Simulation,Study,on,Influence,of,Rake,Angle,in,Gear,Skiving,Processes

，

Department of Mechanical Engineering，Graduate School of Engineering，the University of Tokyo，Tokyo 113⁃8656，Japan

Abstract:Gear skiving is a promising gear cutting technology that can achieve a multiple faster internal gear cutting process than that of gear shaping.However，the kinematic system complicates skiving process，resulting in severe crater wear due to the intense variation of local cutting features.In particular，the negative rake angle near the cut-out is recognized as influential factor affecting the cutter wear progress，which needs the sophisticated simulation approach to elucidate the underlying cutting mechanism.In this research，the influence of the rake angle，e.g.top and side nominal rake angles of the cutter，is studied to further understand its role in the gear skiving process，for seeking the possibility of skiving process improvement by calculating the effective rake angle.As a result，the top and side rake angles of the cutter can both increase the effective rake angle when compared with the case of the none-rake angle，leading to an enhanced skiving process.This work provides fundamental knowledge of the rake angle for the gear skiving research，contributing to the optimization on the cutter parameters by considering the effective rake angle.

Key words：gear skiving；
numerical simulation；
parametric modeling；
cutting characteristics；
cutter geometries

Gear skiving is a promising gear cutting tech⁃nology with high productivity and accuracy，espe⁃cially for internal gear.However，the high wear rate of cutters has been a barrier to its implementation in the past decades［1］.Recently，with the development of electrical gearbox function and rigid of machine tools as well as tool coating technology，gear skiv⁃ing has been treated as the contemporary gear solu⁃tions technology as a powerful alternative to the gear shaping a hobbing in both internal and external gear teeth manufacturing［2］.Therefore，comprehen⁃sive clarification and understanding of the mecha⁃nism of gear skiving process is urgent to support the arising industrial application.

Due to the complex cutting environment in gear skiving，a direct observation on cutting process is difficult.Hence，various of simulation technolo⁃gies have been developed for visualization of the cut⁃ting process，which is especially helpful for under⁃standing the gear generation process and improving the gear skiving performance.Among the targets of simulation，uncut chip geometry（UCG）is one of the most effective ways for visual display of the cut⁃ting process［3-4］.Cutting characteristic，e.g.，rake angle and uncut chip thickness，could further help for comprehensive understanding the cutting pro⁃cess.To the date，computer aided design（CAD）or CAD-based approach［5-7］and numerical calcula⁃tion approach［8-10］were developed with the skiving process clarification.The above approaches are ef⁃fective in generation 2⁃and 3-dimensional UCG and calculation of cutting characteristics.Based on the cutting characteristics rendered UCG，the features of gear skiving process，such as variation of rake an⁃gle could be displayed，providing an effective tool for the studies on the kinematic，cutting and cutter parameters［6，11-13］.

The kinematic characteristics of the gear skiv⁃ing lead to an intense local cutting feature variation during a single-cut from the approach to the recess flank.In particular，the rake angle experiences a change from the positive to the negative during the skiving process，leading to an unappreciated skiving condition subjected to the rake face.Nominal side and top rakes are two cutter parameters that can change the local effective rake angle.By simulating the variation of cutting characteristics with different cutter geometry conditions，the influences of the rake angle could be clarified［14］，enabling an im⁃provement on the gear skiving process as well as the cutter design.

In this paper，we investigated the influence of the rake angle to the skiving process by conducting the simulation assessment on the local effective rake angle.Specifically，two critical cutter rake angles were investigated，e.g.top and side rake angles，as⁃sociated with the skiving process clarification by pro⁃jecting the effective rake angle to the UCG in the parametric space，enabling the improvement on the cutter design and process assessment.Other statue variable aside effective rake angle can be transferred to the assessment by following the research route of this work.The reminder of this paper is constructed as follows：Section 1 presents the basic knowledge and kinematic of gear skiving，i.e.，the configura⁃tion of the cutter and the workpiece，and the infeed technology.Section 2 presents the simulation result of skiving process based on the given cutting edge profile and cutting conditions.The influences of a part of cutter geometries are given in Section 3，fol⁃lowed by discussion and concluding remarks summa⁃rized in Section 4.

For a given set of gearing，gear skiving sys⁃tem is consisted of the cutter and kinematic param⁃eters，which are critical for determining the cutting performance［15-18］.Fig.1 presents the configuration of a typical internal gear skiving.The taper angle is attached to the pinion cutter，also named as coni⁃cal cutter，to avoid the interference between the cutter flank and the tooth gap，which is due to the kinematic nature of the meshing motion of gear skiving kinematic.Basic elements of the kinematic system include the inclination angle and the tile an⁃gle（or sometimes uses offset angle），for generat⁃ing the cutting velocity during meshing and interfer⁃ence-free adjustment，respectively.In particular，both the tilt angle［1，16］and the offset angle are help⁃ful for reducing the interference risk in skiving pro⁃cess，as shown in Fig.1，but are generally applica⁃ble to different scenarios，i.e.，the tilt angle is for conical cutter and the offset angle is for cylindrical cutter.

