Tooth profile design for the manufacture of helical gear
sets with small numbers of teeth
Chien-Fa Chen
a, Chung-Biau Tsay
b,*
aDepartment of Mechanical Engineering, National United University, Miaoli 36003, Taiwan, ROC
b
Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan, ROC Received 16 June 2004; accepted 13 January 2005
Available online 11 May 2005
Abstract
Based on gear theory and generating mechanism, this investigation presents a complete mathematical model of a helical gear set with small number of teeth. The unavoidable tooth-profile undercutting of the gears with small number of teeth is examined by using the developed mathematical model and the conventional method of tooth-profile shifting. Furthermore, an alternative method for lessening the tooth-profile undercutting is also presented by considering a modification of the basic fillet geometry using a modified rack cutter. A third method, combining the aforementioned two methods for the design of helical gears with small number of teeth is also proposed to yield a gear set without tooth undercutting. The mating gear with profile shifting is generated using the pinion as a shaper. The tip fillet and root fillet are modified and a clearance between the pinion and the mating gear is also included in the design. Analysis results indicate that the change of distance between the centers of gear set depends only on gear shifting. Moreover, computer graphs are demonstrated the profile-shifted and the proposed modified gear tooth profiles.
q2005 Elsevier Ltd. All rights reserved.
Keywords: Undercutting; Modified tooth surface; Small number of teeth; Profile shifting
1. Introduction
It is known that spur and helical gears with small number of teeth may exhibit tooth undercutting. Gears with small number of teeth are typically not used in power trans-missions. However, helical gears are extensively used in industry, and may be generated by hobs, shapers and rack cutters. Several researchers[1–5]and AGMA publications
[6,7] have significantly contributed to the design and
manufacturing of this type of gearing. Many researchers have also studied gear design and manufacturing with tooth-profile shifting. Ishibashi et al.[8]derived a mathematical model of the spur gear with two or three pinion teeth, according to the basic geometry. Their mathematical model was used to investigate the design, manufacture and load capacity. Ishibashi and Yoshino[9]also determined the load capacity of Novikov gears with three to five pinion teeth.
Additionally, Komori et al.[10]developed a spur gear with LogiX tooth profiles which have zero relative curvature at many contact points. Arikan[11]determined the maximum possible contact ratios using an x-zero gear pair for spur gears with small number of teeth. Analysis results were also compared with addendum modification coefficients rec-ommended by ISO.
Tooth undercutting occurs at the generated gear tooth surfaces, under certain conditions, such as small number of teeth, small pressure angles and negatively shifted profiles. If tooth undercutting occurs, the tooth thickness near the gear fillets is reduced and the gear bending moment capacity is also decreased. Mabie and Reinholtz [12] considered geometric relationships to study the tooth undercutting of spur gears generated by shaper cutters. Many researchers
[13,14] studied tooth undercutting for various types of
gearing. Litvin [15,16] presented a detailed theory of the gear non-undercutting conditions. Tsai and Tsai [17]
proposed a method of designing high-contact-ratio spur gears with quadratic parametric tooth profiles, that have a short addendum without tooth undercutting.
0890-6955/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2005.01.017
www.elsevier.com/locate/ijmactool
* Corresponding author. Tel./fax: C886 3 572 8450. E-mail address: [email protected] (C.-B. Tsay).
Tooth undercutting may reduce the gear strength and contact ratio. Undercutting is therefore an important problem in gear design. Of course, tooth undercutting can be eliminated by increasing teeth to the gear. However, the use of more teeth is occasionally not allowable. This study aims to obtain a spur or helical gear with small number of teeth, without tooth undercutting. Based on the gear theory and the gear generating mechanism, a mathematical model of the helical gear has been developed. Moreover, the condition for gear non-undercutting is also derived by considering the relative velocity between the rack cutter and the gear blank, and by considering the differentiated equation of meshing. The conventional tooth-profile shift-ing method is used to solve the tooth undercuttshift-ing problem. Nevertheless, this method always increases the tooth thickness of the fillets and reduces the contact ratios. An alternative method, which considers the modification of fillet geometry, is presented to develop a gear set without tooth undercutting. Based on the tooth-profile shifting and
the basic fillet geometry modification methods, a third method that combines these two methods is presented to obtain a gear set with a higher contact ratio and fillet strength without tooth undercutting. Furthermore, the use of a modified pinion with modified tip fillets is proposed to generate the mating gear pair and thus prevents the occurrence of singular points on the generated gear tooth. In such a generation process, tooth-profile shifting and gear clearance have also been considered. Finally, a computer program is developed to generate the complete geometry of the gear, including the involute tooth surfaces, the modified root fillets and the modified tip fillets. Results of this study can be used to design not only spur and helical gears with small number of teeth, but also one-stage high contact ratio gear pairs. Consequently, the total volume of the gearbox can be reduced, the structure of the gear transmission mechanism can be simplified and the gear assembly can also be made easier. Besides, the cost of a on-stage gearbox is cheaper than that of the multi-stage gearbox.
