1、STD-AGMA 94FTMSL-ENGL 1794 Ob87575 0004521 5b4 94FTMSl Computer-Aided Numerical Determination of Hofer, Lewis, Niemann and Colbourne Points by: Chang H. Park George Washington University American Gear Manufacturers Association TECHNICAL PAPER COPYRIGHT American Gear Manufacturers Association, Inc.Li
2、censed by Information Handling ServicesSTDSAGMA 74FTMSL-ENGL 1974 0687575 ClClClLi522 LiTO = Computer-Aided Numerical Determination of Hofer, Lewis, Niemann and Colbourne Points Chang H. Park George Washington University The statements and opinions contained herein are those of the author and should
3、 not be construed as an official action or opinion of the American Gear Manufacturers Association. ABSTRACT In rating the bending strength of the hobbed gear teeth, the critical point located withii the root fillet, where the fracture occurs due to the tensile bending stress, is necessary to be dete
4、rmined Hofer, Lewis, Niemann, and Colboume propose their methods to find this point approximately, in which a numerical iteration method is needed to solve nonlinear onevariable equations established to find their critical points. This paper presents equations expressed easily and in a similar way f
5、or finding these four critical points as well as the general gear tooth profile equations derived based on the vector analysis method. These equations are assuredly solved by the finite difference Newton-Raphson iteration algorithm. Position comparison of their points was achieved with computer-aide
6、d graphical and numerical output. Copyright O 1994 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia. 223 14 October, 1994 ISBN 1-55589442-1 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services STD-AGHA SLIFTMSL-ENGL
7、1774 m Ob87575 OOOLI523 337 m Computer-Aided Numerical Determination of Hofer, Lewis, Niemann, and Colboume Points Chang H. Park Department of Civil, Mechanical and Environmental Engineering The George Washington University Washington, D.C. 20052 Abstract In rating the bending strength of the hobbed
8、 gear teeth, the critical point located within the root fillet, where the fracture occurs due to the tensile bending stress, is necessary to be determined. Hofer, Lewis, Niemann, and Colbourne propose their methods to find this point approximately, in which a numerical iteration method is needed to
9、solve nonlinear one-variable equations established to find their critical points. This paper presents equations expressed easily and in a similar way for finding these four critical points as well as the general gear tooth profile equations derived based on the vector analysis method. These equation
10、s are assuredly solved by the finite difference Newton-Raphson iteration algorithm. Position comparison of their points was achieved with computer-aided graphical and numerical output. Table of Contents Page i. List of Symbols . 2 . Introduction 3. Involute Profile Equations . 4. Tool Trochoid and R
11、oot Fillet Equations . 5. Detection of Undercut 6. Finite Difference Newton-Raphson Method . 7. Determination of Hofer Point 8. Determination of Lewis Point 9. Determination of Niemann Point . 10. Determination of Colbourne Point 11. Numerical Examples . 12. Conclusions . 13. Bibliography .1 .2 .3 .
12、5 .7 .8 .8 .9 . 11 . 11 . 13 . 16 . 16 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSTD.AGMA 94FTMSL-ENGL 1994 H b87575 000452Li 273 N (Y, X ck dev rT d RJ Rb 8 P PO Pi P 0 inv SQRT arctan 1. List of Symbols number of teeth in pinion or gear tool pr
13、essure angle, radians addendum modification coefficient clearance factor for standard gear pair deviation from reference thickness of tool tooth for one side radius of tool tooth tip arc half of tool tooth top land length, O for full-round radius of reference pitch circle radius of base circle of in
14、volute parametric variable (roll angle) of involute vector, radians origin angle of involute, radians starting angle of involute profile, radians end angle of involute profile, radians slope angle of involute, radians load point parameter, radians load angle, radians x-component of involute vector y
15、-component of involute vector parametric variable (roll angle) of root fillet and tool trochoid vectors, radians reference angle of root fillet and tool trochoid vectors, radians starting angle of root fillet and tool trochoid, radians end angle of root fillet and tool trochoid, radians slope angle
16、of root fillet and tool trochoid, radians x-component of tool trochoid vector y-component of tool trochoid vector x-component of root fillet vector y-component of root fillet vector of pinion of gear involute function, inva! = tana! - (Y, radians square root of inverse tangent function, -a/2 - 7r/2
17、1 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services2. Introduction There is a similarity in determining Hofer, Lewis, and Niemann critical points on the root fillet of a gear tooth. These three critical points are commonly associated with the slope angl
