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1、05FTM14 Determining the Shaper Cut Helical Gear Fillet Profile by: G. Lian, Amarillo Gear Company TECHNICAL PAPER American Gear Manufacturers Association Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not
2、 for Resale, 04/18/2007 11:10:40 MDTNo reproduction or networking permitted without license from IHS -,-,- Determining the Shaper Cut Helical Gear Fillet Profile G. Lian, Amarillo Gear Company The statements and opinions contained herein are those of the author and should not be construed as an offi
3、cial action or opinion of the American Gear Manufacturers Association. Abstract Thispaperdescribesarootfilletformcalculatingmethodforahelicalgeargeneratedwithashapercutter. The shapercutterconsideredhasaninvolutemainprofileandellipticalcutteredgeinthetransverseplane. Since the fillet profile cannot
4、be determined with closed form equations, a Newtons approximation method was usedinthecalculationprocedure. Thepaperwillalsoexplorethefeasibilityofusingashapertoolalgorithmfor approximating a hobbed fillet form. Finally, the paper will also discuss some of the applications of fillet form calculation
5、 procedures such as form diameter (start of involute) calculation and finishing stock analysis. Copyright 2005 American Gear Manufacturers Association 500 Montgomery Street, Suite 350 Alexandria, Virginia, 22314 October, 2005 ISBN: 1-55589-862-9 Copyright American Gear Manufacturers Association Prov
6、ided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:10:40 MDTNo reproduction or networking permitted without license from IHS -,-,- 1 Determining the Shaper Cut Helical Gear Fillet Profile By George Lian Amarillo Gear Company 1 Intro
7、duction Analytical methods for determining the gear fillet profile (trochoid) have been well documented. Khiralla 1 described methods for calculating fillet profile of hobbed and shaped spur gears. Colbourne 2 provided equations for calculating the trochoid of both involute and non-involute gears ge
8、nerated by rack or shaper tools. The MAAG Gear Handbook 3 also provided equations for calculating trochoid generated with rack type tools that have circular tool tips. Vijayakar, et al. 4 presented a method of determining spur gear tooth profile using an arbitrary rack. The above mentioned are only
9、samples of many published works. However, the method for determining the trochoid of a helical gear generated with a shaper tool is not widely published. This paper presents an intuitive algorithm where the fillet profile of a shaper tool generated external or internal helical gear can be calculated
10、. A shaper tool generating a gear can be visualized as a gear set meshing with zero backlash. The algorithm in this paper is based on a shaper tool in tight mesh with a semi-finished helical gear. The semi-finished gear geometry was used for calculation because the shaper tool, used as the semi-fini
11、shing tool, is usually the one that generates the trochoid. However, if the shaper cutter is the finishing tool, the algorithm presented will also work by letting the finishing stock equal zero. The trochoid of a spur gear can also be calculated by letting the helix angle equal zero. The shaper tool
12、 used in this algorithm may have a different reference normal pressure angle than that of the gear. A necessary condition for a shaper tool to generate the correct involute profile on a gear is that both the tool and the gear must have equal normal base pitch. This paper stipulates that the axis of
13、the shaper tool and the gear are parallel, which is often true for gear shaping. Consequently, the shaper tool and the gear must also have an equal base helix angle. Although, the algorithm is based on the shaper cutter as a generating tool, the presented method can also be used to calculate a troch
14、oid generated with a hob or a rack type tool if the number of the shaper teeth is large (e.g. 10000). 2 Symbols and Convention The symbols are defined where first used. This paper tries to adhere to the following rules in subscript usage: Symbols related to tool geometry have subscript “0”; No subsc
15、ript is used for symbols related to the gear; Subscript “n” is used for measurements in the normal plane; Subscript “r” is used for symbols related to the semi-finished gear Subscript “g” is used for symbols related to the generating pitch circle When dual signs are used in an equation (e.g.), the u
16、pper sign is for external gears and the lower one for internal gears. Non-italicized upper case symbols are used to designate points on the shaper tool, the gear, or other points of interest. Points are also represented as the coordinates ( , )x y. The length of a vector (e.g.R) is represented asR .