Nomenclature
ai tool setting of rack cutter generating the involute
gear (iZ0, 1)
e total profile shifting coefficient
ei profile shifting coefficient (iZ1, 2, p, g)
C clearance of the mating gears
E center distance of the mating gears (or shaper and generated gear)
E0 operating center distance of the mating gears [Lij] projection transformation matrix (from Sjto Si)
l surface parameter of rack cutter (in mm) [Mij] coordinate transformation matrix (from Sj
to Si)
m12 gear ratio
mn normal module of the gear
nðjÞi unit normal vector of rack cutter surface j (jZ1,
2 and 3, which represent part 1, 2 and 3 of rack cutter normal section (Fig. 1), respectively) represented in coordinate system i (iZc, a) nðjÞ1 unit normal vector of generated gear surface j
(jZ1w3)
RðjÞi position vector of surface j (jZ1, 2 and 3, which represent part 1, 2 and 3 of rack cutter normal section (Fig. 1), respectively, and their corresponding generated gear tooth surfaces) represented in coordinate system i (iZ1, 2, c, a)
r distance between gear rotational center and beginning point of modified region (in mm) ri radius of pitch circle of pinion and gear (iZ1, 2)
(in mm)
rj radius of centrode of pinion and gear (jZp, g)
(in mm)
rr radius of the modified root fillet of shaper (in
mm)
rt radius of the modified tip fillet of shaper (in mm)
Si coordinate system i (iZ1, 2, a, c, h)
T tangent vector to a curve
Ti number of teeth of pinion and gear (iZ1, 2)
xðjÞi ; yðjÞi ; zðjÞi position vector of modified fillet surface j (jZr, t where r represents root fillet and t indicates tip fillet) represented in coordinate system i (iZ1, 2)
u surface parameter of rack cutter (in mm) Vðc1Þtr transfer velocity of the contact point
Vð1Þr relative velocity of the contact point with the
gear
VðcÞr relative velocity of the contact point with the
rack cutter
Sa normal section of rack cutter surface
Se rack cutter surface
jbs operating pressure angle of helical gear
jn normal pressure angle of rack cutter (in degrees)
js transverse pressure angle helical gear
ri radius of tip and root fillets of rack cutter (iZ0,
1) (in mm)
qn variable parameter of root fillet of rack cutter (in
degrees)
zn variable parameter of tip fillet of rack cutter (in
degrees)
fi rotational angle of pinion and gear (iZ1, 2) (in
degrees)
l lead angle of gear (in degrees)
si spanned angle of modified root and tip fillet (iZ
2. Mathematical model of the modified tooth surface
2.1. Rack cutter surfaces
Fig. 1depicts the normal section of rack cutter Saused
to generate involute helical gears. Part 1 of the tip fillet of the generating rack cutter is an arc of radius r0, which
generates the root fillet of the gear; part 2 of the tool profile is a straight line M0M2 that generates the involute profile of
the gear, and part 3 of the bottom fillet of the generating rack cutter is an arc of radius r1, which generates the tip
fillet of the gear. The position vector of the straight line M0M2of the rack cutter normal section can be represented
in the coordinate system Sa(Xa,Ya,Za) by the following
equations. Rð2Þa l cos jnKa0 Gðl sin jnKa0tan jnKb0Þ 0 2 6 4 3 7 5; 0%l%ða0Ca1Þ=cos jn: (1) Similarly, the equation of the tip fillet of the normal section of the generating rack cutter can be expressed as follows:
Rð1Þa Z
Ka0Cr0sin jnKr0sin zn
GðKa0tan jnKb0Kr0cos jnCr0cos znÞ
0 2 6 6 4 3 7 7 5; jn!zn!p=2: (2)
The equation of the bottom fillet of the generating rack cutter normal section are as follows.
Rð3Þa Z
a1Kr1sin jnCr1sin qn
Gða1tan jnKb0Cr1cos jnKr1cos qnÞ
0 2 6 6 4 3 7 7 5; jn!qn!p=2: (3)
The upper signs in Eqs. (1)–(3) refer to the left-side of the rack cutter normal section while the lower signs refer to the right-side of the rack cutter normal section. l, zn, and qn
are the design parameters of the rack cutter surface that determine the location of points on the straight line, tip fillet, and bottom fillet, respectively.
