Research ArticleMathematical Model of Helical Gear Topography Measurementsand Tooth Flank Errors Separation

Huiliang Wang1 Xiaozhong Deng2 Jianhai Han2 Jubo Li2 and Jianjun Yang2

1School of Mechanical Engineering Northwestern Polytechnical University Xirsquoan 710072 China2School of Mechatronics Engineering Henan University of Science and Technology Luoyang Henan 471003 China

Correspondence should be addressed to Xiaozhong Deng dxz01163com

Received 23 June 2015 Revised 7 November 2015 Accepted 29 November 2015

Academic Editor Jean J Loiseau

Copyright copy 2015 Huiliang Wang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

During large-size gear topological modification by form grinding the helical gear tooth surface geometrical shape will be complexand it is difficult for the traditional scanning measurement to characterize the whole tooth surface Therefore in order tocharacterize the actual tooth surfaces an on-machine topography measurement approach is proposed for topological modificationhelical gears on the five-axis CNC gear form grindingmachine that canmeasure themodified gear tooth deviations on themachineimmediately after grinding Combined with gear form grinding kinematics principles the mathematical model of topographymeasurements is established based on the polar coordinate methodThemathematical models include calculating trajectory of thecentre of measuring probe defining gear flanks by grid of points and solving coordinate values of topologymeasurement Finally anumerical example of on-machine topographymeasurement is presented By establishing the topography diagram and the contourmap of tooth error the tooth surface modification amount and the tooth flank errors are separated respectively Research resultscan serve as foundation for topological modification and tooth surface errors closed-loop feedback correction

1 Introduction

Gear transmission is a major form of transmitting machinemovement and power In particular cylindrical helical gearpair is widely used in the high speed and heavy loadmechanical transmission because its transmission is stableand its carrying capacity is strong Precision and quality havebeen the critical issues for manufacturing helical gears as thetooth surface accuracy directly affects the performance ofequipment [1] It is known that gear quality could be affectedby various types of errors and factors such asmachine errorsheat treatment distortions variation of grinding forces andunpredictable factors In practice gear form grinding canbe a very effective means for eliminating tooth flank errorsof heavy-duty or large-size gear On the other hand toothsurface modification is another effective technology that canreduce the vibration and noise of gear transmission deviceimprove meshing state effectively prevent the edge contactand prolong the working life of gear [2ndash4]

There are two methods for gear measurement methodpolar coordinate measuring method [5] and rectangular

coordinate measuring method [6] Zhang et al [7] expressedthe conversion formula between the polar angle and thegenerating angle during measuring tooth flank errors bythe polar coordinate measuring method Gao et al [8 9]proposed a new pretravel quantitative calibration method onthe basis of analyzing the cause of the pretravel which couldimprove the on-machine measurement accuracy using touchtrigger probe Nafi et al [10 11] studied an error separationmethod based onmultistep redundancymethod inwhich theerrors of themeasurementmachine tool standard ball probeand probe tip producing in the measurement process areseparated from calibration results by multistep measuringShih and Chen [12] proposed free-form flank correction inhelical gear grinding using a five-axis computer numericalcontrol gear profile grinding machine but the errors of theactual tooth surface were assessed using the gear measure-ment centre not on-machine measurement Kobayashi et al[13 14] calculated the angle of the grinding wheel installationand optimized contact line between the grinding wheel andthe work gear in order to achieve the accuracy of the gearflank profile Lee et al [15 16] fabricated modified cylindrical

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 176237 10 pageshttpdxdoiorg1011552015176237

2 Mathematical Problems in Engineering

gear drive that could reproduce precisely the predesignedfourth-order polynomial function of transmission error

The traditional tooth surface measurement is measuredin the central line of profile and longitudinal direction withreference to the standard involute helical surface modifica-tion as well as the error Due to tooth surface for topologicalmodification this measure cannot satisfy the modificationdesign and processing requirements The traditional toothsurface measurement process is to measure the tooth flankerrors when the profile error is in the central tooth heightand the longitudinal error is in the central tooth widthrespectively Bymeasuring the topographymeasurement canobtain the modification amount as well as the error withreference to the standard involute helical surface For thetopological modification tooth surface this measure processcannot satisfy the requirements of the modification designand machining Therefore how to implement topographymeasurement and dominate completely the tooth surfaceerror information is the key points of this study for thetopological modification tooth surface

The outline of the remainder of the paper is as followsIn Section 2 an on-machine measurement model of helicalgear based on gear form grinding is developed A numericalexample is the main subject of Section 3 In Section 4 thecharacterization of tooth surface error and modificationamount are shown respectively Finally some conclusions aredrawn in Section 5

2 On-Machine Measurement

21 Coordinate Systems of Measurement Based on the topo-logical modification surface equation as a theory equationhelical gear flanks are defined by a grid of points where anelementary sketch affects the ideal tooth surface Accordingto the measuring probe along the tooth surface measurementpath planning the measured displacement is recorded by adata acquisition system The topography diagram is drawnaccording to the tooth surface errors which can observeclearly the error distribution of 45 points on the tooth surfaceand provide the theoretical basis of the tooth surface errorfeedback correction and closed-loop manufacture of helicalgears

The measurement of the helical gear is inspected alongthe radial workpiece using straight probe which is shown inFigure 1 During machining and measuring the workpiecewhether the locating datum is consistent is the key Thepremise of accurately measuring the tooth surface devi-ation on the machine is that the space position of themeasuring coordinate system and the space position of theworkpiece coordinate system coincide too The origin ofthe measurement coordinate system 119878119862 is locating at point119862(1199091119862 1199101119862 1199111119862)which is the centre of rotary tableWhen geargrinding andmeasuring it is locatedwith datum end face andaxis using rotary table and top respectively The theoreticaltooth surface design coordinate system origin and work-piece coordinate system origin coincide as 119874

1 On-machine

measurement can improve the accuracy of grinding toothsurface due to avoiding the reinstallation of the workpieceThe relative position of the space of the coordinate systems

Probe

TipMeasured gear

Rotary table

Reference axis

Reference faceTip

Zc

Za Z1

X1

Xa

Xc

Y1

Yc

Ya Oc

O1a

L

120593

Figure 1 Coordinate systems of on-machine measurement

is shown in Figure 1 1198781(1198831 1198841 1198851) is workpiece coordinate

system 119878119888(119883119888 119884119888 119885119888) is measuring coordinate system 119871 is

the axial distance of the two coordinate system origins and120593 is tooth surface rotation angle The space position relationbetween coordinate system 1198781 and coordinate system 119878119888 isdetermined by the transformation matrixM1198881

The tooth flank form geometry can be generally repre-sented by a position vector and a unit normal vector in theworkpiece coordinate system 1198781 as

r1= r1(1199061 1205791)

n1 = n1 (1199061 1205791) (1)

here (1199061 1205791) are generating surface parameters

For the accurate detection of actual processing toothsurface the theoretical tooth surface should be transformedto the gear measuring coordinate system 119878

119862 by coordinate

transformation which is obtained as follows

r119888= M1198881r1(1199061 1205791)

n119888 = M1198881n1(1199061 1205791)

(2)

here

M1198881=

[[[[[

[

cos120593 sin120593 0 0

minus sin120593 cos120593 0 0

0 0 1 119871

0 0 0 1

]]]]]

]

(3)

22 Mathematical Model of Measurement The use of a polarcoordinate to inspect gears is becoming increasingly usefulin modern gear manufacturing especially on gear formgrinding On-machine measurement of gear profile error isarranged based on the CNC gear form grinder which hasfive digital servo closed-loop controlled axes three rectilinearmotions (119883119884 and 119885) and two rotational motions (119860119862)Here 119862 is the rotation axis of work gear and 119883 is the radialmotion for feeding thewheel down to tooth depthThe theory

Mathematical Problems in Engineering 3

A

x

X

B

Y

O

x0

rb

D

C

120572x

1205720

120601A 120579A

120579B

O0 O1

R1

120601AB

120601BB998400

Figure 2 Schematic illustration of polar coordinate measurementmethod

involute curve is achieved by axes 119862 and 119883 when they aremoving at the same time Because of the motion limit in axial119884direction polar coordinatemeasurement is amajormethodof on-machine measurement when measuring profile erroron the CNC gear form grinder The mathematical modelof polar coordinate method measurement is establishedaccording to the workpiece rotary movement (119862 axis) andradial feed movement (119883 axis) [17]

On-machine measurement system can not only measureprofile error and helical error but also measure pitch errorof helical gears Using the measurement data analysis ofthese errors which may correct processing parameters andaxial motion equation of polynomial coefficients realizes theclosed-loop feedback processing of tooth surface modifica-tion thus laying the foundation for improving the precisionof tooth surface

The polar coordinates of spherical probemethod are usedto measure the involute tooth profile as shown in Figure 2 119860is the measuring start point 119861 is the measuring point 1198611015840 isthe actual location of measuring point 119861 120601119860119861 is an angle ofrotary motion for C axis which can be expressed as

120601119860119861

= 120601119860+ 120601119861 (4)

where

120601119861 = 120579119861 minus 120579119860

120579119860= tan120572

0minus 1205720=

radic1199092

0minus 1199032

119887minus 1198771

119903119887

minus 1205720

120579119861= tan120572

119909minus 120572119909=

radic1199092 minus 1199032

119887minus 1198771

119903119887

minus 120572119909

(5)

where 119903119887is the radius of the base cylinder variable parameters

1199090and 119909 are coordinate data of probe centre corresponding

to start measuring points 119860 and 119861 respectively 1205720and

120572119909are the pressure angle in the cross section for the base

circle radius corresponding to points 119860 and 119861 in profilerespectively 120579

119860and 120579

119861are the spread angle of an involute

curve of the measured gear profile corresponding to points119860 and 119861 in profile respectively 119877

1is the equivalent radius of

the spherical probe and can be expressed as 1198771= 119877 sdot cos120573

119887

where119877 is the radius of the spherical probe and 120573119887 is the helixangle on the base circle

On the other hand

120601119860= ang119860119874119883 minus ang119861

1015840119874119883

= arccos119903119887

1199090

minus 1205720minus (arccos

119903119887

119909minus 120572119909)

(6)