18、e of the root fillet. Alternatively, Colbourne i introduces a different approach to identify a critical point. Hofer method, known as 30 degree tangent method, determines its critical point using the slope angle of the root fillet only. This point has been used to evaluate the tooth form factor in D
19、eutsches Institut fr Normung (DIN) Standard 3990 21. Lewis simplified bending stress analysis by using a parabola which makes a gear tooth an equal strength cantilever. Lewis point is found where an upward parabola inscribes with the root fillet. This means the slope angles of the parabola and the r
20、oot fillet are equal to each other at Lewis point. This point has been used formally in the American Gear Manufacturers Association (AGMA) Standard 3. Also, AGMA introduces Errichellos 41 numerical procedure for finding this point. He uses the Newton-Raphson method as a numerical iteration method to
21、 find Lewis point from his iteration function. Niemann point is located where a tangent line to the root fillet passes through its specific point. In finding this point numerically, it is easily understood that with replacing the quadratic equation of parabola for Lewis point by its linear tangent l
22、ine equation, it can be determined in a similar way as Lewis point is determined Consequently, the above-mentioned three points are all obtainable independently of stress analysis. However, Colbourne finds the maximum stress value iterating his fairly complete stress function described based on the
23、gear tooth cantilever stress analysis. It is reasonable that Colbourne point, where the maximum stress occurs, can be a good estimated critical point. Equations established to find these four critical points are formed into nonlinear one-variable equations. The Newton-Raphson numerical iteration met
24、hod may be best suited to solve these equations. In general, however, the Newton-Raphson method suffers from obtaining failed or undesired solutions and requiring the first order derivatives of iteration functions as iteration functions become complicated. Two special strategies for finding the four
25、 critical points are described in this paper when the Newton-Raphson method is used. One is finding a good initial guess point and the other is simplifying the numerical iteration algorithm by using the finite difference method. For convenience, throughout this paper, all variables or dimensions hav
26、ing length units, inches or millimeters, are 2 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSTD - AGMA 94FTMSL-ENGL 1994 111 Obi37575 000452b 4b normalized at 1 diametral pitch (D.P.) for the D.P. design system or at 1 module for the module design s
27、ystem. The very common full- depth and external gear tooth system is used with an assumption that the generating pitch line of the cutting tool is higher than the center of its tip arc, which is a usual cutting condition (h x in Figure 3). The origin of the coordinate system is located at the center
28、 of the gear blank. Also, nonundercut spur gear teeth are assumed to be generated by nonprotuberance rack-type cutting tools. For helical gear teeth, designers can refer to DIN 3990 23 and AGMA Standard 31 where equivalent spur gear teeth methods are introduced. 3. Involute Profile Equations cutting
29、 tool I o i4 Figure 1 Involute profile Figure 2 Origin angle of the curve involute profile Figure 1 defines illustratively the involute vector and some fundamental mathematical properties of the involute. A term O, in Figures 1 and 2, the origin angle of the involute, is used efficiently for various
30、 profile analyses. Figure 2 shows the reference positions of gear tooth and tool tooth space which their center lines are coincident with the y-axis and how to determine 6,. The following equations stand for the basic mathematical properties of the involute profiles in gear teeth generatedby nonprot
31、uberance rack-type cutting tools. Origin angle of the involute, radians: O, = 7r/2 - invcu, - 2 (/4 + x tanq - dev)/N (3.1) 3 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Serviceswhere ac = tool pressure angle, radians invac = tana, - ac, radians dev = devi
32、ation from reference thickness (n/4, along the datum line) of tool tooth for one side, usually positive for gear tooth thinning and negative for gear tooth thickening. N = number of teeth in pinion or gear x = addendum modification coefficient The positive amount of addendum modification means the t
33、ool offset between the reference pitch line and the generating pitch line of the cutting tool in the direction of the tool tooth tip when dev is equal to the desired amount of gear tooth thinning for one side. If not equal, refer to the detailed comments in AGMA Information Sheet 51 and 161 on the a