In simulating the rack cutter surface for the helical gear generation, the normal section of the rack cutter Sa, attached
to the coordinate system Sawith its origin Oa, is translated
along the line OaOc as shown in Fig. 2. Therefore, uZ
jOaOcj is also one of the design parameters of the rack cutter
surface, and l is the lead angle of the generated helical gear. The rack cutter surface Scfor helical gear generation can be
represented in the coordinate system Sc(Xc,Yc,Zc) by
applying the following homogeneous coordinate transform-ation matrix equtransform-ation:
RðiÞc Z ½McaRðiÞa ; (4)
where
½Mca Z
1 0 0 0
0 sin l cos l u cos l 0 Kcos l sin l u sin l
0 0 0 1 2 6 6 6 6 4 3 7 7 7 7 5;
and iZ1, 2 and 3. Substituting Eq. (1) into Eq. (4), enables the position vector of the rack cutter surface Sctraced out by
the straight line M0M2 (part 2 inFig. 1), to be represented in
coordinate system Scas
Rð2Þc Z
l cos jnKa0
Gðl sin jnKa0tan jnKb0Þsin l C u cos l
Hðl sin jnKa0tan jnKb0Þcos l C u sin l
2 6 4 3 7 5: (5)
Based on the differential geometry, the unit normal vectors of the above-mentioned rack cutter surface rep-resented in coordinate system Scare
nðiÞc Z vRðiÞc vl ! vRðiÞc vu vRðiÞc vl ! vRðiÞc vu : (6)
2.2. Generated tooth surfaces
Fig. 3illustrates the relationship between rack cutter Sc
and generated gear of the gear generation mechanism. In deriving equations for the gear tooth surface, the coordinate systems Sc(Xc,Yc,Zc), S1(X1,Y1,Z1) and Sh(Xh,Yh,Zh) are
attached to the rack cutter, generated gear and gear housing, respectively. Based on gear theory, the generated gear surface can be obtained by simultaneously considering the locus of the imaginary rack cutter represented in coordinate system S1 and the equation of meshing [3,15,16] of the
cutter and the generated gear. Thus, the mathematical model of the gear tooth surface is
RðiÞ1 Z ½M1cR ðiÞ c; (7) and XðiÞc KxðiÞc nðiÞcx ZY ðiÞ c KyðiÞc nðiÞcy ZZ ðiÞ c KzðiÞc nðiÞcz ; (8) where ½M1c Z
cos f1 Ksin f1 0 r1ðcos f1Cf1sin f1Þ
sin f1 cos f1 0 r1ðsin f1Kf1cos f1Þ
0 0 1 0 0 0 0 1 2 6 6 6 6 4 3 7 7 7 7 5: Symbols XcðiÞ, YcðiÞand ZcðiÞrepresent the coordinates of a
point on the instantaneous axis of gear rotation IKI in coordinate system Sc; xðiÞc , yðiÞc and zðiÞc are coordinates of the
instantaneous contact point on the rack cutter surface Sc;
and nðiÞcx, nðiÞcy and nðiÞcz are the direction cosines of the rack
cutter surface unit normal nðiÞc . Substituting Eqs. (5) and (6)
into Eq. (8), yields the equation of meshing for the rack cutter and generated gear.
f1ðl; u; f1Þ ZG a0 cos jn Cb0sin jnKl sin l C ðr1f1Ku cos lÞsin jnZ 0: (9)
Substituting Eqs. (5) and (9) into Eq. (7) yield the generated helical gear with involute profile tooth surface as follows:
xð2Þ1 Z ðl cos jnKa0Cr1Þcos f1Hða0
Kl cos jnÞcot jnsin l sin f1;
yð2Þ1 Z ðl cos jnKa0Cr1Þsin f1G ða0
Kl cos jnÞcot jnsin l cos f1; and
zð2Þ1 ZGða0Kl cos jnÞcotjntan l sin l
G a0tan jnCb0Kl sin jn cos l Cr1f1tan l: (10)
The following matrix equation gives the unit normal of the tooth surface.
nðiÞ1 Z ½L1cn ðiÞ c ; (11) where ½L1c Z cos f1 Ksin f1 0 sin f1 cos f1 0 0 0 1 2 6 4 3 7 5:
2.3. Tooth undercutting analysis
At any instantaneous contact point of the rack cutter and generated gear, the absolute velocities of the rack cutter and generated gear are the same. Nevertheless, the absolute velocity can be decomposed into components, the relative velocity VðcÞr and Vð1Þr and transfer velocity V
ðcÞ tr and V
ð1Þ tr of
the rack cutter and generated gear, respectively. Therefore, VðcÞr CV c ð Þ tr Z Vð1Þr CV 1 ð Þ tr ; or VðcÞr CV ðcÞ tr KV ð1Þ tr Z VðcÞr CV ðc1Þ tr Z V ð1Þ r : (12)
When tooth undercutting occurs, a singular point appears on the generated gear tooth surface and its surface tangent TZ0 at this singular point. The mathematical definition of singularity of the generated gear can be represented in coordinate system Sc by equation Vð1Þr Z 0. Therefore, Eq.
