Integrated Closed Loop in 5Axis CnC Gear Manufacturing C. Gosselin, ing. Ph.D., Involute Simulation Softwares Inc., Canada J. Thomas, Dr. Ing., ZG Hypoid GmbH, Germany

Abstract CnC manufacturing is becoming increasingly present in the gear industry. It offers versatility in tooth shape and tooling that no dedicated gear cutting machine can offer, although output volume may be lower when compared to dedicated machines. In some cases, such as very large spiral bevel gears, CnC machines may be the only choice available since gear machine makers such as Gleason and Klingelnberg do not offer machines capable of cutting spiral bevel gears beyond ~1 meter in diameter. Any gear cutting machine, whether dedicated or CnC, is inherently affected by build errors which are transmitted to the work piece. The same applies to tools, although to a lesser extent because of their comparatively small size. Such manufacturing errors can be quite significant in how they affect the kinematics of a gear set and must therefore be controlled. The Closed Loop, used to control manufacturing errors, calculates changes in machine settings needed to remove specific errors. With 5Axis CnC machines, specific machine settings must be translated to the architecture of the 5Axis CnC machine used. This paper presents an integrated approach at Closed Loop, or Corrective Machine Settings, applied to 5Axis CnC machines where a Unified model is used to calculate tooth flank topography and generate CnC part programs and CMM target files. The CMM output is then used to calculate machine setting changes minimizing the differences between the manufactured and design tooth flanks. Actual examples of gears cut and corrected on 5Axis CnC machines are presented 1. Introduction Until recently, gear manufacturing has required dedicated machines which are specialized in tooth shapes such as spur/helical, straight bevel, Coniflex™ or spiral bevel gears. Dedicated gear cutting machines use specific tools and offer reliability and repeatability which are fundamental in large production. However, dedicated gear cutting machines can cut only specific tooth shapes.

The 5Axis CnC machine can, by nature, reach any point of a given volume with a totally free tool orientation in reference to a work piece. Therefore, using 5Axis CnC machines to cut gears is a natural consideration. For small lots and/or large dimension gears, 5Axis CnC machines may have a significant economic advantage since they can cut different tooth shapes using different tools, and can be put to other tasks when not cutting gears. However, several issues need to be addressed when using 5Axis machines. First, the movements of the reference tool in the original cutting process must be translated into tool and work piece movements in a 5Axis machine. Second, since a variety of tools may be used, the movements of each tool must be defined in reference to the basic cutting process. Third, while contemporary 5Axis machines are of excellent quality, machine components, machine build and tool tolerances, machine wear and ambient temperature all contribute to locate the tool and work piece in different position and orientation than what is expected. Therefore, a means allowing the control of tooth surface manufacturing errors is essential to allow cutting gear teeth within tolerances, what is customarily referred to as Closed Loop or Corrective Machine Settings, and which has been in use since around 1984 [2] with dedicated spiral bevel gear machines. The output is generally expressed in terms of machine setting changes applied to a specific machine. This paper presents an integrated approach to Closed Loop where a Unified model [1] is used to describe the geometry and movements of a tool in reference to the work piece. The Unified model is then used to pilot the tool in the work piece space. Tools include dish type cutters for Coniflex™ straight bevel gears, Face Mill cutters for spiral bevel gears, and Conical Side Milling Tools – or CoSIMT - End Mill or Ball Mill tools for any tooth shape. CMM measurement is used to compare the target, or designed, tooth flank topography to that obtained after cutting. CMM measurement, coupled to proportional constants, is used to calculate machine setting changes to correct user selected surface errors such as spiral and pressure angle, lengthwise crowning and profile curvature, and tooth warp or bias. The use of the Unified model, coupled to the calculated machine setting changes, then leads to modified 5Axis machine coordinates and angles, and thus a corrected tooth. 2. Unified Model The generating process [1] is based on the concept of a cutter blade whose shape and movement represents one tooth of a theoretical generating gear meshing with the work. The fundamental equation of meshing can be written as:

N • Vr = 0

(1)

Eq. (1) states that, at any point, the relative speed vector of the contacting surfaces between tool and work must be in a plane perpendicular to the common normal to the surfaces. When applied using the reference frames depicted in Fig. 1, Eq. (1) yields an unbounded generated surface in a reference frame attached to the work piece. The surface is a function of the machine settings and three variables, respectively cutter position αc (angular or linear), work piece roll angle α3 and position S of a point along the edge of the cutter blade:

S = f ( α c ,α 3 )

(2)

The solution of Eq. (2) is a series of contact points between the cutter blade and the work which describe a line along the path of the cutter blade. The envelope of a series of such lines yields a generated tooth as shown in Fig 3. Eq. (2) is solved numerically in real time. The Unified model includes work and tool adjustments and movements found in gear cutting machines. In CnC controlled machines, machine settings can be continuously altered during generation, thus allowing for significant improvements in the kinematics of gear sets. Fig. 1 below represents the most general case in the simulation of cutting processes, and is therefore the basis for the general model. The implicit equation of the general tooth surface is:   X = D [α c ]1 [τ ]3 [J ]1 [R ] [L1m ]1 [Γ ] 2 [P ] [α 3 ]3 [Rc

]3

  S cos(φ )    D = 0  ( R ± S sin(φ ))

(3) (4)

 D is the position vector of a point along a Face Mill or Face Hob cutter blade, as given in Eq.  (4). Vector D is rotated by cutter phase angle α c , tilt angle τ and swivel angle J, translated to the origin of the machine by vector R, rotated by cradle angle L1m and root angle Γ , translated to the origin of the work piece by vector P, rotated by roll angle

α 3 and finally

rotated by Rc , the Face Hobbing cutter to work piece timing relationship when applicable. Fig. 2 shows the general representation of a Face Mill or Face Hob cutter blade from which



vector D is obtained as given in Eq. (4).



Similarly, Eq. (5) defines vector N , the normal to the cutter blade at point S, and Eq. (6) gives the transformations needed between the reference frames of the cutter blade and the work piece, where translations R and P have been omitted.

 sin(φ )     N = 0   cos(φ ) 

(5)

  N x = N [α c

(6)

]1 [τ ]3 [J ]1 [L1m ]1 [Γ ] 2 [α 3 ]3 [Rc ]3

In Eq. (3) and (6) above, rotations and translations can be expanded in Taylor series to allow higher order manufacturing flexibility on CnC controlled machines.

Work Tooth Trace Radial

M

Offset

Z1

Z3 ac

Cradle rotation L1m

Cutter Blade D1

S

D3

Fig. 3: Generated Tooth

Z2 Swivel J D2

TopView

Cutter tilt t

D1

αc

D1

D3

Z1

D1

Machine plane

D2

Cutter Blade

Axis of Rotation

I.B.

O.B. φc

φd

D1

Z2

Z3

T

D3

D2

P

N

X1

M S

Sliding base Xb

Machine root angle Γ

X2

a3

D3 D2 Rd Rc

X3

Machine center to back

Fig. 1: Reference Frames for Simulation

SideView

Fig. 2: Cutter Reference Frame

3. Conversion from Dedicated to 5 Axis CnC Machine In the approach proposed here, a Face Mill or Face Hob cutter is replaced by another tool (such as a CoSIMT) which is forced to follow the path of one cutter blade (Fig. 5). This approach is extended to other tools such as End Mill and Ball Mill, and readily lends itself to

the use of Closed Loop described in the next section. Fig. 4 below shows a reference Face Mill cutter in a dedicated machine. By contrast, the current approach in industry uses a cloud of points describing the tooth flanks, such as a STEP file, and the tool is guided along the surfaces by a CAM software. Cradle

Work piece axis

Machine center

Cradle rotation carries the cutter and reorients its axis

Cutter axis

Fig. 4: Ref. Cutter and Work Piece Relative Position

Fig.5: Ref. Cutter and CoSIMT

The conversion from a conventional gear cutting machine to a 5 Axis CnC machine is based on two conditions: i)

maintaining the location of the center of the tool in relation to the origin of the reference frame attached to the work piece, and

ii)

maintaining the relative orientation and phase angle between the axis of the tool and that of the work piece (Fig.4).