In the above equations rotary angle 120601119860119861

is expressed as

120601119860119861 =

radic1199092 minus 1199032

119887

119903119887

minus

radic1199092

0minus 1199032

119887

119903119887

minus arccos119903119887

119909+ arccos

119903119887

1199090

(7)

Generally speaking when a gear is measured on grindingmachine using the polar coordinate method the probe willbe set in the radial direction of the workpiece in which it isdoing rotary motion along its axis As soon as it touches thesurface a signal is sent to the gear form grinding machinecontroller which stopsmovements ofmachine axes such as119862119883 and 119885 So the control system can acquire the real positionof the machine which is converted into coordinates of thecontact point of the probe tip with gear tooth surface Whenthe probe moves one of the contacts breaks and a binarysignal comes out from the probe head Then this binarysignal is converted into the grinding machine controllerimmediately The controller gets the output signal of theprobe and latches actual angle of turntable 1206011015840

119860119861 Comparing

the achieved angle 1206011015840119860119861

and its theoretical value 120601119860119861 error

was calculated for the tooth profileThe deviation of the toothsurface is given by

Δ120588 = 119903119887 (120601119860119861 minus 1206011015840

119860119861) (8)

During grinding process the probe is placed below thewheel so that it cannot be destroyed When measurementis needed to be done grinding is stopped automatically andprobe is stretched out by the air cylinder Then the touchprobe is taken at a position which is at such a distance apartfrom the tooth surface The probe tip will not touch theworkpiece while it moves over the workpiece in 119883 and 119885

directions When the probe touches the tooth surface thecontroller records the coordinate values of axes 119862 119883 and 119885immediately

23 Determining of the Probe Centre Helical gears that trans-form rotation between parallel axes in opposite directionsare in external meshing and are provided with screw toothsurfaces of opposite directions A helical gear tooth surface isgenerated by an involute curve that performs a screwmotionSegmented topologicalmodification crowns the tooth surfacein profile and longitudinal directions at the same time Profilesegment modification is achieved by modifying the grinding

4 Mathematical Problems in Engineering

gyc

za1c

rbxa1

xcR1

u1

M

N

h

C

C0

120572

1205901

ya1

120601

Oa1c

Figure 3 Coordinate systems applied to solve the centre of sphericalprobe coordinates

wheel and longitudinal segment modification is completedby changing the grinding path of the wheel relatively to thegear

As shown in Figure 3 given an involute curve ℎ119892 in thecross section of tooth surfaces of gear if you want to measuretooth profile error of any point such as119872 point you may seta position in normal direction of the involute curve at point119872 That is to say the centre of spherical probe is located inthe extension cord of tangent119873119872 on the base circle

Coordinate systems 1198781(1199091 1199101 1199111) and 119878

119886(119909119886 119910119886 119911119886) are

rigidly connected to the workpiece and rotary table respec-tively So coordinate system 119878

1(1199091 1199101 1199111) is namedworkpiece

coordinate system Coordinate system 119878119888(119909119888 119910119888 119911119888) is the

measurement coordinate system in which the probe rotatesthe workpiece axis The coordinate system 119878

119888initially coin-

cides with 119878119886 Due to no displacement in 119884

1axial direction as

the probe in the measurement the centre of spherical probeonly always can be moved along the119883

1axial direction

Any point such as point 119872(1199091119872 1199101119872 1199111119872) on the tooth

surface measured using on-machine measurement has coor-dinates in the workpiece coordinate system 119878

1(1199091 1199101 1199111)

which can be expressed as

1199091119872

= 119903119887cos (120590

1+ 1199061) + 1199031198871199061sin (120590

1+ 1199061)

1199101119872

= 119903119887sin (120590

1+ 1199061) minus 1199031198871199061cos (120590

1+ 1199061)

1199111119872

= 0

(9)

where 1199061 is the spread angle of an involute 1205901 is half of theangular width of the space on the base circle and 119903119887 is theradius of the base cylinder

Point 119862(1199091119862 1199101119862 1199111119862) the centre of spherical probe maybe expressed in coordinate system 1198781 by

1199091119862 = 1199091119872 + 119877 cos120573119887 sin (1205901 + 1199061)

1199101119862= 1199101119872

minus 119877 cos120573119887cos (120590

1+ 1199061)

1199111119862 = 119877 sin120573119887

(10)

where 120573119887is the helix angle on the base circle

To keep the centre of spherical probe in the coordinateplane 119909

111990011199111 the measurement coordinate system 119878

119888 the

workpiece and the probe should be together rotated by anangle 120601 around the 119885

1axis The transformation matrix 119878

119888to

1198781is expressed as follows

M1119862=

[[[[[

[

cos120601 minus sin120601 0 0

sin120601 cos120601 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(11)

For the rotated point119862 its coordinate point119862(11990911198620

11991011198620

11991111198620

) the centre of spherical probe is expressed in coordinatesystem 119878

1by

[[[[[

[

cos120601 minus sin120601 0 0

sin120601 cos120601 0 0

0 0 1 0

0 0 0 1

]]]]]

]

[[[[[

[

1199091119862

1199101119862

1199111119862

1

]]]]]

]

=

[[[[[

[

11990911198620

11991011198620

11991111198620

1

]]]]]

]

(12)

Because of the centre of spherical probe in the coordinateplane 119909

111990011199111from beginning to end point 119862 is equal to zero

in 1198841direction which can be expressed as

11991011198620

= 0 (13)

Therefore comparing (12) with (13) satisfies the followingrelation

1199091119862

sin120601 + 1199101119862

cos120601 = 0 (14)

So the rotated angle 120601 can be determined as follows

120601 = minus arctan1199101119862

1199091119862

= arctanminus1199101119872

+ 119877 cos120573119887cos (120590

1+ 1199061)

1199091119872

+ 119877 cos120573119887sin (120590

1+ 1199061)

(15)

Consequently the coordinates (11990911198620 11991011198620 and 11991111198620) can

be obtained in coordinate system 1198781 Coordinate system 119878

1is

introduced by the tooth surface generationmodeling and it isrigidly fixed to the gear with 119885

1axis coinciding with the axis

of the gear Hence

11990911198620 = [1199091119872 + 119877 cos120573119887 sin (1205901 + 1199061)] cos120601

minus [1199101119872

minus 119877 cos120573119887cos (120590

1+ 1199061)] sin120601

11991011198620 = 0

11991111198620

= 1199111119872

+ 119877 sin120573119887

(16)

24 Generation of Grid Point of Tooth Surface Formeasuringtheoretical data the tooth surface equation of measured gearmust be known Assuming the grinding parameters of thegear pair are known a series of coordinate transformationand derivation processes are needed to get the tooth surfaceequation Here the topological modification is carried out

Mathematical Problems in Engineering 5

123

451

II

I

2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

1

2

3

4

5j

j

i

i

Figure 4 Definition of tooth flank grid

on the driving gear on the other hand the driven gear is astandard involute gear For the topologicalmodification toothsurface tooth surface measurement areas need to be plannedto get the whole tooth surface error information Accordingto the relevant standards of American gear measurement thedivision of tooth surface grid in the longitudinal directionshould be less than 10 of the tooth width in the profiledirection it should be less than 5 of the working depth butmust be less than or equal to 06mm The density of toothsurfacemeasurement point is generally taken to be 9 columnslong in the longitudinal direction and taken to be 5 rows inthe profile direction a total of 45 points The tooth surfacemeasurement path planning is shown in Figure 4

The plane coordinate system XOZ is an axis of rotationprojection section of the tooth surface where 119874 is centrepoint of base circle and119872lowast is a point in the projection surfaceat any measured point in the tooth surface Given a point by(119909lowast 119911lowast) the following system of nonlinear equations can beused to solve for the surface parameters

radic1199092

119894+ 1199102

119894= 119909lowast

119911119894= 119911lowast

(17)

where the coordinates (119909119894 119910119894 and 119911

119894) are three coordinate

components of tooth surface position vector r(119894)1 119894 = 1 2

45The tooth surface of topological modification is no longer

the involute tooth surface and becomes complex because ofthe existence of machining error So the whole tooth surfaceneeds to be digitally processed to get more accurate toothsurface geometric shape namely dividing grid on the toothsurface and computing the coordinate data of the grid pointFigure 5 shows the partition of the grid point inside gear shaftsection after rotation projection in which the grid points aredivided into 5 rows by 9 columns

In Figure 5 1198601 1198602 1198603 and 1198604 are the four boundarypoints of the tooth surface 1198601015840

1 11986010158402 11986010158403 and 119860

1015840

4are the

four new boundary points corresponding to tooth surfacecontraction grid area Here Δ119897

1is the top contraction Δ119897

2is

the front contraction Δ1198973is the root contraction and Δ119897

4is

the back contraction respectively The coordinate of a node119875119894119895(119894 = 1 sim 5 119895 = 1 sim 9) in the tooth surface grid is (119885

119894119895 119877119894119895)

X

O

1

5

9

A2

A9984001

A1

A9984002 A998400

3

A9984004

A4

ZZij

Rij

Pij

A3

Figure 5 Positions of grid points on one tooth

The coordinates of the tooth surface boundary points1198601

1198602 1198603 and 119860

4in the XOZ plane are respectively shown as

follows

1198601

1198851198601= 0

1198831198601= 119903119891

1198602

1198851198602 = 0

1198831198602 = 119903119886

1198603

1198851198603= 119887

1198831198603= 119903119886

1198604

1198851198604= 119887

1198831198604= 119903119891

(18)

here 119903119891is the tooth root radius 119903

119886is the tooth top radius and

119887 is the tooth width respectively

6 Mathematical Problems in Engineering

The new coordinates of tooth surface boundary points1198601015840

11198601015840211986010158403 and1198601015840

4in the XOZ plane are respectively shown

as follows

1198601015840

1

1198851198601 = Δ1198974

1198831198601 = 119903119891 + Δ1198973

1198601015840

2

1198851198602= Δ1198974

1198831198602= 119903119886minus Δ1198971

1198601015840

3

1198851198603= 119887 minus Δ119897

2

1198831198603= 119903119886minus Δ1198971

1198601015840

4

1198851198604 = 119887 minus Δ1198972

1198831198604 = 119903119891 + Δ1198973

(19)

The coordinates (1198851119895 1198831119895) (119895 = 1 sim 9) of 9 equal-division

points 1198751119895 between119860

1015840

1and1198601015840

4in the XOZ plane are shown as

follows

1198851119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

1198831119895= 119903119891+ Δ1198973

(20)

