study on the cutting time of the hypoid gear tooth flank

Mechanism and Machine Theory xxx (2014) xxx–xxx

MAMT-02255; No of Pages 12

Contents lists available at ScienceDirect

Mechanism and Machine Theory

j ourna l homepage: www.e lsev ie r .com/ locate /mechmt

Study on the cutting time of the hypoid gear tooth flank

Szu-Hung Chen, Zhang-Hua Fong⁎Department of Mechanical Engineering, National Chung-Cheng University, Taiwan, ROC

a r t i c l e i n f o

⁎ Corresponding author at: No. 168, University RdE-mail address: [email protected] (Z.-H. Fong).

0094-114X/$ – see front matter © 2014 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.mechmachtheory.2014.01.

Please cite this article as: S.-H. Chen, Z.-H.(2014), http://dx.doi.org/10.1016/j.mechm

a b s t r a c t

Article history:Received 11 January 2013Received in revised form 14 January 2014Accepted 16 January 2014Available online xxxx

Hypoid-gear cutting is a highly specified and complicated metal cutting process. The degree offreedom (DOF) of a six-axis computer numerical control (CNC) machine tools is sufficiently highto emulate the complex generating motion of conventional hypoid-gear cutting. This studypresents a proposal for a constant material removal rate (CMRR) cutting process for hypoid gearswith the aid of CNC flexibility. The relative spatial relationship between the cutter and theworkpiece is kept strictly identical to the machine settings of the conventional hypoid generator.A hypoid gear cutting plan is developed to implement the CMRR concept by varying the feed rateand cradle roll velocity (CRV) to retain the cutting torquewithin a certain range during the cuttingprocess. A CRV function can be numerically obtained to provide a material-removal polynomial(MRP) curve. The validation of the proposed CMRRprocess is experimentally verified on a six-axisCNC hypoid-gear generator bymeasuring the electrical current of the cutter spindlemotor, whichis proportional to the cutting torque. Machining time is reduced to 38%, and themaximum cuttingtorques is not exceeded.

© 2014 Elsevier Ltd. All rights reserved.

Keywords:Hypoid gearSpiral bevel gear6-Axis CNC machine toolSolidworks APIConstant material removal rate

1. Introduction

Spiral bevel and hypoid gears are extensively used in power transmission between intersecting and crossed axes, respectively.Spiral bevel and hypoid-gear cutting is a highly specified metal cutting process with a demanding accuracy requirement. Thestructure of modern hypoid-gear generators is essentially a six-axis computer numerical control (CNC) machine tool with aservo-controlled cutting spindle, i.e., a six-axis machine tool. The degree of freedom (DOF) of CNC machines is sufficiently high toemulate the complex generating motion of conventional hypoid-gear cutting. However, CNC flexibility of the six-axis machineprovides more control possibilities in implementing generating and corrective motions.

The cutting force on the conventional hypoid generator is governed by the cradle generation-roll velocity and feed-in of thesliding base. The typical machining process of hypoid generators begins with a plunge cutting as roughing and is followed by theup and down cradle roll for semi-finishing and finishing-generation cutting with a constant cradle roll velocity (CRV). A smallsliding base feed-in is between the semi-finishing cutting and finishing cutting for better tooth surface roughness. During theroughing cutting, the cutting torque is primarily dependent on the feed-in velocity of plunge cutting. During the semi-finishingand finishing cutting, the cutting torque relies on the CRV and the amount of sliding base feed-in. The cutting torque is inconstanton the conventional hypoid generator with constant plunge and CRVs. A constant material removal rate (CMRR) cutting process isproposed to maximize the cutting efficiency for hypoid gears with the aid of CNC flexibility.

The mathematical model of the universal face-milling hypoid gear generator for spiral bevel and hypoid gears has beenreported by Fong and Shih [1, 2]. The relative position and orientation between the head cutter and workpiece can be determinedusing these mathematical models and corresponding machine settings. The plunge cutting and generating plan for thecradle-type hypoid generator developed by Gleason Works [3] are nearly a standard practice for gear manufacturers. Thecradle-rotation velocity relies on the roll ratio between the virtual generating gear and the workpiece, and is typically maintained

., Min-Hsiung, Chia-Yi 621, Taiwan ROC. Tel.: +886 52720411x33303; fax: +886 5 2720589.

