1、97FTM6 On the Location of the Tooth Critical Section for the Determination of the AGMA J-Factor by: Jos I. Pedrero, UNED, Carlos Garca-Masi, and Alfonso Fuentes, Universidad de Murcia TECHNICAL PAPER * On the Location of the Tooth Critical Section for the Determination of the AGMAJ-Factor Jos I. Ped
2、rero, UNED, Carlos Garca-Masi and Alfonso Fuentes, Universidad de Murcia The statements and opinions contained herein are those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract The bending strength geometry factor J d
3、epends on the location of the critical section and the tooth thickness at this section. The critical section is that in which the uncorrected bending stress, given by Naviersequation, is maximum and consequently it is defined by the point of tangency of the Lewis Parabola and the tooth profile. This
4、 point of tangency is usually placed over the root fillet, but in some casts the tangency may occur at the involute. For these cases the methods descriied in literature for determining the AGMA J-factor are not suitable. This paper presents the condition for tangency at the involute and a method to
5、determine the J factor under this condition. Copyright O 1997 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 November, 1997 ISBN: 1-55589-700-2 STD-ALMA 77FTMb-ENGL 1777 Ob87575 0005107 Li58 D ON THE LOCATION OF THE TOOTH CRITICAL SECTION FOR THE DETE
6、RMINATION OF THE AGMA J-FACTOR Jos I. Pedrero, Professor (*i Carlos Garca-Masi, Associate Professor (+*I Alfonso Fuentes, Assistant Professor (“1 (“1 (*+) UNED, Departamento de Mecnica, Apdo. 601 49, 28080 Madrid, Spain Universidad de Murcia, Departamento de Ingeniera Mecnica y Energtica, Po Alfonso
7、 XII1 44, 30203 Cartagena, Murcia, Spain Introduction Location of the critical section To evaluate the maximum bending stress arising at any point of the tooth during the entire meshing interval, AGMA introduces the bending strength geometry factor, also known as AGMA J-factor 1 I. According to Navi
8、ers equation, the point of maxi- mum bending stress is located at the point of tangency of the tooth profile and the inscribed equal-stress parabola, or Lewis parabola 21, whose vertex is placed on the intersection of the line of action, at any contact point, and the tooth centerl- ine. The most sev
9、ere conditions correspond with the tooth loaded at the tip, for conventional helical gears, and at the highest point of single tooth contact, for spurs and low axial contact ratio helical gears. Consequently, the critical section of the tooth is defined by the point of tangency of the tooth profile
10、and the Lewis parabola for load acting at the above points, and J factor can be expressed as a function of the height of the Lewis parabola (distance from the critical section to the vertex of the parabola) and the tooth thickness at the critical section. Since the point of tangency is usually place
11、d over the root fillet, existing methods to compute the AGMA J-factor 3,41 involve iterative procedures to find the point of tangency of the parabola and the root trochoid. However, tangency may occur at the involute, as seen in Fig. 1. Under these conditions, results provided by the above methods c
12、ould be inappropriate. In this paper the tangency condition at the involute profile is established. Also a method for computing the AGMA J-factor under this condi- tion is presented. AGMA defines the bending strength geometry factor J as 31 I cos (ur cos cy where C, is the helical overlap factor, Kf
13、 the stress correction factor 51, mN the load sharing ratio, cy, the operating helix angle, cy the standard helix angle, qnL the load angle, nr the operating normal pressure angle, h, the height of the Lewis parabola, s, the tooth thickness at the critical section and C, the helical factor 161. Figu
14、re 1 shows an example of tangency occur- ring at the involute. Involute profile and root trochoid are tangent at point E. This means there is neither undercutting nor tool protuberance (or grinding after generation by shaper cutter with Protuberance), although for small undercutting and/or tool prot
15、uberance, it is possible that the tangency point still remains at the involute, as shown in Fig. 2. However, under this condition of tangency at the involute, the point of tangency shouldnt be considered neither for locating the critical section nor for determining the J factor. Though Naviers stres
16、s is maximum at this point, the stress concentration is very small at the invo- lute, and corrected bending stress is greater at any- point of the root fillet. Therefore, the J factor should be computed considering the section of the root in which Naviers stress is maximum, which -1 - STD-AGHA 77FTM
17、b-ENGL 1777 = b87575 0005108 394 D will be defined by the point of intersection of the root trochoid and the thinnest parabola containing a point of the trochoid. Obviously, the parabola tangent to the trochoid, if it exists, coincides with the above thinnest parabola, hence the mentioned existing m
