j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j m a t p r o t e c
A novel hob cutter design for the manufacture
of spur-typed cutters
Jen-Kuei Hsieh
a, Huang-Chi Tseng
b, Shinn-Liang Chang
b,∗aMechanical Engineering Department, National Chiao Tung University, EE507, 1001 Ta Hsueh Road, Hsinchu 30010, Taiwan, ROC bInstitute of Mechanical and Electro-Mechanical Engineering, National Formosa University, Huwei Township, Yunlin County, Taiwan, ROC
a r t i c l e
i n f o
Article history:Received 23 November 2007 Received in revised form 20 February 2008 Accepted 24 February 2008 Keywords: Hob cutter Spur-typed cutter Undercutting
a b s t r a c t
Spur-typed cutters with multiple cutting angles are important tools frequently used in the manufacture of many machine elements. Due to their complex geometry, the production of these cutters involves milling and several grinding processes. Such a complex manufactur-ing process, coupled with the need for expensive manufacturmanufactur-ing tools, makes the cutters costly.
Undercutting is a phenomenon that causes weakness at the root of a gear. Engineers use shifted gears, modified tooth profiles, or changes in the pressure angle to overcome undercutting in gears. However, in this paper, by utilizing the undercutting phenomenon, a novel design of straight-sided hob cutter with multiple pressure angles is proposed for the manufacture of spur-typed cutters. With the simultaneous application of multiple pres-sure angles, this tool design concept significantly simplifies the manufacturing process. The effects of cutting angles, the degree of undercutting, and the width of the top land of the cutter are studied. The concepts and results proposed in this paper are beneficial as design guidance for tool designers and manufacturers.
© 2008 Elsevier B.V. All rights reserved.
1.
Introduction
Spur-typed cutter, which is formed with multiple cutting angles, is one of the most frequently used tools in the man-ufacture of machine components. The production of these cutters involves milling, rough grinding, and finish grinding. Due to the complex geometry and manufacturing process, var-ious expensive manufacturing tools must be employed, and hence the manufacture of the cutters can be very costly.
The hobbing of gears is the most effective manufactur-ing process found in the gear industry. New suggestions and methods to improve the precision and efficiency of hobbing have been introduced by researchers. Cluff (1987)
investi-∗Corresponding author at: Room 529, Building 2, The New Campus, National Formosa University, No. 64, Wunhua Road, Huwei Township, Yunlin County 632, Taiwan, ROC. Tel.: +886 5 6315429; fax: +886 5 6312110.
E-mail address:[email protected](S.-L. Chang).
gated how the generating accuracy of hob cutter was affected by cutter geometric peculiarities and resharpening errors.
Radhakrishnan et al. (1982)proposed a method to obtain the grinding wheel profile of the twist drill flute in resharpen-ing.Ainoura and Nagano (1987)investigated the conventional hobbing using a hob with its helix running in the direction opposite the gear, and they found it more effective for the high-speed manufacture of comparatively small module gears for automobiles. InKoelsch’s research (1994), hobbing cutters with different coatings are tested in high-speed cutting, and cermets were found to posses the best performance in high-speed dry cutting. More specifically,Phillips (1994)indicated that hob cutter coated by titanium nitride made productivity
0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.02.071
realized. Bouzakis and Antonidais (1995) proposed a com-putational procedure, which enables the determination of optimum values for the shift displacement and for the cor-responding shift amount.
In theory, the hob cutter may be considered to be a worm that is slotted in the axial direction to form a series of cut-ting blades. In Litvin’s publication (1989), the axial section of the worm was considered as the rack. Most of the litera-ture uses rack cutters to simulate the generating process of a hob cutter.Tsay (1988)investigated helical gears with involute shaped teeth, whose mathematical description was derived by straight-sided rack cutter.Chang et al. (1997a)proposed a gen-eral mathematical model of gear generated by CNC hobbing machine. Recent research, relevant to the hobbing process, includes the paper ofChang et al. (1996), in which the manu-facture of elliptical gears was studied.Chang (1996)simulated the hobbing process through a computer numerically con-trolled (CNC) hobbing machine.Chang et al. (1997b)achieved design optimization by tuning parameters of modified heli-cal gear train. More recently,Kapelevich (2000)investigated the hobbing of an unsymmetrical involute tooth profile.Chang et al. (2002)patented a new hob cutter profile that generated a helical cutting tool. The mathematical model of this heli-cal cutting tool is presented in the paper written byLiu and Chang (2003). However, the above-published work presented little systematic and in-depth discussion on the design and generation of the cutting tool tooth profile. In particular, there were no discussions of the design of a hob cutter capable of generating a spur-typed cutter with multi-cutting angles. Therefore, the possibilities for modifying existing designs are limited.
