MXPA06007001A - Gear tooth profile curvature - Google Patents

Abstract

A gear system which includes a pinion and mating-gear forming a gear pair with predetermined gear ratio (mG), center distance (C), face-width (FW), and limiting stresses. The pinion has a pinion tooth number (N1)nd a first plurality of teeth, each tooth having a first-tooth profile. The mating gear has a mating-gear tooth number (N2) satisfying the expression N2=mG . N1 and a second plurality of teeth, each tooth having a second-tooth profile. The relative curvature of the first-tooth profile and the second-tooth profile is given by an expression kc . Fc, where Fc is a relative reference curvature function given by the expression Fc=(N1+ N2)2 /(N1. N2 . C), and kc is a relative curvature multiplier which is a function of the gear ratio (mG), the center distance (C), the face-width (FW), and the limiting stresses.

Description

GEAR TOOT PROFILE CURVATURE CROSS REFERENCE WITH RELATED REQUESTS This application is related to the provisional patent application of E.U.A. No. 60 / 530,752, filed on December 18, 2003, and claims the first filing date of the provisional application which is incorporated herein by reference.

BACKGROUND OF THE INVENTION The present invention relates to the construction of conjugated gear profiles. In particular, the present invention relates to the construction of profiles of conjugate gears having a relative curvature which is a function of gear ratio, face width factor, center distance and limiting stresses. In the patent of E.U.A. 6101892, incorporated herein by reference, three methods were described to specify the curvatures of conjugate tooth profiles. If s, < j > are the polar coordinates of points in the contact path, and pi, p2 are the radii of curvature of the profile at the corresponding points in the tooth profiles, these methods can be established as follows: \ / p2 + \ J p2 = Cons tan te -j \ (l /, + l / p2) / eos f = Cons tan te '^ (3) f s, f, px, p2) = Cons tan te The first method can be described as relative curvature constant. The second method is suitable for straight gears, and is intended to provide constant contact voltage. In the third method, f can be any function, specified by the designer of the profiles of the tooth. For conventional gears, such as circle envelope profile gears, if the load intensity is known, ie the force of the tooth per unit length of the contact curve, then the stresses of the tooth can be found through known conventional methods. However, finding the load intensity for a given torque is more difficult for cog gears - than for those with a circle envelope profile. In a pair of circle envelope profile gears, the contact curves are straight lines and at each point of a contact line the normal to the surface of the tooth points in the same direction, so that for a determined torque the load intensity is inversely proportional to the total length of the contact lines. The maximum load intensity is found when the The total length of the contact lines is minimal, and this occurs when a contact line passes through a corner of the contact region.

In a pair of Convoloid gears, in contrast, the contact curves are not exactly straight, the normal curves do not point exactly in the same direction, and the contact curves break when they pass through the transition zone. Therefore, the load intensity is not inversely proportional to the total length of the contact curves, and the position of the contact curves for maximum load intensity is unknown. Therefore, it is advisable to design Convoloid gear pairs that have a relative curvature for which the maximum voltages are approached but the limiting voltages are not exceeded.