18、ethods are suitable for this case. This occurs for tangency of the Lewis.paraboia at the root and also for tangency at the involute if under- cutting (or tool protuberance) exists. - -._ -. c /“i “ ., Figure 1 : Tangency at the involute profile Figure 2: Tangency at the involute profile for small un
19、dercutting and/or tool protuberance Nevertheless, for the case of tangency at the involute with no undercutting, no tool protuber- ance, tangency of the tangent parabola and the trochoid occurs at an improper point of tangency Pi, beyond the end of the root trochoid, as shown in Fig. 1, and improper
20、 values for the height of the Lewis parabola, h, and for the tooth thickness, s, would be obtained if the above mentioned methods were employed. In this case, the thinnest parabola containing a point of the trochoid is that containing the point of tangency of the root trochoid and the involute (poin
21、t E in Fig. 11, and this point defines the critical section, which should be considered for computing the J factor. Condition for tangency at the involute For the case of no undercutting and no tool protu- berance it is necessary to know if tangency of the parabola and the profile occurs at the invo
22、lute or at the trochoid. This condition will be established in two ways. First, in terms of the results of the iterative procedure to find the point of tangency of the parabola and the trochoid, which will be useful for designers having the procedure implemented in computer programs. Second, in term
23、s of the maxi- mum distance between the contact point and the center of the gear, which can be checked before the iterations. _.- a) In terms of the results of the iterative procedures AGMA 908-B89 31 expresses the fillet coordinates by selecting an angle a, as independent parameter. This angle is d
24、escribed in Fig. 5-8 in 31, and can be defined as the angle between the tangent to both operating pitch circles (of cutter and gear being generated) and the line containing the rolling point and the center of tool tip radius. This angle a, is the parameter corresponding to the root fillet point that
25、 is being generatad. Obviously, this point is also placed over the last line. When point E at the root fillet is being generated, both involute profiles of gear and tool are meshing at the same point, so that it is placed over the line of action. Consequently, the parameter corresponding to point E,
26、 onP should be equal to the generating pressure angle :, which is given by 31 (2) where tp, is the standard normal pressure angle, xs the generating rack shift coefficient, x, the adden- dum modification factor of tool, ne and n the virtual tooth number of tool and gear, and inv denotes the involute
27、 function (3) invtp = tane -e Note that for xg +xo =O or for gears generated by rack (ne = -1, the generating pressure angle is equal to the standard normal one. In other case, can be obtained by iteration t71. -2- STD-AGMA 97FTMb-ENGL 1997 b87575 0005109 220 Figure 3: Location of the point of tange
28、ncy of the root trochoid and the involute From the above considerations, the tangency condition at the root fillet for no undercutting, no tool protuberance can be expressed as an,;* 1 / (41 where an, is the value for a, obtained from the iterative procedures used in 31. b) In terms of the contact r
29、adius The same condition can be formulated in other way, before the iterations. From Fig. 3 and the involute properties we have (5) E = 8, - tan- 8, where rnE is the virtual radius of point E and rnb the virtual base radius of the gear the J factor is being calculated for. The angle u, between rnP a
30、nd the tooth centerline can be derived from a, = 5 -BE 2 (6) where yb is the angle of the both involute profiles of the virtual tooth at the base radius, - - n + 9 4x tan, + 2 inv, (7) -n n Angles u, and a, are described by a: = 8, -, = tan-8, a, = n - a2 I (8) a, = n-a, -a, = E-T Yb The distance fr
31、om the center of the virtual gear to the intersection of the virtual tooth centerline and the tangent to the virtual profile at point E is given by 6, rn sin a, 6 = - rnr = sin a, (91 /= sin , - + Point M in Fig. 3 is the midpoint of the segment M,M, so the distance from the center of the virtual ge
32、ar to point M is 6, = I(d+rnEcosul) 1 (10) From the properties of the parabola, this distance is the maximum value for the virtual load radius for tangency at the root fillet. Consequently, the maximum load angle is given by = cos- a (1 11 as seen in Fig. 4. From the same figure, this condition can
33、be also expressed as the maximum value for the virtual radius of the contact point for tangency at the root fillet, As said above, the virtual radius of the contact point is the virtual outside radius for tip load (con- ventional helical gears), and the virtual radius of the highest point of single
