ORIGINAL ARTICLE
Design and manufacture of plunge shaving cutter
for shaving gears with tooth modifications
Jia-Hung Liu&Ching-Hua Hung&Shinn-Liang Chang
Received: 11 March 2008 / Accepted: 30 September 2008 / Published online: 29 October 2008
# Springer-Verlag London Limited 2008
Abstract Gear plunge shaving is one of the most efficient ways to finish gears. It has been a challenge for a long time to design and manufacture a plunge shaving cutter to induce gear tooth modification. This paper proposes a method for designing plunge shaving cutters to manufac-ture gears with modifications analytically. By integrating B-spline interpolation, differential geometry, and design optimization, the goal is achieved. Efficiency is greatly improved by avoiding the traditional trial and error method.
Keywords Plunge shaving cutter . Gear tooth modification . B-spline interpolation . Design optimization
1 Introduction
Gears are the most important components in transmission systems. Modifications of gear teeth can accommodate errors and deformations encountered in the manufacture, assembly, and operation of gear pairs. Litvin [1] provided a double-crowned gear modified both in lead and in
profile with improved transmission error. Wagaj and Kahraman [2] investigated the durability of helical gear affected by tooth modifications. Kahraman et al. [3] presented a gear wear model analyzing the influences of tooth modification.
Gear shaving is one of the most efficient and economical processes for gear finishing after hobbing or shaping. There are four basic shaving methods: Conventional/transverse, diagonal, and underpass shaving are composed of axial/ longitudinal and radial infeeds, and the parameters can be optimized for better performances [4]. Plunge shaving is the most advanced gear finishing technique, which only needs radial infeed. Its advantages include increased productivity, accuracy, long tool life, and a simple machine structure [5]. The basic meshing condition of 3D crossed-axis helical gear pair was first derived by Litvin [6], and it has been widely adopted as the fundamental assumption for simula-tion of gear shaving.
In recent years, how to induce gear tooth modification by shaving has become an important subject of research. Chang et al. [7] and Hung et al. [8] derived the mathematical model of traditional shaving machine and investigated the influences of parameters on tooth lead modifications. Litvin et al. [9] proposed a method for shaving gears with double crowning by computer numerical control (CNC) shaving machine. For shaving methods other than plunge shaving, the gear tooth modification is accomplished by tooth modifications of the shaving cutter and the coordinated motions between cutter and gear. For the plunge shaving method, however, it only depends on the surface geometry of the plunge shaving cutter. Tradi-tionally, the cutter surface geometry results from a cutter re-sharpening machine by trial and error.
Focusing on the surface geometry of shaving cutter, what is really significant is precision of the region DOI 10.1007/s00170-008-1783-z
J.-H. Liu
:
C.-H. HungMechanical Engineering Department, National Chiao Tung University, EE507, 1001 Ta Hsueh Rd.,
Hsinchu 30010 Taiwan, Republic of China S.-L. Chang (*)
Institute of Mechanical and Electro-mechanical Engineering, National Formosa University,
Room 529, Building 2, The New Campus, No.64, Wunhua Rd., Huwei Township,
Yunlin County 632 Taiwan, Republic of China e-mail: [email protected]
between start of active profile (SAP) and end of active profile (EAP) as shown in Fig.1. In gear shaving, EAP of cutter tooth shaves the gear tooth tip, while SAP shaves the gear tooth root. Koga et al. [10] explored the errors in the cutter’s tooth profile that resulted from the grinding machine setup. Hsu [11] investigated the topographic error of cutters from a re-sharpening machine for shaving involute gears. Radzevich [12–13] proposed a methodol-ogy for calculating the parameters of the plunge shaving cutter and the respective form grinding wheel from measured discrete points of the modified gear. Neverthe-less, without an analytical description of the modified gear
and the plunge shaving cutter, further investigations are still limited.