The coordinates (1198855119895 1198835119895) (119895 = 1 sim 9) of 9 equal-division

points 1198755119895between1198601015840

2and1198601015840

3in the XOZ plane are shown as

follows

1198855119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

1198835119895= 119903119886minus Δ1198971

(21)

The coordinates (1198851198941 1198831198941) (119894 = 1 sim 5) of 9 equal-division

points 1198751198941between1198601015840

1and1198601015840

2in the XOZ plane are shown as

follows1198851198941= Δ1198974

1198831198941 = 119903119891 + Δ1198973 +

119894 minus 1

4(119903119886 minus 119903119891 minus Δ1198971 minus Δ1198973)

(22)

The coordinates (1198851198949 1198831198949) (119894 = 1 sim 5) of 9 equal-division

points 1198751198949between1198601015840

4and1198601015840

3in the XOZ plane are shown as

follows1198851198949 = 119887 minus Δ1198972

1198831198949= 119903119891+ Δ1198973+119894 minus 1

4(119903119886minus 119903119891minus Δ1198971minus Δ1198973)

(23)

From the above derivational process the coordinates(119885119894119895 119883119894119895) (119894 = 1 sim 5 119895 = 1 sim 9) of any point 119875

119894119895in the

tooth surface grid in the XOZ plane can be calculated by theformulae that are shown as follows

119885119894119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

119883119894119895= 119903119891+ Δ1198973+119894 minus 1

4(119903119886minus 119903119891minus Δ1198971minus Δ1198973)

(24)

Gear

Theory profile

Trace of probe

Sensitive direction

Y

XΔEg

R

Mi

M998400i

120601

Ci

ni

re(u1 1205791)

r1(u1 1205791)

Figure 6 Relationships of the probe geometry and theory profile

Solving nonlinear equations to get the tooth surfaceparameters (119906

1 1205791) for eachmeasured point which substitute

into (2) can get the theoretical coordinates (119909(119894)1 119910(119894)1 and 119911(119894)

1)

and the unit normal vector n119894 Hence

r(119894)1= [119909(119894)

1119910(119894)

1119911(119894)

1]119879

n(119894)1= [119899(119894)

1119909119899(119894)

1119910119899(119894)

1119911]119879

(119894 = 1 2 2 times 45)

(25)

25Measurement Error Compensation When a point ismea-sured on the modification gear tooth surface the inspectiondatum of an on-machine measurement is the centre of theprobe To improve the measurement accuracy the proberadius can be considered in calculating the profile errorsFigure 6 schematically shows the effect of the probe sizeon the measurement As the probe always has a certaindimension an errorΔ119864119892 is unavoidable Due to the size effectof the probe the actual contact point of the probe is at point119872119894 instead of1198721015840

119894

According to the envelope surface characteristics thenormal vector is perpendicular to the probe sphere andpassed through the centre of the probe 119862119894 The theoreticalmotion trajectory of probe centre is shown as follows

r119890 (1199061 1205791) = r

1 (1199061 1205791) + 119877 sdot n1 (1199061 1205791) (26)

here r119890(1199061 1205791) is the trace of the probe centre and 119877 is theprobe radius

Strictly speaking the measuring tooth surface deviationis the deviation of the actual contact point 119872119894 in thesensitive direction of the probe so an important factor of themeasurement error is the actual contacting position of theprobe In the measurement the probe is also very close tothe contact pointmicroscopically the nearness of the contactpoint119872

119894can approximately be regarded as a small planeThe

Mathematical Problems in Engineering 7

Table 1 Parameters of the gears in the trials

Items Symbol Unit DataDriving gear 119911

1mdash 30

Driven gear 1199112

mdash 30Normal model 119898

119899mm 65

Pressure angle 120572119899

deg 20Helix angle 120573 deg 13Face width 119887 mm 53Top profile crowning 119886

119898119901(119888119891)1mm 00011

Bottom profile crowning 119886119898119901(119889119890)

1mm 00014Top limit angle 119906

119888rad 0496

Bottom limit angle 119906119889

rad 0196Front longitudinal crowning 119886

119898119897(ℎ119886)1mm 00012

Back longitudinal crowning 119886119898119897(119887119895)

1mm 00012Front limit angle 120579

119886rad 0018

Back limit angle 120579119887

rad 0104

measurement error caused by probe radius is Δ119864119892 and its

geometric relationships with probe radius can be expressedas follows

Δ119864119892= 119877(

1

cos120601minus 1) (27)

According to (27) the larger the probe radius 119877 thegreater the measurement error so the probe radius shouldbe small here the radius 119877 is 2mm The normal direction ofthe measuring points and the sensitivity of the probe are alsoincluded in the XOY plane and experimental results showthat the angle 120601 is very small

From Figure 6 it can be seen that the measured headshould be in contact with the theoretical point 1198721015840

119894 and the

119883 119884 direction of the coordinates of the value of a certainamount of compensation can be guaranteed Assuming that119872119894 coordinates are (119909119894 119910119894)119872

1015840

119894coordinates are (1199091015840

119894 1199101015840

119894) so the

relationship between the two expressions is

1199091015840

119894= 119909119894minus 119877119905119892120601

1199101015840

119894= 119909119894 + 119877119905119892120601 sin120601

(28)

The contact points are in agreement with the theoreticalcontact points after compensationThe contact position of themeasuring head does not affect the measurement results

3 Numerical Example

The parameters of the gears are listed in Table 1 The drivinggear is modified in the profile and longitudinal directionsat the same time On the contrary the tooth surface of thedriven gear is a conventional screw involute surface Thencoordinates on the tooth surface are measured and calculatedusing a MATLAB code

The measurement data of tooth surface can be obtainedby measuring the grid points in the order with the probeSelecting the radius of the probe 119877 = 2mm the probecentre trajectory curve is the envelope surface of measured

90

95

100

105

110

01234

0

20

40

60 Rotated topological pointTrace of the centre

of the probe

minus1minus2

y (mm)

x (mm)

z(m

m)

Figure 7The contrast position of rotated topological point and theprobe centre

tooth surface When the tooth surface is detected using polarcoordinate method the trajectory coordinates of the probecentre and the gear rotating angle 120601 are calculated using thecontrol unitThe relationship of relative position between thetopology points in the rotated tooth surface and the trajectoryof probe centre is shown in Figure 7

4 Tooth Flank Errors Separation

According to the predetermined spacing the tooth surfacedeviation of each measured tooth surface detection pointis point-to-point measured by probe along the profile andlongitudinal directions of the gear r0(1199061 1205791) is the standardinvolute tooth surface vector r1(1199061 1205791) is the theoreticaltopological modification tooth surface vector and r1119904(1199061 1205791)is the actual tooth surface vector after grinding respec-tively Due to the error of the machine tool adjustmentand movement the actual tooth surface often deviates fromthe theoretical tooth surface The tooth surface error Δ119864is usually measured along the direction of the unit normalvector n

1and is presented as

Δ119864 = (r1119904(1199061 1205791) minus r1(1199061 1205791)) sdot n1(1199061 1205791) (29)

According to the standard involute tooth surface vectorr0(1199061 1205791) and themodified tooth surface vector r

1(1199061 1205791) the

topological modification amount 120575 can be obtained which isshown as follows

120575 = (r1(1199061 1205791) minus r0(1199061 1205791)) sdot n0(1199061 1205791) (30)

If the tooth surface topology deviation is expressed asΔ120588the tooth surface error Δ119864 can be expressed by Δ120588 and 120575 asfollows

Δ119864 = Δ120588 minus 120575 (31)

Equation (31) shows that the tooth surface error can beobtained by removing the modification amount out of thetopology deviation of tooth surface

8 Mathematical Problems in Engineering

Table 2 The modification amount and tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00564 minus00436 minus00314 minus00257 minus00239 minus00285 minus00291 minus00385 minus004972 minus00449 minus00387 minus00215 minus00159 minus00145 minus00147 minus00214 minus00289 minus004363 minus00406 minus00302 minus00197 minus00115 minus00027 minus0012 minus00173 minus00237 minus003914 minus00479 minus00421 minus00236 minus00214 minus00136 minus00234 minus00257 minus00356 minus004125 minus00535 minus00509 minus00352 minus00298 minus00172 minus00287 minus00327 minus00492 minus00518

Left flankmm

1 2 3 4 5 6 7 8 95 minus00524 minus00487 minus00301 minus00272 minus00196 minus00264 minus00314 minus00501 minus005444 minus00498 minus00410 minus00198 minus00189 minus00035 minus00197 minus00241 minus00417 minus004633 minus00436 minus00375 minus00221 minus00035 minus00021 minus00046 minus00210 minus00324 minus004522 minus00510 minus00417 minus00183 minus00167 minus00138 minus00153 minus00189 minus00394 minus005311 minus00571 minus00462 minus00294 minus00285 minus00227 minus00249 minus00267 minus00472 minus00592

Table 3 The tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00044 minus00047 minus00046 minus00046 minus00062 minus00062 minus00066 minus00061 minus000612 minus00061 minus00064 minus00063 minus00064 minus00079 minus00079 minus00083 minus00077 minus000793 minus00058 minus00061 minus00062 minus00063 minus00075 minus00076 minus00084 minus00074 minus000754 minus00056 minus00059 minus00058 minus00059 minus00074 minus00074 minus00078 minus00072 minus000745 minus00048 minus00051 minus00050 minus00051 minus00066 minus00066 minus00071 minus00064 minus00066

Left flankmm

1 2 3 4 5 6 7 8 95 minus00053 minus00057 minus00055 minus00056 minus00057 minus00058 minus00087 minus00054 minus000654 minus00062 minus00062 minus00068 minus00072 minus00084 minus00067 minus00086 minus00039 minus000743 minus00049 minus00074 minus00072 minus00058 minus00054 minus00069 minus00094 minus00075 minus000872 minus00072 minus00049 minus00085 minus00063 minus00072 minus00069 minus00064 minus00058 minus000761 minus00088 minus00076 minus00067 minus00053 minus00069 minus00072 minus00073 minus00067 minus00069

The tooth surface topology deviation Δ120588 is shown inTable 2 and the tooth surface error Δ119864 is shown in Table 3respectively