BRIEF DESCRIPTION OF THE INVENTION Briefly stated, one embodiment of the present invention relates to a gear system comprising a pinion and a geared gear. The pinion has a number of pinion teeth (N, a radius of the pinion pitch circle (Rp-?) And a first plurality of teeth, each tooth having a first tooth profile.The gear meshing has a number of teeth of geared gear (N2), a pitch circle radius of the geared gear (Rp2), and a second plurality of teeth, each tooth has a second tooth profile.The pinion and the geared gear form a pair of gears having a relationship of gear (me) equal to N2 / N- ?, a width of the face (Fw) and a width factor of the face (fw) equal to (2 • 2 FW.) The relative curvature of the profile of the first tooth and the second tooth profile is a multiple of a relative reference curvature (K rer), the manifold given by the expression K m • K ref, where y K m is a relative curvature multiplier that is greater than a and j and aa, j + 2d, where d is approximately 0.15 and j, is given by a predetermined relationship between the gear ratio (I? IG) and the face width factor (fw). The predetermined ratio corresponding to at least one relative curvature multiplier value in a table of relative curvature multiplier values has the following properties: Face width factor (fw) 4.0 5.0 6.0 Gear ratio (mG) 1.0 0.41 0.40 0.39 1.5 0.43 0.41 0.40 2.0 0.43 0.43 0.41 2.5 0.48 0.45 0.41 3.0 0.48 0.48 0.44 4.0 0.48 0.48 0.48 6.0 0.46 0.46 0.46 16.0 0.43 0.43 0.43 Another embodiment of the present invention relates to a gear system comprising a pinion and a geared gear. The pinion has a number of pinion teeth (N-t), a pinion pitch circle radius (Rp?) And a first plurality of teeth, each tooth having a first tooth profile. The geared gear has a number of meshing gear teeth (N2), a geared gear pitch circle radius (Rp2), and a second plurality of teeth, each tooth having a second tooth profile. The pinion and the geared gear form a pair of gears that have a central distance (C) equal to (Rp1 + Rp2), gear ratio (p G) equal to N2 / N- ?, a width of the face (Fw) and a face width factor (fw) equal to (2 • Rp2) / Fw. The relative curvature of the first tooth profile and the second tooth profile is given by the expression kc • Fc where Fc is a relative reference curvature function given by the expression Fc = (N? + N2) 2 / (N N2 • C) and kc is a relative curvature multiplier that is greater than bj, rd and less than b and + 2d, where d is approximately 0.439 and bj, j is given by a predetermined relationship between the gear ratio (ITIG) and the factor width of the face (fw). The predetermined relationship corresponds to at least one relative curvature multiplier value in a table of relative curvature multiplier values that has the following properties: Face width factor (fw) 4.0 5.0 6.0 Gear ratio (mG) 1.0 1,199 1,170 1,5 1,257 1,199 1,190 2.0 1,257 1,257 1,199 2.5 1,403 1,316 1,199 3.0 1,403 1,403 1,286 4.0 1,403 1,403 1,403 6.0 1,345 1,345 1,345 16.0 1,257 1,257 1,257 Another embodiment of the present invention relates to a gear system having a predetermined gear ratio (mG), a predetermined central distance (C), a predetermined face width (Fw) and predetermined limit stresses. The gear system comprises a pinion and a geared gear. The pinion has a number of pinion teeth (N-i), and a first plurality of teeth, each tooth having a first tooth profile. The geared gear has a number of gear teeth meshed (N2) that satisfies the expression N2 = ITIG-N-I, and a second plurality of teeth, each tooth has a second tooth profile. The pinion and the geared gear form a pair of gears having a face factor of the face (fw) equal to (2- N2-C) / ((N? + N2> Fw) .The relative curvature of the profile of first tooth and the second tooth profile is a multiple of a relative reference curvature (Kref), the multiple given by the expression Km-Kr? f, where the relative curvature of the profile of the first tooth and the profile of the second tooth is given by the expression kc • Fc where Fc is a function of relative reference curvature given by the expression c = (N? + N2) 2 / (Nr N2-C) and kc is a relative curvature multiplier where kc is determined through a method comprising the following steps: (a) determining a first plurality of load intensities for a predetermined input torque, each load current is associated with a unique angular position of a plurality of angular positions of the pinion , the plurality of angular positions encompasses n an angular pitch of the pinion, each load intensity is based on a test relative curvature multiplier (kc '); (b) determining a plurality of tooth voltages corresponding to a higher load intensity of the plurality of load intensities; (c) setting the highest load current to a graded load current such that a tooth voltage of the plurality of tooth voltages approximates one of the predetermined limiting stresses; (d) determining a limiting torque corresponding to the graded load intensity; (e) repeating steps (a) - (d) for a plurality of test relative curvature multipliers (kc ') within a predetermined scale of relative test curvature multipliers and selecting as a relative curvature multiplier (? c) ) the test relative curvature multiplier (kc ') that corresponds to the limiting torque that has the largest value. Another preferred embodiment of the present invention is a gear system having a predetermined gear ratio (GTIG), a predetermined central distance (C), a predetermined face width (Fw), and predetermined limit stresses. The gear system comprises a pinion and a geared gear. The pinion has a number of pinion teeth (N-i), and a first plurality of teeth, each tooth having a first tooth profile. The geared gear has a number of gear teeth meshed (N2) that satisfies the expression N2 = me Ni, and a second plurality of teeth, each tooth has a second tooth profile. The pinion and the geared gear form a pair of gears that have a face width factor (fw) equal to (2- N2-C) / ((N? + N2 Fw) .The relative curvature of the profile of the first tooth and the second tooth profile is given by an expression kFc where Fc is a function of relative reference curvature given by the expression Fc = (N -? + N2) 2 / (Nr N2-C) and where kc is a Relative curvature multiplier which is a function of gear ratio (mG), face width factor (fw), center distance (C), and one of the limiting stresses.