17、 3 Coordinate System The reference position of a shaper tool generating a gear is depicted in Fig. 1, for external gear shaping, and Fig. 2, for internal. The following coordinate system and sign conventions are followed: Standard Cartesian coordinate system is used. The center of the shaper tool 0
18、O is(0,0). The reference position of the shaper tool is with one of its teeth aligned with the y-axis. The end of the shaper tooth points in the y direction. Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie
19、 Not for Resale, 04/18/2007 11:10:40 MDTNo reproduction or networking permitted without license from IHS -,-,- 2 r g0 r g C g O G External Gear (0,-C g) G Shaper tool (0,0) O 0 x+ y+ Fig. 1 Shaping an external gear The center of the gear, G O, is also on the y- axis with one of the tooth spaces alig
20、ned with the y-axis. The opening of the tooth space is in the +y direction. Angular measures, related to tool or gear rotation or location of a point, are signed. CCW rotation from the reference line is positive, and CW, negative. 4 Shaper Tool and Gear Geometry The following are required tool and g
21、ear data for calculating the trochoid: Shaper tool data: nd0 P is the ref. normal diametral pitch, tool (in-1) 0 n is the number of teeth, tool n0 is the ref. normal pressure angle, tool 0 is the ref. helix angle, tool n0 s is the ref. normal circular thickness, tool (in) a0 d is the outside diamete
22、r, tool (in) 0 is the tool tip radius (in) 0 is the protuberance (in) Gear data: nd P is the ref. normal diametral pitch, gear (in-1) n is the number of teeth, gear n is the ref. normal pressure angle, gear OG G (0, 0) O 0 Internal gear r g0 y+ Shaper tool x+ r g (0, c g) c g Fig. 2 Shaping an inter
23、nal gear is the ref. helix angle, gear n s is the ref. normal circular thickness, gear (in) s is the stock allowance per flank, gear (in), defined on the reference pitch circle (not along the base tangent). 4.1 Basic Shaper Tool and Gear Geometry The following equations calculate the basic tool and
24、gear geometry: Standard transverse pressure angle of tool, 0 n0 0 0 tan arctan() cos = (1) Standard reference pitch radius of tool, 0 r (in) 0 0 nd00 2cos n r P = (2) Base radius of tool, b0 r (in) b000 cosrr= (3) Ref. transverse circular thickness of tool, 0 s (in) n0 0 0 cos s s = (4) Copyright Am
25、erican Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:10:40 MDTNo reproduction or networking permitted without license from IHS -,-,- 3 Transverse base pitch of tool, b0 p (in) b0 b0 0 2 r p n
26、= (5) Normal base pitch of tool, nb0 p (in) n0 nb0 nd0 cos p P = (6) Base helix angle of tool, b0 nb0 b0 b0 arccos() p p = (7) Base circular thickness of tool, b0 s (in) 0 b0b00 0 2(inv) 2 s sr r =+ (8) where inv is the involute function of an angle invtan = Standard reference pitch radius of gear,
27、r (in) nd 2cos n r P = (9) Base radius of semi-finished gear, br r (in) brb0 0 n rr n = (10) The helix angle at standard pitch radius of semi- finished gear, r b0 r br tan arctan() r r = (11) Transverse pressure angle at reference pitch radius of semi-finished gear, r br r arccos() r r = (12) Transv
28、erse circular thickness of semi-finished gear, r s (in) ns r r 2 cos s s + = (13) Base circular thickness of semi-finished gear, br s (in) r brbrr 2(inv) 2 s sr r = (14) 4.2 Center of Tool Tip on a Shaper Tool A shaper tool for gear semi-finishing usually has protuberance. It generates undercut on a
29、 gear, so that the finishing tool only needs to machine the involute profile of the gear. To obtain the designed amount of protuberance on a shaper tool, the tool tip is made tangent to the involute profile that is temporarily formed by increasing the shaper tooth thickness to include the protuberan
30、ce (Fig. 3). The tangent point, common to the tool tip and the involute profile, will be referred to as the profile tangent point, 0 P. When the temporarily formed involute profile is removed, the shaper tool will have the designed amount of protuberance. s b0_pr r b0 S0 P0 Tube of radius 0 Involute
31、 profile including protuberance Cutter profile Normal plan view P n0 Pn0 A View “A-A“ S 0 y-axis r S0 Transverse plan view 0 P090 0 P 0 A r P0 P0 P0 2 r b0 0 inv P0 cos b0 Fig. 3 Tool tip of a shaper tool The shaper tool tip is also made tangent to the outside diameter of the tool (Fig. 4) so that t
32、he transition from the outside diameter to the tool tip will be smooth. The common tangent point on the Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:10:40 MDTNo reproductio