ll rights reserved.012

Fong, Study on the cutting time of the hypoid gear tooth flank, Mech. Mach. Theoryachtheory.2014.01.012

http://dx.doi.org/10.1016/j.mechmachtheory.2014.01.012

mailto:[email protected]


http://www.sciencedirect.com/science/journal/0094114X


2 S.-H. Chen, Z.-H. Fong / Mechanism and Machine Theory xxx (2014) xxx–xxx

as a constant during generation cutting. However, Ko and Kim [4] proposed an adaptive feed-rate control algorithm for theball-milling process with a NURBS interpolator that considered the CMRR, and which results in a constant cutting force in thecutting tool and workpiece. This CMRR concept is not easy to implement in hypoid-gear cutting because of the relatively complexcutting motion. The cutting simulation and crash examination for face-milling spiral-bevel gears on a general six-axis CNC havebeen developed under VERICUT by Xu Yanwei et al. [5] and Sun Yin-Sheng et al. [6]. The relationship between chip creation andthe force on the tool was studied by Christian Brecher and Fritz Klocke et al. [7]. They provide an analysis of the cutting force onthe cutting edge of the tool and rolling feed.

A hypoid gear-cutting plan is proposed to implement the CMRR concept by varying the feed rate and CRV to maintain thecutting torque within a certain range during the cutting process. Because of machine stiffness, the relative forced deflectionbetween the head cutter and the workpiece is typically proportional to the cutting torque. The machined tooth-surface accuracycan be improved by limiting the maximum cutting torque. The cycle time is reduced by maintaining the material removal rate inthe high range for as long as possible while the machined tooth surface is within the accuracy requirement. The relative spatialposition relationship between the cutter and workpiece is kept strictly identical as defined by the machine settings of theconventional hypoid generator.

With the aid of SolidWork© API functions, the total removal volume and corresponding cradle angle are fitted to a polynomialcurve called the material-removal polynomial (MRP) curve. The material-removal rate is obtained by differentiating the MRPcurve with the virtual cradle roll increment. By adjusting the CRV and sliding base feed-in, a CRV function can be numericallyobtained to provide a stepwise MRP curve during the generation-cutting process. The maximum allowable cutting torques forroughing and finishing cutting within the range of tooth surface accuracy and the corresponding power limitation of the spindlemotor are experimentally obtained on the experimental cutting machine. Once the maximum allowable cutting torque is defined,the optimized CRV function is calculated by averaging the MRP with the threshold values corresponding to the maximum cuttingtorques. The validation of the proposed CMRR process is experimentally verified on a six-axis CNC hypoid gear generator bymeasuring the electrical current of the cutter-spindle motor, which is proportional to the cutting torque. The cutting time isreduced to 40%, and the maximum allowable cutting torques is not exceeded.

2. Conversion of six-axis machine settings from the cradle-type hypoid generator

The mathematical model [1] of a cradle-type hypoid gear generator for spiral and hypoid gears is adopted, which is compatibleto the most traditional cradle-type hypoid machines, as shown in Fig. 1. The coordinate systems St and S1 are rigidly connected tothe cutter and the workpiece, respectively. The homogenous transformation matrices, as shown in Eq. (1), define the relativeposition of the head cutter with respect to the workpiece. The matrix is composed of universal machine settings and generatingmotion parameters, such as the cradle angle ϕc and the workpiece rotation angle ϕ1.

where

Pleas(201

M1t ϕ1ð Þ ¼ M17M76M65M54M43M32M2t

¼a11 ϕcð Þ a12 ϕcð Þ a13 ϕcð Þ a14 ϕcð Þa21 ϕcð Þ a22 ϕcð Þ a23 ϕcð Þ a24 ϕcð Þa31 ϕcð Þ a32 ϕcð Þ a33 ϕcð Þ a34 ϕcð Þ

0 0 0 1

2664

3775

ð1Þ

a11 ¼ − cosμg cosi cosγm sin j−qð Þ þ sini sinγmð Þ− cos j−qð Þ cosγm sinμg

Fig. 1. Coordinate systems for the cradle-type hypoid generator.