The deviation distribution (error and modificationamount) of the tooth surface obtained by an on-machinemeasurement is shown in Figure 8 The dashed parts (thesuperposition of error and modification) in regions 2 46 and 8 of the tooth surface are obviously presented as aparabolic shape in the profile and longitudinal directionsrespectivelyThemaximumvalue of deviation at the tooth topis minus00571mm and the error in the centre of tooth surfacearea 1 is minus00049mm The trend is gradually increasedtowards both sides presenting the shape where the middle ishigh and both sides are low

The measurement using the topological modificationtooth surface equation is shown in Figure 9 The measuredresult reflects the actual tooth surface error the distributionof which has unobvious trend The maximum error in thetooth top is minus00088mm and the error in the centre of righttooth surface area 1 is minus00054mm

To separate flank form deviations from modificationamount and to improve the perception of deviation dia-grams all flank deviations are referenced by the modificationamount The characterization of the tooth surface error andmodification amount are shown in Figures 10 and 11 Themaximum error is about minus88 um in the upper and lower

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus564

minus535

minus524

minus571

minus497

minus518

minus544

minus592

Top land

Figure 8 Topography diagram of modification amount and error(unit um)

surface but the minimum error is about minus54 um which isconcentrated in the central area of the profile and longitudinaldirections The normal error of any intersection of profiledirection and longitudinal direction is obtained by the errorof the contour map

Mathematical Problems in Engineering 9

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus44

minus48

minus56

minus88

minus61

minus66

minus65

minus69

Top land

Figure 9 Topography diagram of error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0minus002minus004minus006minus008

Mod

ifica

tion

amou

nt

and

erro

r (m

m)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)Longitudinal (mm)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

Mod

ifica

tion

amou

nt

and

erro

r (m

m) 0

minus002minus004minus006minus008

(b) Left flank

Figure 10 The contour map of modification amount and error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

times10minus3

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

Longitudinal (mm)

minus4

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

minus9

times10minus3

(b) Left flank

Figure 11 The contour map of error (unit um)

5 Conclusions

In this study an on-machine profile measurement systemalong with a five-axis CNC gear form grinding machineis developed to improve accuracy of grinding This system

includes the following steps (1) calculate trajectory of thecentre of spherical probe (2) define gear flanks by a grid ofpoints (3) obtain the coordinate values of topology measure-ment points Grinding experiments are performed to verifythe accuracy and efficiency of the topographymeasurements

10 Mathematical Problems in Engineering

With contour map the profile and longitudinal directionsof the error changes are easily seen Using statistical processcontrol techniques to monitor the grinding process cantimely attain the detection of changes in the product errorexceptions and take the necessary measures to prevent theoccurrence of waste As the numerical examples show on-machine measurement method can inspect tooth qualityAn additional advantage of the approach is to characterizemodification amount and tooth surface error respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the National Natural ScienceFoundation of China for its financial support Part of thiswork was performed under Contracts no 51375144 and no51405135 and the Priority Project of Research in Universitiesin Henan Province (Grant no 15A460021)

References

[1] H L Wang X Z Deng and J J Yang ldquoForm grinding andexperiment on segment topographicmodification gearrdquo Journalof Aerospace Power vol 29 no 12 pp 3000ndash3008 2014

[2] H-Y You P-Q Ye J-S Wang and X-Y Deng ldquoDesign andapplication of CBN shape grinding wheel for gearsrdquo Interna-tional Journal of Machine Tools and Manufacture vol 43 no12 pp 1269ndash1277 2003

[3] Q Fan R SDafoe and JW Swanger ldquoHigher-order tooth flankform error correction for face-milled spiral bevel and hypoidgearsrdquo Journal ofMechanical DesignmdashTransactions of theASMEvol 130 no 7 Article ID 072601 2008

[4] J Argyris M De Donno and F L Litvin ldquoComputer programin visual basic language for simulation of meshing and contactof gear drives and its application for design of worm gear driverdquoComputer Methods in Applied Mechanics and Engineering vol189 no 2 pp 595ndash612 2000

[5] Z L Zhang Y Fu Q R Yin and Y Zeng ldquoStudy on the polarmethod of measuring of gear profile errorsrdquo Chinese Journal ofMechanical Engineering vol 37 no 4 pp 70ndash72 2001

[6] C H Gao K Cheng and D Webb ldquoInvestigation on samplingsize optimisation in gear tooth surface measurement using aCMMrdquo The International Journal of Advanced ManufacturingTechnology vol 24 no 7-8 pp 599ndash606 2004

[7] Z L Zhang Y Fu and Y Zeng ldquoExpressing gear involuteerror by polar angle amp generating angle in polar coordinatemeasuring methodrdquo Tool Engineering vol 34 no 4 pp 39ndash402000

[8] F Gao B H Zhao and Y Li ldquoNovel pre-travel calibrationmethod of touch trigger probe based on error separationrdquoChinese Journal of Scientific Instrument vol 34 no 7 pp 1581ndash1587 2013

[9] F Gao Y Li S Tian Y Huang L Hao and J Wang ldquoStudy onthe on-machine measurement method of NC wheel gear formgrinding machinerdquo Chinese Journal of Scientific Instrument vol29 no 3 pp 540ndash544 2008

[10] A Nafi J R R Mayer and A Wozniak ldquoNovel CMM-basedimplementation of the multi-step method for the separation ofmachine and probe errorsrdquo Precision Engineering vol 35 no 2pp 318ndash328 2011

[11] A Nafi J R RMayer andAWozniak ldquoReduced configurationset for the multi-step method applied to machine and probeerror separation on a CMMrdquo Measurement vol 45 no 10 pp2321ndash2329 2012

[12] Y-P Shih and S-D Chen ldquoFree-formflank correction in helicalgear grinding using a five-axis computer numerical control gearprofile grindingmachinerdquo Journal ofManufacturing Science andEngineering vol 134 no 4 Article ID 041006 2012

[13] Y Kobayashi N Nishida Y Ougiya and H Nagata ldquoToothtrace modification processing of helix gear by form grindingmethodrdquo Transactions of the Japan Society of Mechanical Engi-neers Part C vol 61 no 590 pp 4088ndash4093 1995

[14] Y Kobayashi N Nishida and Y Ougiya ldquoEstimation ofgrinding wheel setting error in helical gear processing byform grindingrdquo Transactions of the Japan Society of MechanicalEngineers Part C vol 63 no 612 pp 2852ndash2858 1997

[15] C-K Lee ldquoManufacturing process for a cylindrical crown geardrive with a controllable fourth order polynomial function oftransmission errorrdquo Journal of Materials Processing Technologyvol 209 no 1 pp 3ndash13 2009

[16] C K Lee and C K Chen ldquoMathematical models meshinganalysis and transmission design for robust cylindrical gear setgenerated by double blade-disks with parabolic cutting edgesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 218 no 12 pp1539ndash1553 2004

[17] Z Y Shi and Y Ye ldquoResearch on the generalized polar-coordinate method for measuring involute profile deviationsrdquoChinese Journal of Scientific Instrument vol 22 no 2 pp 140ndash142 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

gear drive that could reproduce precisely the predesignedfourth-order polynomial function of transmission error

The traditional tooth surface measurement is measuredin the central line of profile and longitudinal direction withreference to the standard involute helical surface modifica-tion as well as the error Due to tooth surface for topologicalmodification this measure cannot satisfy the modificationdesign and processing requirements The traditional toothsurface measurement process is to measure the tooth flankerrors when the profile error is in the central tooth heightand the longitudinal error is in the central tooth widthrespectively Bymeasuring the topographymeasurement canobtain the modification amount as well as the error withreference to the standard involute helical surface For thetopological modification tooth surface this measure processcannot satisfy the requirements of the modification designand machining Therefore how to implement topographymeasurement and dominate completely the tooth surfaceerror information is the key points of this study for thetopological modification tooth surface

The outline of the remainder of the paper is as followsIn Section 2 an on-machine measurement model of helicalgear based on gear form grinding is developed A numericalexample is the main subject of Section 3 In Section 4 thecharacterization of tooth surface error and modificationamount are shown respectively Finally some conclusions aredrawn in Section 5

2 On-Machine Measurement

21 Coordinate Systems of Measurement Based on the topo-logical modification surface equation as a theory equationhelical gear flanks are defined by a grid of points where anelementary sketch affects the ideal tooth surface Accordingto the measuring probe along the tooth surface measurementpath planning the measured displacement is recorded by adata acquisition system The topography diagram is drawnaccording to the tooth surface errors which can observeclearly the error distribution of 45 points on the tooth surfaceand provide the theoretical basis of the tooth surface errorfeedback correction and closed-loop manufacture of helicalgears

The measurement of the helical gear is inspected alongthe radial workpiece using straight probe which is shown inFigure 1 During machining and measuring the workpiecewhether the locating datum is consistent is the key Thepremise of accurately measuring the tooth surface devi-ation on the machine is that the space position of themeasuring coordinate system and the space position of theworkpiece coordinate system coincide too The origin ofthe measurement coordinate system 119878119862 is locating at point119862(1199091119862 1199101119862 1199111119862)which is the centre of rotary tableWhen geargrinding andmeasuring it is locatedwith datum end face andaxis using rotary table and top respectively The theoreticaltooth surface design coordinate system origin and work-piece coordinate system origin coincide as 119874

1 On-machine

measurement can improve the accuracy of grinding toothsurface due to avoiding the reinstallation of the workpieceThe relative position of the space of the coordinate systems

Probe

TipMeasured gear

Rotary table

Reference axis

Reference faceTip

Zc

Za Z1

X1

Xa

Xc

Y1

Yc

Ya Oc

O1a

L

120593

Figure 1 Coordinate systems of on-machine measurement

is shown in Figure 1 1198781(1198831 1198841 1198851) is workpiece coordinate

system 119878119888(119883119888 119884119888 119885119888) is measuring coordinate system 119871 is

the axial distance of the two coordinate system origins and120593 is tooth surface rotation angle The space position relationbetween coordinate system 1198781 and coordinate system 119878119888 isdetermined by the transformation matrixM1198881

The tooth flank form geometry can be generally repre-sented by a position vector and a unit normal vector in theworkpiece coordinate system 1198781 as

r1= r1(1199061 1205791)

n1 = n1 (1199061 1205791) (1)

here (1199061 1205791) are generating surface parameters

For the accurate detection of actual processing toothsurface the theoretical tooth surface should be transformedto the gear measuring coordinate system 119878