Numerical approximation and interpolation are com-monly used to construct an analytical description of discrete data points. Piegl and Tiller [14] provided basic theories for numerical approximations and interpolations. Wang et al. [15] studied the geometric relationship and conjugate motion of digital gear tooth surface composed of discrete points. Barone [16] used B-spline curve fitting and swept surface to model gears with profile modifications. In this paper, the analytical description of the gear with tooth modifications is first constructed by B-spline surface fitting. Then, the grinding wheel profile is parameterized and optimized for minimizing the surface deviations of theoret-ical and ground (from re-sharpening machine) tooth surfaces of the plunge shaving cutter. Efficiency is greatly improved by avoiding the traditional trial and error method, and the constructed mathematical model of plunge shaving cutter can be further utilized as the base for extending researches.
2 Surface interpolation of the modified gear tooth surface
To integrate the modified gear tooth surfaces into the analytical process, especially for those with both lead and profile modifications, B-spline surface interpolation is Fig. 1 SAP and EAP of a shaving cutter tooth
Fig. 2 Model of gear tooth modifications [2]
selected for its ease of manipulation. In practice, the sampling points of the modified surface can be obtained by coordinate measuring machine or from other sources. In this paper, for studying purposes, the most commonly used numerical model is adopted for generation of interpolating points: The tooth flank is modified in profile (root to tip) and lead (side to side) directions independently as shown in Fig. 2 [2]. The magnitude at the tip at and the gear roll
angle at the startαtdefine the boundaries of the tip relief.
Between these two points, the profile follows a linear trajectory. Similarly, the magnitude ar and the starting roll
angle αr define the starting point of a root modification.
The amount of lead crowning is denoted by h.
A B-spline representation enables the simulation of surface irregularities and the control of small tooth
geometric modifications, such as rounding and reliving. Given a grid of sampling points Dk;‘ (0≤k≤m and 0≤‘≤n) and orders p and q (degrees p−1 and q−1), it can be represented as below [14]: Dk;‘¼ Xm i¼0 Xn j¼0 Ni;pð ÞNsk j;qð ÞPt‘ i;j ð1Þ
where sk’s and tl’s are the chosen parameter values; Ni,p(sk)
(Nj,q(t‘)) is the i-th (j-th) B-spline basis function of order p
(q); and pi,j’s (0≤i≤m and 0≤j≤n) are the control points.
Once the numbers of sampling points m and n are selected, then the B-spline orders p and q are limited by Eq.2: p3mþ 1
mþ 1 ; q
3nþ 1
nþ 1 ð2Þ
that is, p≥3 and q≥3. Table 1 Basic data of the target surfaces
Gear data
Diameter of base circle db2 119.618 mm
Diameter of addendum circle dadd2 126.71 mm
Diameter of root circle dr2 119.33 mm
Diameter of pitch circle dp2 123.915 mm
Normal pressure angle in pitch circleαpn2 14.5°
Face width fw2 18 mm
Gear tooth number Z2 79
Helix angle in pitch circleβp2 17°
Normal circular tooth thickness spn2 2.32mm
Table 2 Data of gear tooth modifications of the target surfaces
Parameter ΣS2 ΣL2 ΣD2 at 0 0 6e−3 mm αt N/A N/A 31.8° ar 0 0 6e−3 mm αr N/A N/A 28.2° h 0 6e−3 mm 6e−3 mm
Fig. 3 Flowchart of interpola-tion error analysis
Solving Eq.1, Pi,j’s can be obtained, and the interpolated
B-spline surfacePI
2 can be represented as follows:
XI 2ðu; vÞ ¼ Xm i¼0 Xn j¼0 Ni;pð ÞNu j;qð ÞPv i;j ð3Þ
where u and v denote the surface parameters; 2 denotes the surface of gear tooth; and I denotes the surface obtained by interpolation.
