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Research Papers

# Virtual Model of Gear Shaping—Part I: Kinematics, Cutter–Workpiece Engagement, and Cutting ForcesPUBLIC ACCESS

[+] Author and Article Information
Andrew Katz

Precision Controls Laboratory,
Department of Mechanical and
Mechatronics Engineering,
University of Waterloo,

Kaan Erkorkmaz

Precision Controls Laboratory,
Department of Mechanical and Mechatronics
Engineering,
University of Waterloo,
e-mail: kaane@uwaterloo.ca

Fathy Ismail

Precision Controls Laboratory,
Department of Mechanical and Mechatronics
Engineering,
University of Waterloo,

1Corresponding author.

Manuscript received July 28, 2017; final manuscript received March 5, 2018; published online April 16, 2018. Assoc. Editor: Laine Mears.

J. Manuf. Sci. Eng 140(7), 071007 (Apr 16, 2018) (15 pages) Paper No: MANU-17-1483; doi: 10.1115/1.4039646 History: Received July 28, 2017; Revised March 05, 2018

## Abstract

Gear shaping is, currently, the most prominent method for machining internal gears, which are a major component in planetary gearboxes. However, there are few reported studies on the mechanics of the process. This paper presents a comprehensive model of gear shaping that includes the kinematics, cutter–workpiece engagement (CWE), and cutting forces. To predict the cutting forces, the CWE is calculated at discrete time steps using a tridexel discrete solid modeler. From the CWE in tridexel form, the two-dimensional (2D) chip geometry is reconstructed using Delaunay triangulation (DT) and alpha shape reconstruction. This in turn is used to determine the undeformed chip geometry along the cutting edge. The cutting edge is discretized into nodes with varying cutting force directions (tangential, feed, and radial), inclination angles, and rake angles. If engaged in the cut during a particular time-step, each node contributes an incremental force vector calculated with the oblique cutting force model. Using a three-axis dynamometer on a Liebherr LSE500 gear shaping machine tool, the cutting force prediction algorithm was experimentally verified on a variety of processes and gears, which included an internal spur gear, external spur gear, and external helical gear. The simulated and measured force profiles correlate closely with about 3–10% RMS error.

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## Introduction

Gear shaping is one of the prominent methods of manufacturing cylindrical gears. It is a generating process which uses a modified cylindrical gear as a tool that axially reciprocates up and down to cut the teeth in the workpiece (shown in Fig. 1). The cutter and the workpiece continuously rotate during the cutting action which simulates the rolling of two gears and, at the beginning of the process, the cutter is radially fed into the workpiece until it reaches the final depth of cut. Compared to gear hobbing (which uses a cutter that resembles a worm gear), gear shaping is generally not as productive, however is more versatile [1]. For example, gear hobbing is unable to generate internal gears or gears with geometric constraints which would interfere with a gear hob. Furthermore, gear shaping may be used as a finishing operation of hardened gears [2]. Therefore, it is important to have an understanding of the mechanics of the operation to improve productivity and the quality of the machined gears. The objective of this paper is to present research in the kinematics, cutter–workpiece engagement (CWE), and cutting forces in gear shaping which will serve as a basis for machining simulation tools that allow process planners to optimize their programs.

###### Literature Review.

In machining research, increasing the productivity (material removal rate), while maintaining or improving the quality of the machined workpiece is of utmost priority. Some of the major limiting factors for achieving these goals are the following:

• Process stability (forced and chatter vibrations lead to poor surface quality and tool breakage/wear due to unstable cutting forces).

• Tool/machine rigidity (elastic deflections of the tool relative to the workpiece due to cutting forces lead to dimensional errors of the finished workpiece).

• Tool and workpiece overheating.

In classical machining operations (e.g., milling, turning, and drilling), these phenomena have been thoroughly researched and, in industry, the research is now methodically being applied to improve productivity and quality. In gear machining, however, this area of research is still in its infancy due to the complex nature of the processes used to machine gears.

In order to be able to predict the process stability and the machined part quality in any machining operation, it is essential to be able to predict the chip geometry and cutting forces. In classical machining processes, the chip geometry can usually be calculated with analytical expressions as a function of the process parameters and tool geometry. The cutting forces can be predicted by determining the varying chip thickness and width along the cutting edge and calculating incremental cutting forces with an orthogonal or oblique cutting model. For example, in the simple case of turning, the chip geometry is a function of the axial feedrate, radial depth of cut, and the geometry of the turning tool. For calculation of the cutting forces, the chip is analyzed in two sections: the tool nose radius zone and the straight edge zone [35]. The chip thickness varies along the tool nose radius zone and so the cutting force is calculated by discretizing the chip into small segments and integrating the incremental cutting forces. Although more complicated, similar methods are applied to the calculation of cutting forces in milling [68], multipoint thread turning [9], drilling [10,11], and broaching [12,13].

In the recent literature, computer-aided design (CAD) software has been used to calculate the cutter–workpiece engagement in more complicated processes. In milling, for example, cutting forces can be calculated analytically if the axial depth of cut, feedrate, and tool immersion is known. However, for complicated workpieces machined with long computer numerical control (CNC) programs, CAD software is often needed to determine the depth of cut and tool immersion at discrete time steps. To do this, the intersection of the tool and workpiece is calculated and the workpiece is continuously updated. This method is shown in Refs. [1416] for three-axis milling, Ref. [17] for five-axis milling, Ref. [18] for broaching, and Refs. [19] and [20] for gear hobbing.

In gear shaping, there has been some study in chip formation during cutting. There exist several two-dimensional (2D) models which can predict the 2D generated cross section of the workpiece. By tracing the trochoidal path of the shaper cutter on a plane (considering the tool geometry and process parameters), the generated tooth profile and 2D chip geometry can be mathematically determined. This model is shown in Ref. [21] which focused on the effects of asymmetric teeth and tip fillets, in Ref. [22] which focused on the effect of protuberance and semitopping, in Ref. [23] which focused on the effect of the gear ratio in internal gear generation, and in Ref. [24] which focused on the generation of noncircular gears. König and Bouzakis [25] studied the typical 2D chip cross sections found in a gear shaping process and analyzed their effects on the chip flow and tool wear. They concluded that based on the width of the tip of the teeth on the gear shaper cutter, there are different cross sections of chip which will result in better tool life characteristics due to the way the chips flow off the rake face.

While earlier studies have focused only on a specific aspect of gear shaping at a time, it is clear that there is a lack of a complete model for three-dimensional chip geometry and cutting forces in gear shaping. Cutting force prediction in internal spur gear shaping has been presented in Ref. [26], however is extended in this paper to predict forces in any time of cylindrical gear shaping, including helical and external gears. Furthermore, a method to calibrate cutting coefficients from in-process force measurements is presented. Following this introduction, Sec. 2 describes the kinematics of the gear shaping process. Section 3 presents the cutting force prediction model, the cutting coefficient calibration method, and experimental validation. Finally, Sec. 4 will conclude the presented research. In the second part of this paper [27], a model to predict elastic deflections in gear shaping as well as profile deviation is presented.

## Kinematics

The kinematics of gear shaping can be considered a superimposition of three different motion components: the reciprocating motion, rotary feed motion, and radial feed motion. The reciprocating feed moves the cutter up and down, the rotary feed rotates the workpiece and tool proportionally to their gear ratio, and the radial feed moves the cutter into the workpiece until the nominal depth of cut is reached. Figures 2(a) and 2(b) show these components for the generation of an internal gear.