BRIEF DESCRIPTION OF THE DRAWINGS The above description, as well as the following detailed description of the invention, will be better understood when read together with the accompanying drawings. For purposes of illustrating the invention, the modalities that are currently preferred are shown in the drawings. However, it should be understood that the invention is not limited to the precise arrangements and mediations shown. In the drawings: Fig. 1 is a cross-sectional view of gear tooth profiles of a pinion tooth engaged with a gear tooth meshed in accordance with a preferred embodiment of the present invention; Figure 1A is an enlarged view of the transition zones in Figure 1; and Figure 2 is a diagram of a preferred method for determining the relative curvature multiplier for the relative curvature of the first and second tooth profiles of the pinion and teeth of the geared gear of Figure 1.

DETAILED DESCRIPTION OF THE INVENTION Referring to FIGS. 1-1A, the gear tooth profiles of a gear system, generally designated as 10, and hereinafter referred to as the gear system 10 according to the invention, are shown in a transverse plane. present invention. A preferred embodiment of the gear system 10 having a relative curvature consistent with the present invention comprises a pinion 100 and a geared gear 200. The pinion 100 has a first or plurality of teeth. Each tooth of the first plurality of teeth has a profile of first tooth 102. The profile of first tooth 102 has a central line of profile of first tooth 104 and intersects with a circle, of pinion passage 106 at the point of passage of the tooth. first tooth profile 108. The pitch circle of the pinion 106 has a pinion pitch circle radius (Rp?). The first tooth profile 102 includes a first transition zone 140 disposed in a first concave portion 134 extending within the height of the foot 130 of the pinion 100 and a first convex portion 124 extending within the height of the foot 120 of the pinion 100. The pinion 100 has a number of pinion teeth (Ni) corresponding to the number of teeth in the first plurality of teeth. The geared gear 200 has a second plurality of teeth. Each tooth of the second plurality of teeth has a profile of second tooth 202. The profile of second tooth 202 has a center line of second tooth 204 and intersects with a pitch circle of geared gear 206 at the point of passage of the tooth profile. second tooth 208. The pitch circle of the geared gear 206 has a gear pitch circle radius (Rp2). The second tooth profile 202 includes a second transition zone 240 disposed between a second concave portion 234 extending within the height of the foot 230 of the geared gear 200 and a second convex portion 224 extending within the height of the foot 220 of the geared gear 200. The second concave portion 234 is conjugated with the first convex portion 124 of the first tooth profile 102 of the first plurality of teeth of the pinion 100. The second convex portion 224 is conjugated with the first concave portion 134 of the first tooth 102 of the first plurality of teeth of the pinion 100. The geared gear 200 has a number of gear teeth meshed (N2) corresponding to the number of teeth in the second plurality of teeth. The pinion 100 and the geared gear 200 form a pair of gears having a gear ratio (mG) equal to N- | / N2, a central distance (C), a face width (Fw), and a factor of face width (fw) equal to (2- Rp2) / Fw. The relative curvature for the first and second-tooth profiles 102, 202 can be represented by the following equation: *, + k2 = Ae ~ B ?? (4) where? = S sin < j > / mn (5) ?? and? 2 are the curvatures of the profile, equal to the reciprocals of the radii of curvature, A and B are constants selected by the user, mn is the normal module, and? is the coordinate without dimension along the line of centers, originating at the point of passage. The user specifies the curvatures relative to? = - 1,? = 0, and? = 1, so there will be a pair of values for A and B at the height of the pinion foot, and a different pair at the height of the pinion. head. Studies have been conducted to determine the optimal input values for many pairs of gears. Because the results for a pair of gears can be increased or reduced to scale, the center distance is not a factor. The pairs of gears have been specified by their number of teeth and by their face width factor, which is defined as the pitch diameter of the gear divided by the width of the face. The results of the studies show that the lowest load intensities are found when three relative input curves are either all equal, or have a very similar value. For this reason, the function given previously in equation (4) is no longer used, and the relative curvature is specified as a constant during the entire gearing cycle. Accordingly, the relative curvature of the first tooth profile 102 and the second tooth profile 202 is a multiple of a first relative reference relative curvature (Kref), the multiple given by the expression Km-Kref where Km is a curvature multiplier. relative.