Undercutting is a phenomenon that causes weakness at the root of a gear. Various methods have been used to overcome this problem. In this paper, a novel design for a straight-sided hob cutter with multiple pressure angles is pro-posed, and the undercutting phenomenon is used to facilitate the manufacture spur-typed cutters. The proposed hob cut-ter can generate the multiple cutting angles simultaneously, which significantly reduces time and cost required in the tra-ditional process. The multiple pressure angles of hob cutter include a large pressure angle that generates the involute sur-face of the spur-typed cutter, which acts as the main body. A smaller pressure angle of, possibly, less than 5◦, undercuts the spur-typed cutter and forms the radial rake angle, which is a special application of undercutting in gear geometry. A third pressure angle, which is the largest one in the hob cutter, gen-erates the relief and clearance angles. Characteristics of the spur-typed cutter, including cutting angles, top land width, and full undercutting, are all studied.
The main theme of this paper is the concept of using multiple pressure angles to generate the complex multiple cutting angles. The developed mathematical model and the conducted analyses contribute a lot both in designing and manufacturing the spur-typed cutters.
2.
A novel design for a hob cutter
A hob cutter with straight-sided cutting face is commonly used in the manufacture of involute gear. An improper
param-Fig. 1 – Normal section of hob cutter.
eter design of the cutter will cause the undercutting at the roots of the gear. However, by appropriately designing the parameters of hob cutter, the undercutting can be controlled and become beneficial.Fig. 1shows the normal section of the novel type hob cutter, which is also considered as the profile of a rack cutter. The cutting face ofFig. 1can be divided into six regions, i.e. left cutting face I, right cutting face II, fillet cutting faces III and IV, top land cutting face V, and chamfering cutting face VI. The profile is almost similar to an ordinary hob cutter for involute gears, except for the two large pressure angles, i.e. regions I and VI, and a small pressure angle, i.e., region II. The cutting face II, with its small pressure angle, generates the cut-ting face and radial rake angle of the cutter by undercutcut-ting. The cutting face I, with its large pressure angle, generates the main body of the cutter. The cutting face VI, with the largest pressure angle, generates the clearance and relief angles of the cutter. The origin of the coordinate system Sa(Xa, Ya, Za) is located at the middle of the rack cutter body. The positive Xa axis is set upwards; positive Yais directed to the right, while the Zaaxis can be determined by the right-hand rule. InFig. 1, represents the pitch line of the rack cutter, Lis the pressure angle of the left straight-sided cutting face, Ris the pressure angle of the right straight-sided cutting face, and 3 is the largest pressure angle cutting face VI. HKW is the addendum of the rack cutter and HFW is the dedendum of the rack cut-ter, while 2b0and P0represent the tooth thickness and pitch of the rack cutter, respectively. In this paper, the theory of gearing proposed byLitvin (1989), which considers the locus equation and meshing equation simultaneously, is used to generate the tooth profile of the cutter.