Fig.1 Configuration of internal gear skiving

The rotation speed relationship between the cutter and the workpiece when the cutter provides the incremental movement is as follows［15］

wherevais the axial feed speed；
ZcandZware the tooth numbers of the cutter and the workpiece，re⁃spectively；
andβwis the helix angle of the work⁃piece.The gear skiving process requires attentive experimental trails for a moderate cutting perfor⁃mance.Therefore，an effective numerical simula⁃tion approach was proposed to visualize the cutting process and quantify the cutting characteristics.

The numerical simulation is applied to obtain the cutting process and results based on a given set of cutter，gear and cutting conditions.Fig.2 shows the flowchart of simulation process，which is inte⁃grated into our in-house developed gear skiving simulation software，enabling us to obtain the lo⁃cal cutting feature and parameter optimization.The cutting edge sweep surface could be obtained based on the cutter geometries and cutting condi⁃tions including module，tooth number，infeed，etc.The calculation was carried out based on the two adjacent sweep surfaces.The output of simula⁃tion approach includes the UCG，rake angle，un⁃cut chip thickness，as well as the parameters indi⁃cating the workpiece quality，such as the surface roughness.In addition，based on the simulation output，the cutter geometries or cutting conditions could be redesigned for achieving a better cutting performance.

Fig.2 Flowchart of the proposed simulation approach

2.1 Theory and modeling

The modeling method of the sweep surface and other cutting characteristics are presented as follows.Firstly，the cutting edge could be parameterized for detailed understanding of cutting process as follows

whereu∈Iu=［0，1］.The workpiece is assumed to be fixed and the sweep surface resulting from the rel⁃ative motion of the cutting edge on the workpiece can be expressed by the following equation

wherew∈Iw=［0，1］.Tis the homogeneous rota⁃tion matrix andG（w）the trajectory ofS.Parameterθis the function of the parameterwthat defined by the kinematic motion of the skiving process.When the simulated tooth gap of the gear rotates back to the simulation position，the cutting is performed with axial infeed movement，which needs to be not⁃ed that the cutting may not be strictly conducted by the same tooth of the cutter but is determined the gear ratio.This fact is important for understanding the underlying mechanism of the eccentricity error of the cutter on the surface formation.

The gear skiving is featured with the evolving cutting conditions during a single⁃cut，e.g.，effec⁃tive rake angle，uncut chip thickness，and clearance angle.Therefore，UCG is of great importance for comprehensive understanding the cutting processes.Generally，the local cutting feature varies with the meshing moment during a single-cut.Oblique cut⁃ting model was used to derive the cutting feature from the differential cutting edge［9］.

2.2 Case study

A case study was carried out based on the above modeling process.A tapered cutter with invo⁃lute tooth profile was employed to process an inter⁃nal spur gear.The module of the cutter is 2 mm and the tooth number is 25.Detail parameters are listed in Table 1.Two rough passes and one finishing pass were conducted to complete the whole gear gap.

Table 1 Cutter geometries and cutting conditions

Fig.3 presents the calculated sweep surface in Cartesian coordinate system based on the cutting edge and the cutting conditions of the 1st pass.A curved cutting trajectory of the center point of thecutting edge is formed.The generated gear gap could be clearly seen fromX-Yview.By employing the cloud-basedZ-map method［9］or a level counter method［19］，the gear gap on the workpiece and UCG could be obtained.Here，theZ-map method was em⁃ployed to illustrate the result.TheX-axis values on the sweep surface and the workpiece were com⁃pared.The meshing size is 40 μm × 40 μm.Noted that，the finite element model based numerical calcu⁃lation is not applied to this work since the geometri⁃cal feature related the skiving process alteration is concerned.

Fig.3 Sweep surface of the 1st pass

Figs.4 and 5 show UCGs of three passes ren⁃dered by the effective rake angles and the uncut chip thicknesses，respectively，depicted in the paramet⁃ric space for better understanding the cutting charac⁃teristics on the cutting edge.The horizontal axis of Figs.4，5 is the parameteruof the cutting edge，while the vertical axis indicates the trajectory param⁃eter ofw，which could be regarded as the cutting time.Color coding is conducted to indicate the part of the cutting edge，i.e.，cutting edge on right flank，tooth top，and left flank，respectively.For the effective rake angle，all three passes show the same trend that the effective rake angle decrease from cut-in to cut-out.In three different passes，it could be seen that as the later of cut-out，i.e.，the larger ofw，the smaller rake angle becomes，indi⁃cating a better cutting performance when the cutting finished earlier.The obvious boundary between the different color blocks is caused by the discontinuous of the cutting edge at the tooth top and flanks.Therefore，a continuous curve could reduce abrupt variation of cutting conditions between the tooth flanks and tooth top.