e cite this article as: S.-H. Chen, Z.-H. Fong, Study on the cutting time of the hypoid gear tooth flank, Mech. Mach. Theory4), http://dx.doi.org/10.1016/j.mechmachtheory.2014.01.012


where

3S.-H. Chen, Z.-H. Fong / Mechanism and Machine Theory xxx (2014) xxx–xxx

Pleas(201

a12 ¼ − cos j−qð Þ cosγm cosμg þ cosi cosγm sin j−qð Þ þ sini sinγm

�sinμg

�

a13 ¼ − cosγm sini sin j−qð Þ þ cosi sinγm

a14 ¼ −ΔAþ SR cosq cosγm−ΔB sinγm

a21 ¼ cosμgð cosi cos j−qð Þ cosϕ1 þ ð cosγm sini‐ cosi sin j−qð Þ sinγmÞ sinϕ1Þ− cosϕ1 sin j−qð Þ þ cos j−qð Þ sinγm sinϕ1

�sinμg

�

a22 ¼ ‐ cosμg cosϕ1 sin j−qð Þ þ cos j−qð Þ sinγm sinϕ1ð Þ− cosγm sini‐ cosi sin j−qð Þ sinγmð Þ sinϕ1− cosϕ1 sin j−qð Þ þ cos j−qð Þ sinγm sinϕ1ð Þ sinμg

a23 ¼ cos j−qð Þ cosϕ1 sini−ð cosi cosγm þ sini sin j−qð Þ sinγmÞ sinϕ1

a24 ¼ SR cosq sinγm sinϕ1 þ cosϕ1 Em−SR sinqð Þ þ ΔB cosγm sinϕ1

a31 ¼ cosμgð− cosγm cosϕ1 siniþ cosið cosϕ1 sin j−qð Þ sinγm þ cos j−qð Þ sinϕ1ÞÞþ cos j−qð Þ cosϕ1 sinγm− sin j−qð Þ sinϕ1Þ sinμg

�

a32 ¼ cosγm cosϕ1 sini sinμg− sin j−qð Þð cosμg sinϕ1 þ cosi cosϕ1 sinγm sinμgÞ þ cos j−qð Þð cosμg cosϕ1− cosi sinϕ1 sinμg

a33 ¼ cosϕ1 cosi cosγm þ sini sin j−qð Þ sinγmð Þ þ cos j−qð Þ sini sinϕ1

a34 ¼ sinϕ1 Em−SR sinqð Þ− cosϕ1 SR cosq sinγm þ ΔB cosγmð Þ

A mathematical model of a six-axis Cartesian-type CNC hypoid gear generator is presented in [2]. Fig. 2 shows the coordinatesystems for the six-axis Cartesian-type CNC machine. The coordinate systems Sts and S1

s are rigidly connected to the cutter and theworkpiece, respectively, and the transformation matrices from St

s to S1s define the spatial relationship between the cutter and the

workpiece in the coordinate system S1s , as shown in Eq. (2).

Mspt ¼ Ms

pdMsdeM

semM

smhM

sht ¼

S11 S12 S13 S14S21 S22 S23 S24S31 S32 S33 S340 0 0 1

2664

3775 ð2Þ

S11 ¼ cosϕSB cosϕ

SC

S12 ¼ cosϕSB cosϕ

SC

S13 ¼ sinϕSB

S14 ¼ Mt þ Ct þ Tt þ cosϕSBΔ

SX− sinϕS

CΔSZ

Fig. 2. Six-axis Cartesian-type CNC hypoid gear generator.




Pleas(201

S21 ¼ − cosϕSC sinϕ

SAsinϕ

SB− cosϕS

AsinϕSC

S22 ¼ − cosϕSA cosϕ

SC− sinϕS

AsinϕSBsinϕ

SC

S23 ¼ cosϕSB sinϕ

SA

S24 ¼ cosϕSA− sinϕS

A sinϕSBΔ

SX þ cosϕS

BΔSZ−Et

� �

S31 ¼ − cosϕSA cosϕ

SC sinϕ

SB þ sinϕS

AsinϕSC

S32 ¼ − cosϕSC sinϕ

SA− cosϕS

AsinϕSBsinϕ

SC

S33 ¼ cosϕSA sinϕ

SB

S34 ¼ − sinϕSAΔ

SY− cosϕS

A sinϕSBΔ

SX þ cosϕS

BΔSX−Et

� �

Fig. 3. The flowchart diagram of the cutting simulation.



Fig. 4. Double roll-mill generated cycle [8].