2.1. Equations for the rack cutter
The equations for the six regions of the cutting face shown in the Sacoordinate system can be represented as follows:
(1) Left cutting face I:
The coordinates of the origin point eIof the cutting face I inFig. 1can be shown as
x(eI)
y(eI)
a = b0− R
tan(45+ L/2)− HKW tan L+ R cos L (2)
When parameter (I) indicates the position on the cut-ting face I, the equation of this cutcut-ting face can thus be represented in the Sacoordinate system as the following equation: r(I)a =
⎡
⎢
⎢
⎢
⎣
R − HKW − R sin L+ (I)cos L
b0− R
tan(45+ L/2)− HKW tan L+ R cos L+
(I)sin L 0 1
⎤
⎥
⎥
⎥
⎦
(3)The unit normal to this cutting face is obtained as
n(I)a = sin Lia− cos Lja (4)
(2) Right cutting face II:
The coordinates of the origin point eIIof the cutting face II inFig. 1can be shown as
x(eII)
a = R − HKW − R sin R (5)
y(eII)
a = −b0+ R
tan(45+ R/2)+ HKW tan R− R cos R (6)
When parameter (II) indicates the position on the cut-ting face II, the equation of this cutcut-ting face can thus be represented in the Sacoordinate system as
r(II)a =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
R − HKW − R sin R+ (II)cos R −b0+ R
tan(45+ R/2) +HKW tan R− R cos R− (II)sin R
0 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(7)The unit normal to this cutting face is obtained as
n(II)a = −sin Ria− cos Rja (8)
(3) Fillet cutting face III:
InFig. 1, parameter R is the radius of fillet. The geometry shows that CI= HWK tan Land bI= R/tan(45 + L/2). Coor-dinates of the origin OIof the fillet cutting face III can be shown as x(OI) a = R − HKW (9) y(OI) a = b0− R tan(45+ L/2)− HKW tan L (10)
Parameter Iindicates the position on the fillet cutting face III, which defines the cutting face position with the Xaaxis for 0≤ I≤ (90 − L). The equation of this cutting face can thus be represented in the Sa coordinate system as the
following equation: ra(III)=
⎡
⎢
⎢
⎢
⎢
⎣
R − HKW − R cos I) b0− Rtan(45+ L/2)− HKW tan L+ R sin I 0 1
⎤
⎥
⎥
⎥
⎥
⎦
(11)The unit normal to this cutting face is obtained as
n(III)a = cos Iia− sin Ija (12) (4) Fillet cutting face IV:
If parameter R defines the radius of the fillet,Fig. 1shows that CII= HKW tan R and bII= R/tan(45 + R/2). Coordi-nates of the origin OII of the fillet cutting face IV can be shown as x(OII) a = R − HKW (13) y(OII) a = −b0+ R tan(45+ R/2)+ HKW tan R (14)
Parameter II indicates the position on the fillet cutting face III, which defines the cutting face position with the Xa axis for 0≤ II≤ (90 − R). The equation of this cutting face can thus be represented in the Sacoordinate system as
r(IV)a =
⎡
⎢
⎢
⎢
⎢
⎣
R − HKW − R cos II −b0+ Rtan(45+ R/2)+ HKW tan R− R sin II 0 1
⎤
⎥
⎥
⎥
⎥
⎦
(15)The unit normal of this cutting face is obtained as
n(IV)a = −cos IIia− sin IIja (16)
(5) Top land cutting face V:
The top land cutting face V is formed by two tangential points dII and dI. The coordinates of point dII in the Sa coordinate system are
x(dII) a = −HKW (17) y(dII) a = −b0+ R tan(45+ R/2)+ HKW tan R (18)
And the coordinates of point dI in the Sa coordinate system are x(dI) a = −HKW (19) y(dI) a = b0− R tan(45+ L/2)− HKW tan L (20)
Parameter indicates the position of point dIIalong the Ya direction and 0≤ ≤ (YdI
a − YadII). The equation of this cutting face can thus be represented in the Sacoordinate
Fig. 2 – Coordinate system relationship of rack cutter and generated gear.
system as the following equation:
r(V)a =
⎡
⎢
⎢
⎢
⎢
⎣
−HKW −b0+ R tan(45+ R/2)+ HKW tan R+ 0 1⎤
⎥
⎥
⎥
⎥
⎦
(21)The unit normal of this cutting face is obtained as
n(V)a = ia (22)
(6) Chamfering cutting face VI:
The coordinates of the starting point q on this cutting face inFig. 1can be shown as
xa(q)= HFW − e (23)
y(q)a = b0+ (HFW − e) tan L (24)
Parameter (VI)indicates the position on the cutting face VI. The equation of this cutting face can thus be represented in the Sacoordinate system as the following equation:
r(VI)a =
⎡
⎢
⎢
⎢
⎣
HFW− e + (VI)cos 3 b0+ (HFW − e) tan L+ (VI)sin 30 1
⎤
⎥
⎥
⎥
⎦
(25)The unit normal to this cutting face is obtained as
n(VI)a = sin 3ia− cos 3ja (26)
2.2. Locus equations of rack cutter
When a rack cutter is used to cut gears, the coordinate sys-tems’ relationship can be represented as inFig. 2. In this figure, the theoretical pitch surface of the rack cutter is tangent to the pitch surface of the spur-typed cutter to be cut when the shifted distance C = 0.0. As the theoretical pitch surface of the rack cutter, i.e., the Ya− Za plane, moves toward the spur-typed cutter, C becomes negative. The mathematical model proposed here can then be used to simulate different spur-typed cutters with different shifted distances manufactured by the same rack cutter or hob cutter. In the manufacturing
of a spur-typed cutter, the active pitch surface Yd− Zd trans-lates towards the left while the generated spur-typed cutter rotates in a counter-clockwise direction. The locus equation of the rack cutter represented in the S1coordinate system and attached to the spur-typed cutter represents the cutting pro-cess. The generated profile can thus be obtained by solving the locus equation and the meshing equation [(6) and (16)] simultaneously. The meshing equation correlates the surface parameter of the cutting face to the motion parameter of the generation process.