As seen in Fig. 2(c) which shows the various coordinate systems for the process, gear shaping can be described with four variables: the axial rotation of the cutter $ϕct$, the axial rotation of the gear workpiece $ϕgt,$ the center-to-center distance between the cutter and workpiece $rt$, and the vertical position of the tool $zt$. For generalization of the process, the origin of the machine coordinate system (MCS) can be assumed to be coincident with the origin of the workpiece coordinate system (WCS). Therefore, the homogenous transformations of the WCS and tool coordinate system (TCS) relative to the MCS can be defined as Display Formula

(1)$TWCSMCSt=cosϕgt−sinϕgt00sinϕgtcosϕgt0000100001,TTCSMCSt=cosϕct−sinϕct0−rtsinϕctcosϕct00001zt0001$

For the cutting simulation, it is convenient to keep the workpiece stationary and represent the tool in the WCS. The transformation between the tool and the workpiece can then be calculated as Display Formula

(2)$TTCSWCSt=TWCSMCS−1TTCSMCS=cosϕcgt−sinϕcgt0−rtcosϕgtsinϕcgtcosϕcgt0rtsinϕgt001zt0001$

Above, $ϕcgt=ϕct−ϕgt$ is the relative angular position between the cutter and the gear. In Secs. 2.12.4, the kinematics of each of the three motion components are described in detail and experimentally verified.

###### Reciprocating Feed.

The reciprocating motion moves the cutter up and down which cuts the teeth in the workpiece. During the return stroke, there is back-off motion which prevents the tool cutting edge from rubbing against the workpiece while moving up. Although the back-off motion is important for ensuring good quality of the finished gear, the motion does not affect the position of the tool during cutting, and therefore, is not included in the kinematic model. In gear shaping machines, the reciprocating motion is typically accomplished using a slider-crank mechanism as illustrated in Fig. 3(a).

The connecting rod length $dcon$ is a constant which is a parameter of the machine. The length of the crank rod governs the cutting stroke length; hence, the crank rod length is $0.5dstroke$. The stroke length is a function of the workpiece face width and tool overruns ($dstroke=b+dtop+dbottom$) as illustrated in Fig. 3(b). The cutting stroke feed ($fcut$) is normally defined in units of DS/min in industry (double strokes per minute); therefore, the cutting stroke frequency ($ωs$) expressed in radians per second is Display Formula

(3)$ωs=fcut·2π60$

Using trigonometric relationships, the vertical position of the tool can be expressed as Display Formula

(4)$zt=dtop−0.5dstroke1−cosωst+dcon2−0.5dstrokesinωst2−dcon$

This can also be simplified (with about 0.25% error [28]) to a pure sinusoidal equation without the effect of the slider-crank mechanism Display Formula

(5)$zt≅dtop−0.5dstroke1−cosωst$

###### Rotary Feed.

The cutter and workpiece both have rotary feeds that are proportional to their gear ratio. The rotary feeds can be expressed as a function of the rotary feedrate ($frotary$), which is given in units of mm/DS (millimeters per double stroke). The rotational velocity (in radians per second) of the workpiece ($ωg$) can be solved as Display Formula

(6)$ωg=frotary·fcut60rpg$

Here, $rpg$ is the pitch radius of the gear. Therefore, the rotational position of the workpiece ($ϕc$) is simply Display Formula

(7)$ϕgt=ωgt$

The rotational position of the tool ($ϕc$) is a function of the workpiece rotation and the vertical position of the tool. The tool must rotate as the workpiece rotates to emulate the rolling of the gears. For helical gears, the tool must also rotate while reciprocating, following the profile of the helical teeth. The rotation of the cutter can be expressed as Display Formula

(8)$ϕct=−ϵωgRt︸rollingofgears+zttanβrpc︸helicalcomponent,R=NcNg,ϵ=+1forexternalgear−1forinternalgear$

Here, $R$ is the gear ratio between the cutter and the workpiece, $Nc$ is the number of teeth on the cutter, $Ng$ is the number of teeth on the gear, $ϵ$ is a gear type modification factor, and $rpc$ is the pitch radius of the cutter.

A gear shaping process consists of one or more cutting passes (usually one or two roughing passes and one finishing pass). At the beginning of each cutting pass, the radial feed slowly immerses the cutter into the workpiece to avoid overloading the tool. Each pass removes some radial depth of cut which can be defined using the center-to-center radial distances $rstart$ and $rend$. Table 1 shows how these radial distances are determined for the $ith$ cutting pass in $n$ number of passes.

Here, $dcut$ is the depth of cut specified for the pass, $rscrape=rag+ϵrac$ is the scraping distance (the radial distance at which the addendum of the cutter just touches the addendum of the workpiece), and $rgc=rpg+ϵrpc$ is the final nominal center-to-center distance (in practice, $rgc$ is often manually overridden to correct for tooth thickness errors in the finished gear). Typically, the infeed motion is defined by radial feed at start of infeed ($fradial,start$) and radial feed at end of infeed ($fradial,end$) given in units of mm/DS. These can be translated into respective linear velocities ($vstart$, and $vend$) with the formulas Display Formula

(9)$vstart=−ϵfradial,start·fcut60,vend=−ϵfradial,end·fcut60$

Usually, $vr,end to prevent overloading of the tool since the chip area will increase as the cutter approaches the final center-to-center distance. After the infeed is complete, the workpiece is rotated an additional $360deg$ to complete the cutting pass. Although the infeed can be defined using the four parameters ($rstart$, $rend$, $fradial,start$, and $fradial,end$), different gear shaping machines may use different velocity and acceleration profiles during the infeed. For example, sophisticated machines may use jerk limited trajectory planning where the rate of change in acceleration is limited [29]. This results in a piecewise profile in which the kinematics of the radial feed is described by the below equations: Display Formula

(10)$rt=rstart+vstartt+12astartt2+aend−astart6tinfeedt3for0≤t
Here, the value of the accelerations would be determined by the machine's proprietary algorithms. For example, if the machine has a predetermined radial acceleration at the end of infeed ($aend$), then the total time required for the infeed ($tinfeed$) can be calculated by solving a quadratic equation, and then the radial acceleration at start of infeed ($astart$) can be determined Display Formula
(11)$tinfeed=2vend+vstart±2vend+vstart2−6aendrend−rstartaendastart=2vend−vstarttinfeed−aend$

When reconstructing a machine tool's actual motion profile, logic may be used to determine the sign of the plus–minus in the quadratic formula. If only one of the answers is positive, then the positive answer is correct. If both answers are positive, then the answer that yields the observed profile is chosen.

###### Experimental Validation.

In order to validate the kinematic model, servoposition commands have been captured at 9 ms sampling period from the Siemens 840D CNC servocontroller in a Liebherr LSE500 gear shaping machine tool. The Liebherr LSE500 machine is a CNC-based machine that is capable of producing cylindrical gears of all types up to a diameter of 500 mm. Commanded position data from the four important axes ($rt$, $ϕct$, $ϕgt$, and $zt$) were captured from the machine during a cutting pass. Table 2 shows the gear data and the process data for the cutting pass. The machine has its own proprietary optimization algorithms which may slightly alter the nominal process parameters to achieve better quality gears. For example, there is temperature compensation that automatically adjusts the center-to-center distance to take into account thermal deformations of the machine. It is difficult to reverse engineer the machine's proprietary compensation algorithms, so some of the parameters of the feed profile were manually adjusted to improve the alignment of the measured and simulated profiles, as shown in the bold values in Table 2.