The relative spatial position of the cutter and the workpiece should be identical, irrespective of whether the work gear isproduced on a cradle-type or a Cartesian-type generator. Each element of the transformation matrix between the head cutter andworkpiece should be the same on both generators. The displacements and rotation angle of the six-axis is calculated bycomparing the transformation matrix in Eqs. (3) and (4), as shown below.

Pleas(201

M1t ϕ1ð Þ ¼ M1t i; j; θc; SR; Em;ΔA;ΔB;γm;ϕcð Þ ð3Þ

MSpt ¼ M1t ϕ1ð Þ ¼

a11 ϕcð Þ a12 ϕcð Þ a13 ϕcð Þ a14 ϕcð Þa21 ϕcð Þ a22 ϕcð Þ a23 ϕcð Þ a24 ϕcð Þa31 ϕcð Þ a32 ϕcð Þ a33 ϕcð Þ a34 ϕcð Þ

0 0 0 1

2664

3775 ð4Þ

The motion of each axis of the Cartesian-type CNC machine can be expressed as a function of the machine settings of thecradle-type generator, as follows:

ϕSA ¼ tan−1 a23 ϕcð Þ

a33 ϕcð Þ ð5Þ

Fig. 5. Material-removal polynomial curves.



Fig. 6. Roll rates and material-removal rates.


Pleas(201

ϕSB ¼ sin−1a13 ϕcð Þ ð6Þ

ϕSC ¼ tan−1 a12 ϕcð Þ

a11 ϕcð Þ ð7Þ

ΔSX ¼ a14 ϕcð Þ−Ct−Mt−Ttð Þ cosϕS

B þ Et−a34 ϕcð ÞcϕSA−a24 ϕcð ÞsϕS

A

� �sinϕS

B ð8Þ

ΔSY ¼ a24 ϕcð Þ cosϕS

A−a34 ϕcð Þ sinϕSA ð9Þ

ΔSZ ¼ Et−a34 ϕcð Þ cosϕS

A−a24 ϕcð Þ sinϕSA

� �cosϕS

B þ −a14 ϕcð Þ þ Ct þMt þ Ttð Þ sinϕSB ð10Þ

Displacement of each axis of the Cartesian-type hypoid generator is then calculated from the start cradle roll angle to the endcradle roll angle with each given increment of the cradle roll.

3. Cutting simulation on the SolidWorks API

We developed computer software to simulate the hypoid-gear cutting generated on a universal cradle-type hypoid generatorwith the aid of SolidWorks® API. The flowchart of the software is shown in Fig. 3. First, the solid models of the workpiece blankand the head cutter are generated automatically according to the given gear and cutter data. The spatial relationship between theworkpiece and the head cutter is calculated according to the transfer functions defined in Eqs. (5) through (10) from the start rollposition to the end roll position for each given cradle roll angle increment. The overlap portion of the workpiece and cutter isremoved sequentially from the workpiece by the SolidWorks® API substrate Boolean function; the total overlap volume isrecorded corresponding to the sliding base and cradle roll position.

After completing the Boolean substrate operation, the tooth form is shaped on the remaining solid modeling of the workpiece.The basic mathematics of this software relies on the correct spatial relationship between the workpiece and head cutter andBoolean substrate operation of solid modeling. These mathematical operations are time consuming but simple, and can be visuallyinspected on the monitor. They are numerically stable, compared to solving simultaneous system equations of meshing. The

Fig. 7. The CRV and corresponding MRR.



Fig. 8. Cycle time for roughing.


developed software is suitable for simulating the actual cutting hypoid-gear process, and for verifying the accuracy of the machinesettings and the conversion from universal cradle-type hypoid generators to general-purpose Cartesian-type six-axis CNCmachine tools.

The total removal volume from start position to the current sliding base and cradle roll position ϕc is recorded and fitted as anMRP curve. The instantaneous material-removal rate is obtained by differentiating the MRP curve from the sliding base or cradleroll angle ϕc. By adjusting the CRV and sliding base feed-in, a CRV function can be numerically established to provide a modifiedMRP curve during the generation-cutting process.

4. Roll rates and removal volume rates

As shown in Fig. 4, a double-roll generating cutting plan [8] for the Gleason SGDH method is used to show the hypoidgear-cutting simulation and analysis methodology. The Gleason face-milling SGDH method is a duplex helical method for cuttingspiral-bevel gears; both right and left flanks are cut with the same cutter andmachine settings. As shown in Fig. 4, the Z-axis is thecutter position and the V-axis is the virtual cradle angle. The cutter plunges into the set-in position from the index position. Theset-in position determines the stock allowance for finishing. After the plunging process, the virtual cradle (V-axis) moves from thestart roll position (RP) to the end second RP using different roll rates. The roughing process is from the start RP to the first end RP,and the finishing process is from the first end RP to the second end RP. In the roughing process, four roll rates and positions areused to plan the roll velocity. The roll rate of the fishing process is constant.

