form a pair of helical teeth. Meshing start point X_sFrom meshing end point X_eEach meshing rigidity value is calculated until the calculation result is output through the output unit 8, and then the routine is terminated.
[0150]
As a result, as shown in FIG. 17, a meshing rigidity value that matches the theoretical value of the meshing rigidity value calculated using the equations (1) to (6) very well is obtained.
[0151]
According to such an embodiment, the first and second branch points at which the behavior of the meshing rigidity curve changes are calculated, and the meshing rigidity curve of the section from the first branching point to the second branching point is approximated. By setting the equation and the approximate expression of the meshing stiffness curve in the section from the meshing start point to the first branch point and from the second bifurcation point to the meshing end point to be different functions, the actual meshing stiffness curve is very good. The approximate expression that coincides can be set, and by using the approximate expression set in this way, the meshing rigidity value of the helical tooth pair can be grasped with high accuracy by simple arithmetic processing.
[0152]
Also, by comparing the magnitude relationship between the front meshing rate and the overlapping meshing rate of the helical tooth pair, and by setting the approximate equation of the meshing stiffness curve to a different function according to the comparison result, the actual meshing stiffness curve and It is possible to set an approximate expression that matches very well, and by using the approximate expression set in this way, it is possible to grasp the meshing rigidity value of the helical tooth pair with high accuracy by simple calculation processing.
[0153]
In these cases, by setting the approximate expression of the section from the first branch point to the second branch point to a function having quadratic and quaternary terms, an approximation that closely matches the actual mesh stiffness curve An expression can be set. At that time, if the front meshing rate is smaller than the overlapping meshing rate, an approximate function created based on the meshing rigidity curve of a predetermined tooth width / height ratio (for example, BH = 6) is based on the coordinates of the branch point. By deforming and setting the approximate expression, the approximate expression can be set with high accuracy by simple processing.
[0154]
Further, when the front meshing rate is larger than the overlapping meshing rate, an approximate expression of the meshing rigidity curve in the section from the meshing start point to the first branch point and from the second branch point to the meshing end point is expressed by the twist angle. By setting different functions in accordance with the comparison result of the magnitude relationship between and the preset twist angle, the approximate expression can be set with high accuracy. Specifically, when it is determined that the twist angle is smaller than the set twist angle, the approximate expression can be set with high accuracy by setting the approximate expression to a linear function. On the other hand, when it is determined that the twist angle is larger than the set twist angle, an exponential function of a predetermined order defined based on the twist angle and a function of a predetermined order defined based on the twist angle is provided. By setting to the function, the approximate expression can be set with high accuracy. At that time, the comparison of the twist angle and the set twist angle is performed by using the twist angle equivalently converted with a predetermined tooth width / height ratio (for example, BH = 6), so that the function to be used is selected under a certain index. Can be done uniformly. Further, when the torsion angle is determined to be larger than the set torsion angle, the order is defined using a torsion angle that is equivalently converted with a predetermined tooth width ratio (for example, BH = 6). It is possible to define a favorable order with a simple operation.
[0155]
Further, when the front meshing rate is smaller than the overlapping meshing rate, an approximate expression of the meshing rigidity curve of the section from the meshing start point to the first branch point and from the second branch point to the meshing end point is expressed by the twist angle. The approximate expression can be set with high accuracy by setting an exponential function of a predetermined order defined on the basis of the above and a function having a function of a predetermined order defined based on the twist angle. At that time, by defining the order using a torsion angle that is equivalently converted with a predetermined tooth width ratio (for example, BH = 6), it is possible to define a favorable order with uniform calculation. .
[0156]
Based on information obtained by performing various analyzes such as the meshing natural frequency and rotational vibration using the meshing rigidity value obtained from the approximation formula set with high accuracy in this way, the helical gear pair specification design, etc. By doing so, a helical gear pair excellent in silence and strength can be obtained.
[0157]
【The invention's effect】
As described above, according to the present invention, it is possible to grasp the meshing rigidity value of the helical tooth pair with high accuracy with a simple calculation process for a wide range of gear specifications.
[Brief description of the drawings]
FIG. 1 is a flowchart showing a meshing rigidity calculation routine for a pair of helical teeth.
FIG. 2 is a schematic configuration diagram of a helical gear pair engagement rigidity calculation device.
FIG. 3 is a schematic view showing an example of a computer for realizing a meshing rigidity calculating device for a pair of helical teeth.
FIG. 4 is an explanatory diagram showing an action plane of a helical gear pair.
FIG. 5 is a chart showing a meshing rigidity curve of a helical tooth pair in each case.
FIG. 6 is an explanatory diagram showing a meshing branch point in a case (case 1) in which the front meshing rate is larger than the overlapping meshing rate.
FIG. 7 is an explanatory diagram showing a meshing branch point in a case (case 2) in which the front meshing rate is equal to the overlapping meshing rate.
FIG. 8 is an explanatory diagram showing a meshing branch point in a case (case 3) in which the front meshing rate is smaller than the overlapping meshing rate.
FIG. 9 is a chart showing a meshing rigidity curve between branch points in case 1;
FIG. 10 is a chart showing a meshing rigidity curve between branch points in case 3;
FIG. 11 is a chart showing each meshing rigidity curve of FIG. 10 after coordinate conversion.
FIG. 12 is a chart showing a meshing rigidity curve outside the branching point in case 1 when the branching point is close to the meshing start point and the meshing end point.
FIG. 13 is a chart showing a meshing rigidity curve outside the branching point in case 1 when the branching point is close to the meshing center point.
FIG. 14 is a chart showing a meshing rigidity curve outside a branch point in case 2;
FIG. 15 is a chart showing a comparison result between a theoretical value and an approximate value of a meshing rigidity curve in case 1;
FIG. 16 is a chart showing comparison results between theoretical values and approximate values of meshing rigidity curves in Case 2;
FIG. 17 is a chart showing a comparison result between the theoretical value and approximate value of the meshing rigidity curve in case 3;
[Explanation of symbols]
1 ... meshing rigidity calculation device
6... Calculation unit (branch point calculation means, engagement rate comparison means, first approximate expression setting means, second approximate expression setting means, torsion angle comparison means)