Linear acceleration profiles were considered in the reconstruction of the axis movement in the simulation, where $ainfeed,end$ of $0.0075$ was determined manually from the captured position profile. Figure 4(a) shows the captured and simulated position of the $rt$ axis during the infeed portion of the cutting pass, along with the velocity and acceleration determined by numerical differentiation. It can be seen that the profiles based on linear acceleration closely emulate the movement of the gear shaping machine. Additionally, comparisons for the position of all four of the axes can be seen in Fig. 4(b) during the whole process and during a zoomed in portion at the beginning of the cutting pass. The only postprocessing applied to the captured profiles is shifting to account for the initial position of the axes and a first-order low pass filter with 10 Hz cutoff frequency in the $rt$ acceleration profile. As seen, all four axes in the simulated profiles match closely to the captured commanded position data. The additional cutter rotation due to the helical engagement of the cutter and workpiece can be seen in the captured $ϕc$ data.

## Cutting Force Prediction

To predict the cutting forces, the tool cutting edge is first discretized into nodes where each node represents an oblique cutting force model with varying cutting force directions, local inclination angle, and local normal rake angle. Then, at each time-step, the cutter–workpiece engagement is calculated using a tridexel modeler called ModuleWorks.2 The two-dimensional chip geometry is reconstructed along the tool edge using alpha shape reconstruction, and, finally, the force contribution from each node is summated to achieve the total force vector. A similar model had been presented in Ref. [26] for internal spur gears. In this paper, the approach has been generalized and extended to be applicable to any type of cylindrical gear (spur or helical, as well as internal or external).

###### Oblique Cutting Force Model.

The oblique cutting force model [30] is commonly used when the cutting edge is not perpendicular to the cutting velocity, as seen in Fig. 5. In this model, there are three force components that are generated: the tangential force $Ft$ which is parallel and opposite in direction to the cutting velocity $Vc$, the feed force $Ff$ which is perpendicular to the cutting edge and cutting velocity and is directed outward from the workpiece surface, and the radial force $Fr$ which is perpendicular to the tangential and feed force. Each of the cutting force components is expressed as a linear function with respect to the undeformed chip area ($a=bh$), and undeformed chip width $b$. The cutting coefficients ($Ktc$, $Kte$, $Kfc$, $Kfe$, $Krc$, and $Kre$) are typically identified through cutting experiments Display Formula

(12)$Ft=Ktca+Kteb,Ff=Kfca+Kfeb,Fr=Krca+Kreb$

The angle between the cutting edge and the radial direction is defined as the inclination/oblique angle $i$. The normal rake angle $αn$ is defined as the angle between the feed direction and the rake face measured on the plane orthogonal to the cutting edge (herein, called the cutting plane instead of the normal plane to avoid confusion with the gear normal plane).

A more comprehensive cutting model is the orthogonal to oblique model [30] which is characterized by the shear angle $ϕ$, average friction angle (herein, denoted by $λ$ instead of $β$ to avoid confusion with the gear helix angle), and shearing stress $τ$. This model allows cutting coefficients that are estimated through orthogonal cutting tests to be applied to tools with oblique angles. To obtain the oblique cutting coefficients, the orthogonal to oblique transformation is applied [30] Display Formula

(13)$Ktc=τsinϕcosλ−αn+tanitanηsinλcos2ϕ+λ−αn+tan2ηsin2λKfc=τsinϕcosisinλ−αncos2ϕ+λ−αn+tan2ηsin2λKrc=τsinϕcosλ−αntani−tanηsinλcos2ϕ+λ−αn+tan2ηsin2λ$

Here, the chip flow angle $η$ is normally assumed to be equal to the inclination angle ($η=i$) [32]. The tangential and feed edge coefficients ($Kte,Kfe$) are assumed to be equal to the edge coefficients identified in the orthogonal cutting tests, and the radial edge coefficient $Kre$ is typically approximated as zero.

###### Tool Edge Discretization.

The cutting tool edge is discretized into points called nodes where each node contributes a three-dimensional force component calculated with the oblique cutting force model. To discretize the cutting edge, the mathematical model of the rake face as well as the cutting velocity must be known. Afterward, the cutting force directions (tangential, feed, and radial), local inclination angle, and local normal rake angle can be determined for each node.

###### Rake Face Model.

The geometry of the rake face is different for spur and helical gear shaping cases. However, in both cases, the tool edge is first discretized by generating points on the transverse plane of the cutter based on the gear data. Then, the nodes on the transverse plane are modified based on the model of the rake face to obtain the nodes on the cutting edge.

In spur gear shaping, the rake face is modeled as a downward facing cone where the angle between the transverse plane and the rake face is the cutter global rake angle ($α$) as shown in Fig. 6(a). The equation of the cone is simply Display Formula

(14)$z=−x2+y2tanα$

Starting from the generated transverse gear profile, the cutting edge can be obtained by the vertical projection of each point onto the conical rake face (shown in Fig. 7(a)). Denoting the location of a node on the transverse profile (in the tool coordinate system) as $pt=xtyt0T$, the corresponding point on the rake face is Display Formula

(15)

In the helical gear shaper case, each tooth has its own rake face which includes the effects of the helix angle and global rake angle of the cutter as seen in Fig. 6(b). The transverse nodes are first orthogonally projected onto the normal plane of the tooth to obtain the normal nodes, and then vertically projected onto the rake face to obtain the rake face nodes as seen in Fig. 7(b).

The equation of the normal plane for a tooth is given by Display Formula

(16)$n̂normal·pn=0$

Here, $pn$ is any point on the plane, and $n̂normal$ is the vector normal to the normal plane calculated as Display Formula

(17)$n̂normal=Rtoothn̂normal,0=Rtoothv̂helix×x̂=cosγ−sinγ0sinγcosγ00010−cosβsinβ×100=−sinβsinγsinβcosγcosβ$

Here, $β$ is the helix angle of the cutter. $Rtooth$ is the rotation matrix about the $z$ axis for the particular tooth; hence, $γ$ defines the angle of the tooth on the transverse plane measured from the $x$ axis to the tip of the tooth as illustrated in Fig. 7(c). $n̂normal,0$ is the normal vector of the normal plane for a tooth where $γ=0$. This is determined by the cross product of $v̂helix=0−cosβsinβT$ and the $x$ axis, which are both vectors that are coincident with the normal plane (illustrated in Fig. 7(b)).

Accordingly, the orthogonal projection of the transverse nodes onto the normal plane is performed Display Formula

(18)$pn=xnynznT=pt−pt·n̂normaln̂normal$

The equation of the rake plane of the tooth that also has the effect of the cutter's global rake angle ($α$) is given by Display Formula

(19)$n̂rake·pr=0$

Here, $pr$ is any point on the rake plane, and $n̂rake$ is the vector normal to the rake plane that is determined by Display Formula

(20)$n̂rake=Rtoothn̂rake,0=Rtoothv̂helix×v̂rake=cosγ−sinγ0sinγcosγ00010−cosβsinβ×cosα0−sinα=cosβcosγsinα−cosαsinβsinγcosαcosγsinβ+cosβsinαsinγcosαcosβ=nrake,xnrake,ynrake,z$
Similar to the normal plane, $n̂rake,0$ is the normal vector of the rake plane for a tooth with $γ=0$ which is determined by the cross product of two vectors coincident with the plane ($v̂helix=0−cosβsinβT$ and $v̂rake=cosα0−sinαT$, as shown in Fig. 7(b)).