The plunging process is applied to only the large module hypoid gear. For the small module gear, the plunge cutting can beeliminated and begin cutting directly from roughing and finishing roll processes. The feed rates of the sliding base in the plungingprocess, the amount of set-in, and the roll rates for the roughing and finishing can be modified to adjust the cutting force. Theaccuracy of the finished tooth surface relies primarily on the relative deflection between the cutter and the workpiece because ofthe cutting force in the roughing and finishing roll process. Therefore, the plunging process is ignored at this moment becauseimproving the cutting efficiency is being attempted with acceptable tooth surface accuracy. It is assumed that the roughing roll

Table 1Basic parameters of the work gear in the experiment.

Gear data

Shaft angle Σ (deg) 90Offset V (mm) 0Mean normal module mn (mm) 2.5Mean radius rm (mm) 50Mean spiral angle βm (deg) 35Number of teeth z (−) 25Pitch cone angle δ (deg) 45Face width F (mm) 11Pressure angle αn (deg) 20Whole tooth height h (mm) 5.25Normal backlash jn (mm) 0.05–0.10

Cutter data

Convex Concave

Number of cutter starts z0 (−) 8Blade angle α0 (deg) 24 16Cutter radius r0 (mm) 44.45 44.45

Please cite this article as: S.-H. Chen, Z.-H. Fong, Study on the cutting time of the hypoid gear tooth flank, Mech. Mach. Theory(2014), http://dx.doi.org/10.1016/j.mechmachtheory.2014.01.012


Table 2Machine settings for the universal cradle-type hypoid generator.

Tilt angle i (deg) 2.210000Swivel angle j (deg) 180.750000Initial cradle angle setting θc (deg) 83.590000Radial setting SR (mm) 36.640060Vertical offset Em (mm) 0.000000Increment of machine Center to back ΔA (mm) −0.156410Sliding base feed setting ΔB (mm) 1.394180Machine root angle γm (deg) 40.440000Roll ratio Ra 1.404096


process starts from toe-to-heel with cradle up-roll, and is followed by the cradle down-roll to finish the cutting. The CRV can becontinuously controlled by the CNC controller.

Typical outputs for the cutting simulation are shown in Figs. 5 and 6. Figs. 5(a) and (b) show the MRP curve of roughing andfinishing, respectively. The cutting simulation is performed from 40° to −40° for the cradle roughing up-roll, and from −40° to40° for the cradle finishing down-roll. According to the MRP curves, the actual cutting starts at the cradle roll angle where theslope of the MRP curve deviates from zero, and the cutting finishes at the cradle angle where the slope of the MRP curve returns tozero. The start and first RP are determined according to the start and end cradle angles determined by the MRP curves. As shownin Fig. 6(a), three roll rates are in the roughing process and only one in the finishing process. Roll rates from one RP to the nextvary linearly from one roll rate to the next avoiding excessive cutting torque in the rolling process. The first and second variableRP are at 15% and 70% of the cradle angle, respectively. The end second RP is the same as the start RP.

The MRR fm can be expressed as

Pleas(201

f m ¼ ΔvΔt

¼ ΔvΔϕ

f r ð11Þ

Δv is the difference of the MRP curves between the two steps and Δt is the roll time, which equals to interval angle Δϕ
wheredivided by roll rate fr. The MRR at every cradle-RP can then be calculated, and is summarized as the MRR curve shown in Fig. 6(b).The maximum allowable MRR for roughing and finishing cuttings should depend on the cutting machine stiffness.
5. Optimizing the feed rates on the six-axis CNC hypoid gear generator

A hypoid gear-cutting plan is proposed to improve cutting efficiency and reduce cutting time by varying the feed rate and CRV.In the generating cut process, the cutting torque should be as high as possible for higher production efficiency. The MRR fm shouldbe maintained at a high margin for as long as possible. The high margin of MRR fm can be experimentally determined on an actualcutting machine by checking the surface roughness and accuracy of the generated tooth surface with varying material-removalrates.

