1、05FTM14Determining the Shaper Cut HelicalGear Fillet Profileby: G. Lian, Amarillo Gear CompanyTECHNICAL PAPERAmerican Gear Manufacturers AssociationDetermining the Shaper Cut Helical Gear FilletProfileG. Lian, Amarillo Gear CompanyThe statements and opinions contained herein are those of the author
2、and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractThis paper describes a root fillet form calculating method for a helical gear generated with a shaper cutter. Theshaper cutter considered has an involute main profile and elliptical cut
3、ter edge in the transverse plane. Sincethe fillet profile cannot be determined with closed form equations, a Newtons approximation method wasused in the calculation procedure. The paper will also explore the feasibility of using a shaper tool algorithm forapproximating a hobbed fillet form. Finally,
4、 the paper will also discuss some of the applications of fillet formcalculation procedures such as form diameter (start of involute) calculation and finishing stock analysis.Copyright 2005American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2005I
5、SBN: 1-55589-862-91 Determining the Shaper Cut Helical Gear Fillet Profile By George Lian Amarillo Gear Company 1 Introduction Analytical methods for determining the gear fillet profile (trochoid) have been well documented. Khiralla 1 described methods for calculating fillet profile of hobbed and sh
6、aped spur gears. Colbourne 2 provided equations for calculating the trochoid of both involute and non-involute gears generated 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
7、 al. 4 presented a method of determining spur gear tooth profile using an arbitrary rack. The above mentioned are only 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 intu
8、itive algorithm where the fillet profile of a shaper tool generated external or internal helical gear can be calculated. 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-finishe
9、d helical gear. The semi-finished gear geometry was used for calculation because the shaper tool, used as the semi-finishing 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 stoc
10、k equal zero. The trochoid of a spur gear can also be calculated by letting the helix angle equal zero. The shaper tool 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
11、on a gear is that both the tool and the gear must have equal normal base pitch. This paper stipulates that the axis of 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 a
12、lgorithm is based on the shaper cutter as a generating tool, the presented method can also be used to calculate a trochoid 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 t
13、ries to adhere to the following rules in subscript usage: Symbols related to tool geometry have subscript “0”; No subscript 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 Subscri
14、pt “g” is used for symbols related to the generating pitch circle When dual signs are used in an equation (e.g. ), the upper 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 o
15、f interest. Points are also represented as the coordinates (, )x y . The length of a vector (e.g. R ) is represented as R. 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 coordinat
16、e system and sign conventions are followed: Standard Cartesian coordinate system is used. The center of the shaper tool 0O 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. 2 rg0rgCgOGExternal
17、 Gear(0,-Cg)GShaper tool(0,0)O0x+y+Fig. 1 Shaping an external gear The center of the gear, GO , is also on the y-axis with one of the tooth spaces aligned 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
18、, 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 gear data for calculating the trochoid: Shaper tool data: nd0P is the ref. normal diametral pitch, tool (in-1) 0n is the number of teeth, tool n0 is t
19、he ref. normal pressure angle, tool 0 is the ref. helix angle, tool n0s is the ref. normal circular thickness, tool (in) a0d is the outside diameter, tool (in) 0 is the tool tip radius (in) 0 is the protuberance (in) Gear data: ndP is the ref. normal diametral pitch, gear (in-1) n is the number of t