The relative curvature of reference (Kret) is the relative curvature at the point of passage of a pair of straight gears with a pressure angle of 20 degrees, which have the same number of teeth and center distance as the pair of gears that is considered, and is given through the equation of Euler-Savary. Therefore, The procedure shown in Figure 2, discussed later in detail, has been used to calculate the limiting torques corresponding to the relative curvature multiplier (Km) for the following case combinations based on the relative curvature of the reference previous (Kref): Gear ratios (mG): 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 6.0, 16.0.

Face width factors (fw): 4.0, 5.0, 6.0.

For each combination of gear ratios previous (mG) and face width factors (fw), the number of pinion teeth (N-i) and the relative curvature multiplier (Km), which they gave the largest limiting torque. The relative curvature multipliers (Km) are shown in table 1.

TABLE 1 Relative curvature multipliers (km) Face width factor (fw) 4.0 5.0 6.0 Gear ratio (mG) 1.0 0.41 0.40 0.39 1.5 0.43 0.41 0.40 2.0 0.43 0.43 0.41 2.5 0.48 0.45 0.41 3.0 0.48 0.48 0.44 4.0 0.48 0.48 0.48 6.0 0.46 0.46 0.46 16.0 0.43 0.43 0.43 The numbers of pinion teeth (N-i) that correspond to the multipliers presented in table 1 are shown in table 2.