34、dditional amount of the tool offset for both sides of a gear tooth. Components of the involute vector I = (Ix, Iy): + (3.2a) (3.2b) where Rb = radius of base circle of involute = N/2 cosac 8 = parametric variable (roll angle) of the involute, z O, radians 2 Magnitude of the involute vector: I = SQRT
35、(1 + 8 ) (3.3) where SQRT = square root of When I, distance from the center of the gear blank to a specific point on the involute profile, is known, 8 is determined by using equation (3.3) as follow: End angle of the involute profile: (3.5) where R( = radius of outside circle Slope angle of the invo
36、lute, radians: 8, - 8 + 8, (3.6) Since 8, simply increases as 8 does, the involute profile is a concave curve. Load angle, radians: $L - 8, - n/2 = 8 + 8, - m/2 (3 7) 4 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSTD.AGMA SLlFTMSL-ENGL 1994 Ob87575
37、 0004528 919 4. Tool Trochoi and Root Fillet Equations Unlike involute equations, the root fillet equations are complicated and difficult to use. However, the root fillet can be analyzed efficiently by the tool trochoid defined as path of the center of the tool tooth tip arc (Figure 3). The relation
38、 between their position vectors are illustrated in Figure 4. This relation can be defined in terms of “contour“. Their basic mathematical properties are described by the following expressions. 96 of Yb I iY inerating pitch line tooth cutting tool - ab = ob / I Figure 4 Relation between tool trochoid
39、 and root fillet vectors Figure 3 Tool trochoid vector -* Components of the tool trochoid vector, H = ( ?r/2 + q - po - pf - do, radians The nonundercut gear tooth profiles can be drawn by plotting SI R, (equations (4.8a) and (4.8b) and I, I, (equations (3.2a) and (3.2b) between pi and pf, and Oi an
40、d Of with root and top circular arcs. Note that + expressed by equation (5.1) is independent of dev, which means that dev cannot affect undercutting occurrence or avoidance but the location and slope of the intersection between the root fillet and the involute profile. The smoothness condition other
41、 than x. z &, is also given by equation (5.5). (5.5) 6. Finite Difference Newton-Raphson Method The Newton-Raphson method is used very often to solve non- linear equations because of its fast algorithm to obtain the solutions. It, however, finds also failed or undesired solutions very quickly due to
42、 the initial guess point given improperly. Fortunately, the initial guess point set to (pi + pf)/2 commonly leads to the successful solutions to the equations for finding the four critical points introduced in this paper. Another disadvantage is that it requires the first order derivatives of the it
43、eration functions, which makes the numerical system complicated. The iteration process is simplified by using the first order derivative approximated by the finite difference method 81. To solve f(p) = O, the finite difference iteration process is as follow: Ap f(p) P=P- f(p + Ap) - f(p) Ap can be a
44、n arbitrary small value, say 0.00001, which does not mean a desired error in f (p) = O. This method is especially efficient in finding the solutions from equations for the four critical points described in this paper or in different fashions. Unless otherwise noted, this method is supposed to be use
45、d to find the four critical points for this paper. Errors will be zero or less than 1.0e-10 by enhanced machine accuracy with the above initial guess and the following stop criteria. The iteration process should be terminated before f (p) becomes equal to f (p + Ap) with machine accuracy or when the
46、 prescribed limitation number of iterations, say 5 or more (10) for Colbourne point, is reached. 7. Determination of Hofer Point (5.4) As shown in Figure 5, only the slope angle of the root fillet can be used to determine Hofer point. This point can be determined 8 COPYRIGHT American Gear Manufactur
47、ers Association, Inc.Licensed by Information Handling ServicesSTD-ALMA 74FTMSL-ENGL 1774 Ob87575 0004532 3ttT W Figure 5 Layout of Hofer point independently of loading conditions, load angle or load point. A numerical iteration method is needed to solve for p from equation (7.1). After finding p, th
48、e coordinates of Hofer point are determined by equations (4.8a) and (4.8b). From equation (7.2), Hofer point does not fail to lie between the starting angle pi and the end angle pf of the root fillet. In equation (7.2), usually A 2 d, h x or x is limited to such a value that generates tip narrowingi
49、n the gear tooth tip. Thus, cy, should be greater than /6 (30 degrees) for a positive number of teeth N. This critical situation possibly happens when a, = 30 degrees and N = 00 which is a rack with 30 degree pressure angle. Critical condition for Hofer point on the tangent point: Equations (5.21, (5.31, (5.51, and (7.1) give K + 2 (h-x)/tana, - 2 d N=2R,= a, - /6 (7.2) 8. Determination of Lewis Point As shown in Figure 6, the k slope angle of the root fillet also can be used to determine Lewis point where an upward parabola inscribes within the root fillet. This