20、eeth, gear n is the ref. normal pressure angle, gear OGG(0, 0)O0Internal gearrg0y+Shapertoolx+rg(0, cg)cgFig. 2 Shaping an internal gear is the ref. helix angle, gear ns is the ref. normal circular thickness, gear (in) s is the stock allowance per flank, gear (in), defined on the reference pitch cir
21、cle (not along the base tangent). 4.1 Basic Shaper Tool and Gear Geometry The following equations calculate the basic tool and gear geometry: Standard transverse pressure angle of tool, 0 n000tanarctan( )cos=(1) Standard reference pitch radius of tool, 0r (in) 00nd0 02cosnrP=(2) Base radius of tool,
22、 b0r (in) b00 0cosrr= (3) Ref. transverse circular thickness of tool, 0s (in) n000cosss =(4) 3 Transverse base pitch of tool, b0p (in) b0b002 rpn= (5) Normal base pitch of tool, nb0p (in) n0nb0nd0cospP= (6) Base helix angle of tool, b0 nb0b0b0arccos( )pp= (7) Base circular thickness of tool, b0s (in
23、) 0b0b0 002( inv)2ssrr=+ (8) where inv is the involute function of an angle inv tan= Standard reference pitch radius of gear, r (in) nd2cosnrP=(9) Base radius of semi-finished gear, brr (in) br b00nrrn= (10) The helix angle at standard pitch radius of semi-finished gear, r b0rbrtanarctan( )rr= (11)
24、Transverse pressure angle at reference pitch radius of semi-finished gear, r brrarccos( )rr= (12) Transverse circular thickness of semi-finished gear, rs (in) nsrr2cosss+ =(13) Base circular thickness of semi-finished gear, brs (in) rbrbr r2( inv)2ssrr= (14) 4.2 Center of Tool Tip on a Shaper Tool A
25、 shaper tool for gear semi-finishing usually has protuberance. It generates undercut on a 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
26、 is temporarily formed by increasing the shaper tooth thickness to include the protuberance (Fig. 3). The tangent point, common to the tool tip and the involute profile, will be referred to as the profile tangent point, 0P . When the temporarily formed involute profile is removed, the shaper tool wi
27、ll have the designed amount of protuberance. sb0_prrb0S0P0Tube of radius 0Involute profile including protuberanceCutter profileNormalplanviewPn0Pn0AView “A-A“S0y-axisrS0Transverseplan view0P0900P0ArP0P0P02 rb00invP0cosb0Fig. 3 Tool tip of a shaper tool The shaper tool tip is also made tangent to the
28、 outside diameter of the tool (Fig. 4) so that the transition from the outside diameter to the tool tip will be smooth. The common tangent point on the 4 E0S0da0290Tube of radius 0E0rE0y-axisEn00En00S0View “A-A“AE0rS0Fig. 4 End of tool tip (with helix angle exaggerated) shaper tool tip and the outsi
29、de diameter of the tool will be referred to as the end tangent point, 0E . The following are the required data for calculating the center of the shaper tool tip: a0d is the outside diameter, tool (in) b0s is the base circular thickness, tool (in) 0 is the tool tip radius (in) 0 is the protuberance (
30、in) b0 is the base helix angle, tool 0 is the ref. helix angle, tool The base circular thickness of the involute profile, formed by increasing the shaper tool tooth thickness to include the protuberance, b0_prs 0b0_pr b0b02cosss=+(15) Coordinates of the center of tool tip, 0S 0S0 S0S0 S0S(sin, cos)r
31、r= (16) where S0ris the tool radius to center of tool tip (in) 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, 0P Pn000 0 0 Pn00cosPS( ,sin )cos=+ (17) where Pn0 is the auxiliary angle that locates0P . The angl
32、e is measured in the normal plane, CW from the horizontal axis of the tool tip. Pn0 will usually have a negative value. Tool radius to profile tangent point, P0r (in) P0 0Pr = (18) Transverse pressure angle, P0 , at 0P b0P0P0arccos( )rr= (19) The tangent angle, P0 , at 0P (the derivation of Eq.20, i
33、s given in Annex A) 0P0Pn0cosarctan( )tan =(20) The angle between the y-axis and the radius to the profile tangent point,P0 b0_prP0 P0b0inv2sr = (21) Coordinates of the end tangent point, 0E En000 0 0 En00cosES( ,sin )cos=+ (22) where En0 is the auxiliary angle that locates 0E . The angle is measure
34、d in the normal plane, CW from the horizontal axis of the tool tip. En0 will usually have a negative value. The angle of tangent, E0 , at the end tangent point, 0E 5 0E0En0cosarctan( )tan=(23) Tool radius to end tangent point, E0r (in) E0 0Er = (24) The following are conditions for the tool tip to p