119862 by coordinate

transformation which is obtained as follows

r119888= M1198881r1(1199061 1205791)

n119888 = M1198881n1(1199061 1205791)

(2)

here

M1198881=

[[[[[

[

cos120593 sin120593 0 0

minus sin120593 cos120593 0 0

0 0 1 119871

0 0 0 1

]]]]]

]

(3)

22 Mathematical Model of Measurement The use of a polarcoordinate to inspect gears is becoming increasingly usefulin modern gear manufacturing especially on gear formgrinding On-machine measurement of gear profile error isarranged based on the CNC gear form grinder which hasfive digital servo closed-loop controlled axes three rectilinearmotions (119883119884 and 119885) and two rotational motions (119860119862)Here 119862 is the rotation axis of work gear and 119883 is the radialmotion for feeding thewheel down to tooth depthThe theory

Mathematical Problems in Engineering 3

A

x

X

B

Y

O

x0

rb

D

C

120572x

1205720

120601A 120579A

120579B

O0 O1

R1

120601AB

120601BB998400

Figure 2 Schematic illustration of polar coordinate measurementmethod

involute curve is achieved by axes 119862 and 119883 when they aremoving at the same time Because of the motion limit in axial119884direction polar coordinatemeasurement is amajormethodof on-machine measurement when measuring profile erroron the CNC gear form grinder The mathematical modelof polar coordinate method measurement is establishedaccording to the workpiece rotary movement (119862 axis) andradial feed movement (119883 axis) [17]

On-machine measurement system can not only measureprofile error and helical error but also measure pitch errorof helical gears Using the measurement data analysis ofthese errors which may correct processing parameters andaxial motion equation of polynomial coefficients realizes theclosed-loop feedback processing of tooth surface modifica-tion thus laying the foundation for improving the precisionof tooth surface

The polar coordinates of spherical probemethod are usedto measure the involute tooth profile as shown in Figure 2 119860is the measuring start point 119861 is the measuring point 1198611015840 isthe actual location of measuring point 119861 120601119860119861 is an angle ofrotary motion for C axis which can be expressed as

120601119860119861

= 120601119860+ 120601119861 (4)

where

120601119861 = 120579119861 minus 120579119860

120579119860= tan120572

0minus 1205720=

radic1199092

0minus 1199032

119887minus 1198771

119903119887

minus 1205720

120579119861= tan120572

119909minus 120572119909=

radic1199092 minus 1199032

119887minus 1198771

119903119887

minus 120572119909

(5)

where 119903119887is the radius of the base cylinder variable parameters

1199090and 119909 are coordinate data of probe centre corresponding

to start measuring points 119860 and 119861 respectively 1205720and

120572119909are the pressure angle in the cross section for the base

circle radius corresponding to points 119860 and 119861 in profilerespectively 120579

119860and 120579

119861are the spread angle of an involute

curve of the measured gear profile corresponding to points119860 and 119861 in profile respectively 119877

1is the equivalent radius of

the spherical probe and can be expressed as 1198771= 119877 sdot cos120573

119887

where119877 is the radius of the spherical probe and 120573119887 is the helixangle on the base circle

On the other hand

120601119860= ang119860119874119883 minus ang119861

1015840119874119883

= arccos119903119887

1199090

minus 1205720minus (arccos

119903119887

119909minus 120572119909)

(6)

In the above equations rotary angle 120601119860119861

is expressed as

120601119860119861 =

radic1199092 minus 1199032

119887

119903119887

minus

radic1199092

0minus 1199032

119887

119903119887

minus arccos119903119887

119909+ arccos

119903119887

1199090

(7)

Generally speaking when a gear is measured on grindingmachine using the polar coordinate method the probe willbe set in the radial direction of the workpiece in which it isdoing rotary motion along its axis As soon as it touches thesurface a signal is sent to the gear form grinding machinecontroller which stopsmovements ofmachine axes such as119862119883 and 119885 So the control system can acquire the real positionof the machine which is converted into coordinates of thecontact point of the probe tip with gear tooth surface Whenthe probe moves one of the contacts breaks and a binarysignal comes out from the probe head Then this binarysignal is converted into the grinding machine controllerimmediately The controller gets the output signal of theprobe and latches actual angle of turntable 1206011015840

119860119861 Comparing

the achieved angle 1206011015840119860119861

and its theoretical value 120601119860119861 error

was calculated for the tooth profileThe deviation of the toothsurface is given by

Δ120588 = 119903119887 (120601119860119861 minus 1206011015840

119860119861) (8)

During grinding process the probe is placed below thewheel so that it cannot be destroyed When measurementis needed to be done grinding is stopped automatically andprobe is stretched out by the air cylinder Then the touchprobe is taken at a position which is at such a distance apartfrom the tooth surface The probe tip will not touch theworkpiece while it moves over the workpiece in 119883 and 119885

directions When the probe touches the tooth surface thecontroller records the coordinate values of axes 119862 119883 and 119885immediately

23 Determining of the Probe Centre Helical gears that trans-form rotation between parallel axes in opposite directionsare in external meshing and are provided with screw toothsurfaces of opposite directions A helical gear tooth surface isgenerated by an involute curve that performs a screwmotionSegmented topologicalmodification crowns the tooth surfacein profile and longitudinal directions at the same time Profilesegment modification is achieved by modifying the grinding

4 Mathematical Problems in Engineering

gyc

za1c

rbxa1

xcR1

u1

M

N

h

C

C0

120572

1205901

ya1

120601

Oa1c

Figure 3 Coordinate systems applied to solve the centre of sphericalprobe coordinates

wheel and longitudinal segment modification is completedby changing the grinding path of the wheel relatively to thegear

As shown in Figure 3 given an involute curve ℎ119892 in thecross section of tooth surfaces of gear if you want to measuretooth profile error of any point such as119872 point you may seta position in normal direction of the involute curve at point119872 That is to say the centre of spherical probe is located inthe extension cord of tangent119873119872 on the base circle

Coordinate systems 1198781(1199091 1199101 1199111) and 119878

119886(119909119886 119910119886 119911119886) are

rigidly connected to the workpiece and rotary table respec-tively So coordinate system 119878

1(1199091 1199101 1199111) is namedworkpiece

coordinate system Coordinate system 119878119888(119909119888 119910119888 119911119888) is the

measurement coordinate system in which the probe rotatesthe workpiece axis The coordinate system 119878

119888initially coin-

cides with 119878119886 Due to no displacement in 119884

1axial direction as

the probe in the measurement the centre of spherical probeonly always can be moved along the119883

1axial direction

Any point such as point 119872(1199091119872 1199101119872 1199111119872) on the tooth

surface measured using on-machine measurement has coor-dinates in the workpiece coordinate system 119878

1(1199091 1199101 1199111)

which can be expressed as

1199091119872

= 119903119887cos (120590

1+ 1199061) + 1199031198871199061sin (120590

1+ 1199061)

1199101119872

= 119903119887sin (120590

1+ 1199061) minus 1199031198871199061cos (120590

1+ 1199061)

1199111119872

= 0

(9)

where 1199061 is the spread angle of an involute 1205901 is half of theangular width of the space on the base circle and 119903119887 is theradius of the base cylinder

Point 119862(1199091119862 1199101119862 1199111119862) the centre of spherical probe maybe expressed in coordinate system 1198781 by

1199091119862 = 1199091119872 + 119877 cos120573119887 sin (1205901 + 1199061)

1199101119862= 1199101119872

minus 119877 cos120573119887cos (120590

1+ 1199061)

1199111119862 = 119877 sin120573119887

(10)

where 120573119887is the helix angle on the base circle

To keep the centre of spherical probe in the coordinateplane 119909

111990011199111 the measurement coordinate system 119878

119888 the

workpiece and the probe should be together rotated by anangle 120601 around the 119885

1axis The transformation matrix 119878

119888to

1198781is expressed as follows

M1119862=

[[[[[

[

cos120601 minus sin120601 0 0

sin120601 cos120601 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(11)

For the rotated point119862 its coordinate point119862(11990911198620

11991011198620

11991111198620

) the centre of spherical probe is expressed in coordinatesystem 119878

1by

[[[[[

[

cos120601 minus sin120601 0 0

sin120601 cos120601 0 0

0 0 1 0

0 0 0 1

]]]]]

]

[[[[[

[

1199091119862

1199101119862

1199111119862

1

]]]]]

]

=

[[[[[

[

11990911198620

11991011198620

11991111198620

1

]]]]]

]

(12)

Because of the centre of spherical probe in the coordinateplane 119909

111990011199111from beginning to end point 119862 is equal to zero

in 1198841direction which can be expressed as

11991011198620

= 0 (13)

Therefore comparing (12) with (13) satisfies the followingrelation

1199091119862

sin120601 + 1199101119862

cos120601 = 0 (14)

So the rotated angle 120601 can be determined as follows

120601 = minus arctan1199101119862

1199091119862

= arctanminus1199101119872

+ 119877 cos120573119887cos (120590

1+ 1199061)

1199091119872

+ 119877 cos120573119887sin (120590

1+ 1199061)

(15)

Consequently the coordinates (11990911198620 11991011198620 and 11991111198620) can

be obtained in coordinate system 1198781 Coordinate system 119878

1is

introduced by the tooth surface generationmodeling and it isrigidly fixed to the gear with 119885

1axis coinciding with the axis

of the gear Hence

11990911198620 = [1199091119872 + 119877 cos120573119887 sin (1205901 + 1199061)] cos120601

minus [1199101119872

minus 119877 cos120573119887cos (120590

1+ 1199061)] sin120601

11991011198620 = 0

11991111198620

= 1199111119872

+ 119877 sin120573119887

(16)

24 Generation of Grid Point of Tooth Surface Formeasuringtheoretical data the tooth surface equation of measured gearmust be known Assuming the grinding parameters of thegear pair are known a series of coordinate transformationand derivation processes are needed to get the tooth surfaceequation Here the topological modification is carried out