34、tooth contact for spur and low axial contact ratio helical gears, which are given by rnc = rno (mF l) (131 (cl +nmncosn) 2 +z* (mF 1 rnc = cl =(f+fnb2)tan-(02-Zb) where rnc is the virtual radius of the contact point (load point), r, is the virtual outside radius, mn the normal module, m, the axial c
35、ontact ratio and subindex 2 denotes the mating gear. It is clear that the condition for tangency at the root trochoid can be expressed as rn fntrnsx (14) To determine rnCmu from the above method it is necessary to know the virtual radius of the point in which the root trochoid is tangent to the invo
36、lute profile rnP Since point E is placed on the involute, it meshes during the generation with point Eo of the tool, in which the involute of the tool is tangent to the tool tip circumference, as seen in Fig. 5. The sum of the curvature radii of both involute profiles at any contact point is constan
37、t, so that r, can be expressed as a function of In and follow; if not, follow without changes. If the condition for tangency at the involute is checked before the iterations and it is verified, the J factor can be computed without iterations. As said above. the critical section of the tooth should b
38、e the one defined by the point of the root fillet contained in the thinnest parabola containing a point of the fillet. For the case of no undercutting and no tool protuberance this point is point E, the tangency point of the trochoid and the involute, as derived from Fig. 1. Since the coordinates of
39、 point E can be expressed analytically, the tooth thickness at this point s, and the height of the Lewis parabola h, can be also expressed analytically. The virtual radius of the load point rnC may be obtained from equations (131. The load angle qnL can be expressed as a function of this radius, as
40、seen in Fig. 4, (18) 1 IT - 3 tan, - inv9, 2n n and, from the same figure, the load radius r, is given by (1 9) The height of the parabola and the tooth thick- ness at the critical section can be derived from h, = rnL -rn,cos 2 - Reference normal circular thickness s, = 1 .5708; 3.C24 Tooth thinning
41、 for backlash. Gearsets sparating on a standard center distance. Sum of addendum modification coefficients x, +x, =O. Axial contact ratio m,=2. Addendum modification coefficient xo =O; Amount of protuberance 6, =O. -6- trochoid parameter u,=24.609 deg. This value is smaller than that for the generat
42、ing pressure angle (25,007 deg is obtained), so the result is not valid. The method presented herein provides a value for the J factor of 0.43806, which is 0.02% greater. Really the differences are rather insignificant taking into account that there are many other factors in gear design which will m
43、ask the effect of an error less than 1 % in the value of the geometry factor. However, the method presented herein is more correct, and the results it provides more accurate. Conclusions Gears generated with neither undercutting nor tool protuberance may present the tangency of the inscribed parabol
44、a and the tooth profile at the involute. Under these conditions, the methods described in literature to determine the AGMA J- factor, in which the critical section is described by the point of tangency of the constant stress parab- ola and the root trochoid, are not suitable. In this paper, the tang
45、ency condition at the involute profile is established and a method for calculating the J factor for these cases is presented. In this method the critical section is defined by the highest point of the root fillet, in which the trochoid is tangent to the involute. In this section Naviers bending stre
46、ss is smaller than that in the section defined by the point of tangency, but the stress correction factor is much greater. Also the thinnest parabola containing a point of the trochoid contains the highest one. Since the coordinates of this point can be easily obtained, the method allows to determin
47、e the J factor analytically, without iterations. Computer programs having implemented the method described in i31 could easily take into account the above considerations: 1. Follow the method until calculating the fillet parameter of the critical section u,. 2. Check the conditions for tangency at t
48、he root fillet, equation (41, no undercutting, equation (1 71, and no tool protuberance. 2.1. If any of them are satisfied, follow without changes. Acknowledgment Thanks are expressed to the Spanish Council for Technical and Scientific Research for the financial support for the research project PB95
49、-0876-C02, “Generation of Conjugate Profiles for Gear Teeth. Development of the Behaviour Models for Bending and Wear“. References AGMA Standard 2001 -C95, “Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth“, American Gear Manufacturers Association, Alexandria, VA (1 995). W. Lewis, “Investigation of the Strength of Gear Teeth,“. Proc. of the Engineers Club, 16-23, Philadelphia, PA (1 893). AGMA information Sheet 908-B89, “Geometry Factors for Determining the P