Using the equation of the plane, the vertical projection of the normal nodes onto the rake plane can be performed to obtain the rake face nodes Display Formula

(21)$pr=xryrzrT=xnyn−nrake,xxn+nrake,yynnrake,zT$

###### Cutting Velocity.

The cutting velocity $Vc$ of the tool is a combination of the three kinematic components Display Formula

(22)$Vct=Vradt︸radialfeed+Vrott︸rotaryfeed+Vrect︸reciprocating$

In spur gear shaping, the radial and rotary velocities are 2–3 orders of magnitude smaller than the reciprocating motion. Therefore, the radial and rotary feed can be ignored and the cutting velocity can be approximated as Display Formula

(23)$Vct≅Vrect=00dztdtT$

Since cutting only occurs during the down stroke, the unit vector of the cutting velocity ($V̂c$) is constant for every node location Display Formula

(24)$V̂c=VcVc=00−1Tforspurcase$
In helical gear shaping, there is additional tool rotation due to the helical engagement of the tool and workpiece. In this case, the magnitude of the helical component is significant. The rotation of the cutter relative to the gear workpiece is Display Formula
(25)$ϕcgt=ϕct−ϕgt=−ϵωgRt︸rotaryfeedofcutter+zttanβrpc︸helicalcomponent−ωgt︸rotaryfeedofgear$

The angular velocity of the cutter relative to the gear is then Display Formula

(26)$ωcgt=dϕcgtdt=−ϵωgR+dztdttanβrpc−ωg≅dztdttanβrpc$

For rake node location $pr=xryrzrT$, the tangential velocity due to the rotation of the tool is Display Formula

(27)$Vrott=xr2+yr2︸radialdistancetonodelocationdztdttanβrpc︸angularvelocity−yrxr0Txr2+yr2︸tangentialunitvector=dztdttanβrpc−yrxr0T$

Then, the total cutting velocity is Display Formula

(28)$Vct≅Vrott+Vrect=dztdttanβrpc−yrxr0T︸tangentialvelocityduetotoolrotation+00dztdtT︸velocityduetoreciprocatingmotion=dztdt−yrtanβrpcxrtanβrpc1T$

Therefore, the unit vector of the cutting velocity will be constant regardless of the value of $dzt/dt$ (in the tool coordinate system), but different for each node location. Since cutting always occurs in the negative $z$ direction, the unit vector of the cutting velocity is Display Formula

(29)$V̂c=VcVc=yrtanβrpc−xrtanβrpc−1yrtanβrpc2+−xrtanβrpc2+1forhelicalcase$
As just mentioned, the tangential velocity due to tool rotation is different for each node location. Nodes that are farther away from the axis of rotation will have larger tangential velocity than nodes which are closer. Therefore, each node has a unique and different cutting velocity vector.

###### Tangential, Feed, and Radial Direction Calculation.

In both the helical and spur cases, the cutting directions for each node are calculated with the following procedure, given the knowledge of the cutting velocity and rake node location. Considering Fig. 8, the location of the current rake node is denoted by $pr,0$, the location of the next node by $pr,1$, and the location of the previous node by $pr,−1$.

$pr,0.5$ and $pr,−0.5$ are midpoint nodes which are calculated as follows: Display Formula

(30)$pr,0.5=pr,1+pr,02pr,−0.5=pr,0+pr,−12$

The tangent direction $t̂$ is defined opposite to the cutting velocity unit vector Display Formula

(31)$t̂=−V̂c$

An edge vector $e¯$ and a unit edge vector $ê$ are defined which adjoin the two midpoints Display Formula

(32)$e=pr,0.5−pr,−0.5,ê=ee$

With the assumption that the node locations are ordered in a clockwise manner around the axis of rotation, the feed direction $f̂$ is the cross-product between the edge vector and the tangent direction Display Formula

(33)$f̂=ê×t̂$

Next, an inclination vector $î$ is defined as orthogonal to the feed vector and edge vector Display Formula

(34)$î=f̂×ê$

Accordingly, the inclination angle $i$ can be calculated as the angle between the inclination vector and the tangent direction Display Formula

(35)$i=acosî·t̂$

The radial direction $r̂$ is calculated as the cross-product between the tangential and feed directions, however, care must be taken to flip the radial direction if necessary to ensure that it points in the same direction of the inclination Display Formula

(36)$r̂=t̂×f̂,ifr̂·î<0⇒r̂=−r̂$

The edge width $b$ is calculated as the length of the edge vector projected onto the radial direction Display Formula

(37)$b=ecosi$

Finally, the normal rake angle $αn$ is defined as the angle between the feed direction and the intersection of the rake face and the cutting plane, called the rake vector $ĝ$. The cutting plane is the plane orthogonal to the cutting edge. The normal rake angle is then calculated as the angle between the rake vector and the feed direction, with care being taken to also handle the negative rake angle case, should it arise Display Formula

(38)$αn=acosĝ·f̂,ifĝ·î<0⇒αn=−αn$

Figure 9 shows how the local rake angle and inclination angles vary along the cutting edge in a spur and helical gear shaper case. As one would expect, there is symmetry in both the inclination and rake angles in the case of spur gears cutter. There is low inclination angle and high rake angle at the tip and root of the teeth while there is high inclination angle and low rake angle on the flanks of the teeth. In the helical case, the rake angle drops to negative at the left root and increases substantially at the right root.

###### Cutter–Workpiece Engagement.

As evident in the cutting force model, knowledge of the chip geometry is a prerequisite for calculating the cutting force. Specifically, the undeformed chip area $a$, thickness $h$, and width $b$ must be known along the cutting edge. To obtain this information in simulation, the cutter–workpiece engagement (interference) is calculated using a solid modeler called ModuleWorks.2 The ModuleWorks engine is a highly optimized solid modeler specifically developed for material removal simulation that uses the tridexel representation to discretely model the workpiece. The tridexel representation is a method of modeling surfaces and volumes with arrays of parallel line segments (called “nails”) [32], which is generally more computationally efficient and robust than boundary representation. The nails have points on them which represent where material begins and ends.

Within the ModuleWorks engine, the cutter is represented by a single transverse plane at the bottom of the cutter as seen in Fig. 10(a). Using a thin plane at the bottom of the cutter as a representation of the tool allows for the cutter clearance angles to be omitted from the CAD model which decreases simulation time. To perform the cutter–workpiece engagement calculation with the ModuleWorks engine, the workpiece must be kept stationary and the tool is swept from a starting position to an end position. Therefore, the position of the cutter at the beginning and end of each time-step is calculated in the workpiece coordinate system using the transformation defined in Eq. (2). Then, a cutting operation is performed in the engine to obtain the removed material in tridexel representation as depicted in Fig. 10(b). The nails in the removed material are then analyzed to determine the two-dimensional chip geometry and cutting force, as explained further in Sec. 3.4.

###### Cutting Force Calculation.

To obtain the undeformed chip area along the cutting edge, the chip geometry must be analyzed on a two-dimensional plane. The chip cross section is constructed on the plane normal to the Z-axis at the dexel height closest to the midpoint of the tool movement during the time-step. As seen in Fig. 11 which shows a typical chip generated during a helical shaping case, the chip cross section can change considerably throughout the movement of the tool during a time-step. Therefore, analyzing the chip at the middle of the tool movement gives the best representation of the average chip cross section.