(12) becomes
VðcÞr CVðtrc1ÞZ 0: (13)
RecallingFig. 3, the conditions under which a singular point may appear on the working surface of the helical gear generated by the straight line (line M0M2shown inFig. 1) of the rack cutter are considered here. The relative tranfer velocity between the gear blank and the rack cutter,
represented in coordinate system Sc, can be obtained as follows: Vðc1Þc Z u1 r1f1Kyð2Þc xð2Þc 0 2 6 6 4 3 7 7 5: (14)
Notably, Eq. (5) specified the position vector of the rack cutter surface, Rð2Þc . Eq. (13) can thus be rewritten by
vRð2Þc vl dl dtC vRð2Þc vu du dt Z KV ðc1Þ tr ; (15)
Differentiating the equation of meshing, Eq. (9), yields vf1 vl dl dtC vf1 vu du dtZ K vf1 vf1 df1 dt : (16)
Eqs. (15) and (16) form a system of four linear equations in two unknowns, dl/dt and du/dt. This system of equations has a unique solution for the unknowns if the matrix
A Z vRð2Þc vl vRð2Þc vu KV ðc1Þ tr vf1 vl vf1 vu K vf1 vf1 df1 dt 2 6 6 4 3 7 7 5 (17)
has a rank of two. This yields the following four determinants equal zero:
D1Z vxð2Þc vl vxð2Þc vu KV ðc1Þ x;tr vyð2Þc vl vyð2Þc vu KV ðc1Þ y;tr vf1 vl vf1 vu K vf1 vf1 df1 dt Z 0; (18) D2Z vxð2Þc vl vxð2Þc vu KV ðc1Þ x;tr vzð2Þc vl vzð2Þc vu KV ðc1Þ z;tr vf1 vl vf1 vu K vf1 vf1 df1 dt Z 0; (19) D3Z vyð2Þc vl vyð2Þc vu KV ðc1Þ y;tr vzð2Þc vl vzð2Þc vu KV ðc1Þ z;tr vf1 vl vf1 vu K vf1 vf1 df1 dt Z 0; (20) and D4Z vxð2Þc vl vxð2Þc vu KV ðc1Þ x;tr vyð2Þc vl vyð2Þc vu KV ðc1Þ y;tr vzð2Þc vl vzð2Þc vu KV ðc1Þ z;tr Z 0: (21)
Eq. (21) is the same as the equation of meshing, and it is satisfied since the points of tangency of the rack cutter and the generated tooth surfaces are considered. Thus, only Eqs. (18)–(20) should be applied to determine the conditions of singularity for the generated tooth surface. Therefore, a sufficient condition for the occurrence of a singularity of the generated tooth surface is Gðl; u; f1Þ Z D 2 1CD 2 2CD23Z 0: (22)
Eq. (22) yields the condition of tooth undercutting as follows: l Z 1 cos jn a0K r1sin2jn B ; (23) where
B Z sin2jnCcos2jnsin2l:
2.4. Generating nonstandard gears by rack cutters
Nonstandard gears can be generated by a standardized tool used for generation of standard gears but with modified tool settings with respect to the gear being generated. Usually, modified tool settings are applied to prevent tooth undercutting of the generated gear. Referring to Fig. 4 and Eq. (23), the rack cutter will not undercut the gear tooth if the following inequality is satisfied. epmn cos jn Rjlj Z 1 cos jn a0K r1sin2jn B : (24) Thus, the normal shifting coefficient of the rack cutter setting ep, which corresponds to gear tooth
non-under-cutting is epR a0 mn K T1 2 sin l sin2jn B ; (25)
where T1represents the number of teeth on the generated
gear; mnis the normal module, and a0is the standardized
parameter of the rack cutter, as shown in Fig. 1. Example 1. Nonstandard gears generated by different profile shifting coefficients.
Table 1 lists some important design parameters for
helical gears with small number of teeth. Fig. 5 plots the profiles of the pinion with shifting coefficients ep of
0.5 and 1.0. Tooth undercutting on the gear root is found to be reduced by increasing the profile shifting coefficient. However, the thickness of the gear tooth is increased compared with the standard gear for nonstan-dard gears.