TABLE 2 Number of pinion teeth (Ni) Face width factor (fw) 4.0 5.0 6.0 Gear ratio (mG) 1.0 18 22 28 1.5 12 16 18 2.0 14 11 14 2.5 14 10 12 3.0 14 13 10 4.0 13 13 12 6.0 13 13 13 16.0 11 11 12 For pairs of gears whose gear ratio (mG) and face width factor (fw) are among the numbers in the table, the relative curvature multiplier (Km) can be determined through linear interpolation. For example, if the gear ratio (mG) is 1.4 and the face width factor (fw) is 4.3, the relative curvature multiplier (Km) can be found as follows: (Km) = 0.2 (0.7) * 0.41 + 0.3 * 0.40) + 0.8 (0.7 * 0.43 + 0.3 * 0.41) = 0.4206 For pairs of gears with a face width factor (fw) less than 4.0, the relative curvature multiplier (Km) fits the same to the value that would be obtained if the face width factor (fw) were 4.0. For pairs of gears with a face width factor (fw) of more than 6.0, the relative curvature multiplier (fw) is set equal to the value that would be obtained if the face width factor (fw) were 6.0. For any pair of gears with a gear ratio (mG) greater than 16.0, the relative curvature multiplier (Km) is set equal to 0.43. For pairs of gears with pinion teeth numbers (N-which are different from those in Table 2, even the relative curvature multipliers (Km) can be used in Table 1. The result will not be optimal, since the torque of limiting torque will be less than the value when using the pinion tooth numbers (Ni) of table 2. For multipliers of relative curvature (Km) greater than those given in table 1, the limiting torque decreases slowly. Relative curvature (Km) lower than those in Table 1, the limiting torque sometimes decreases, or alternatively, the limiting torque can increase pers the profile contact ratio falls below 1.0. A profile contact ratio of less than 1.0 is generally not considered acceptable, but because the gears are helical, they may be adequate, as they provide an or a constant angular velocity ratio 5. It is evident that the relative curvature multipliers (Km) above and below those in table 1 can be used to design satisfactory gear pairs. For this reason, this description covers a scale of relative curvature multipliers (Km), which extends from 0.15 below the values in Table 1, to 0.30 per . above the values in the box. For the lower curvature (Km) lower limit multipliers shown above, the profile contact ratio is less than 0.85, which means that the gear pair is more likely to be unacceptable. For multipliers of relative curvature (Km) greater than upper limit, the limiting torque is 80% or less of the limiting torque when using the relative curvature multipliers (Km) given in table 1. For the above reasons, it has been determined that the scale of possible multipliers of relative curvature (Km) is greater than aaj-¿> e 0 lower than aj + 2d, where < Ss approximately 0.15 and aj corresponds to at least one relative curvature multiplier value in Table 1; The relative curvature of the profile of first tooth 102 and the profile of second tooth 202 can also be given alternatively through the expression ks Fc where Fc is a function of relative reference curvature given by the expression Fc = (Ni + N2) 2 / (N N2- C) and kc is a multiplier of relative curvature that is greater than b - d and less than b, j + 2d, where d is approximately 0.439 and b, j is given through a predetermined relationship between the gear ratio (mG) and the face width factor (fw). The The predetermined ratio corresponds to at least one relative curvature multiplier value in a table of relative curvature multiplier values that has the following properties: TABLE 3 Relative curvature multipliers. { kc) Face width factor (fw) 4.0 5.0 6.0 Gear ratio (mG) 1.0 1.199 1.170 1.140 1.5 1.257 1.199 1.170 2.0 1.257 1.257 1.199 2.5 1.403 1.316 1.199 3.0 1.403 1.403 1.286 4.0 1.403 1.403 1.403 6.0 1.345 1.345 1.345 16.0 1.257 1.257 1.257 For gear pairs with a face width factor (fw) of less than 4.0, the relative curvature multiplier (kc) is set equal to the value that would be obtained if the relative curvature multiplier (kc) were 4.0. For pairs of gears with a face width factor (fw) of more than 6.0, the relative curvature multiplier (kc) is set equal to the value that would be obtained if the relative curvature multiplier (kc) were 6.0. For any pair of gears with a gear ratio greater than 16.0, the relative curvature multiplier (kc) is set equal to 1.257. For gear pairs with a gear ratio or face width factor that falls between the numbers in the frame, interpolation is used, preferably linear based on at least one or two relative curvature multiplier values to determine the relative curvature multiplier (kc). As an alternative to determine the value for the relative curvature multiplier (kc) by consulting the frame when the values for the width of the face (Fw), number of teeth of the pinion (Ni), radius of the pitch circle of the Sprocket (Rp?), number of teeth of the gear meshing (N2) and radius of pitch circle of the geared gear (Rp2), the relative curvature multiplier (kc) can be determined through the five-step procedure discussed below. Referring to Figure 2, the determination of the relative curvature multiplier (kc) comprises a multi-step procedure involving known methods for determining tooth profiles, contact curves, load intensities, tooth tensions and the like. The methods by which these characteristics or parameters are determined have been described by Buckingham, ANALYTICAL MECHANICS OF GEARS, McGraw-Hill, 1949, republished by Dover, NY, 1963, incorporated in its entirety to this and the US patent 6,101, 892, also incorporated in its entirety to the present and not discussed in this document for reasons of brevity. Because you can use a procedure similar to the procedure below to determine the relative curvature multipliers (Km), for brevity, the determination of the relative curvature multipliers (Km) is not discussed. With reference to Figure 2, a first step 310 for determining the relative curvature multiplier (Kc) comprises determining a plurality of load intensities for a predetermined input torque. Each load intensity is associated with a unique angular position of a plurality of angular positions of the pinion 100. The plurality of angular positions encompasses an angular pitch of the pinion 100. Each load intensity is based on the number of teeth of the pinion (Ni) and a test relative curvature multiplier (k'c). More specifically, in step 310, the face width factor iw = (2 • Rp2) / Fw and the relative reference curvature function Fc = (Ni + N2) 2 / (Ni • N2 • C). A value is assumed for the test relative curvature multiplier (k'c) within a predetermined scale, for example, 0.7 <; k'c < 2.3, and the profiles of first and second teeth 102, 202 are determined in accordance with the teachings of the U.S. patent. No. 6,101, 892. A value is assumed, for example 11, 530 kg-cm, for an input torque (input Z) - A plurality of angular positions encompassing an angular pitch of the pinion 100. For each angular position of the plurality of angular positions, the positions of the contact curves are calculated and the contact curves are divided into a plurality of small increments. Preferably, the number of increments in the plurality of small increments is greater than 200 increments and less than 500 increments in each contact curve and is based on the desired accuracy of the calculations. An arbitrary value, for example 17,850 kg / m, is assumed for the load intensity and a contribution of torque of a contact force in each increment is determined. The torque contributions are summed to obtain a total torque corresponding to the load intensity assumed. The assumed load current is subsequently adjusted so that the total torque is equal to the assumed input torque (ü input) - The procedure for determining a graded load intensity that corresponds to the total torque is repeated for each angular position of the pinion and the highest load intensity is selected for further processing. A second step 320 comprises determining a plurality of tooth tensions, for example, a contact voltage, an engagement tension of the pinion, and a geared gear engagement tension, which corresponds to a higher load intensity of the plurality of intensities of cargo. A third step 330 comprises setting the highest load intensity to a graded load current so that a tooth voltage of the plurality of tooth tensions approaches a predetermined limiting voltage which is a characteristic of the material from which it is manufactured. the pair of gears. A fourth step 340 comprises determining a limiting torque corresponding to the graded load intensity. The corresponding limiting torque is equal to the limiting torque for the pair of gears for the test relative curvature multiplier (k'c) assumed in the first step 310. A fifth step 350 comprises repeating from the first to the fourth steps 310-340 for a plurality of test relative curvature multipliers (k'c) within the predetermined scale of test relative curvature multipliers and selecting as the relative curvature multiplier (kc) the test relative curvature multiplier (k'c) which corresponds to the limiting torque that has the largest value. As an alternative to determine the value for the relative curvature multiplier (kc) by consulting the frame when the values for gear ratio (mG), center distance (C), face width (Fw), and limiting stresses, the relative curvature multiplier (kc) can be determined through the five-step procedure discussed above as modified below. In the first step 310, the number of pinion teeth (Ni) is set equal to a number of teeth of the test pinion (N'i).