To reconstruct the chip cross section from the tridexel format, a point cloud is generated on the construction plane which includes the end points of each X and Y nail on the plane, the intersection of the Z nails on the plane, and the engaged nodes along the tool edge (Fig. 12(a)). The engaged nodes include any nodes on the tool which are within a certain distance to any of the X and Y nail endpoints or Z nail intersections. The distance threshold is chosen to be the dexel resolution ($ddexel$) which is the distance between two adjacent nails. Here, the transverse node locations are used since the tool is represented with a transverse plane in the CWE calculation. However, since the node geometry is defined in the TCS and the CWE is in the WCS, the node locations are transformed into the WCS Display Formula

(39)$ptWCS1︸transversenodelocationinWCS=TTCSWCS︸transformbetweentoolandworkpieceptTCS1︸transversenodelocationinTCS$

From the point cloud on the plane, the chip geometry is reconstructed using the alpha shape method [33]. Alpha shape is a method for determining the shape of a set of points on a plane. To determine the alpha shape of the point cloud, the Delaunay triangulation (DT) is first calculated. The DT is defined as the nonoverlapping triangulation of the set of points in which the circumscribed circle of each triangle does not contain any other point [34]. This results in the triangulation with the fewest number of thin triangles. There are many algorithms for calculating the DT which offer varying levels of speed and robustness. In this case, the Bowyer–Watson algorithm [35] has been implemented due to its simplicity and robustness.

Once the DT of the point cloud is obtained, the alpha shape is determined by removing any triangles whose circumscribed circle's radius is larger than the alpha distance $dα$. In this case, an alpha threshold of $dα=2ddexel$ is used as it was found to produce an accurate reconstruction of the two-dimensional chip geometry. Figure 12(b) shows the circumscribed circles of the triangles that are part of the alpha shape for that particular case. The triangles in the alpha shape form the geometry of the chip cross section (Fig. 12(c)).

Subsequently, each triangle in the alpha shape is associated with the engaged tool node that is closest to the centroid of that triangle as shown in Fig. 12(d).

In the cutting force model, the undeformed chip characteristics (area, thickness, and width) are defined on the plane orthogonal to the cutting velocity. In the helical gear shaping case, the cutting velocity is not coincident with the Z axis, however, the alpha shape is constructed on the plane normal to the Z axis. Therefore, the triangles of the alpha shape must be orthogonally projected onto the plane orthogonal to the cutting velocity as depicted in Fig. 13 before calculating the undeformed chip area.

The summation of the area of each triangle ($A$) associated with a node comprises the undeformed chip area which is used to calculate the incremental tangential, feed, and radial forces ($dFt$, $dFf$, and $dFr$) with the oblique cutting force model Display Formula

(40)$dFt,f,r=K(t,f,r)c(∑ Aassociatedtriangles︸a,chiparea)+K(t,f,r)eb$

Finally, the total cutting force vector for a time-step is determined by integrating the incremental cutting forces from each node Display Formula

(41)$FWCS=∑engagednodesdFtt̂WCS+dFff̂WCS+dFrr̂WCS$

In Sec. 3.2, the tangential, feed, and radial directions were defined in the TCS, therefore, they must be rotated into the WCS Display Formula

(42)$t̂WCS︸tangentialvectorinWCS=RTCSWCS︸rotationmatrixbetweentoolandworkpiecet̂TCS︸tangentialvectorinTCS,f̂WCS=RTCSWCSf̂TCS,r̂WCS=RTCSWCSr̂TCS$

###### Experimental Validation.

Experimental validation of the cutting force prediction algorithm has been carried out by defining several case studies designed to validate the model for different types of gears, processes, and materials. In this paper, three case studies are presented which include an internal spur gear single-pass process with small module, an external spur gear two-pass process with large module, and an external helical gear two-pass process with medium module. The gear data and process parameters can be found in Fig. 14. The first case uses an AISI 1141 steel workpiece material, while the second and third cases use an AISI 8620 steel workpiece material. All of the tools are PM-HSS with Balinit® Alcrona Pro coating and have a global rake angle of 5 deg.

###### Cutting Coefficient Determination.

While cutting coefficients are commonly calibrated through orthogonal turning tests, cutting coefficients can also be estimated mechanistically with knowledge of the process kinematics, tool/workpiece geometry, and in-process force measurements [7]. In the developed model, the cutting coefficients have been estimated directly from cutting force measurements gathered from the gear shaping machine. The orthogonal to oblique model consists of six parameters: the shear stress $τ$, shear angle $ϕ$, friction angle $λ$, and edge coefficients ($Kte$, $Kfe$, $andKre$). To predict the shear stress, shear angle, and friction angle from experimental data, a cubic search space is first defined, where $600≤τ≤850$, $20deg≤λ≤40deg$, and $20deg≤ϕ≤40deg$. From the cubic search space, many candidate points are chosen and the orthogonal to oblique transformation (Eq. (13)) is used to determine the cutting coefficients based on the average local inclination and rake angle of the cutter. Then, based on experimentally measured forces and simulated chip characteristics, a least squares problem is formulated that solves for the edge coefficients. Afterward, the error for each candidate set of coefficients is evaluated and the candidate with the least error is chosen.

The least squares problem is formulated as follows where one data point is taken at the midpoint of each stroke during a process: Display Formula

(43)$FxFyFzs=1⋮FxFyFzs=S︸measuredforces=∑engagednodesat̂∑engagednodesaf̂∑engagednodesar̂∑engagednodesbt̂∑engagednodesbf̂∑engagednodesbr̂s=1⋮∑engagednodesat̂∑engagednodesaf̂∑engagednodesar̂∑engagednodesbt̂∑engagednodesbf̂∑engagednodesbr̂s=S︸regressors(fromsimulation)KtcKfcKrcKteKfeKre︸cuttingcoefficients$

Above, $s$ is the stroke number, $S$ is the total number of strokes in the process, $t̂$, $f̂$, $andr̂$ are, respectively, the tangential, feed, and radial unit vectors for each engaged tool node, $a$ is the chip area determined by the alpha shape reconstruction for each engaged node, and $b$ is the chip width for each engaged node. This can be simplified to the following form: Display Formula

(44)$Y=ϕCϕeθcθe$

Here, $Y$ is the vector of measured forces, $ϕc$ is the matrix of regressors pertaining to the cutting coefficients, $ϕe$ is the matrix of regressors pertaining to the edge force coefficients, $θc=KtcKfcKrcT$ is the vector of cutting coefficients, and $θe=KteKfeKreT$ is the vector of edge coefficients. Subsequently, the edge force coefficients can be solved by linear regression Display Formula

(45)$θe=pinvϕeY−ϕcθc$

Above, $pinv{ϕe}$ represents the left-pseudo inverse of $ϕe$. Since the influence of the radial edge coefficient is typically assumed to be negligible ($Kre≅0$), the third column of $ϕe$ is set to zero. The error for each candidate set of coefficients is evaluated using the RMS error of normalized forces Display Formula

(46)$Error=13S∑s=1Sex2+ey2+ez2ex=Fx,measured−Fx,simulatedFxmax,ey=Fy,meas−Fy,simFymax,ez=Fz,meas−Fz,simFzmax$

The identified best set of coefficients for each material is given in Table 3. The AISI 1141 cutting coefficients were determined from the internal spur gear single-pass process, and the AISI 8620 steel coefficients were determined from the external spur gear with a single-pass process. A contour plot that shows how the error changes based on the shear angle and friction angle for the identified shear stress can be seen for the AISI 1141 steel in Fig. 15.