3. Modification of the root fillet surfaces of the pinion In a one-stage gear set with a high gear ratio, one of the gears must be extremely small. Tooth undercutting on the gear root may occur when the small number of teeth are used. This work proposes a modified curvilinear root profile, rather than an ordinary gear root profile, to improve
the bending strength of the pinion root. Fig. 6 shows the cross section of the pinion tooth surface on plane z1Z0 mm.
Since tooth undercutting occurs on the involute profile of the generated pinion tooth surface, the mathematical model of the involute profile (refer to Eq. (10)) of the first quadrant is expressed as follows:
x1Z ðl cos jnKa0Cr1Þcos f1K ðl cos jn
Ka0Þcot jnsin l sin f1;
y1Z ðl cos jnKa0Cr1Þsin f1C ðl cos jn
Ka0Þcot jnsin l cos f1;
(26)
Table 1
Some major design parameters for helical gears with small number of teeth
Parameters Pinion Gear Notes
Normal module mn 1.75 mm/teeth Given
Pressure angle 208
Lead angle 608
Teeth number 2 60
Shifting coefficient e1Z0.7, e2Z0.7 epZe1Z0.7 egZKe1Ce2ZK1.4
Total shifting coefficient eZepCegZe2ZK0.7 Calculated
Center distance 61.106 mm
l corresponding to undercutting 0.236 mm l corresponding to point C 0.246 mm Radius of root fillet rr 1.045 mm
Spanned angle of root fillet sr K60.148%sr%08
Clearance C 0.4 mm
Radius of tip fillet rt 0.332 mm
Spanned angle of tip fillet st 08%st%26.698
and
z1Z Kða0tan jnKl sin jnÞcos l
C a0 cos jnsin jn K l sin jn tan l sin l K b0 cos l Cr1f1tan l:
Closely examining Fig. 6 reveals that the curve EF is an involute profile generated by the straight line of the rack cutter, and point F is the beginning point at which tooth undercutting occurs. Notably, the parameter l corresponding to point F can be obtained using Eq. (23). Consider a point C, which is close to point F, where the parameter l of the rack cutter profile (Fig. 1) that corresponds to point C slightly exceeds that of the rack cutter profile that corresponds to point F. xðCÞ1 and yðCÞ1 are the components of the line O1C represented in
coordinate system S1 and are given by Eq. (26). An
expression for the length jO1CjZ r should be derived
firstly as follows: r Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxðCÞ1 Þ2C ðyðCÞ 1 Þ 2 q Z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r21K r2
1K ½ðl cos jnKa0Þð1Ccot2jnsin2lÞ Cr12
1 Ccot2j nsin2l
s
: (27) The line BC is the tangent to the involute curve EF at point C, while the line AC is the normal to the involute curve EF at point C. Angle A0O1E depends on the number
of teeth T1, and is expressed as follows:
:A0O1E Z
p T1
: (28)
In this study, the length AC is the proposed radius of the modified root fillet: jACjZrr: Applying the law of
sine to triangle AO1C yields the following equation:
rr
sin ar
Z r sin br
: (29)
Differentiating Eq. (26) yields the tangent vector to the involute profile at point C
T Zdx1 dl i1C
dy1
dl j1: (30)
According to Eq. (28), the unit vector of the slope of line O1A is m Zcosp T1 i1Csin p T1 j1: (31)
Taking the dot product of vectors T and m yields the angle formed by lines BA and BC:
qrZ cosK1 cosp T1 dx1 dl C sinTp 1 dy1 dl ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx1 dl 2 C dydl1 2 q : (32)
According to Fig. 6, the angle br equals p/2Kqr.
Substituting br into Eq. (29) yields the radius rr of the
modified root fillet.
With reference to Fig. 6, the modified circular arc CH with a radius of rrand a center at point A is proposed for the
modified root fillet when the pinion exhibits serious tooth undercutting. The screw motion of the modified circular arc performs the helical pinion root-fillet surface. Conse-quently, the cross sections of the helicoids corresponding to z1Z0 and z1Zconstant, represent the same plane curve in
two positions. One cross section coincides with the other after a rotation about the z1 axis through an angle h in a
screw motion. The proposed modified root fillet of the pinion can be represented in coordinate system S1 as
follows.
xðrÞ1 Z xðCÞ1 cos hHy ðCÞ
1 sin h CrrsinðdrGhÞ KrrsinðsrGhÞ;
yðrÞ1 Z xðCÞ1 sin hGy ðCÞ
1 cos hHrrcosðdrGhÞGrrcosðsrGhÞ;
and zðrÞ1 Z r1h tan l;
(33) where drZp2K
p
T1Kbr; dr%sr%drCbr; and sr represents
the spanned angle of the modified root fillet.