B-spline interpolation can be implemented using the MATLAB Spline Toolbox by specifying a set of data points and either knot sequences or orders of the interpolated surface. The process of obtaining the B-spline surface is illustrated in Fig. 3. For example, to interpolate the gear tooth surfaces described in Tables1and 2, including three target gear tooth surfaces (standard PS2, lead crowned
P
L2, and double crowned
P
D2), three sets of sampling
points must be acquired from the model shown in Fig.2. In order to obtain gridded data points for interpolations, the target surfaces are sampled in the cylindrical coordinate system (represented by R2,θ2, and Z2) for the nature of the
cylindrical gear, as shown in Fig.4.
By specifying the number of sampling points m and n in the R2and Z2coordinates as well as the respective orders p
and q, three interpolated tooth surfaces can be expressed as PI S2ðR2; Z2Þ, PI L2ðR2; Z2Þ, and PI D2ðR2; Z2Þ. Three sets
of data points (30×30 in R2and Z2coordinates for each set)
are also sampled from the three target surfaces for error analysis. The interpolation error EIis defined as a 30×30 matrix composed of values of arc length between the respective points (with the same values of radius and at the same axial cross-section) uniformly sampled from the target surfaceP2and the interpolated surface
PI
2. Each element
eI
i;jof EIcan be expressed as:
eIi;j¼ ri;j tan1 yi;j xi;j tan1 yIi;j xI i;j !! ; i; j ¼ 1; 2; . . . ; 30 ð4Þ where ri,j denotes the value of radius; xi,j, yi,j, and xIi;j; yIi;j
denote the points on P2 and
PI
2 , respectively, and the
maximum eI
max and the mean eImean can be obtained.
Based on the error analysis in this section, EI is not sensitive at all to the parameters in the Z2 coordinates,
including n and q. Even with lead modifications, the non-linearity remains quite small compared with that caused by profile modifications. Table 3 presents six conditions of PI
S2. Increasing m and p improve the values of eImean and
Fig. 4 Cylindrical coordinate used for sampling data points
Table 3 Conditions and errors of B-spline surface interpolationPIS2
Parameter Condition 1 2 3 4 5 6 m (R2coordinate) 3 4 4 5 5 5 n (Z2coordinate) 3 3 3 3 3 3 p (R2coordinate) 3 3 4 3 4 4 q (Z2coordinate) 3 3 3 3 3 4 eI meanð103mmÞ 9.805 1.721 1.193 0.645 0.185 0.185 eI maxð103mmÞ 15.6745 4.55 1.795 1.8539 0.0679 0.0679
Fig. 5 Interpolation error ofPI
eI
max, which are commonly required to be less than
10−3 mm. Although the errors are the same in conditions
5 and 6, the parameters in condition 6 are preferable for higher differentiability. Errors of surfacePI
L2are similar to
that of PI
L2. Nevertheless, for surface
PI
D2, under
conditions similar to condition 6, m must increase to 14, as shown in Fig.5, because of higher non-linearity due to profile modifications.
The interpolated surfaces PIL2and PID2 are compared with PIS2 for validations as shown in the topographic charts (Figs.6 and7). The topographic errors between top and root on the left and right tooth flanks are calculated for
points of specific radiuses and Z cross-sections. PI S2 is
presented in straight solid lines as the base of comparison, while PI
L2 and
PI
D2 are presented in dashed lines. In
Fig. 6,PI
L2 is shown with lead modification only, and in
Fig. 7, PI
D2 is shown with both lead and profile
modifications. Note that the values of the topographic differences are all represented in arc length.
3 Topographic errors of gear plunge shaving cutter The tooth surface of the shaving cutter is usually finished last using a cone-grinding wheel on the shaving cutter re-sharpening machine. Because the topographic accuracy of the plunge shaving cutter maps directly onto the work gear, it is important to identify the topographic error of the ground tooth surfacesPG
S1,
PG
L1, and
PG
D1in comparison
to the theoreticalPTS1,PTL1, andPTD1. Here, 1 denotes the surface of shaving cutter tooth; G and T denote the surfaces derived from the re-sharpening machine and the interpolat-ed surfaces, respectively.