###### Results.

Using a three-axis Kistler 9255A dynamometer, the cutting forces in each of the cases were measured on the Liebherr LSE500 gear shaping machine. Figure 16 shows a comparison of the simulated and measured cutting forces for the three case studies with entire process profiles and zoomed in sections. The accuracy of the simulated cutting forces are evaluated by calculating the RMS error of the forces taking one data point from each stroke. Each data point is determined by calculating the average simulated and measured force over the cutting stroke. Table 4 shows the RMS error for each of the cases along with percentage error calculated based on the maximum force in each direction.

Examining the internal gear one-pass process, several trends can be seen in the cutting forces profiles. During the radial infeed of the tool, the cutting forces slowly increase until the peak force is seen at the end of the infeed where there is a combination of the radial feed and rotary feed which results in the maximum chip area. After the infeed is complete, the forces remain relatively steady as each of the teeth is cut into the workpiece. Of course, the magnitude in the $x$ and $y$ directions is changing sinusoidally as the dynamometer is rotating with the gear. Throughout the entire cutting pass, a slight wavy pattern can be observed as seen in the zoomed in profile. This is due to the repetitive teeth engagement pattern of the gear shaping process. Each time a new tooth on the tool starts its engagement in the workpiece, a maximum chip area will soon occur after which results in a peak cutting force. After the steady-state cutting phase, the completion of the gear occurs, which begins when the tool comes back around to the section of the workpiece where the infeed first began. During this stage, the cutting forces slowly decrease until they become zero and the workpiece is complete. The simulated and measured cutting forces show very good correlation throughout the entire process with an RMS error less than 5% (based on single point evaluation from the midpoint of each stroke).

Examining the second case, similar trends can be seen in each of the passes as with the internal gear case. The infeed, steady-state and completion phases can be seen, however, the repetitive pattern due to the varying teeth engagement is much more pronounced in this case. This is due to the module being larger and the workpiece being an external gear which results in a smaller contact ratio. In this case, there are only 2–3 tool teeth in contact with the workpiece at once, whereas in the internal gear case there were 4–5 teeth in contact at once. This results in less uniform chip engagement throughout the process and, therefore, less uniform cutting forces. As can be seen, the simulated and measured profiles correlate well, however, there is more error compared to the internal spur one-pass process (5–10% compared to 3–5%). This can be seen particularly in the magnitude of the peak forces at the beginning of the repeating waves. It is hypothesized that this error may be caused by thermal effects; due to the cutting forces being large, the workpiece may be undergoing temperature increase which would change the cutting properties of the material.

Examining the third case, it can be seen that the overall trends of the profiles correlate well; however, there is some noteworthy error in the roughing pass which shows underprediction of the $z$ forces and spikes at the end of the cutting strokes in the $x$ and $y$ directions. It is hypothesized that this error is attributed due to rubbing/interference of the tool during cutting which is evident by gouges and scratches appearing in the finished workpiece (Fig. 17). The interference would cause additional friction as the cutter moves down the workpiece thus increasing the cutting force in the $z$ direction. Once the cutter finished the cutting stroke, then the built-up elastic deformation from the tool interference is released which may explain the measured cutting force spikes in the $x$ and $y$ directions. Another possible source of error is tool wear on the helical gear shaper which can be visibly seen on the bottom of Fig. 14. Further experiments are needed to investigate the cause of the discrepancy. For example, designing a helical cutting test where there is no resulting gouges or scratches on the workpiece would help differentiate whether the error is due to tool rubbing. Such experiments are planned as future work. Nevertheless, there is still good correlation between the measured and simulated cutting forces.

## Conclusions

In this paper, an algorithm to predict the cutting forces in gear shaping was presented. Using an experimentally verified kinematic model of the process and a mathematical model of the cutting tool rake face, the cutting edge is discretized into nodes with varying cutting force direction (tangential, feed, and radial), inclination angle, and rake angle. At each time-step, the cutter–workpiece engagement is obtained with a tridexel solid modeler and refined using alpha shapes to determine the varying chip geometry along the cutting edge. Finally, incremental cutting forces are determined and summed along the cutting edge. The developed model also allows for cutting force coefficients to be directly estimated from experimental gear shaping force measurements.

Using a three-axis dynamometer mounted on the gear workpiece table, experimental cutting forces were recorded and compared against simulated forces for three different case studies. The simulated cutting forces show good correlation with the measured forces (3–10% RMS error). The most discrepancy occurred in the helical external gear case in which it is hypothesized that these discrepancies are due to rubbing of the tool during cutting and tool wear on the cutter. More experiments are needed to verify these sources of error. Nevertheless, the model serves as a foundation for prediction of elastic tool deflection, vibrations, and part quality.

## Acknowledgements

The authors would like to acknowledge the support of Ontario Drive & Gear Ltd. as well as Liebherr-Verzahntechnik GmbH.

## Funding Data

• Natural Sciences and Engineering Research Council of Canada (CANRIMT2 Network, CRD-46581-14, RGPIN-5312).

• Ontario Centres of Excellence (22638).