In the fifth step 350a, it is repeated from the first to the fourth steps 310-340 for a plurality of test relative curvature multipliers (k'c) within a predetermined scale of test relative curvature multipliers and the test relative curvature multiplier (k'c) which corresponds to the limiting torque that has the largest value as a relative test curvature multiplier related to the number of pinion teeth (k "c). step 360. The sixth step 360 repeats from the first to the fifth steps 310-350 for a plurality of numbers of test pinion teeth (N'i) within a predetermined scale of test pinion tooth numbers, for example 10 <N'i <30, and the test relative curvature multiplier related to the number of pinion teeth (k "c) is selected corresponding to the limiting torque having the largest value as the relative curvature multiplier (kc). Those skilled in the art will understand that changes can be made to the modalities described above without departing from the broad inventive concept thereof. Thus, it is understood that this invention is not limited to the particular embodiments described, but is intended to encompass the modifications within the spirit and scope of the present invention as defined in the appended claims.

Claims (24)

NOVELTY OF THE INVENTION CLAIMS

1. - A gear system comprising: a pinion having a number of pinion teeth (Ni), a radius of the pinion pitch circle (Rp) and a first plurality of teeth, each tooth having a first tooth profile; and a geared gear having a number of meshing gear teeth (N2), a pitch circle radius of the geared gear (Rp2), and a second plurality of teeth, each tooth having a second tooth profile, wherein the pinion and the geared gear form a pair of gears having a gear ratio (mG) equal to N2 / N ?, a face width (Fw) and a face width factor (f) equal to (2 • p2) / Fw, and where the relative curvature of the profile of the first tooth and the profile of the second tooth is a multiple of a relative curvature of reference (K ref), the multiple given by the expression K m • K ref, where? Ref and K m is a relative curvature multiplier that is greater than aa-y and less aa, j + 2d, where d is approximately 0.15 and j is given by a predetermined relationship between the gear ratio (mG) and the factor of face width (fw), the default ratio that corresponds to at least one multiplier of relative curvature in a table of values of relative curvature has the following properties: Width factor of I; to face (fw) 4.0 5.0 6.0 Gear ratio (mG) 1.0 0.41 0.40 0.39 1.5 0.43 0.41 0.40 2.0 0.43 0.43 0.41 2.5 0.48 0.45 0.41 3.0 0.48 0.48 0.44 4.0 0.48 0.48 0.48 6.0 0.46 0.46 0.46 16.0 0.43 0.43 0.43

2. - The gear system according to claim 1, further characterized in that the relative curvature multiplier (kc) when the face width factor (fw) is less than 4.0, corresponds to the relative curvature multiplier (kc) when the face width factor (fw) is equal to 4.0.

3. The gear system according to claim 1, further characterized in that the relative curvature multiplier (km) when the face width factor (fw) is greater than 6.0, corresponds to the relative curvature multiplier (km) when the face width factor (fw) is equal to 6.0.

4. The gear system according to claim 1, further characterized in that the relative curvature multiplier (km) corresponds to an interpolated value based on at least two relative curvature multiplier values in the table of relative curvature multiplier values .

5.- The system of gears of conformity. with claim 1, further characterized in that d is equal to 0.