When the cutting simulation of the universal cradle-type hypoid generator was performed, the material-removal volume inthe cutting process was observed from the MRP curve. The start and end roll angles can be precisely defined at where the changesof the material-removal volumes equal zero. This can ensure that the generating process is complete and reduce the dry-run time.

A CMRR process is proposed by reducing the roll rate fr at a large removal-volume position to avoid excessive cutting torqueand increasing the roll rate fr at the smaller removal-volume position to reduce the machining time. In the proposed CMRR

Fig. 9. The MRP and design CRV graph.



Fig. 10. The MRR and cycle time.


process, the MRR fm is set as a constant in Eq. (11), and the roll rate fr is calculated by f r ¼ f m=ΔυΔϕ ;, where Δv/Δϕ is obtained from

the MRR curve. However, the roll rate fr has limitations on the actual machine. If the calculated fr is greater than the defaultmaximum roll rate, fr is set to the maximum roll rate. In the start RP, the roll rate is linearly raised by 0.7 of the maximum roll rateto either the maximum roll rate or until a high margin of MRR is reached. In the end RP, the roll rate linearly decreases from themaximum roll rate to 0.7 of the maximum roll rate.

The high margin of MRR fm relies on the cutting machine stiffness and the minimum chip thickness without rubbing thecutting edge; it should be decided by the experiments. The roll rate with this CMRR cutting plan is compared to the SGDH cuttingplan depicted in the Gleason handbook in Fig. 7(a), and the corresponding MRR curve is shown in Fig. 7(b). The figure shows thatthe high margin of MRR fm for both two-cutting plans is identical, but the average roll rate of CMRR is significantly higher than inthe SGDH-cutting plan. The total machining time is decreased and the high margin of MRR fm is maintained by the proposedCMRR roll rates. As shown in Fig. 8, the cycle time for cutting one tooth slot by the proposed CMRR and SGDH process is 10.958and 19.858 s, respectively. The machining time is 8.396 and 13.456 s, respectively. The cycle time and machining time of theCMRR cutting plan are reduced by 45% and 38%, respectively, compared to the SGDH-cutting plan.

6. Experiment and discussion

The proposed CMRR cutting plan is compared to the SGDH-cutting plan using the general-purpose six-axis machine tool bycutting the same spiral bevel gear. The basic dimension of the target gear and cutter data are shown in Table 1, and the machinesettings of the SGDHmethod are shown in Table 2. Because the typical module of the work gear is only 2.5 mm, the plunge cuttingis omitted, and a one-tooth slot is cut by cradle rolling cut from toe-to-heel (40° to −40°) for roughing and heel-to-toe (−40° to40°) for finishing. The convex and concave sides of the gear flanks are finished using the same machine settings. The machinesetting is transformed from the universal cradle-type hypoid generator to the six-axis CNCmachine tools with corresponding feedrates by Eqs. (5)–(10). The cutting torque is obtained by acquiring the TCMD data of the cutter motor using FANUC servo-guidesoftware.

After performing the cutting simulation with the aid of the CAD system, the MRP curve of the CMRR cutting plans for roughingand finishing is shown in Fig. 9(a). According to the MRP curves, the actual cutting begins from cradle angle 38° to −27°. In therough process, the high margin of MRR is 18 mm3, and the maximum roll rate is 9°/s. In the finish process, the highmargin of MRR

Fig. 11. The MRR of the math model.



Fig. 12. Instantaneous TCMD of the cutting-spindle motor.


is 7 mm3, and the maximum roll rate is 9°/s. The roll rates of the CMRR cutting plan are shown in Fig. 9(b), and the correspondingMRR and cycle time are shown in Fig. 10(a) and (b), respectively.

The simulated MRR with respect to time is shown in Fig. 11. The high margin of MRR for roughing (h1) is 18 mm3/s, and theduration time (t1) is 4.947 s. The high margin of MRR for finishing (h2) is 7 mm3/s, and the duration time (t2) is 2.015 s. The ratiomh′ of h1 to h2 is 2.57, and the ratio mt′ of t1 to t2 is 2.455.

Fig. 12 shows the instantaneous TCMD of the cutting-spindle motor recorded by the FANUC servo-guide. Take the average per1 s of the TCMD data, and the result is shown in Fig. 13. The maximum TCMD for roughing (h3) is 0.563%, and the duration time(t3) is 3.363 s. The maximum TCMD for roughing (h4) is 0.237%, and the duration time (t4) is 1.33 s. The ratio mh of h3 to h4 is2.379, and the ratio mt of t3 to t4 is 2.529.