(1) Locus equation of left cutting face I:
The locus equation of the left cutting face I shown in the generated cutter coordinate system S1can be obtained by transforming the cutting face equation from Sato S1. The transformation matrix and locus equation are as follow:
[M1a]=
⎡
⎢
⎢
⎢
⎣
cos 1 −sin 1 0 r11sin 1+ (r1− c) cos 1 sin 1 cos 1 0 −r11cos 1+ (r1− c) sin 1
0 0 1 0 0 0 0 1
⎤
⎥
⎥
⎥
⎦
(27) r(I)1 = [M1a]· r(I)a (28)(2) Locus equation of right cutting face II:
r(II)1 = [M1a]· r(II)a (29)
(3) Locus equation of fillet cutting face III:
r(III)1 = [M1a]· r(III)a (30)
(4) Locus equation of fillet cutting face IV:
r(IV)1 = [M1a]· r(IV)a (31)
(5) Locus equation of top land cutting face V:
r(V)1 = [M1a]· r(V)a (32)
(6) Locus equation of chamfering cutting face VI:
r(VI)1 = [M1a]· r(VI)a (33)
3.
Profile of spur-typed cutter
The mathematical model of the generated tooth profile can be obtained by considering the locus equation and the meshing equation [(6) and (16)] simultaneously. The loci equations are shown in the previous section while the meshing equation is represented by the following equation:
n(n)a · V(12)= 0 (34)
where n(n)a is common unit vector normal to the two contact surfaces; V(12)is the relative velocity between these two con-tact surfaces.
The meshing equation means that the common normal is perpendicular to the relative velocity at the contact point. It is independent of the selected coordinate system. For conve-nience, the common unit normal and relative velocity will be represented in the Shcoordinate system in this paper. From Fig. 2, the common normals of the cutting faces represented in the Sacoordinate system are the identical to those repre-sented in the Shcoordinate system, i.e., na= nh. The velocity of the rack cutter is therefore
V(F)h = −r1ω1jh (35)
The velocity of the contact point of the generated cutter is
V(1)h = OaOh× 1+ 1× ra (36)
where
OaOh= (−r1+ c)ih+ r11jh (37)
The relative velocity is obtained and expressed as
V(F1)h = V(F)
h − V
(1)
h (38)
(1) The relative velocity between the cutting face I and the generated cutter is as follows:
V(F1)h = −ω1
b0− R
tan(45+ L/2)− HKW tan L + R cos L+ (I)sin L− r11
ih+ ω1(R − HKW −R sin L+ (I)cos L− c)jh (39)
Substituting the common unit normal equation(4)and the relative velocity equation(39)into Eq.(34), the meshing equation becomes:
(I) = −(R − HKW − c) cos
L− [b0− R cot(45 + L/2) − HKW tan L− r11] sin L (40)
Solving this meshing equation(40)and the locus equation
(28)simultaneously, the generated tooth profile by cutting face I can thus be obtained.
(2) The relative velocity between the cutting face II and gen-erated cutter is as follows:
V(F1)h = −ω1
−b0+ R
tan(45+ R/2)+ HKW tan R − R cos R− (II)sin R
− r11
ih+ ω1(R − HKW − R sin R+ (II)cos R− c)jh (41)
Substituting the common unit normal equation (8)and relative velocity equation(41)into Eq.(34), the meshing equation becomes:
(II) = (−R + HKW + c) cos
R+ [−b0+ R cot(45 + R/2) +HKW tan R− r11] sin R (42)
Solving this meshing equation(42)and the locus equation
(29)simultaneously, the generated tooth profile by cutting face II can thus be obtained.