The basic meshing condition for the crossed helical gear set [6] is used to calculate the basic geometric data for the shaving cutter. It needs the following eight basic items: the tooth number Z1, the normal circular tooth thickness spn1,
the helix angleβp1of the shaving cutter; the tooth number
Z2, the normal circular tooth thickness spn2, the helix angle
βp2 of the work gear; and the normal module mpn and
pressure angle αpn of the shaving cutter and work gear.
Figure8shows the coordinate system of the CNC shaving machine [11]. Considering coordinate transformation
r1¼ x½ 1 y1 z1 1T ¼ M12ð Þf2 r2ðR2; Z2Þ n1¼ nx1 ny1 nz1
T
¼ L12ð Þf2 n2ðR2; Z2Þ
ð5Þ and meshing equation
f Rð 2; Z2; f2Þ ¼nhvð Þh12 ¼ 0 ð6Þ
simultaneously, the theoretical tooth surface of shaving cutters PT
S1 ,
PT
L1 , and
PT
D1 can be derived from
surfacesPI S2,
PI
L2, and
PI
D2. M12and L12are matrices
for transforming position and unit normal vectors (r2 and
n2) of surfaces PT S1 , PT L1 , and PT D1 from coordinate
system S2(gear) to S1(shaving cutter), and v 12 ð Þ
h denotes the
vector of relative velocity on auxiliary coordinate system Sh.
Likewise, as shown in Fig. 9, the ground surfaces of shaving cutters PG S1, PG L1, and PG D1 can be obtained by
coordinate transformations from Sg (grinding wheel) to Ss
(shaving cutter) rs¼ x½ s ys zs 1T ¼ Msgrg ns¼ nxs nys nzs T ¼ Lsgng ð7Þ Fig. 6 Validations of interpolated surfaces (PIS2vs.
PI
L2)
Fig. 7 Validations of interpolated surfaces (PI S2vs.
PI
and the meshing equation g ug; qg; f
¼nmvð Þmsg ¼ 0 ð8Þ
An example is provided for illustration.
Example 1 For interpolated gear tooth surfaces PI S2 and
PI
D2(see Tables1and2), the corresponding data of plunge
shaving cutter and grinding wheel are presented in Tables4. Obtaining theoretical tooth surfacesPT
S1and
PT
D1 as well
as ground tooth surfacesPGS1andPGD1through Eqs.5,6,7, and8, the topographic errors are calculated by
eTopoi;j ¼ ri;j tan1 y
T i;j xT i;j ! tan1 yGi;j xG i;j !! ; i¼ 1; 2 . . . 5; j ¼ 1; 2; . . . ; 9 ð9Þ
and illustrated in Figs.10and11by 5×9 grids of specified radiuses and Z cross-sections, in which theoretical surfaces are presented in straight solid lines, while the corresponding ground ones are shown in dashed lines. From SAP to EAP of the cutter tooth, errors between PT
D1 and
PG
D1
(Fig. 11) are less than those between PT S1 and
PG
S1
(Fig.10). This is because of the nature of a grinding wheel with cone angle; that is, it presents a parabolic shape on the pitch line of a shaving cutter tooth, so that a specific amount of lead modification can be compensated.
4 Design optimization of the cone-grinding wheel Compared with the setting parameters of the shaving cutter re-sharpening machine, the cone angle θc of the grinding
wheel results in more influences on topographic errors of the final product. Traditionally, θc is modified back and
forth for the desired accuracy, which is time-consuming. In this section, two examples are provided for illustrating design optimization by adjusting θc (Example 2) and the
profile of the grinding wheel (Example 3).