4. Modification of the tip fillet surface of the shaper The pointed teeth are an important issue in gear design and manufacture, especially for helical gears with small number of teeth. Since the gear teeth are generated by a pinion-type shaper, the tip fillet of the shaper must be considered to avoid the occurrence of singular points in the generation process. Fig. 7depicts the cross section of the shaper on the plane z1Z0 mm. Point J is the
beginning point of the tip fillet that lies on the involute profile, and the coordinates of point J are xðJÞ1 and yðJÞ1 ; point K is the point of intersection of the X1axis and the
vector T that is tangent to the involute tooth profile at
point J. OJ is the point of intersection of the X1axis and
the normal vector that is perpendicular to the involute profile at point J, and point OJ is also the center of the
modified tip fillet. The length OJJ is the proposed radius
of the modified tip fillet, jOJJjZ rt. Eq. (26) determines
the coordinates of point J. Following the procedure similar to that of previous section, the angle formed by the axis X1 and line JK can be expressed by
qtZ cos K1 dx1 dl ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx1 dl 2 C dydl1 2 q : (34)
According toFig. 7, angle btequals p/2Kqt. Applying
the law of sine to triangle JOJL yields the radius of the
modified tip fillet, rtZ y ðJÞ 1 =sin bt.
Similarly, the proposed modified tip fillet of the shaper can be represented in coordinate system S1as follows:
xðtÞ1 Z xðJÞ1 cos h K rtcos btcos h C rtcosðstHhÞ;
yðtÞ1 Z xðJÞ1 sin h K rtcos btsin hHrtsinðstHhÞ;
and zðtÞ1 Z r1h tan l
(35)
where stis the spanned angle of the modified tip fillet and
0%st%bt.
The radius of the modified tip fillet, rt, is determined by
the location of point J. Point J is selected as the point that at which the modified tip fillet makes a clearance between the addendum of the pinion and the dedendum of the generated gear, as shown inFig. 7. AGMA’s fundamental formula for the recommended clearance of fine pitch gears is suggested as follow:
C Z 0:2mnC0:05; (36) where C represents the clearance and mn is the normal
module of the gear.
5. Mathematical model of the gear generated by shapers
Fig. 8 displays the kinematic relationship between the
shaper and the generated gear. Coordinate systems S1(X1,Y1,Z1) and S2(X2,Y2,Z2) are rigidly attached to the
shaper cutter and the gear, respectively. The position vector of the shaper cutter can be transformed from coordinate system S1 to S2 by applying the following homogeneous
coordinate transformation matrix equation.
RðiÞ2 Z ½M21R ðiÞ
1 ; (37)
where
½M21 Z
cosðf1Cf2Þ sinðf1Cf2Þ 0 KE cos f2
Ksinðf1Cf2Þ cosðf1Cf2Þ 0 E sin f2
0 0 1 0 0 0 0 1 2 6 6 6 6 4 3 7 7 7 7 5; and E is the center distance between the shaper and the generated gear. The equation of meshing between the shaper and the generated gear is
X1ðiÞKxðiÞ1 nðiÞ1x Z Y1ðiÞKyðiÞ1 nðiÞ1y Z Z1ðiÞKzðiÞ1 nðiÞ1z ; (38) where X1ðiÞ, Y1ðiÞand Z1ðiÞare coordinates of the instantaneous contact point I, and xðiÞ1 , yðiÞ1 , zðiÞ1 and nðiÞ1x, nðiÞ1y, nðiÞ1z are the coordinates and the normal vector components of an instantaneous point on the shaper profile, respectively, represented in coordinate system S1. According to gear
theory, the profile of a generated gear can be obtained by simultaneously considering the equation of meshing and the loci of the shaper cutter represented in the gear’s coordinate system. Thus, the involute gear profile generated by the shaper (i.e. involute profile shown inFig. 7) is expressed as follows:
xð2Þ2 Z ðl cos jnKa0Cr1Þcos f2
Gða0Kl cos jnÞcot jnsin l sin f2KE cos f2;
yð2Þ2 ZKðl cos jnKa0Cr1Þsin f2Gða0Kl cos jnÞ
!cot jnsin l cos f2CE sin f2 and
zð2Þ2 ZGða0Kl cos jnÞcot jntan l sin l
G a0tan jnCb0Kl sin jn cos l Cr1m12f2tan l; ð39Þ where m12ZTT2 1Z f1 f2:
Similarly, the modified gear root fillet generated by the modified tip of the shaper (Fig. 7) can be represented as follows: xðtÞ2 Z ðxðJÞ1 Krtcos btÞcosðf1Cf2KhÞ Crtcos½srHðh K f1Kf2Þ K E cos f2; yðtÞ2 Z KðxðJÞ1 Krtcos btÞsinðf1Cf2KhÞ Hrtsin½srHðh K f1Kf2Þ C E sin f2; zðtÞ2 Z r1h tan l; (40) and f2ðtÞZ Kr1sinðstHhGf1Þ C ðx ðJÞ 1 Krtcos btÞsin stZ 0: (41) Similarly, the modified gear tip fillet generated by the modified root fillet of the shaper (Fig. 6) is as follows:
xðrÞ2 Z xðCÞ1 cosðf1Cf2KhÞGyðCÞ1 sinðf1Cf2KhÞ Crrsin½drG ðh K f1Kf2Þ
Krrsin½srG ðh K f1Kf2Þ K E cos f2;
yðrÞ2 Z KxðCÞ1 sinðf1Cf2KhÞGyðCÞ1 cosðf1Cf2KhÞ
Hrrcos½drG ðh K f1Kf2Þ
Grrcos½srG ðh K f1Kf2Þ C E sin f2;
zðrÞ2 Z r1h tan l;
(42) and
f2ðrÞZK r1cosðsrGhHf1Þ C xðCÞ1 cos srCyðCÞ1 sin sr
CrrsinðdrKsrÞ Z 0:
(43)
6. Designing a nonstandard gear generated by shapers The center distance of gear pairs depends on the profile-shifting coefficient. The total profile-profile-shifting coefficient of the gear pairs is eZepCeg when the profile-shifting
coefficients are ep and eg for the pinion and gear,
respectively. Thus, the operating center distance[16]is
E0Zðr1Cr2Þcos js cos jbs
; (44)
where jsis the transverse pressure angle, and the operating
pressure angle, jbs, is given by the following involute
function[16]. invjbsZ 2 tan js epCeg T1CT2 Cinvjs; (45)
where ep and eg represent the normal profile shifting
coefficients of the rack cutter and the generated gear, respectively, and T1and T2represent the number of teeth of
the pinion and gear. The involute function can be
alternatively expressed as follows[16].
invjbsZ tan jbsKjbs: (46)
The operating center distance can be obtained by substituting Eqs. (45) and (46) into Eq. (44). Notably, the operating center distance depends on the sum of profile shifting coefficients. Fig. 9 presents the profile shifting relationship of the pinion and the gear during their generation. The pinion is generated using a rack cutter with a positive profile shifting coefficient epZe1, and then
using the pinion as the shaper (i.e. the same pinion-type shaper) to generate the mating gear with a negative profile shifting coefficient e2. This generating procedure changes
the instantaneous center of rotation from point I to point I0, such that the profile shifting coefficient of the generated gear is egZKe1Ce2. Hence, the total profile shifting coefficient
of the mating gear pair is eZepCegZe2. Restated, the
change of the center distance depends only on the profile shifting coefficient e2. The change of center distance does
not impact the gear ratio m12, but does affect the radii of
pinion and gear centrodes. The new pitch radii of the pinion and the gear[16]are
rpZ E0 1 C m12 and rgZ E0 1 C 1=m12 : (47)
Example 2. Computer graphs of the pinion and gear with profile shifting coefficients.
Table 1lists the major parameters of the pinion and gear.
Table 1also shows all other corresponding design parameters
calculated according to the proposed tip and fillet modifi-cation method. Based on the developed pinion and gear mathematical models expressed in Eqs. (10), (33) and (39)–(47),Fig. 10displays the computer graphs of the pinion and the gear with profile shifting and modified tooth surfaces.
7. Discussion
The conventional method of applying positive profile-shifted cutting is widely used in industry to manufacture helical gears with small number of teeth, to avoid tooth undercutting. However, positive profile-shifted cutting causes the increase of gear fillet thickness as the number of teeth decreases. The increase of fillet thickness results in the decrease of gear addendum. This study proposes an alternative method that modifies the geometry of the fillet. A third method, combining the profile-shifted cutting and fillet modification, is also proposed, to yield an improved result. Fig. 11 compares the tooth profiles of the pinion obtained by applying the methods of the tooth-profile shifting and the combination of profile-shifted cutting and tooth fillet modification, respectively. The parameters are the same as those shown in Table 1. Differences between the tooth surfaces obtained by these two methods are observed.