Claims (10)

An approximation formula of the meshing stiffness curve is set based on the specifications of the helical tooth pair meshing with each other, and each meshing stiffness value between the meshing start point and the meshing end point is calculated based on the approximate formula. A device for calculating the meshing rigidity of a tooth pair,
A branch point calculating means for calculating the first and second branch points of the meshing rigidity curve of the helical tooth pair;
First approximation formula setting means for setting an approximation formula of the meshing rigidity curve in a section from the first branch point to the second branch point;
Second approximation formula setting means for setting an approximation formula of the meshing rigidity curve in a section from the meshing start point to the first branch point and from the second branch point to the meshing end point ;
A meshing rate comparison means for comparing the magnitude relationship between the front meshing rate of the helical tooth pair and the overlapping meshing rate ;
The first and second approximate expression setting means set the approximate expression of the meshing rigidity curve in each section to a different function according to a comparison result between the front meshing rate and the overlapping meshing rate. A feature is a device for calculating meshing rigidity of a pair of helical teeth.   The first approximate expression setting means sets the approximate expression to a function having a variable on the equivalent action line of the helical tooth pair and having a quadratic term and a quadratic term. The apparatus for calculating the meshing rigidity of the helical tooth pair according to claim 1.   When it is determined by the mesh rate comparison means that the front mesh rate is smaller than the overlap mesh rate,
  The first approximate expression setting means sets the approximate expression by deforming an approximate function created based on a meshing rigidity curve having a predetermined tooth width / height ratio, based on the coordinates of the branch point. The apparatus for calculating the meshing rigidity of the helical tooth pair according to claim 1 or 2.   A torsion angle comparison that compares the magnitude relationship between the torsion angle of the helical tooth pair and a preset set torsion angle when the intermeshing rate comparing means determines that the front engagement rate is greater than the overlapping engagement rate. Having means,
  The second approximate expression setting means sets the approximate expression to a different function according to a comparison result of the torsion angle comparison means. The helical rigidity calculation device for helical pairs.   5. The helical tooth pair according to claim 4, wherein the torsion angle comparing means compares the magnitude relationship with the set torsion angle by using a torsion angle equivalently converted at a predetermined tooth width / height ratio. Meshing rigidity calculation device.   When the twist angle comparison means determines that the twist angle is smaller than the set twist angle,
  The second approximate expression setting means sets the approximate expression to a linear function having coordinates on the equivalent action line of the helical tooth pair as a variable. A device for calculating the meshing rigidity of a pair of helical teeth.   When it is determined by the twist angle comparison means that the twist angle is larger than the set twist angle,
  The second approximate expression setting means is a function having the approximate expression as a variable with coordinates on the equivalent action line of the helical tooth pair as a variable, and an exponential function of a predetermined order defined based on the twist angle. 6. The apparatus according to claim 4, wherein the function is set to a function of a predetermined order defined based on the torsion angle.   8. The calculation of the meshing rigidity of a pair of helical teeth according to claim 7, wherein the second approximate expression setting means defines the order using a torsion angle equivalently converted at a predetermined tooth width / height ratio. apparatus.   When it is determined by the mesh rate comparison means that the front mesh rate is smaller than the overlap mesh rate,
  The second approximate expression setting means is a function having the approximate expression as a variable with coordinates on the equivalent action line of the helical tooth pair as a variable, and an exponential function of a predetermined order defined based on the twist angle. 9. The apparatus according to claim 1, wherein the function is set to a function of a predetermined order defined based on the torsion angle.   10. The calculation of meshing rigidity of a pair of helical teeth according to claim 9, wherein the second approximate expression setting means defines the order using a torsion angle obtained by equivalent conversion with a predetermined tooth width / height ratio. apparatus.

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