## References

Klocke, F. , and Kobialka, C. , 2000, “ Reducing Production Costs in Cylindrical Gear Hobbing and Shaping,” Gear Technol., Mar./Apr., pp. 26–31.
Klocke, F. , and Köllner, T. , 1999, “ Hard Gear Finishing With a Geometrically Defined Cutting Edge,” Gear Technol., Nov./Dec., pp. 24–29.
Armarego, E. , and Uthaichaya, M. , 1977, “ A Mechanics of Cutting Approach for Force Prediction in Turning Operations,” J. Eng. Prod., 1(1), pp. 1–18.
Reddy, R. G. , Kapoor, S. G. , and DeVor, R. E. , 2000, “ A Mechanistic Force Model for Contour Turning,” ASME J. Manuf. Sci. Eng., 122(3), pp. 398–405.
Meyer, R. , Köhler, J. , and Denkena, B. , 2012, “ Influence of the Tool Corner Radius on the Tool Wear and Process Forces During Hard Turning,” Int. J. Adv. Manuf. Technol., 58(9–12), pp. 933–940.
Altintas, Y. , and Lee, P. , 1996, “ A General Mechanics and Dynamics Model for Helical End Mills,” CIRP Ann.—Manuf. Technol., 45(1), pp. 59–64.
Budak, E. , Altintas, Y. , and Armarego, A. , 1996, “ Prediction of Milling Force Coefficients From Orthogonal Cutting Data,” ASME J. Manuf. Sci. Eng., 118(2), pp. 216–224.
Omar, O. , El-Wardany, T. , and Elbestawi, M. , 2007, “ An Improved Cutting Force and Surface Topography Prediction Model in End Milling,” Int. J. Mach. Tools Manuf., 47(7–8), pp. 1263–1275.
Khoshdarregi, M. R. , and Altintas, Y. , 2015, “ Generalized Modeling of Chip Geometry and Cutting Forces in Multi-Point Thread Turning,” Int. J. Mach. Tools Manuf., 98, pp. 21–32.
Chandrasekharan, V. , Kapoor, S. , and DeVor, R. , 1998, “ A Mechanistic Model to Predict the Cutting Force System for Arbitrary Drill Point Geometry,” ASME J. Manuf. Sci. Eng., 120(3), pp. 563–570.
de Lacalle, L. L. , Rivero, A. , and Lamikiz, A. , 2009, “ Mechanistic Model for Drills With Double Point-Angle Edges,” Int. J. Adv. Manuf. Technol., 40(5–6), pp. 447–457.
Sutherland, J. , Salisbury, E. , and Hoge, F. , 1997, “ A Model for the Cutting Force System in the Gear Broaching Process,” Int. J. Mach. Tools Manuf., 37(10), pp. 1409–1421.
Ozturk, O. , and Budak, E. , 2003, “ Modeling of Broaching Process for Improved Tool Design,” ASME Paper No. IMECE2003-42304.
Imani, B. , Sadeghi, M. , and Elbestawi, M. , 1998, “ An Improved Process Simulation System for Ball-End Milling of Sculptured Surfaces,” Int. J. Mach. Tools Manuf., 38(9), pp. 1089–1107.
Merdol, S. D. , and Altintas, Y. , 2008, “ Virtual Cutting and Optimization of Three-Axis Milling Processes,” Int. J. Mach. Tools Manuf., 48(10), pp. 1063–1071.
Spence, A. D. , Abrari, F. , and Elbestawi, M. , 2000, “ Integrated Solid Modeller Based Solutions for Machining,” Comput.-Aided Des., 32(8–9), pp. 553–568.
Budak, E. , Ozturk, E. , and Tunc, L. , 2009, “ Modeling and Simulation of 5-Axis Milling Processes,” CIRP Ann.—Manuf. Technol., 58(1), pp. 347–350.
Hosseini, A. , and Kishawy, H. , 2013, “ Parametric Simulation of Tool and Workpiece Interaction in Broaching Operation,” Int. J. Manuf. Res., 8(4), pp. 422–442.
Klocke, F. , Gorgels, C. , Schalaster, R. , and Stuckenberg, A. , 2012, “ An Innovative Way of Designing Gear Hobbing Processes,” Gear Technol., May, pp. 48–53.
Tapoglou, N. , and Antoniadis, A. , 2012, “ CAD-Based Calculation of Cutting Force Components in Gear Hobbing,” ASME J. Manuf. Sci. Eng., 134(3), p. 031009.
Fetvaci, C. , 2010, “ Generation Simulation of Involute Spur Gears Machined by Pinion-Type Shaper Cutters,” Strojniski Vestnik—J. Mech. Eng., 56(10), pp. 644–652.
Tsay, C.-B. , Liu, W.-Y. , and Chen, Y.-C. , 2000, “ Spur Gear Generation by Shaper Cutters,” J. Mater. Process. Technol., 104(3), pp. 271–279.
Shunmugam, M. S. , 1982, “ Profile Deviations in Internal Gear Shaping,” Int. J. Mach. Tool Des. Res., 22(1), pp. 31–39.
Chang, S.-L. , and Tsay, C.-B. , 1998, “ Computerized Tooth Profile Generation and Undercut Analysis of Noncircular Gears Manufactured With Shaper Cutters,” ASME J. Mech. Des., 120(1), pp. 92–99.
König, W. , and Bouzakis, K. , 1977, “ Chip Formation in Gear-Shaping,” Ann. CIRP, 26(1), pp. 17–20.
Erkorkmaz, K. , Katz, A. , Hosseinkhani, Y. , Plakhotnik, D. , Stautner, M. , and Ismail, F. , 2016, “ Chip Geometry and Cutting Forces in Gear Shaping,” CIRP Ann.—Manuf. Technol., 65(1), pp. 133–136.
Katz, A. , Erkorkmaz, K. , and Ismail, F. , 2018, “ Virtual Model of Gear Shaping Part II: Elastic Deformations and Virtual Gear Metrology,” ASME J. Manuf. Sci. Eng., in press.
Katz, A. , 2017, “ Cutting Mechanics of the Gear Shaping Process,” Ph.D. thesis, UWSpace, Waterloo, ON, Canada.
Erkorkmaz, K. , and Altintas, Y. , 2001, “ High Speed CNC System Design—Part I: Jerk Limited Trajectory Generation and Quintic Spline Interpolation,” Int. J. Mach. Tools Manuf., 41(9), pp. 1323–1345.
Altintas, Y. , 2012, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University Press, Cambridge, UK.
Stabler, G. V. , 1964, “ The Chip Flow Law and Its Consequences,” Fifth Machine Tool Design and Research Conference, Birmingham, UK, Sept., pp. 243–251.
Benouamer, M. O. , and Michelucci, D. , 1997, “ Bridging the Gap Between CSG and Brep Via a Triple Ray Representation,” SMA'97 Fourth ACM Symposium on Solid Modeling and Applications, Atlanta, GA, May 14–16, pp. 68–79.
Edelsbrunner, H. , Kirkpatrick, D. , and Seidel, R. , 1983, “ On the Shape of a Set of Points in the Plane,” IEEE Trans. Inf. Theory, 29(4), pp. 551–559.
Berg, M. D. , Cheong, O. , Kreveld, M. V. , and Overmars, M. , 2008, Computational Geometry: Algorithms and Applications, 3rd ed., Springer-Verlag TELOS, Santa Clara, CA.
Watson, D. , 1980, “ Computing the n-Dimensional Delaunay Tessellation With Application to Voronoi Polytopes,” Comput. J., 24(2), pp. 167–172.
Topics: Gears , Cutting , Kinematics
View article in PDF format.