The MRR graph is similar to the experimental results. By comparing ratios mh and mh′, a difference of 1.60% is shown, and thedifference between mt and mt′ is 3.10%. This proves that the TCMD of the cutter motor is proportional to the MRR, and proves thefeasibility of the CMRR cutting plan.

7. Conclusion

This paper proposes a CMRR cutting plan for hypoid-gear generation to reduce the cycle time of gear cutting. The MRP issimulated by the mathematical model for machine-setting conversions with the aid of SolidWorks® API.

Fig. 13. Averaged TCMD of the cutting-spindle motor.




According to the MRP curve and MRR graph, the start and end roll angle was decided precisely to reduce the dry-run time. Theexperiments showed that the TCMD of the cutter-spindle motor is proportional to the MRR. The consistency between thesimulated MRR graph and the averaged TCMD of the cutter motor proves the feasibility of the proposed CMRR cutting plan. In theexperiments, 40% of machining time was reduced by the proposed feed rates under the same cutting torque; the machining timeis also helpful to protect the tools and machine from excessive cutting torque. The gear can be manufactured with higherefficiency on the six-axis CNC hypoid-gear generator with the proposed CMRR method. It is also applied in the grinding process.

Nomenclatureϕc cradle angleϕ1 workpiece rotation angleγm machine root angleμg cutter rotation anglei tilt anglej swivel angleq cradle rotation angleΔA increment of machine center to backΔB sliding base feed settingSR radial settingEm vertical offsetϕA,B,C = rotational axes of the six-axis machine, as shown in Fig. 2Mt = workpiece mounting distance, as shown in Fig. 2Ct = arbor distance, as shown in Fig. 2Tt = distance between rotational center of machine root angle and workpiece mounting plane, as shown in Fig. 2Cd = distance between the pitch apex of work gear and the origin Oc, as shown in Fig. 2ΔA,B,C = axis displacement of the six-axis machine, as shown in Fig. 2Et = offset between rotational axis of machine root angle and workpiece spindlerm mean radiusγ pitch cone angleγo face cone angleγr root cone angleLo face apex beyond crossing pointLr root apex beyond crossing pointαco outside blade angleαci inside blade anglercm cutter radiuswc cutter point widthhco cutter tooth heightΔϕ interval angle

Superscripts 6-axis CNC machine

subscripta workpiece rotation axesb workpiece spindle pivot axesc cutter rotation axest cutter head coordinate system1 workpiece coordinate system

References

[1] Z.-H. Fong, Mathematical model of universal hypoid generator with supplemental kinematic flank correction motion, ASME J. Mech. Des. 122 (2000)136–142.

[2] Y.-P. Shin, Flank correction for spiral bevel and hypoid gears on a six-axis CNC hypoid generator, ASME J. Mech. Des. 130 (2008) 431–440.[3] H.J. Stadtfeld, Operating Instructions for the Gleason PHOENIX® 175HC Computer Controlled Hypoid Cutting Machine, The Gleason Works, Rochester, NY,

2000.[4] Tae Jo Ko, Hee Sool Kim, Sung Ho Park, Machineability in NURBS interpolator considering constant material removal rate, INT J. Mach. Tool Manu. 45 (2005)

665–671.


http://refhub.elsevier.com/S0094-114X(14)00013-5/rf0005









[5] Y.-W. Xu, L.-H. Zhang, W. Wei, L.-P. Wang, Virtual simulation machining on spiral bevel gear with new type spiral bevel gear milling machine, ICMTMA 2(2009) 432–435.

[6] Y.-S. Sun, Z.-J. Xu, NC machining simulation of spiral bevel gear based on Vericut, Coal Mine Mach. vol. 31 (no. 09) (2010).[7] Christian Brecher, Fritz Klocke, Tobias Schroder, Uwe Rutjes, Analysis and simulation of different manufacturing processes for bevel gear cutting, JAMDSM 2

(1) (2008) 165–172.[8] H.J. Stadtfeld, Operating instructions for the Gleason no.116 hypoid generator, The Gleason Works, Rochester, New York, 1976.









study on the cutting time of the hypoid gear tooth flank

Documents