Gear tooth profile curvature

A gear system comprising:

• a pinion having a pinion tooth number (N1), a pinion pitch circle radius (Rp1) and a first plurality of teeth, each tooth having a first-tooth profile, the pinion;
• and
• a mating gear having a mating-gear tooth number (N2), a mating-gear pitch circle radius (Rp2) and a second plurality of teeth, each tooth having a second-tooth profile;
• wherein the pinion and the mating gear form a gear pair having a center distance (C) equal to (Rp1+Rp2), a gear ratio (mG) equal to N2/N1, a face width (Fw), and a face-width factor (fw) equal to (2·Rp2)/Fw, and
• wherein the relative curvature of the first-tooth profile and the second-tooth profile is defined by the expression kc·Fc where Fc is a relative reference curvature function defined by the expression Fc=(N1+N2)2/(N1·N2·C) and kc is a relative curvature multiplier that is more than bi,j−δ and less than bi,j+2δ where δ is between 0 and 0.439 ″ and bi,j is defined by a predetermined relationship between the gear ratio (mG) and the face-width factor (fw), the predetermined relationship corresponding to at least one relative curvature multiplier value in a relative curvature multiplier value table having the following properties:
• Face Width Factor (fw)
• Gear Ratio (mG)
• 4.0
• 5.0
• 6.0
• 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.

A gear system having a predetermined gear ratio (mG), a predetermined center distance (C), a predetermined face width (Fw), and predetermined limiting stresses, the gear system comprising:

• a pinion having a pinion tooth number (N1) and a first plurality of teeth, each tooth having a first-tooth profile;
• and
• a mating gear having 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,
• wherein the pinion and the mating gear form a gear pair having a face-width factor (fw) equal to (2·N2·C)/((N1+N2)·Fw), and
• wherein the relative curvature of the first-tooth profile and the second-tooth profile is defined by an expression kc·Fc where Fc is a relative reference curvature function defined by the expression Fc=(N1+N2)2/(N1·N2·C), and kc is a relative curvature multiplier where kc is determined by a process comprising the following steps:
• (a) determining a plurality of load intensities for a predetermined input torque, each load intensity being associated with a unique angular position of a plurality of angular positions of the pinion, the plurality of angular positions spanning one angular pitch of the pinion, each load intensity based on a trial relative curvature multiplier (k′c);
• (b) determining a plurality of tooth stresses corresponding to a greatest load intensity of the plurality of load intensities;
• (c) scaling the greatest load intensity to a scaled load intensity such that one tooth stress of the plurality of tooth stresses approaches one of the predetermined limiting stresses;
• (d) determine a limiting torque corresponding to the scaled load intensity;
• and
• (e) repeating steps (a)–(d) for a plurality of trial relative curvature multipliers (k′c) within a predetermined range of trial relative curvature multiplier values and selecting as relative curvature multiplier (kc) the trial relative curvature multiplier (k′c) corresponding to the limiting torque having the greatest value.

A gear system having a predetermined gear ratio (mG), a predetermined center distance (C), a predetermined face width (Fw), and predetermined limiting stresses, the gear system comprising:

• a pinion having a pinion tooth number (N1), and a first plurality of teeth, each tooth having a first-tooth profile;
• and
• a mating gear having 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,
• wherein the pinion and the mating gear form a gear pair having a face-width factor (fw) equal to (2·N2·C)/((N1+N2)·Fw), and
• wherein the relative curvature of the first-tooth profile and the second-tooth profile is defined by an expression kc·Fc,
• where Fc is a relative reference curvature function defined by the expression
• Fc=(N1+N2)2/(N1·N2·C),
• and
• where kc is a relative curvature multiplier which is a function of the gear ratio (mG), the face-width factor (fw), the center distance (C), and one of the limiting stresses.