Mathematical Problems in Engineering 5

123

451

II

I

2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

1

2

3

4

5j

j

i

i

Figure 4 Definition of tooth flank grid

on the driving gear on the other hand the driven gear is astandard involute gear For the topologicalmodification toothsurface tooth surface measurement areas need to be plannedto get the whole tooth surface error information Accordingto the relevant standards of American gear measurement thedivision of tooth surface grid in the longitudinal directionshould be less than 10 of the tooth width in the profiledirection it should be less than 5 of the working depth butmust be less than or equal to 06mm The density of toothsurfacemeasurement point is generally taken to be 9 columnslong in the longitudinal direction and taken to be 5 rows inthe profile direction a total of 45 points The tooth surfacemeasurement path planning is shown in Figure 4

The plane coordinate system XOZ is an axis of rotationprojection section of the tooth surface where 119874 is centrepoint of base circle and119872lowast is a point in the projection surfaceat any measured point in the tooth surface Given a point by(119909lowast 119911lowast) the following system of nonlinear equations can beused to solve for the surface parameters

radic1199092

119894+ 1199102

119894= 119909lowast

119911119894= 119911lowast

(17)

where the coordinates (119909119894 119910119894 and 119911

119894) are three coordinate

components of tooth surface position vector r(119894)1 119894 = 1 2

45The tooth surface of topological modification is no longer

the involute tooth surface and becomes complex because ofthe existence of machining error So the whole tooth surfaceneeds to be digitally processed to get more accurate toothsurface geometric shape namely dividing grid on the toothsurface and computing the coordinate data of the grid pointFigure 5 shows the partition of the grid point inside gear shaftsection after rotation projection in which the grid points aredivided into 5 rows by 9 columns

In Figure 5 1198601 1198602 1198603 and 1198604 are the four boundarypoints of the tooth surface 1198601015840

1 11986010158402 11986010158403 and 119860

1015840

4are the

four new boundary points corresponding to tooth surfacecontraction grid area Here Δ119897

1is the top contraction Δ119897

2is

the front contraction Δ1198973is the root contraction and Δ119897

4is

the back contraction respectively The coordinate of a node119875119894119895(119894 = 1 sim 5 119895 = 1 sim 9) in the tooth surface grid is (119885

119894119895 119877119894119895)

X

O

1

5

9

A2

A9984001

A1

A9984002 A998400

3

A9984004

A4

ZZij

Rij

Pij

A3

Figure 5 Positions of grid points on one tooth

The coordinates of the tooth surface boundary points1198601

1198602 1198603 and 119860

4in the XOZ plane are respectively shown as

follows

1198601

1198851198601= 0

1198831198601= 119903119891

1198602

1198851198602 = 0

1198831198602 = 119903119886

1198603

1198851198603= 119887

1198831198603= 119903119886

1198604

1198851198604= 119887

1198831198604= 119903119891

(18)

here 119903119891is the tooth root radius 119903

119886is the tooth top radius and

119887 is the tooth width respectively

6 Mathematical Problems in Engineering

The new coordinates of tooth surface boundary points1198601015840

11198601015840211986010158403 and1198601015840

4in the XOZ plane are respectively shown

as follows

1198601015840

1

1198851198601 = Δ1198974

1198831198601 = 119903119891 + Δ1198973

1198601015840

2

1198851198602= Δ1198974

1198831198602= 119903119886minus Δ1198971

1198601015840

3

1198851198603= 119887 minus Δ119897

2

1198831198603= 119903119886minus Δ1198971

1198601015840

4

1198851198604 = 119887 minus Δ1198972

1198831198604 = 119903119891 + Δ1198973

(19)

The coordinates (1198851119895 1198831119895) (119895 = 1 sim 9) of 9 equal-division

points 1198751119895 between119860

1015840

1and1198601015840

4in the XOZ plane are shown as

follows

1198851119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

1198831119895= 119903119891+ Δ1198973

(20)

The coordinates (1198855119895 1198835119895) (119895 = 1 sim 9) of 9 equal-division

points 1198755119895between1198601015840

2and1198601015840

3in the XOZ plane are shown as

follows

1198855119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

1198835119895= 119903119886minus Δ1198971

(21)

The coordinates (1198851198941 1198831198941) (119894 = 1 sim 5) of 9 equal-division

points 1198751198941between1198601015840

1and1198601015840

2in the XOZ plane are shown as

follows1198851198941= Δ1198974

1198831198941 = 119903119891 + Δ1198973 +

119894 minus 1

4(119903119886 minus 119903119891 minus Δ1198971 minus Δ1198973)

(22)

The coordinates (1198851198949 1198831198949) (119894 = 1 sim 5) of 9 equal-division

points 1198751198949between1198601015840

4and1198601015840

3in the XOZ plane are shown as

follows1198851198949 = 119887 minus Δ1198972

1198831198949= 119903119891+ Δ1198973+119894 minus 1

4(119903119886minus 119903119891minus Δ1198971minus Δ1198973)

(23)

From the above derivational process the coordinates(119885119894119895 119883119894119895) (119894 = 1 sim 5 119895 = 1 sim 9) of any point 119875

119894119895in the

tooth surface grid in the XOZ plane can be calculated by theformulae that are shown as follows

119885119894119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

119883119894119895= 119903119891+ Δ1198973+119894 minus 1

4(119903119886minus 119903119891minus Δ1198971minus Δ1198973)

(24)

Gear

Theory profile

Trace of probe

Sensitive direction

Y

XΔEg

R

Mi

M998400i

120601

Ci

ni

re(u1 1205791)

r1(u1 1205791)

Figure 6 Relationships of the probe geometry and theory profile

Solving nonlinear equations to get the tooth surfaceparameters (119906

1 1205791) for eachmeasured point which substitute

into (2) can get the theoretical coordinates (119909(119894)1 119910(119894)1 and 119911(119894)

1)

and the unit normal vector n119894 Hence

r(119894)1= [119909(119894)

1119910(119894)

1119911(119894)

1]119879

n(119894)1= [119899(119894)

1119909119899(119894)

1119910119899(119894)

1119911]119879

(119894 = 1 2 2 times 45)

(25)

25Measurement Error Compensation When a point ismea-sured on the modification gear tooth surface the inspectiondatum of an on-machine measurement is the centre of theprobe To improve the measurement accuracy the proberadius can be considered in calculating the profile errorsFigure 6 schematically shows the effect of the probe sizeon the measurement As the probe always has a certaindimension an errorΔ119864119892 is unavoidable Due to the size effectof the probe the actual contact point of the probe is at point119872119894 instead of1198721015840

119894

According to the envelope surface characteristics thenormal vector is perpendicular to the probe sphere andpassed through the centre of the probe 119862119894 The theoreticalmotion trajectory of probe centre is shown as follows

r119890 (1199061 1205791) = r

1 (1199061 1205791) + 119877 sdot n1 (1199061 1205791) (26)

here r119890(1199061 1205791) is the trace of the probe centre and 119877 is theprobe radius

Strictly speaking the measuring tooth surface deviationis the deviation of the actual contact point 119872119894 in thesensitive direction of the probe so an important factor of themeasurement error is the actual contacting position of theprobe In the measurement the probe is also very close tothe contact pointmicroscopically the nearness of the contactpoint119872

119894can approximately be regarded as a small planeThe

Mathematical Problems in Engineering 7

Table 1 Parameters of the gears in the trials

Items Symbol Unit DataDriving gear 119911

1mdash 30

Driven gear 1199112

mdash 30Normal model 119898

119899mm 65

Pressure angle 120572119899

deg 20Helix angle 120573 deg 13Face width 119887 mm 53Top profile crowning 119886

119898119901(119888119891)1mm 00011

Bottom profile crowning 119886119898119901(119889119890)

1mm 00014Top limit angle 119906

119888rad 0496

Bottom limit angle 119906119889

rad 0196Front longitudinal crowning 119886

119898119897(ℎ119886)1mm 00012

Back longitudinal crowning 119886119898119897(119887119895)

1mm 00012Front limit angle 120579

119886rad 0018

Back limit angle 120579119887

rad 0104

measurement error caused by probe radius is Δ119864119892 and its

geometric relationships with probe radius can be expressedas follows

Δ119864119892= 119877(

1

cos120601minus 1) (27)

According to (27) the larger the probe radius 119877 thegreater the measurement error so the probe radius shouldbe small here the radius 119877 is 2mm The normal direction ofthe measuring points and the sensitivity of the probe are alsoincluded in the XOY plane and experimental results showthat the angle 120601 is very small

From Figure 6 it can be seen that the measured headshould be in contact with the theoretical point 1198721015840

119894 and the

119883 119884 direction of the coordinates of the value of a certainamount of compensation can be guaranteed Assuming that119872119894 coordinates are (119909119894 119910119894)119872

1015840

119894coordinates are (1199091015840

119894 1199101015840

119894) so the

relationship between the two expressions is

1199091015840

119894= 119909119894minus 119877119905119892120601

1199101015840

119894= 119909119894 + 119877119905119892120601 sin120601

(28)

The contact points are in agreement with the theoreticalcontact points after compensationThe contact position of themeasuring head does not affect the measurement results

3 Numerical Example

The parameters of the gears are listed in Table 1 The drivinggear is modified in the profile and longitudinal directionsat the same time On the contrary the tooth surface of thedriven gear is a conventional screw involute surface Thencoordinates on the tooth surface are measured and calculatedusing a MATLAB code

The measurement data of tooth surface can be obtainedby measuring the grid points in the order with the probeSelecting the radius of the probe 119877 = 2mm the probecentre trajectory curve is the envelope surface of measured

90

95

100

105

110

01234

0

20

40

60 Rotated topological pointTrace of the centre

of the probe

minus1minus2

y (mm)

x (mm)

z(m

m)

Figure 7The contrast position of rotated topological point and theprobe centre

tooth surface When the tooth surface is detected using polarcoordinate method the trajectory coordinates of the probecentre and the gear rotating angle 120601 are calculated using thecontrol unitThe relationship of relative position between thetopology points in the rotated tooth surface and the trajectoryof probe centre is shown in Figure 7

4 Tooth Flank Errors Separation

According to the predetermined spacing the tooth surfacedeviation of each measured tooth surface detection pointis point-to-point measured by probe along the profile andlongitudinal directions of the gear r0(1199061 1205791) is the standardinvolute tooth surface vector r1(1199061 1205791) is the theoreticaltopological modification tooth surface vector and r1119904(1199061 1205791)is the actual tooth surface vector after grinding respec-tively Due to the error of the machine tool adjustmentand movement the actual tooth surface often deviates fromthe theoretical tooth surface The tooth surface error Δ119864is usually measured along the direction of the unit normalvector n