Referring to Eq. (3), the position of the center point of a Face Mill or Face Hob cutter, in the work piece reference frame, is given as: 0   X c = 0  [R ] [L1m 0 

]1 [Γ ]2 [P ] [α 3 ]3

(7)

Referring to Eq. (6), the orientation of the axis of a Face Mill or Face Hob cutter, in the work piece reference frame, is given as: 1     N c = 0  [τ 0 

]3 [J ]1 [L1m ]1 [Γ ]2 [α 3 ]3

(8)

The axis of rotation of the work piece being: 0     X 3 = 0  1 

(9)

Angle Λ between the axis of the cutter and that of the work piece is therefore:

  Λ = X 3 • Nc

(10)

5Axis CnC lathe and milling machines can come in different variants: i) a tilting and rotating work piece table (AC, AB types); ii) a tilting tool and a rotating work piece table (BC type), or iii) a tilting and swiveling tool with fixed work piece table (BA type), iv) and other variants, all of which are used to accommodate angle Λ between the tool and work piece axes. Tools include dish type cutters for Coniflex™ straight bevel gears, Face Mill cutters compatible with the CnC machine size, Conical Side Milling Tool cutters (or CoSIMT) such as Sandvik’s InvoMill, End Mill and Ball Mill tools. Basically, End Mill, Ball Mill and CoSIMT tools can cut any tooth shape, including Face Hobbed gears. Each tool type requires a specific axis definition for Eq. (7) and (8) in order to calculate angle Λ in Eq. (10).

4. Closed Loop The Closed Loop, fundamental in any gear manufacturing software, calculates changes in machine settings to remove surface errors and match the manufactured tooth surface with the designed tooth surface [2, 3, 4], what is termed Surface Matching. Surface errors fall in two broad categories: •

1st order : pressure and spiral angles,

•

2nd order: lengthwise crowning, profile curvature, bias (or warp)

Surface errors can be calculated as follows:

•

pressure angle error:

 i ε i , j − ε1, j   ∑ j  row =1 yi , j − y1, j  ∑ i Φ = col =1 j

i

•

spiral angle error:

Ψ =

∑

row =1

i

•

lengthwise crowning error:

•

bias error:

Ξ =

∑

 j ε i , j − ε i ,1  ∑  col =1 xi , j − xi ,1  j i

(11)

(12)

(2ε i , mid − (ε i ,1 + ε i , j )) 2

row =1

ζ = Φ1 − Φ j

(13)

i

(14)

•

profile curvature error:

j

(2ε mid , j − (ε1, j + ε i , j ))

col =1

2

∑

ξ =

(15)

j

Where i is the index of row data, along the tooth flank, j is the index of column data across the tooth flank, mid is the index of the mid-column or mid-row data, εi,j is the error value at point ij of the measurement grid, xi,j is the distance between points along the tooth flank, and

yi,j is the distance between measurement points across the tooth flank. Surface Matching, the algorithm used in Closed Loop, is based on the response of the error surface, i.e. the difference between the simulated and measured tooth surfaces, Fig. 6, to changes in selected machine settings. Proportional changes are obtained by changing machine settings, recalculating the error surface and solving Eq. (11) to (15).

Toe to Heel bias error

Spiral angle error

Crowning error Concave-OB -0.0323

-0.0017

Profile error 0.0171 Toe [mm]

-0.0043 Top

Pressure angle error

Convex-IB

Fig. 6: Typical Error Surface A combination of machine settings is sought such that the theoretical surface matches the measured surface. To do so, the following objective functions are satisfied:

Φ(mi ) −T1 ≤ L1

(16)

Ψ (mi ) −T2 ≤ L2

(17)

Ξ(mi ) −T3 ≤ L3

(18)

ζ (mi ) −T4 ≤ L4

(19)

ξ (mi ) −T5 ≤ L5

(20)

Where mi are the considered machine settings, Φ and Ψ are the averaged pressure and spiral angle errors, Ξ and ζ are the lengthwise crowning and bias error values, ξ is the profile curvature error, Ti are target surface deviations and Li are the tolerances within which the objective functions can be considered satisfied. A Newton-Raphson based solution is used to

solve the above functions. The solution yields new machine settings for a dedicated machine which are converted to a 5Axis CnC machine. 5. Applications The following examples demonstrate the effectiveness of the proposed Closed Loop using different geometries, tools, and 5Axis CnC machines. (This is not possible with commercial CAM software based on clouds of points such as STEP files describing the tooth surfaces.) 5.1: Application 1: 32 Tooth, 1.25 mm module, Coniflex Pinion; dish type cutter. The pinion was cut on an AC type 5Axis CnC machine. The first cut yielded the results of Fig. 7. Helix (fb) and pressure (fa) angle errors are visible on both tooth flanks. Table 1: Averaged Surface Errors after 1st Cut AVERAGE ERRORS

(Right)

(Left)

Tooth Thickness

:

0.0087

Pressure Angle

:

1.57.35

-1.27.45

Helix Angle

:

0.12.59

-0.12.53

Crowning

:

-0.0029

-0.0050

Profile Curvature

:

-0.0024

-0.0032

Fig. 7: Error Surface after 1st Cut Fig. 8 shows the same part after the 1st Closed Loop iteration. Pressure and helix angle errors have all but disappeared and the Contact Pattern is well located on the tooth flank.