(3) The relative velocity between the cutting face III and the generated cutter is as follows:
V(F1)h = −ω1[b0− R
tan(45+ L/2)− HKW tan L
+R sin I− r11]ih+ ω1(R − HKW − R cos I− c)jh
(43)
Substituting the common unit normal equation(12)and the relative velocity equation(43)into Eq.(34), the meshing equation becomes: I= tan−1
−b0+ R cot(45 + L/2) + HKW tan L+ r11 R − HKW − c
(44)
Solving this meshing equation(44)and the locus equation
(30)simultaneously, the generated tooth profile by cutting face III can thus be obtained.
(4) The relative velocity between the cutting face IV and gen-erated cutter is as follows:
V(F1)h = −ω1
−b0+ R tan(45+ R/2)+ HKW tan R −R sin II−r11ih+ ω1(R − HKW − R cos II− c)jh (45)
Substituting the common unit normal equation(16)and relative velocity equation (45)into Eq.(34), the meshing equation becomes: II= tan−1
−b0+ R cot(45 + R/2) + HKW tan R− r11] R − HKW − c
(46)
Solving this meshing equation (46)and the locus equa-tion(31)simultaneously, the generated tooth profile by the cutting face IV can thus be obtained.
(5) The relative velocity between the cutting face V and the generated cutter is as follows:
V(F1)h = −ω1
−b0+ R tan(45+ R/2)+ HKW tan R+ − r11ih+ ω1(−HKW − c)jh (47)
Substituting the common unit normal equation(22)and relative velocity equation (47)into Eq.(34), the meshing equation becomes: =
b0− R cot 45+ R 2− HKW tan R+ r11
(48)
Solving this meshing equation(48)and the locus equation
(32)simultaneously, the generated tooth profile by cutting face V can thus be obtained.
Table 1 – Parameters of rack cutter and spur-typed cutter
Parameters of shifted spur-typed cutter
Circular pith (cp, mm) 2.8
Module (m, mm) 0.89126
Number of teeth (T) 12
Height of chamfering (e, mm) 0.3 Whole depth (HKW + HFW, , mm) 1.77 Length of cutter (mm) 30 Pitch diameter (mm) 5.348 Shifted distance (c, mm) 0.2 0.0 −0.2 Outside diameter (D, mm) 11.791 11.391 10.991 Root diameter (d, mm) 8.251 7.851 7.451 Parameters of rack cutter
Addendum (HKW, mm) 1.422
Dedendum (HFW, mm) 0.348
Tooth thickness (2b0, mm) 1.9
Tip radius (r, mm) 0.15
Pressure angle of face I ( L,◦) 48
Pressure angle of face II ( R,◦) 3
Pressure angle of face VI ( 3,◦) 57
(6) The relative velocity between the cutting face VI and gen-erated cutter is as follows:
V(F1)h = −ω1[b0+ (HFW − e) tan L+ (VI)sin 3− r11]i h +ω1(HFW− e + (VI)cos 3− c)jh (49) The same procedure of substituting the common unit nor-mal equation(26)and relative velocity equation(49)into Eq.(34)gives a meshing equation of
(VI)= −(HFW − e − c) cos 3− [b0+ (HFW − e)
tan L− r11] sin 3 (50)
Solving this meshing equation(50)and the locus equation
(32)simultaneously, the generated tooth profile by cutting face VI can thus be obtained.
Example 1. A spur-typed cutter has a circular pitch cp = 2.8 mm, with 12 flutes, an outside diameter of 11.391 mm, and a root diameter of 7.851 mm. The relevant parameters are shown inTable 1. The profile of the rack cutter is first designed as shown inFig. 3. Using the mathematical model developed, the generated spur-typed cutter is shown inFig. 4.Fig. 4also shows that the developed mathematical model matches the simulation of the generated rack cutter. The result proves that
Fig. 3 – Profile of the novel-design rack cutter.
Fig. 4 – Tooth profile and generation simulation of spur-typed cutter (c =−0.2).
Fig. 5 – Generated tooth profile with different shift amount.
the novel design of using the proposed rack cutter (hob cutter) is an effective and efficient way of manufacturing a spur-typed cutter.Fig. 5shows the transverse section of the spur-typed cutter. This figure reveals that the same rack cutter with a different shift can produce different spur-typed cutters.Fig. 6
shows the transverse section of a spur-typed cutter with the same outside diameters.
4.