1and is presented as

Δ119864 = (r1119904(1199061 1205791) minus r1(1199061 1205791)) sdot n1(1199061 1205791) (29)

According to the standard involute tooth surface vectorr0(1199061 1205791) and themodified tooth surface vector r

1(1199061 1205791) the

topological modification amount 120575 can be obtained which isshown as follows

120575 = (r1(1199061 1205791) minus r0(1199061 1205791)) sdot n0(1199061 1205791) (30)

If the tooth surface topology deviation is expressed asΔ120588the tooth surface error Δ119864 can be expressed by Δ120588 and 120575 asfollows

Δ119864 = Δ120588 minus 120575 (31)

Equation (31) shows that the tooth surface error can beobtained by removing the modification amount out of thetopology deviation of tooth surface

8 Mathematical Problems in Engineering

Table 2 The modification amount and tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00564 minus00436 minus00314 minus00257 minus00239 minus00285 minus00291 minus00385 minus004972 minus00449 minus00387 minus00215 minus00159 minus00145 minus00147 minus00214 minus00289 minus004363 minus00406 minus00302 minus00197 minus00115 minus00027 minus0012 minus00173 minus00237 minus003914 minus00479 minus00421 minus00236 minus00214 minus00136 minus00234 minus00257 minus00356 minus004125 minus00535 minus00509 minus00352 minus00298 minus00172 minus00287 minus00327 minus00492 minus00518

Left flankmm

1 2 3 4 5 6 7 8 95 minus00524 minus00487 minus00301 minus00272 minus00196 minus00264 minus00314 minus00501 minus005444 minus00498 minus00410 minus00198 minus00189 minus00035 minus00197 minus00241 minus00417 minus004633 minus00436 minus00375 minus00221 minus00035 minus00021 minus00046 minus00210 minus00324 minus004522 minus00510 minus00417 minus00183 minus00167 minus00138 minus00153 minus00189 minus00394 minus005311 minus00571 minus00462 minus00294 minus00285 minus00227 minus00249 minus00267 minus00472 minus00592

Table 3 The tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00044 minus00047 minus00046 minus00046 minus00062 minus00062 minus00066 minus00061 minus000612 minus00061 minus00064 minus00063 minus00064 minus00079 minus00079 minus00083 minus00077 minus000793 minus00058 minus00061 minus00062 minus00063 minus00075 minus00076 minus00084 minus00074 minus000754 minus00056 minus00059 minus00058 minus00059 minus00074 minus00074 minus00078 minus00072 minus000745 minus00048 minus00051 minus00050 minus00051 minus00066 minus00066 minus00071 minus00064 minus00066

Left flankmm

1 2 3 4 5 6 7 8 95 minus00053 minus00057 minus00055 minus00056 minus00057 minus00058 minus00087 minus00054 minus000654 minus00062 minus00062 minus00068 minus00072 minus00084 minus00067 minus00086 minus00039 minus000743 minus00049 minus00074 minus00072 minus00058 minus00054 minus00069 minus00094 minus00075 minus000872 minus00072 minus00049 minus00085 minus00063 minus00072 minus00069 minus00064 minus00058 minus000761 minus00088 minus00076 minus00067 minus00053 minus00069 minus00072 minus00073 minus00067 minus00069

The tooth surface topology deviation Δ120588 is shown inTable 2 and the tooth surface error Δ119864 is shown in Table 3respectively

The deviation distribution (error and modificationamount) of the tooth surface obtained by an on-machinemeasurement is shown in Figure 8 The dashed parts (thesuperposition of error and modification) in regions 2 46 and 8 of the tooth surface are obviously presented as aparabolic shape in the profile and longitudinal directionsrespectivelyThemaximumvalue of deviation at the tooth topis minus00571mm and the error in the centre of tooth surfacearea 1 is minus00049mm The trend is gradually increasedtowards both sides presenting the shape where the middle ishigh and both sides are low

The measurement using the topological modificationtooth surface equation is shown in Figure 9 The measuredresult reflects the actual tooth surface error the distributionof which has unobvious trend The maximum error in thetooth top is minus00088mm and the error in the centre of righttooth surface area 1 is minus00054mm

To separate flank form deviations from modificationamount and to improve the perception of deviation dia-grams all flank deviations are referenced by the modificationamount The characterization of the tooth surface error andmodification amount are shown in Figures 10 and 11 Themaximum error is about minus88 um in the upper and lower

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus564

minus535

minus524

minus571

minus497

minus518

minus544

minus592

Top land

Figure 8 Topography diagram of modification amount and error(unit um)

surface but the minimum error is about minus54 um which isconcentrated in the central area of the profile and longitudinaldirections The normal error of any intersection of profiledirection and longitudinal direction is obtained by the errorof the contour map

Mathematical Problems in Engineering 9

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus44

minus48

minus56

minus88

minus61

minus66

minus65

minus69

Top land

Figure 9 Topography diagram of error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0minus002minus004minus006minus008

Mod

ifica

tion

amou

nt

and

erro

r (m

m)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)Longitudinal (mm)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

Mod

ifica

tion

amou

nt

and

erro

r (m

m) 0

minus002minus004minus006minus008

(b) Left flank

Figure 10 The contour map of modification amount and error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

times10minus3

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

Longitudinal (mm)

minus4

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

minus9

times10minus3

(b) Left flank

Figure 11 The contour map of error (unit um)

5 Conclusions

In this study an on-machine profile measurement systemalong with a five-axis CNC gear form grinding machineis developed to improve accuracy of grinding This system

includes the following steps (1) calculate trajectory of thecentre of spherical probe (2) define gear flanks by a grid ofpoints (3) obtain the coordinate values of topology measure-ment points Grinding experiments are performed to verifythe accuracy and efficiency of the topographymeasurements

10 Mathematical Problems in Engineering

With contour map the profile and longitudinal directionsof the error changes are easily seen Using statistical processcontrol techniques to monitor the grinding process cantimely attain the detection of changes in the product errorexceptions and take the necessary measures to prevent theoccurrence of waste As the numerical examples show on-machine measurement method can inspect tooth qualityAn additional advantage of the approach is to characterizemodification amount and tooth surface error respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the National Natural ScienceFoundation of China for its financial support Part of thiswork was performed under Contracts no 51375144 and no51405135 and the Priority Project of Research in Universitiesin Henan Province (Grant no 15A460021)

References

[1] H L Wang X Z Deng and J J Yang ldquoForm grinding andexperiment on segment topographicmodification gearrdquo Journalof Aerospace Power vol 29 no 12 pp 3000ndash3008 2014

[2] H-Y You P-Q Ye J-S Wang and X-Y Deng ldquoDesign andapplication of CBN shape grinding wheel for gearsrdquo Interna-tional Journal of Machine Tools and Manufacture vol 43 no12 pp 1269ndash1277 2003

[3] Q Fan R SDafoe and JW Swanger ldquoHigher-order tooth flankform error correction for face-milled spiral bevel and hypoidgearsrdquo Journal ofMechanical DesignmdashTransactions of theASMEvol 130 no 7 Article ID 072601 2008

[4] J Argyris M De Donno and F L Litvin ldquoComputer programin visual basic language for simulation of meshing and contactof gear drives and its application for design of worm gear driverdquoComputer Methods in Applied Mechanics and Engineering vol189 no 2 pp 595ndash612 2000

[5] Z L Zhang Y Fu Q R Yin and Y Zeng ldquoStudy on the polarmethod of measuring of gear profile errorsrdquo Chinese Journal ofMechanical Engineering vol 37 no 4 pp 70ndash72 2001

[6] C H Gao K Cheng and D Webb ldquoInvestigation on samplingsize optimisation in gear tooth surface measurement using aCMMrdquo The International Journal of Advanced ManufacturingTechnology vol 24 no 7-8 pp 599ndash606 2004

[7] Z L Zhang Y Fu and Y Zeng ldquoExpressing gear involuteerror by polar angle amp generating angle in polar coordinatemeasuring methodrdquo Tool Engineering vol 34 no 4 pp 39ndash402000

[8] F Gao B H Zhao and Y Li ldquoNovel pre-travel calibrationmethod of touch trigger probe based on error separationrdquoChinese Journal of Scientific Instrument vol 34 no 7 pp 1581ndash1587 2013

[9] F Gao Y Li S Tian Y Huang L Hao and J Wang ldquoStudy onthe on-machine measurement method of NC wheel gear formgrinding machinerdquo Chinese Journal of Scientific Instrument vol29 no 3 pp 540ndash544 2008

[10] A Nafi J R R Mayer and A Wozniak ldquoNovel CMM-basedimplementation of the multi-step method for the separation ofmachine and probe errorsrdquo Precision Engineering vol 35 no 2pp 318ndash328 2011

[11] A Nafi J R RMayer andAWozniak ldquoReduced configurationset for the multi-step method applied to machine and probeerror separation on a CMMrdquo Measurement vol 45 no 10 pp2321ndash2329 2012

[12] Y-P Shih and S-D Chen ldquoFree-formflank correction in helicalgear grinding using a five-axis computer numerical control gearprofile grindingmachinerdquo Journal ofManufacturing Science andEngineering vol 134 no 4 Article ID 041006 2012

[13] Y Kobayashi N Nishida Y Ougiya and H Nagata ldquoToothtrace modification processing of helix gear by form grindingmethodrdquo Transactions of the Japan Society of Mechanical Engi-neers Part C vol 61 no 590 pp 4088ndash4093 1995

[14] Y Kobayashi N Nishida and Y Ougiya ldquoEstimation ofgrinding wheel setting error in helical gear processing byform grindingrdquo Transactions of the Japan Society of MechanicalEngineers Part C vol 63 no 612 pp 2852ndash2858 1997

[15] C-K Lee ldquoManufacturing process for a cylindrical crown geardrive with a controllable fourth order polynomial function oftransmission errorrdquo Journal of Materials Processing Technologyvol 209 no 1 pp 3ndash13 2009

[16] C K Lee and C K Chen ldquoMathematical models meshinganalysis and transmission design for robust cylindrical gear setgenerated by double blade-disks with parabolic cutting edgesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 218 no 12 pp1539ndash1553 2004