Table 2: Averaged Surface Errors after Correction AVERAGE ERRORS

(Right)

(Left)

Tooth Thickness

:

0.0087

Pressure Angle

:

0.15.49

-0.01.09

Helix Angle

:

0.04.45

-0.02.19

Crowning

:

-0.0024

-0.0049

Profile Curvature

:

0.0017

0.0019

Fig. 8: Error Surface after Correction

5.2. Application 2: 26 Tooth, 1.5 mm module, Duplex Helical Spiral Bevel Pinion The pinion was cut on an AB type 5Axis CnC machine using a 2” Face Mill cutter. The 1st cut yielded the results of Fig. 9. Pressure (fa), spiral (fb) angle and bias errors, are visible. Fig. 10 shows the error surface after the 1st Closed Loop iteration. Clearly, pressure, spiral and bias errors have been corrected.

Fig. 9: Error Surface after 1st Cut

Fig. 10: Error Surface after Correction

5.3. Application 3: 28 Tooth, 12.68 mm module, Duplex Helical Spiral Bevel Gear The gear was cut on an AC type 5Axis CnC machine using a 160 mm diameter Conical Side Milling Tool. The 1st cut yielded the results in Fig. 11. Slight pressure angle errors (fa) are visible; spiral angle error (fb) is close to zero; some lengthwise crowning is present on the Concave flank, some bias is visible on both tooth flanks, and thickness error is significant at +0.1031 mm – see Table 3. Table 3: Averaged Surface Errors after 1st Cut AVERAGE ERRORS

(I.B.)

(O.B.)

Tooth Thickness

:

0.1031

Pressure Angle

: -0.01.49

0.09.05

Spiral Angle

: -0.00.07

-0.00.40

Crowning

:

0.0025

-0.0077

Profile Curvature

:

0.0051

0.0070

Warp Factor

: -0.01.36

0.03.54

Fig. 11: Error Surface after 1st Cut

The results shown in Fig. 12 are obtained after correction. Table 4 lists the residual errors. The initial pressure angle and bias (warp) errors have disappeared.

However, correcting bias on a DH part often introduces a measure of crowning, as is visible in Table 4.

Table 4: Averaged Surface Errors after Correction AVERAGE ERRORS

(I.B.)

(O.B.)

Tooth Thickness

:

-0.0454

Pressure Angle

:

0.01.04

-0.01.36

Spiral Angle

:

0.00.01

0.00.34

Crowning

:

0.0055

-0.0124

Profile Curvature

:

0.0046

0.0038

Warp Factor

:

0.00.15

0.00.13

Fig. 12: Error Surface after Correction 6. Conclusion An integrated Closed Loop approach has been presented. A Unified model for the simulation of gear cutting processes allows the generation of part programs for 5Axis CnC machines for various tooth and tool shapes. [By contrast, the current approach in industry uses a cloud of points to guide the tool along the tooth surfaces using a CAM software]. Using CMM output, the Surface match algorithm is used to calculate the machine setting changes needed to eliminate manufacturing errors. The new machine settings are then fed the Unified model to generate a new part program, a one minute job, to cut the corrected tooth surfaces. Applications to Coniflex™ and spiral bevel gears, using different tools on different 5Axis CnC machines, show that the algorithm is effective and that excellent tooth flank topography is obtained after one Closed Loop iteration.

References [1] Gosselin C., Thomas J., A Unified Approach to the Simulation of Gear Manufacturing and Operation, International Conference on Gears, T.U.M., Munich, October 7-9 2013. [2] Krenzer T.J., Computer Aided Corrective Machine Settings for Manufacturing Bevel and Hypoid Gear Sets, AGMA Paper 84-FTM-4, October 1984. [3] Gosselin C., Shiono Y., Kagimoto H., Aoyama N., Corrective Machine Settings of Spiral Bevel and Hypoid Gears with Profile Deviations, World Congress on Gearing, Paris, March 16-18 1999. [4] Gosselin C., Nonaka T., Shiono Y., Kubo A., Tatsuno T., Identification of the Machine Settings of Real Hypoid Gear Tooth Surfaces, ASME Journal of Mechanical Design, Vol. 120, September 1998.