Example 2 The objective of this example is to minimize the topographic errors between the theoretical tooth surface and the ground one. The flowchart is shown in Fig.12, and the process is integrated by the MATLAB Optimization Toolbox. The problem is formulated as:
Fig. 8 Coordinate systems of gear shaving machine [10]
findθcthat minimizesP 5 i¼1 P9 j¼1e Topo i;j ð Þqc
subject to eTopoi;j < 103mm, i=1,2,...,5, j=1,2,...,9, and 1°≤θc≤30°.
Considering 45 inequality constraints and one boundary constraint, the design variableθc is modified iteratively to
obtain the optimum of topographic error. The process of
calculating eTopoi;j based onθcis considered as eTopoi;j ð Þ, andqc
the sequential quadratic programming algorithm is adopted, where information of finite difference is used instead of gradients. For standard surfaces PT
S1 and
PG
S1 shown in
Fig. 13, the topographic errors are minimized when θc is
modified from 10° (initial value) to 1° (optimum value),
Fig. 10 Topographic errors between theoretical and ground shaving cutter tooth surfaces (PT
S1vs.
PG
S1)
Table 4 Data of cutter and grinding wheel for Example 1 Plunge shaving cutter
Normal circular tooth thickness spn1 2.464 mm
Tooth number Z1 139
Helix angle in pitch circleβp1 20°
Face width fw1 20 mm
Diameter of start of active profile (SAP) 225.922 mm Diameter of end of active profile (EAP) 219.537 mm
Operating center distance Eo 173.04 mm
Operating crossed angleγo 3.002°
Grinding wheel and grinding machine
Operating cone pitch radius Rc 350 mm
Cone angleθc 10°
Pressure angleα 4.675°
Operating radius ro 111.03 mm
which is active for the boundary constraint 1°≤θc≤30°. In
practice, it is very difficult to make a grinding wheel with cone angle less than 1° so that the topographic error cannot be improved any more by only adjustingθc. For
double-crowned tooth surfacesPT D1 and
PG
D1, the errors are still
large at SAP and EAP (Fig. 14) due to profile modifica-tions. To eliminate the errors, the profile of the grinding wheel also needs to be modified.
Example 3 Based on Fig.9, the profile of grinding wheel is parameterized as shown in Fig.15. Coordinate system Sg0 is attached to the unmodified profile. Within the effective length LA+LA(LA=4.2 mm, LB=5.5 mm), the profile with
four sections are defined by wA, hA, wB, and hB. Represented
in S0g, the four points A2, A1, B1, and B2are fitted by a
B-spline curve with order 4. To improve the topographic errors between PTD1 and PGD1, the process of optimization is divided into two levels. The problem formulation for level 1: findθc
that minimizesP
9 j¼1e
Topo
j ð Þ on the pitch line of tooth surfaceqc
subject to eTopoj < 103mm, j=1,2,…, 9 on the pitch line of tooth surface,
and 0.5°≤θc≤30°.
Fig. 14 Topographic errors between theoretical and ground shaving cutter tooth surfaces optimized byθc(
PT
D1vs.
PG
D1)
Fig. 13 Topographic errors between theoretical and ground shaving cutter tooth surfaces optimized byθc(
PT
S1vs.
PG
S1)
Fig. 12 Flowchart of design optimizations of the cone grinding wheel Fig. 11 Topographic errors between theoretical and ground shaving cutter tooth surfaces (PT
D1 vs.
PG
The first level converges efficiently toθc=2.382° and the
problem formulation of level 2: find x=[wA, hA, wB, hB] that minimizesP 5 i¼1 P9 j¼1e Topo i;j ð Þx
subject to eTopoi;j < 103mm, i=1,2,…,5, j=1,2, …, 9 and 10−14<wA<LA−10−4;0<hA<3;10−4<wB<LB−10−4;0<
hB<3 (unit, mm).
Following the similar concepts of programming in Example 2, the optimum design is presented in Table 5.