35、osition properly on a shaper tool tooth: 1) The profile tangent point,0P , on the tool tip must also be a point on the involute profile that includes the protuberance, thus P0 P0 P002+= (25) 2) The angle, P0 , subtended by one half of the transverse circular thickness of the involute curve (include
36、the tool protuberance) at 0P , must equal the angle formed by the y-axis and the line connecting the center of the tool to 0P . P0P0P0arcsin( ) 0xr = (26) where P0x is the x-coordinate of profile tangent point, 0P (in) 3) The end tangent point must also be a point on the outside diameter of the shap
37、er tool, thus a0E002dr = (27) 4) The tangent angle,E0 , at the end tangent point, 0E , must equal the angle formed by the y-axis and the line connecting the center of the tool to 0E E0E0E0arcsin( ) 0xr = (28) Eq.25-28, must all be satisfied for the tool tip to be correctly positioned on a shaper too
38、l tooth. The variables to be determined are S0r , S0 , Pn0 , 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 simpli
39、city, rewrite Eq.25-28, as generic vector equations in the form F(X)=0 (29)where T1234TF(X)=(f (X),f (X),f (X),f (X)=(Eq.25, Eq.26, Eq.27, Eq.28)(30) T0=(0,0,0,0) (31)T1234TS0 S0 Pn0 En0X=( , , , )(, , , )xxxxr=(32) The Newtons iteration equation 6 is written as X1=X+ X (33) where X satisfies the fo
40、llowing system of linear equations JX=-F(X)i (34) where 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 11 112 422124414ff fffJ=ffx xxxxx x nullnullnullnullnullnullnullnul
41、lnull(35) 6 ijfxis the partial derivative of the thiequation with respect to the thjvariableThe partial derivatives in the Jacobian matrix can be approximated using the finite differences ijiijjf(X X) f(X)fxx+ (36) where i is the ithrow of the Jacobian matrix j is the jthcolumn of the Jacobian matri
42、x Xjis a vector with its jthelement equals the jthelement of the current Newtons step, X, and all remaining elements equal 0For each iteration, the sum of the absolute values of the functions (errors) is calculated 4ii=1ERR(X1) f (X1)=(37) The Newtons iteration procedure is terminated when the error
43、 (Eq. 37) becomes smaller than a predetermined tolerance, or when a predetermined number of iterations has been reached. The Newtons iteration procedure is described below: 1) Select a set of initial guess values for the new root, X1 . The following are the suggested values: a01S0 02S03Pn0 n04En0()2
44、( ) 0.0175()( ) 1.4835dxrxxx=2) Select the initial Newtons steps, X . The following values work satisfactorily: TX=(0.01,0.01,0.01,0.01) 3) Evaluate the system of non-linear equations (Eq.30) at the new root, F(X1) . 4) Calculate the error ERR(X1) (Eq.37). 5) The iteration is terminated, if 10ERR(X1
45、) 10 , or a predetermined number of iterations (30 should be sufficient) have been reached. Otherwise, continue with the next step. 6) Save the new roots as the current roots, so that a new set of roots can be calculated X=X1 (38) 7) Calculate the Jacobian matrix, column by column, starting with col
46、umn one using Eq.36. Repeat the calculation procedure for the remaining columns until the Jacobian matrix is completed (Eq.35). 8) Solve the system of linear equations (Eq.34) for the next set of the Newtons steps, X . 9) Calculate new roots, X1, using Eq.33. 10) Repeat steps 3-9, until step 5 is sa
47、tisfied. The system of linear equations in step 8 (Eq.34) can be solved by inverting the Jacobian matrix, or using one of many numerical root finding algorithms, such as Gaussian elimination method 7. 5 Generating Pressure Angle and Center Distance The generating pressure angle and the center distan
48、ce are based on tight meshing a shaper tool with a semi-finished gear. The involute function of the generating pressure angle, ginv , is given by the following equation (the derivation of Eq.39, is given in Annex A): b0br b0gb0 rbinv2(r )sspr+ =(39) where b0s is the base circular thickness of tool (
49、in) brs is the base circular thickness of semi-finished gear (in) b0p is the transverse base pitch of tool (in) b0r is the base radius of tool (in) brr is the base radius of semi-finished gear (in) The generating pressure angle, g , can be calculated by taking the arc of the involute function 5. Generating center distance, gc (in) 7 brb0ggcosrrc=(40) The gene