Characteristics of the spur-typed cutter
In the preceding section, the mathematical model of the generated spur-typed cutter has been derived. However, the cutting angles and the width of the top land of the cutter will also significantly affect the cutting performance. In this section, these important factors are studied.
4.1. Full undercutting on the spur-typed cutter
As previously mentioned, the right cutting face of the pro-posed spur-typed cutter is generated by undercutting the circular edge of the rack cutter. A poorly designed rack cut-ter will generate part of an involute curve on a typical cutting edge. When the cutting edge is fully undercut, the start-ing point of undercuttstart-ing lies beyond the circle of outside
Fig. 6 – Generated tooth profile with different shift amount (outside diameter is constant).
diameter. Hence, the x- and y-components of the equation depicting the right side of the cutting face in Eqs.(29) and (42), respectively, are the same as the x- and y-components of the equation depicting the undercut portion shown in Eqs.
(31) and (46).
Example 2. InExample 1, when the shifted distance c = 0.0, the radius of the intersection point is rP= 5.725, which is larger than the radius of the outside diameter, r = 5.696. This confirms that the design of the rack cutter has satisfied the requirement of a spur-typed cutter.
4.2. Determination of the cutting angles
The cutting angles of the end section profile shown inFig. 7
affect the cutting performance significantly. In this section, the cutting angles of a spur-typed cutter manufactured by the novel hob cutter are investigated. Point A is the intersection point of the undercut portion (region IV) and the curve on the top land of the spur-typed cutter, point B is the intersection point of the chamfered angle edge and the top land curve, and point E is the intersection point of the left cutting edge and the chamfered angle cutting edge. TAand Taare the tan-gential and positional vectors of point A in the S1coordinate system, respectively. TBis the tangential vector of point B in
Fig. 7 – Definitions of the cutting angles.
Fig. 8 – The relief angle at point B.
the S1coordinate system, and TBis the tangential vector of the chamfered angle cutting edge at point B in the S1coordinate system. The radial rake angle Ais the angle between TAand TA, the relief angle Bis the angle between TBand TB, and the clearance angle Eis the angle between TBand TE.
4.2.1. Analysis of the radial rake angle A
Since point A is the intersection point of the undercut portion (region IV) and the curve on the top land of the spur-typed cutter, the following equation is established:
x2
A+ y2A= r1+ HFW (51) where (xA, yA) are the coordinates of point A and r1is the pitch radius of the spur-typed cutter. Substituting the point A Eqs.
(31) and (46)into the above equation, parameter 1is obtained. The positional vector TAand tangential vector TAof point A can thus be obtained. Vector dot production is then performed to attain Abetween these two vectors.
4.2.2. Analysis of the relief angle B
From the involute curve property(17), the involute curve is extended from the base circle. InFig. 8, the normal vector of the involute curve at point B is tangent to the base circle. The directional vector −→OG is thus the same as the tangential vector
TBat point B. The pressure angle at point B, i.e., Bis thus obtained as follows:
rBcos B= rb (52)
where rBis the position vector of point B and rbis the radius of the base circle.
As vectors TB and TB are perpendicular to −BO and→ −→BG, respectively, the relief angle Bis obtained as
B=
2− B (53)
4.2.3. Analysis of the clearance angle E
InFig. 9, TEis the tangential vector at point E while TEis the vector perpendicular to the position vector rE. Applying the theory of involutometry, the clearance angle Eis obtained as
Fig. 9 – Relationship between clearance anglesEand at
point E.
Fig. 10 – The cutting angles when outside diameters are changed according to the shifted amount.
follows:
E=
2+ − E (54)
where
= inv B− inv E (55)
4.2.4. Effect of shifted distance on the cutting angles
By substituting the values fromTable 1, the various cutting angles are obtained and shown inFig. 10where the outside diameters are changed according to the shifted distance. If the outside diameters are kept constant, the cutting angles are as shown inFig. 11.
Fig. 11 – The cutting angles when the outside diameters are kept constant.
4.3. Width of top land of spur-typed cutter
The width of the top land of the spur-typed cutter can be obtained by solving for the coordinates of points A and B shown inFig. 12. The coordinates of point A were solved in the previous section. If the polar angle of point B is solved, the x-component of point B is Bx= rBcos and the y-component is By= rBsin . From the geometric relationship shown inFig. 12, can be expressed as
= ∠KOE + ∠EOB (56)
where
∠EOB = inv B− inv E (57)
Due to pure rolling between the pitch line of the rack cutter and the pitch circles of the generated gears, the arc length of NJ shown inFig. 12is equal to the corresponding distance
Fig. 13 – Rolling position of the largest pressure angle for the rack cutter.