The profile of the grinding wheel is considered straight sided initially. When it reaches optimum, the profile is modified for conjugation to the shaving cutter. The topographic errors betweenPTD1 andPGD1 are shown in Fig.16, where the errors are all controlled below 10−3mm.
Experiment of Example 3 has been conducted for validation of the proposed method. Figure 17 shows the plunge shaving cutter with an enlarged view of the cutting edges. Figure 18 shows the pre-shaved gear with an enlarged view of the obviously scalloped tooth surfaces measured as shown in Fig.19. Firstly, the shaving cutter is ground by the grinding wheel on the re-sharpening machine (Fig.20), on which the grinding wheel are modified first by the dresser according to the calculated cone angle and profile parameters. Then, the gear is plunge shaved on NACHI shaving machine with the setup shown in Fig. 21. Materials of gear and cutter are SCM435 and M2, and the operating speed as well as plunge infeed are set as 150 rpm and 1 mm/min, respectively. The shaved gear is measured as shown in Fig. 22, and the mean values of tooth modification are recorded in Table 6. It is found that: 1. Most of the scallops are eliminated and the surface
roughness is greatly improved, especially in the lead direction;
2. The amounts of modifications, though with little deviations, are close to the original design values;
Fig. 17 Plunge shaving cutter used in the experiment Fig. 16 Topographic errors between theoretical and ground shaving
cutter tooth surfaces by profile optimization (PT D1vs.
PG
D1)
Fig. 15 Parameterized profile of grinding wheel for optimization
Table 5 Results of the second level optimization in Example 3 Initial design Optimum design (mm)
wA 0.1 0.561
hA 0 0.056
wB 0.1 0.538
3. Larger modifications are induced on left flank com-pared with right flank; this is because the left flank is the driving one in shaving with larger cutting force; 4. Efficiency is greatly improved by adopting the
pro-posed method instead of trial and error.
5 Conclusions
Design and manufacture of the plunge-typed gear shaving cutters has always been a challenge, especially for those
used to manufacture gears with modifications. This paper proposes a method for design and manufacture of the plunge shaving cutter for gears with modifications analyt-ically, rather than trial and error. By integrating B-spline interpolation, differential geometry, and design
optimiza-Fig. 21 NACHI shaving machine used in the experiment
Fig. 20 Plunge shaving cutter ground by the grinding wheel of the re-sharpening machine
Fig. 19 Measured data of the pre-shaved gear used in the experiment Fig. 18 Pre-shaved gear used in the experiment
tion, the goal is achieved. To interpolate gears with both lead and profile modifications (double crowned), more sampling points are needed in the radial direction. In manufacturing a shaving cutter, the lead modification can be compensated by adjusting the cone angle, and the profile modification can be implemented by modifying the profile of the grinding wheel. Efficiency is greatly improved by avoiding traditional trial and error method through the proposed one. Besides, an analytical description of the modified gear tooth surface is also constructed, which can be utilized for extending research on serrations and shaving process.
Acknowledgment The authors wish to acknowledge the help from Luren Precision Co., Ltd. for providing all the necessities for the experiments.
References
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Fig. 22 Measured data of the shaved gear after the experiment
Table 6 Achieved tooth modifications of gear after the experiment for validating Example 3
Left flank (10−3mm) Right flank (10−3mm)
Design Real Design Real
Profile
Tip 6 6.9 6 5.2
Root 6 6.1 6 5.1
![Fig. 2 Model of gear tooth modifications [ 2 ]](https://thumb-ap.123doks.com/thumbv2/9libinfo/7571064.125218/2.892.113.395.84.315/fig-model-gear-tooth-modifications.webp)
![Fig. 8 Coordinate systems of gear shaving machine [ 10 ]](https://thumb-ap.123doks.com/thumbv2/9libinfo/7571064.125218/6.892.350.814.80.596/fig-coordinate-systems-gear-shaving-machine.webp)