33、n or networking permitted without license from IHS -,-,- 4 E0 S0 d a0 2 90 Tube of radius 0 E0 r E0 y-axis E n0 0 En0 0 S 0 View “A-A“ A E 0 r S0 Fig. 4 End of tool tip (with helix angle exaggerated) shaper tool tip and the outside diameter of the tool will be referred to as the end tangent point, 0
34、 E. The following are the required data for calculating the center of the shaper tool tip: a0 d is the outside diameter, tool (in) b0 s is the base circular thickness, tool (in) 0 is the tool tip radius (in) 0 is the protuberance (in) b0 is the base helix angle, tool 0 is the ref. helix angle, tool
35、The base circular thickness of the involute profile, formed by increasing the shaper tool tooth thickness to include the protuberance, b0_pr s 0 b0_prb0 b0 2 cos ss =+ (15) Coordinates of the center of tool tip, 0 S 0S0S0S0S0 S(sin,cos)rr= (16) where S0 r is the tool radius to center of tool tip (in
36、) S0 is the offset angle of tool tip. For a shaper tool with full tip radius, S0 will equal zero. Coordinates of the profile tangent point, 0 P Pn0 0000Pn0 0 cos PS(,sin) cos =+ (17) where Pn0 is the auxiliary angle that locates 0 P. The angle is measured in the normal plane, CW from the horizontal
37、axis of the tool tip. Pn0 will usually have a negative value. Tool radius to profile tangent point, P0 r (in) P00 Pr= (18) Transverse pressure angle, P0 , at 0 P b0 P0 P0 arccos() r r = (19) The tangent angle, P0 , at 0 P (the derivation of Eq.20, is given in Annex A) 0 P0 Pn0 cos arctan() tan = (20
38、) The angle between the y-axis and the radius to the profile tangent point, P0 b0_pr P0P0 b0 inv 2 s r = (21) Coordinates of the end tangent point, 0 E En0 0000En0 0 cos ES(,sin) cos =+ (22) where En0 is the auxiliary angle that locates 0 E. The angle is measured in the normal plane, CW from the hor
39、izontal axis of the tool tip. En0 will usually have a negative value. The angle of tangent, E0 , at the end tangent point, 0 E Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:
40、10:40 MDTNo reproduction or networking permitted without license from IHS -,-,- 5 0 E0 En0 cos arctan() tan = (23) Tool radius to end tangent point, E0 r (in) E00 Er= (24) The following are conditions for the tool tip to position properly on a shaper tool tooth: 1) The profile tangent point, 0 P, on
41、 the tool tip must also be a point on the involute profile that includes the protuberance, thus P0P0P0 0 2 += (25) 2) The angle, P0 , subtended by one half of the transverse circular thickness of the involute curve (include the tool protuberance) at 0 P, must equal the angle formed by the y-axis and
42、 the line connecting the center of the tool to 0 P. P0 P0 P0 arcsin()0 x r = (26) where P0 x is the x-coordinate of profile tangent point, 0 P (in) 3) The end tangent point must also be a point on the outside diameter of the shaper tool, thus a0 E0 0 2 d r= (27) 4) The tangent angle, E0 , at the end
43、 tangent point, 0 E, must equal the angle formed by the y- axis and the line connecting the center of the tool to 0 E E0 E0 E0 arcsin()0 x r = (28) Eq.25-28, must all be satisfied for the tool tip to be correctly positioned on a shaper tool tooth. The variables to be determined are S0 r, S0 , Pn0 ,
44、and En0 . Since the systems of the equations are transcendental and cannot be solved directly, the Newtons method is used to calculate the roots for Eq.25-28. 4.3 Solving the System of Non-linear Equations for Center of Tool Tip For simplicity, rewrite Eq.25-28, as generic vector equations in the fo
45、rm F(X)=0 (29) where T 1234 T F(X)=(f (X),f (X),f (X),f (X) =(Eq.25, Eq.26, Eq.27, Eq.28) (30) T 0=(0,0,0,0) (31) T 1234 T S0S0Pn0En0 X=( ,) (,) x xx x r= (32) The Newtons iteration equation 6 is written as X1=X+ X (33) where X satisfies the following system of linear equations JX=-F(X)i (34) where
46、X1 is the vector of the new roots for the next iteration X is the vector of current roots X is the vector of Newtons steps for the next iteration J is the Jacobian matrix where 111 124 22 12 44 14 fff ff J= ff xxx xx xx ? ? ? ? (35) Copyright American Gear Manufacturers Association Provided by IHS u
47、nder license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:10:40 MDTNo reproduction or networking permitted without license from IHS -,-,- 6 i j f x is the partial derivative of the th i equation with respect to the th j variable The partial derivatives in the Jacobian matrix can be approximated using the finite differences iji i jj f (XX )f (X) f xx + (36) where i is the ith row of the Jacobian matrix j is the jth column of the Jacobian matrix Xj is a vector with its jth element equals the jth element