Fig. 14 – Width of top land of spur-typed cutter with constant outside diameter, and variable outside diameter according to the shifted amount, respectively.
shown inFig. 13. The angle ı = ∠NOJ is therefore
ı =
r1 (58)
where r1is the radius of the pitch circle of the generated spur-typed cutter.
Therefore, angle is obtained as
= inv B− inv J+ ı (59)
Fig. 14shows the width of the top land of the spur-typed cut-ter when the outside diamecut-ters are kept constant, and when the outside diameters are changed according to the shifted distance, respectively.
5.
Conclusion
A novel design for a hob cutter capable of generating a spur-typed cutter in one hobbing process has been proposed. The cutting edge in the normal direction has been designed as three straight lines with different pressure angles and two arcs. By applying the equations of the rack profiles of the cutting edges, the principle of coordinate transformation, the
theory of differential geometry, and the theory of gearing, the mathematical models of the spur-typed cutter profile, includ-ing the radial cuttinclud-ing angle, relief angle, and clearance angle have been derived. A computer simulation involving a para-metric study of the three angles was also carried out. The major characteristics of the generated spur-typed cutter were also studied in this paper. The proposed method can be used as design guidance in designing novel hob-type cutters to gener-ate spur-typed cutters. Moreover, it is expected that the results from this paper will contribute to the improvement of the manufacture of plain mill cutters and provide the tool indus-try with a reference for designing and machining similar tools. The results can also act as a basis for researchers to optimise and improve their tool designs.
Acknowledgments
The work outlined in this paper was supported by the national science council under grants NSC91-2212-E-150-022 and NSC92-2212-E-150-033.
r e f e r e n c e s
Ainoura, M., Nagano, K., 1987. The effect of reverse hobbing at a high speed. Gear Technol. 4 (March/April (2)), 8–15.
Bouzakis, K.D., Antonidais, A., 1995. Optimizing of tangential tool shift in gear hobbing. Ann. CIRP 44 (1), 75–78.
Chang, S.L., 1996. Gear hobbing simulation of CNC gear hobbing machines. Dissertation for Doctoral Degree. National Chaio Tung University, Hsinchu, Taiwan, ROC.
Chang, S.L., Liu, J.Y., Hsieh, L.C., 2002. Design of hob cutters for generating helical cutting tools with multi-cutting angles. Chinese Patent 185894.
Chang, S.L., Tsay, C.B., Nagata, S., 1997a. A general mathematical model for gear generated by CNC hobbing machine. Trans. ASME J. Mech. Design 119, 108–113.
Chang, S.L., Tsay, C.B., Tseng, C.H., 1997b. Kinematic
optimization of a modified helical simple gear train. Trans. ASME J. Mech. Design 119, 307–314.
Chang, S.L., Tsay, C.B., Wu, L.I., 1996. Mathematical model and undercutting analysis of elliptical gears generated by rack cutters. Mech. Mach. Theory 31 (7), 879–890.
Cluff, B.W., 1987. Effects of hob quality and resharpening errors on generating accuracy. Gear Technol. 4 (September/October (5)), 37–46.
Kapelevich, A., 2000. Geometry and design of involute spur gears with asymmetric teeth. Mech. Mach. Theory 35, 117–130. Koelsch, J.R., 1994. Hobs in high gear. Manuf. Eng. (July), 67–69. Liu, J.Y., Chang, S.L., 2003. Design of hob cutters for generating helical cutting tools with multi-cutting angles. Int. J. Mach. Tools Manuf. 43 (12), 1185–1195.
Litvin, F.L., 1989. Theory of Gearing. NASA Publication, Washington, DC.
Phillips, R., 1994. New innovations in hobbing. Part I. Gear Technol. 11 (September/October (5)), 16–20.
Radhakrishnan, T., Kawlra, R.K., Wu, S.M., 1982. A mathematical model of the grinding wheel profile required for a specific twist drill flute. Int. J. Mach. Tool Design Res. 22, 239– 251.
Tsay, C.B., 1988. Helical gears with involute shaped teeth: geometry, computer simulation, tooth contact analysis, and stress analysis. Trans. ASME J. Mech. Transm. Autom. Design 110, 482–491.