Fig.4 Effective rake angles subjected to UCG in parameter space

Fig.5 Uncut chip thicknesses subjected to UCG in parameter space

As for the uncut chip thickness，it is far less than the infeed amount in the 1st pass and the 2nd pass.This may indicate that a smaller value of radial and axial in feed dominates the main of the uncut chip thickness.The results reveal that which part of the cutting edge is involved in the cutting process and more prone to wear.In the current cutter geom⁃etries and cutting conditions，the larger uncut chip thickness indicates that the tool wear is remarkable in the tooth top and the flank is aroundu=0.6.Similar with that in effective rake angle，the discon⁃tinues of the uncut chip thickness atu=0.4 andu=0.6 are caused by the discontinues of the cut⁃ting edge.In addition，combining with the effective rake angle result，the uncut chip thickness is large while the effective rake angle is negative in cut-out，indicating the poor cutting conditions.In sum，the right side of the cutting edge can be considered as being subjected to a severe skiving process，result⁃ing in a faster wear rate.

Moreover，based on the above results，the main cutting characteristics affecting the cutting per⁃formance could be researched on the cutting edge.Therefore，based on the distribution of cutting char⁃acteristics，the improvement of the cutting perfor⁃mance could be achieved by redesigning the cutter geometries or the cutting conditions.

Fig.6 shows the influence of the side rake an⁃gle（Σ）on distribution of the effective rake angle，where the side rake angle is set as 0°，15°，and 30°，respectively.The top rake angle is 0°，and the other conditions are the same as those in Table 1.The minimum effective rake of the tooth top，i.e.，u=0.4—0.6，increases slightly while the other part of the cutting edge improves greatly.Especially，the effective rake angle on the right cutting edge，i.e.，u<0.4，is larger than that on the left cutting edge，i.e.，u>0.6，in Fig.6（b）；
however，it be⁃comes smaller than that in Fig.6（d），showing the existence of a value that balances both sides of the cutting-edge.The result reveals that the side rake angle is a critical parameter for determination of the effective rake angle.The side rake angle between 15°—30° shows balanced effective rake angle distri⁃bution on UCG，indicating a better cutting perfor⁃mance.However，the lower limit of the effective rake angle on the tooth bottom is difficult to be changed.

Fig.6 Distribution of effective rake angle on UCG of the1st pass with different side rake angles Σ

Fig.7 shows the influence of the top rake angle（γ）on the distribution of the effective rake angle.The top rake angle of the cutter is set as 0°，5°，and 10°，respectively.The side rake angle is set as 20° according to the above discussion.The other conditions are the same as those in Table 1.With the increase of the top rake angle，the effective rake angle in the cutting process also becomes larg⁃er overall，indicating large top rake angle is helpful for improvement the poor cutting conditions at the cut-out position.

Fig.7 Distribution of effective rake angle on UCG of the 1st pass with different top rake angles γ

Based on the above discussion，the effective rake angle distribution could be improved by em⁃ploying the proper top rake angle and the side rake angle.Especially higher top rake angle could make the effective rake angle positive at the positive of cut-out.However，the larger top rake angle will de⁃crease the strength of the tip of the cutter.There⁃fore，threshold of the cutting force should be consid⁃ered as the limitation of the top rake angle.

In gear skiving，both the UCG and cutting characteristics depend highly on the initial cutter ge⁃ometries and cutting conditions.Due to the various parameters，it is difficult to derive a universal con⁃clusion of cutter geometries or cutting conditions for high skiving performance.Therefore，a conve⁃nient，easy-to-understand，and universal simulation approach is especially helpful for verifying the cut⁃ting process and improving the cutting performance in practice.In this research，a systematic modeling process of gear skiving was introduced.For given cutter geometries and cutting conditions，paramet⁃ric modeling of the cutting edge and cutting condi⁃tions were applied to obtain UCGs and cutting char⁃acteristics.The cutting characteristics rendered UC⁃Gs present clearly the cutting process，which helps engineer or researchers make a comprehensive un⁃derstanding and improvement on the skiving pro⁃cess.

Moreover，an increase of the top rake angle may bring the tool tip strength concern，meaning the improvement on the local rake angle needs to consider the possible cutter failure type，i.e.chip⁃ping，to balance the rake angle incasement and the cutter strength，as shown in Figs.6 and 7，a sudden rake angle variation（aboutu=0.6）could lead to the force concentration，resulting in the chipping phenomenon even the rest of the cutting edge still remains integrity.

The concluding remarks for delivering the transferrable knowledge are summarized as follows：

（1）The parametric modeling process on the gear profile and the skiving process are effective for obtaining the UCGs and cutting characteristics.

（2）The effective rake angle on different parts of the cutting edge shows decreasing trend.The neg⁃ative effective rake angle in the cutting out is an un⁃favorable result.

（3）The uncut chip thickness is zero from the cut-in position and increases gradually to the cut-out position，indicating a upward milling liked the cut⁃ting process.

（4）The influences of part of cutter geometries on the effective rake angle in the cutting process are clarified by using different rake angles and side rake angles of the cutter.