6. - The gear system according to claim 5, further characterized in that the relative curvature multiplier (km) corresponds to an interpolated value based on at least two relative curvature multiplier values in the relative curvature multiplying value table.

7. The gear system according to claim 1, further characterized in that the first tooth profile includes a first transition zone disposed between a first concave portion that is within the height of the foot of the pinion and a first convex portion which is within the height of the pinion head, and the second tooth profile includes a second transition zone disposed between a second concave portion which is within the height of the geared gear foot, the second concave portion is conjugated With the first convex portion of the first tooth profile of the first plurality of teeth of the pinion, and a second convex portion lying within the height of the geared gear head, the second convex portion is conjugated with the first concave portion of the tooth. first tooth profile of the first plurality of teeth of the pinion.

8. The gear system according to claim 7, further characterized in that the curvature multiplier (Km) corresponds to an interpolated value based on at least two values of relative curvature multipliers in the table of relative curvature multiplier values.

9. - The gear system according to claim 7, further characterized in that .5 is equal to zero.

10. The gear system according to claim 9, further characterized in that the relative curvature multiplier corresponds to an interpolated value based on at least two values of relative curvature multipliers in the table of relative curvature multiplying values.

11. A gear system comprising: a pinion having a number of pinion teeth (Ni), a radius of the pitch circle of the pinion (Rp?) And a first plurality of teeth, each tooth having a profile of first tooth, the pinion; and a geared gear having a number of meshing gear teeth (N), a pitch circle radius of the geared gear (Rp2), and a second plurality of teeth, each tooth having a second tooth profile, wherein the pinion and the geared gear form a pair of gears having a central distance (C) equal to (RP? + Rp2), gear ratio (mG) equal to N2 / N ?, a width of the face (Fw) and a factor width of the face (fw) equal to (2 • RP2) / FW, and where the relative curvature of the first tooth profile and the second tooth profile is defined by the expression kc • Fc where Fc is a curvature function relative reference given by the expression Fc = (N? + N2) 2 / (N N2 • C) and kc is a relative curvature multiplier that is greater than by-d and smaller ab¡ + 2d, where d is approximately 0.439 and by is defined by a predetermined relationship between the gear ratio (mG) and the face width factor (fw), the ratio pre determined that corresponds to at least one relative curvature multiplier value in a table of relative curvature multiplier values has the following properties: Face width factor (f) 4.0 5.0 6.0 Gear ratio (mG) 1.0 1,199 1,170 1,132 1,5 1,259 1,199 1,170 2.0 1,257 1,257 1,199 2.5 1,403 1,316 1,199 3,0 1,403 1,403 1,286 4.0 1,403 1,403 1,403 6.0 1,345 1,345 1,345 16.0 1,257 1,257 1,257

12. - The gear system according to claim 11, further characterized in that the relative curvature multiplier (kc) when the face width factor (fw) is less than 4.0, corresponds to the relative curvature multiplier (kc) when the factor width of the face (fw) is equal to 4.0.

13. The gear system according to claim 11, further characterized in that the relative curvature multiplier (kc) when the face width factor (fw) is greater than 6.0, corresponds to the relative curvature multiplier (kc) when the face width factor (fw) is equal to 6.0.

14. The gear system according to claim 11, further characterized in that the relative curvature multiplier (kc) corresponds to an interpolated value based on at least two relative curvature multiplier values in the Relative Curve Multiplier Table .

15. The gear system according to claim 11, further characterized in that equal to 0.

16. The gear system according to claim 15, further characterized in that the relative curvature multiplier (kc) corresponds to a interpolated value based on at least two relative curvature multiplying values in the relative curvature multiplying value table.

17. The gear system according to claim 11, further characterized in that the first tooth profile includes a first transition zone disposed between a first concave portion that is within the height of the pinion foot and a first convex portion which is within the height of the pinion head, and the second tooth profile includes a second transition zone disposed between a second concave portion which is within the height of the geared gear foot, the second concave portion is conjugated With the first convex portion of the first tooth profile of the first plurality of teeth of the pinion, and a second convex portion lying within the height of the geared gear head, the second convex portion is conjugated with the first concave portion of the tooth. first tooth profile of the first plurality of teeth of the pinion.

18. - The gear system according to claim 17, further characterized in that the bending multiplier (kc) corresponds to an interpolated value based on at least two relative curvature multiplying values in the Relative Curve Multiplier Table.