Applying the positive profile-shifting coefficients with values of 0.3 and 0.5 reduces but does not eliminate the tooth undercutting problem, when the tooth-profile shifting method is applied. According toFig. 11, when the shifting coefficient equals 0.7, the tooth profiles generated by these two methods are very similar. When the shifting coefficient equals 0.9, the generated root fillets of the profile-shifted pinion are higher than those generated by the combination of profile shifting and tooth modification method. In other words, by applying the conventional profile-shifted method, an increase in the shifting coefficient leads to a decrease of the gear addendum, and thus reducing the gear contact ratio. Furthermore, the change of the center distance between the pinion and the gear depends on the profile shifting coefficient of the shaper when the gear is generated by shapers according to the proposed method. According to the simulated results, a combination of the tooth modification method and the tooth-profile shifting method can solve the tooth undercutting problem. Additionally, the clearance between the pinion and the gear can be determined during the gear generation process. According to the proposed method, the change of the center distance depends only on gear shifting coefficient when the gear is generated by shapers. Mathematical models developed in this study can be used in designing spur and helical gear sets with small number of teeth.Fig. 12displays the pinion and the gear that designed and manufactured by
Fig. 10. Computer graphs of the pinion and gear with profile shifting and modified tooth surface.
using the proposed method and the gear pair has already been used to a motor-driven wheelchair.
8. Conclusion
A mathematical model of the modified helical gear with small number of teeth has been developed by tooth-profile shifting and basic geometry modification. The condition of tooth undercutting for the involute profile gears has been investigated using the developed mathematical models. Computer graphs of the pinion and gear profiles generated by various methods are displayed for comparison.Figs. 10,
11 and 12have shown the verification and validation of the
proposed methods and their corresponding gear tooth mathematical models. The proposed methods and devel-oped mathematical models of the modified helical gear can be helpful to facilitate to design and manufacture of spur and helical gears with small number of teeth.
References
[1] E. Buckingham, Analytical Mechanics of Gears, Dover Publications Inc, New York, 1949.
[2] D.W. Dudley, Practical Gear Design, McGraw-Hill Book Co, New York, 1982.
[3] F.L. Litvin, C.B. Tsay, Helical gears with circular arc teeth: simulation of conditions of meshing and bearing contact, Transactions of the ASME, ASME Journal of Mechanisms Transmissions Automation Design 107 (1985) 556–564.
[4] F.L. Litvin, Methods for generation of gear tooth surface and basic principals of computer aided tooth contact analysis, Proceeding of Computers in Engineering 1 (1985) 556–564.
[5] J.R. Colbourne, The Geometry of Involute Gears, Springer, New York, 1987.
[6] AGMA, Information Sheet-Geometry Factors for Determining the Strength of Spur, Helical, Herringbone and Bevel Gear Teeth. AGMA, 226.01, 1970.
[7] AGMA, Design Guide for Vehicle Spur and Helical Gears. AGMA, 170.01, 1976.
[8] A. Ishibashi, H. Yoshino, I. Nakashima, Design and manufacturing processes and load carrying capacity of cylindrical gear pairs with 2 to 4 pinion teeth for high gear ratios (1st report design and manufacture and surface durability of gears with 2 to 3 pinion teeth), Transactions of the Japan Society of Mechanical Engineers, Series C 47 (416) (1981) 507–515.
[9] A. Ishibashi, H. Yoshino, Design, manufacture and load carrying capacity of Novikov gears with 3–5 pinion teeth for high gear ratios (1st report, design, manufacture and power transmission efficiency), Transactions of the Japan Society of Mechanical Engineers, Series C 49 (447) (1983) 2039–2047.
[10] T. Komori, Y. Ariga, S. Nagata, A new gears profile having zero relative curvature at many contact points (LogiX Tooth Profile), Transactions of the ASME, Journal of Mechanical Design 112 (1990) 430–436.
[11] M.A.S. Arikan, Determination of maximum possible contact ratios for spur gear drives with small number of teeth, Proceedings of the ASME Design Technical Engineering Conferences 82 (1) (1995) 569–576. [12] H.H. Mabie, C.F. Reinholtz, Mechanisms and Dynamics of
Machinery, 4th ed., Wiley, New York, 1987.
[13] V. Kin, Limitation of Worm and Worm gear surfaces in order to avoid undercutting, Gear Technology 1990; 33–35.
[14] Z.H. Fong, C.B. Tsay, The undercutting of circular-cut spiral bevel gears, Transactions of the ASME, ASME Journal of Mechanical Design 114 (1992) 317–325.
[15] F.L. Litvin, Theory of Gearing, NASA Publication RP-1212, Washington DC, 1989.
[16] F.L. Litvin, Gear Geometry and Applied Theory, Prentice-Hall, New Jersey, 1994.
[17] M.H. Tsai, Y.C. Tsai, Design of high-contact-ratio spur gears using quadratic parametric tooth profiles, Mechanism and Machine Theory 33 (5) (1998) 551–564.