## References

Klocke, F. , and Kobialka, C. , 2000, “ Reducing Production Costs in Cylindrical Gear Hobbing and Shaping,” Gear Technol., Mar./Apr., pp. 26–31.
Klocke, F. , and Köllner, T. , 1999, “ Hard Gear Finishing With a Geometrically Defined Cutting Edge,” Gear Technol., Nov./Dec., pp. 24–29.
Armarego, E. , and Uthaichaya, M. , 1977, “ A Mechanics of Cutting Approach for Force Prediction in Turning Operations,” J. Eng. Prod., 1(1), pp. 1–18.
Reddy, R. G. , Kapoor, S. G. , and DeVor, R. E. , 2000, “ A Mechanistic Force Model for Contour Turning,” ASME J. Manuf. Sci. Eng., 122(3), pp. 398–405.
Meyer, R. , Köhler, J. , and Denkena, B. , 2012, “ Influence of the Tool Corner Radius on the Tool Wear and Process Forces During Hard Turning,” Int. J. Adv. Manuf. Technol., 58(9–12), pp. 933–940.
Altintas, Y. , and Lee, P. , 1996, “ A General Mechanics and Dynamics Model for Helical End Mills,” CIRP Ann.—Manuf. Technol., 45(1), pp. 59–64.
Budak, E. , Altintas, Y. , and Armarego, A. , 1996, “ Prediction of Milling Force Coefficients From Orthogonal Cutting Data,” ASME J. Manuf. Sci. Eng., 118(2), pp. 216–224.
Omar, O. , El-Wardany, T. , and Elbestawi, M. , 2007, “ An Improved Cutting Force and Surface Topography Prediction Model in End Milling,” Int. J. Mach. Tools Manuf., 47(7–8), pp. 1263–1275.
Khoshdarregi, M. R. , and Altintas, Y. , 2015, “ Generalized Modeling of Chip Geometry and Cutting Forces in Multi-Point Thread Turning,” Int. J. Mach. Tools Manuf., 98, pp. 21–32.
Chandrasekharan, V. , Kapoor, S. , and DeVor, R. , 1998, “ A Mechanistic Model to Predict the Cutting Force System for Arbitrary Drill Point Geometry,” ASME J. Manuf. Sci. Eng., 120(3), pp. 563–570.
de Lacalle, L. L. , Rivero, A. , and Lamikiz, A. , 2009, “ Mechanistic Model for Drills With Double Point-Angle Edges,” Int. J. Adv. Manuf. Technol., 40(5–6), pp. 447–457.
Sutherland, J. , Salisbury, E. , and Hoge, F. , 1997, “ A Model for the Cutting Force System in the Gear Broaching Process,” Int. J. Mach. Tools Manuf., 37(10), pp. 1409–1421.
Ozturk, O. , and Budak, E. , 2003, “ Modeling of Broaching Process for Improved Tool Design,” ASME Paper No. IMECE2003-42304.
Imani, B. , Sadeghi, M. , and Elbestawi, M. , 1998, “ An Improved Process Simulation System for Ball-End Milling of Sculptured Surfaces,” Int. J. Mach. Tools Manuf., 38(9), pp. 1089–1107.
Merdol, S. D. , and Altintas, Y. , 2008, “ Virtual Cutting and Optimization of Three-Axis Milling Processes,” Int. J. Mach. Tools Manuf., 48(10), pp. 1063–1071.
Spence, A. D. , Abrari, F. , and Elbestawi, M. , 2000, “ Integrated Solid Modeller Based Solutions for Machining,” Comput.-Aided Des., 32(8–9), pp. 553–568.
Budak, E. , Ozturk, E. , and Tunc, L. , 2009, “ Modeling and Simulation of 5-Axis Milling Processes,” CIRP Ann.—Manuf. Technol., 58(1), pp. 347–350.
Hosseini, A. , and Kishawy, H. , 2013, “ Parametric Simulation of Tool and Workpiece Interaction in Broaching Operation,” Int. J. Manuf. Res., 8(4), pp. 422–442.
Klocke, F. , Gorgels, C. , Schalaster, R. , and Stuckenberg, A. , 2012, “ An Innovative Way of Designing Gear Hobbing Processes,” Gear Technol., May, pp. 48–53.
Tapoglou, N. , and Antoniadis, A. , 2012, “ CAD-Based Calculation of Cutting Force Components in Gear Hobbing,” ASME J. Manuf. Sci. Eng., 134(3), p. 031009.
Fetvaci, C. , 2010, “ Generation Simulation of Involute Spur Gears Machined by Pinion-Type Shaper Cutters,” Strojniski Vestnik—J. Mech. Eng., 56(10), pp. 644–652.
Tsay, C.-B. , Liu, W.-Y. , and Chen, Y.-C. , 2000, “ Spur Gear Generation by Shaper Cutters,” J. Mater. Process. Technol., 104(3), pp. 271–279.
Shunmugam, M. S. , 1982, “ Profile Deviations in Internal Gear Shaping,” Int. J. Mach. Tool Des. Res., 22(1), pp. 31–39.
Chang, S.-L. , and Tsay, C.-B. , 1998, “ Computerized Tooth Profile Generation and Undercut Analysis of Noncircular Gears Manufactured With Shaper Cutters,” ASME J. Mech. Des., 120(1), pp. 92–99.
König, W. , and Bouzakis, K. , 1977, “ Chip Formation in Gear-Shaping,” Ann. CIRP, 26(1), pp. 17–20.
Erkorkmaz, K. , Katz, A. , Hosseinkhani, Y. , Plakhotnik, D. , Stautner, M. , and Ismail, F. , 2016, “ Chip Geometry and Cutting Forces in Gear Shaping,” CIRP Ann.—Manuf. Technol., 65(1), pp. 133–136.
Katz, A. , Erkorkmaz, K. , and Ismail, F. , 2018, “ Virtual Model of Gear Shaping Part II: Elastic Deformations and Virtual Gear Metrology,” ASME J. Manuf. Sci. Eng., in press.
Katz, A. , 2017, “ Cutting Mechanics of the Gear Shaping Process,” Ph.D. thesis, UWSpace, Waterloo, ON, Canada.
Erkorkmaz, K. , and Altintas, Y. , 2001, “ High Speed CNC System Design—Part I: Jerk Limited Trajectory Generation and Quintic Spline Interpolation,” Int. J. Mach. Tools Manuf., 41(9), pp. 1323–1345.
Altintas, Y. , 2012, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University Press, Cambridge, UK.
Stabler, G. V. , 1964, “ The Chip Flow Law and Its Consequences,” Fifth Machine Tool Design and Research Conference, Birmingham, UK, Sept., pp. 243–251.
Benouamer, M. O. , and Michelucci, D. , 1997, “ Bridging the Gap Between CSG and Brep Via a Triple Ray Representation,” SMA'97 Fourth ACM Symposium on Solid Modeling and Applications, Atlanta, GA, May 14–16, pp. 68–79.
Edelsbrunner, H. , Kirkpatrick, D. , and Seidel, R. , 1983, “ On the Shape of a Set of Points in the Plane,” IEEE Trans. Inf. Theory, 29(4), pp. 551–559.
Berg, M. D. , Cheong, O. , Kreveld, M. V. , and Overmars, M. , 2008, Computational Geometry: Algorithms and Applications, 3rd ed., Springer-Verlag TELOS, Santa Clara, CA.
Watson, D. , 1980, “ Computing the n-Dimensional Delaunay Tessellation With Application to Voronoi Polytopes,” Comput. J., 24(2), pp. 167–172.

## Figures

Fig. 1

Gear shaping process

Fig. 2

Kinematic components and coordinate systems in gear shaping process: (a) reciprocating feed, (b) rotary and radial feed, and (c) coordinate systems

Fig. 3

Reciprocating motion kinematics: (a) slider-crank mechanism and (b) stroke length and tool overrun

Fig. 4

Experimental validation of feed drive axis kinematic model: (a) position, velocity, and acceleration of r(t) and (b) position of r(t), ϕc(t), ϕg(t), and z(t)

Fig. 5

Oblique cutting force model

Fig. 6

Rake face model of (a) spur and (b) helical gear shapers

Fig. 7

Projection of transverse nodes onto rake face and definition of tooth angle: (a) projection of nodes for spur shaper, (b) projection of nodes for helical shaper, and (c) definition of tooth angle (γ)

Fig. 8

Cutting direction calculation

Fig. 9

Distribution of inclination and rake angles on single gear tooth with cutter rake angle of 5deg and helical angle of 25deg in helical gear shaper case

Fig. 10

Cutter–workpiece engagement using dexel representation: (a) cutter–workpiece engagement and (b) chip in dexel representation

Fig. 11

Typical chip geometry in helical gear shaping case

Fig. 12

Reconstruction of two-dimensional chip cross section: (a) Delaunay triangulation, (b) alpha shape reconstruction, (c) 2D chip geometry, and (d) triangle-node association

Fig. 13

Projection of triangles onto plane normal to tangential direction

Fig. 14

Experimental case studies

Fig. 15

Error contour plot for AISI 1141 steel at τ=805.6 N/mm2

Fig. 16

Simulated and experimental cutting forces

Fig. 17

Gouges and scraping as seen on the external helical gear (after roughing pass)

## Tables

Table 1 Cutting pass radial distances
Table 2 Gear data and cutting pass parameters for kinematic model experimental validation
Table 3 Identified orthogonal to oblique coefficients
Table 4 RMS error for predicted cutting force in each case study

## Discussions

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