19. The gear system according to claim 17, further characterized in that it equals zero.

20. The gear system according to claim 19, further characterized in that the relative curvature multiplier (kc) corresponds to an interpolated value based on at least two values of relative curvature in the table of relative curvature multipliers. .

21. A gear system having a predetermined gear ratio (mG), a predetermined central distance (C), a predetermined face width (Fw), and predetermined limiting stresses, the gear system comprising: a pinion that has a number of pinion teeth (Ni) and a first plurality of teeth, each tooth has a first tooth profile; and a geared gear having a number of meshing gear teeth (N2) that satisfies the expression N2 = mG-Ni and a second plurality of teeth, each tooth having a second tooth profile, wherein the pinion and the geared gear form a pair of gears having a face width factor (fw) equal to (2-N2-C) / ((N? + N2) -Fw); and wherein the relative curvature of the first tooth profile and the second tooth profile is defined by a kc-Fc expression where Fc is a relative reference curvature function defined by the expression Fc = (N? + N2) 2 / ( N1 N2-C), and kc is a relative curvature multiplier where Kc is determined through a method comprising the following steps: (a) determining a plurality of load intensities for a predetermined input torque, each load intensity is associated with a unique angular position of a plurality of angular positions of the pinion, the plurality of angular positions encompassing an angular pitch of the pinion, each load intensity is based on a test relative curvature multiplier (k'c); (b) determining a plurality of tooth voltages corresponding to a higher load current of the plurality of load intensities; (c) setting the highest load current to a graduated load current so that a tooth voltage of the The plurality of tooth voltages approximates one of the predetermined limiting stresses, (d) determining a limiting torque corresponding to the graded load intensity, and (e) repeating steps (a) - (d) for a plurality of of test relative curvature multipliers (k'c) within a predetermined scale of relative curvature values of the test relative curvature and -select as the relative curvature multiplier (kc) the corresponding relative curvature of test (k'c) multiplier to the limiting torque that has the largest value.

22. - The gear system according to claim 21, further characterized in that the predetermined scale of test relative curvature multipliers is approximately 0.7 to 2.3.

23. A gear system having a predetermined gear ratio (mG), a predetermined central distance (C), a predetermined face width (Fw), and predetermined limiting stresses, the gear system comprising: a pinion that has a number of pinion teeth (Ni) and a first plurality of teeth, each tooth has a first tooth profile; and a geared gear having a number of meshing gear teeth (N2) that satisfies the expression N2 = mG-Ni and a second plurality of teeth, each tooth having a second tooth profile, wherein the pinion and the geared gear form a pair of gears that have a face width factor (fw) equal to (2-N2-C) / ((N? + N2) -Fw); and wherein the relative curvature of the first tooth profile and the second tooth profile is defined by a kc-Fc expression where Fc is a relative reference curvature function defined by the expression Fc = (N? + N2) 2 / ( N1 N2-C), and where kc is a relative curvature multiplier that is a function of gear ratio (mG), face width factor (fw), center distance (C), and one of the limiting tensions.

24. The gear system according to claim 23, further characterized in that the predetermined ratio is determined through a method comprising the following steps: (a) determining a plurality of load intensities for an input torque predetermined, each load intensity is associated with a unique angular position of a plurality of angular positions of the pinion, the plurality of angular positions encompassing an angular pitch of the pair of gears, each load intensity is based on a number of pinion teeth of test (N'i) and a test relative curvature multiplier (k'c); (b) determining a plurality of tooth voltages corresponding to a higher load current of the plurality of load intensities; (c) setting the highest load current to a graduated load current so that a tooth voltage of the plurality of tooth voltages approximates a predetermined limiting voltage; (d) determining a limiting torque corresponding to the graded load intensity; (e) repeating steps (a) - (d) for a plurality of test relative curvature multipliers (k'c) within a predetermined scale of test relative curvature multipliers and selecting as a relative curvature multiplier of test related to the number of pinion teeth (k "c) the test relative curvature multiplier (k'c) that corresponds to the limiting torque that has the largest value, and (f) repeat steps (a) - (e) for a plurality of test pinion tooth numbers (N ') within a predetermined scale of test pinion tooth numbers and selecting as the relative curvature multiplier (kc) the test relative curvature multiplier related to the number of pinion teeth (k "c) that corresponds to the limiting torque that has the largest value.