[17] Z Y Shi and Y Ye ldquoResearch on the generalized polar-coordinate method for measuring involute profile deviationsrdquoChinese Journal of Scientific Instrument vol 22 no 2 pp 140ndash142 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

A

x

X

B

Y

O

x0

rb

D

C

120572x

1205720

120601A 120579A

120579B

O0 O1

R1

120601AB

120601BB998400

Figure 2 Schematic illustration of polar coordinate measurementmethod

involute curve is achieved by axes 119862 and 119883 when they aremoving at the same time Because of the motion limit in axial119884direction polar coordinatemeasurement is amajormethodof on-machine measurement when measuring profile erroron the CNC gear form grinder The mathematical modelof polar coordinate method measurement is establishedaccording to the workpiece rotary movement (119862 axis) andradial feed movement (119883 axis) [17]

On-machine measurement system can not only measureprofile error and helical error but also measure pitch errorof helical gears Using the measurement data analysis ofthese errors which may correct processing parameters andaxial motion equation of polynomial coefficients realizes theclosed-loop feedback processing of tooth surface modifica-tion thus laying the foundation for improving the precisionof tooth surface

The polar coordinates of spherical probemethod are usedto measure the involute tooth profile as shown in Figure 2 119860is the measuring start point 119861 is the measuring point 1198611015840 isthe actual location of measuring point 119861 120601119860119861 is an angle ofrotary motion for C axis which can be expressed as

120601119860119861

= 120601119860+ 120601119861 (4)

where

120601119861 = 120579119861 minus 120579119860

120579119860= tan120572

0minus 1205720=

radic1199092

0minus 1199032

119887minus 1198771

119903119887

minus 1205720

120579119861= tan120572

119909minus 120572119909=

radic1199092 minus 1199032

119887minus 1198771

119903119887

minus 120572119909

(5)

where 119903119887is the radius of the base cylinder variable parameters

1199090and 119909 are coordinate data of probe centre corresponding

to start measuring points 119860 and 119861 respectively 1205720and

120572119909are the pressure angle in the cross section for the base

circle radius corresponding to points 119860 and 119861 in profilerespectively 120579

119860and 120579

119861are the spread angle of an involute

curve of the measured gear profile corresponding to points119860 and 119861 in profile respectively 119877

1is the equivalent radius of

the spherical probe and can be expressed as 1198771= 119877 sdot cos120573

119887

where119877 is the radius of the spherical probe and 120573119887 is the helixangle on the base circle

On the other hand

120601119860= ang119860119874119883 minus ang119861

1015840119874119883

= arccos119903119887

1199090

minus 1205720minus (arccos

119903119887

119909minus 120572119909)

(6)

In the above equations rotary angle 120601119860119861

is expressed as

120601119860119861 =

radic1199092 minus 1199032

119887

119903119887

minus

radic1199092

0minus 1199032

119887

119903119887

minus arccos119903119887

119909+ arccos

119903119887

1199090

(7)

Generally speaking when a gear is measured on grindingmachine using the polar coordinate method the probe willbe set in the radial direction of the workpiece in which it isdoing rotary motion along its axis As soon as it touches thesurface a signal is sent to the gear form grinding machinecontroller which stopsmovements ofmachine axes such as119862119883 and 119885 So the control system can acquire the real positionof the machine which is converted into coordinates of thecontact point of the probe tip with gear tooth surface Whenthe probe moves one of the contacts breaks and a binarysignal comes out from the probe head Then this binarysignal is converted into the grinding machine controllerimmediately The controller gets the output signal of theprobe and latches actual angle of turntable 1206011015840

119860119861 Comparing

the achieved angle 1206011015840119860119861

and its theoretical value 120601119860119861 error

was calculated for the tooth profileThe deviation of the toothsurface is given by

Δ120588 = 119903119887 (120601119860119861 minus 1206011015840

119860119861) (8)

During grinding process the probe is placed below thewheel so that it cannot be destroyed When measurementis needed to be done grinding is stopped automatically andprobe is stretched out by the air cylinder Then the touchprobe is taken at a position which is at such a distance apartfrom the tooth surface The probe tip will not touch theworkpiece while it moves over the workpiece in 119883 and 119885

directions When the probe touches the tooth surface thecontroller records the coordinate values of axes 119862 119883 and 119885immediately

23 Determining of the Probe Centre Helical gears that trans-form rotation between parallel axes in opposite directionsare in external meshing and are provided with screw toothsurfaces of opposite directions A helical gear tooth surface isgenerated by an involute curve that performs a screwmotionSegmented topologicalmodification crowns the tooth surfacein profile and longitudinal directions at the same time Profilesegment modification is achieved by modifying the grinding

4 Mathematical Problems in Engineering

gyc

za1c

rbxa1

xcR1

u1

M

N

h

C

C0

120572

1205901

ya1

120601

Oa1c

Figure 3 Coordinate systems applied to solve the centre of sphericalprobe coordinates

wheel and longitudinal segment modification is completedby changing the grinding path of the wheel relatively to thegear

As shown in Figure 3 given an involute curve ℎ119892 in thecross section of tooth surfaces of gear if you want to measuretooth profile error of any point such as119872 point you may seta position in normal direction of the involute curve at point119872 That is to say the centre of spherical probe is located inthe extension cord of tangent119873119872 on the base circle

Coordinate systems 1198781(1199091 1199101 1199111) and 119878

119886(119909119886 119910119886 119911119886) are

rigidly connected to the workpiece and rotary table respec-tively So coordinate system 119878

1(1199091 1199101 1199111) is namedworkpiece

coordinate system Coordinate system 119878119888(119909119888 119910119888 119911119888) is the

measurement coordinate system in which the probe rotatesthe workpiece axis The coordinate system 119878

119888initially coin-

cides with 119878119886 Due to no displacement in 119884

1axial direction as

the probe in the measurement the centre of spherical probeonly always can be moved along the119883

1axial direction

Any point such as point 119872(1199091119872 1199101119872 1199111119872) on the tooth

surface measured using on-machine measurement has coor-dinates in the workpiece coordinate system 119878

1(1199091 1199101 1199111)

which can be expressed as

1199091119872

= 119903119887cos (120590

1+ 1199061) + 1199031198871199061sin (120590

1+ 1199061)

1199101119872

= 119903119887sin (120590

1+ 1199061) minus 1199031198871199061cos (120590

1+ 1199061)

1199111119872

= 0

(9)

where 1199061 is the spread angle of an involute 1205901 is half of theangular width of the space on the base circle and 119903119887 is theradius of the base cylinder

Point 119862(1199091119862 1199101119862 1199111119862) the centre of spherical probe maybe expressed in coordinate system 1198781 by

1199091119862 = 1199091119872 + 119877 cos120573119887 sin (1205901 + 1199061)

1199101119862= 1199101119872

minus 119877 cos120573119887cos (120590

1+ 1199061)

1199111119862 = 119877 sin120573119887

(10)

where 120573119887is the helix angle on the base circle

To keep the centre of spherical probe in the coordinateplane 119909

111990011199111 the measurement coordinate system 119878

119888 the

workpiece and the probe should be together rotated by anangle 120601 around the 119885

1axis The transformation matrix 119878

119888to

1198781is expressed as follows

M1119862=

[[[[[

[

cos120601 minus sin120601 0 0

sin120601 cos120601 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(11)

For the rotated point119862 its coordinate point119862(11990911198620

11991011198620

11991111198620

) the centre of spherical probe is expressed in coordinatesystem 119878

1by

[[[[[

[

cos120601 minus sin120601 0 0

sin120601 cos120601 0 0

0 0 1 0

0 0 0 1

]]]]]

]

[[[[[

[

1199091119862

1199101119862

1199111119862

1

]]]]]

]

=

[[[[[

[

11990911198620

11991011198620

11991111198620

1

]]]]]

]

(12)

Because of the centre of spherical probe in the coordinateplane 119909

111990011199111from beginning to end point 119862 is equal to zero

in 1198841direction which can be expressed as

11991011198620

= 0 (13)

Therefore comparing (12) with (13) satisfies the followingrelation

1199091119862

sin120601 + 1199101119862

cos120601 = 0 (14)

So the rotated angle 120601 can be determined as follows

120601 = minus arctan1199101119862

1199091119862

= arctanminus1199101119872

+ 119877 cos120573119887cos (120590

1+ 1199061)

1199091119872

+ 119877 cos120573119887sin (120590

1+ 1199061)

(15)

Consequently the coordinates (11990911198620 11991011198620 and 11991111198620) can

be obtained in coordinate system 1198781 Coordinate system 119878

1is

introduced by the tooth surface generationmodeling and it isrigidly fixed to the gear with 119885

1axis coinciding with the axis

of the gear Hence

11990911198620 = [1199091119872 + 119877 cos120573119887 sin (1205901 + 1199061)] cos120601

minus [1199101119872

minus 119877 cos120573119887cos (120590

1+ 1199061)] sin120601

11991011198620 = 0

11991111198620

= 1199111119872

+ 119877 sin120573119887

(16)

24 Generation of Grid Point of Tooth Surface Formeasuringtheoretical data the tooth surface equation of measured gearmust be known Assuming the grinding parameters of thegear pair are known a series of coordinate transformationand derivation processes are needed to get the tooth surfaceequation Here the topological modification is carried out

Mathematical Problems in Engineering 5

123

451

II

I

2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

1

2

3

4

5j

j

i

i

Figure 4 Definition of tooth flank grid

on the driving gear on the other hand the driven gear is astandard involute gear For the topologicalmodification toothsurface tooth surface measurement areas need to be plannedto get the whole tooth surface error information Accordingto the relevant standards of American gear measurement thedivision of tooth surface grid in the longitudinal directionshould be less than 10 of the tooth width in the profiledirection it should be less than 5 of the working depth butmust be less than or equal to 06mm The density of toothsurfacemeasurement point is generally taken to be 9 columnslong in the longitudinal direction and taken to be 5 rows inthe profile direction a total of 45 points The tooth surfacemeasurement path planning is shown in Figure 4

The plane coordinate system XOZ is an axis of rotationprojection section of the tooth surface where 119874 is centrepoint of base circle and119872lowast is a point in the projection surfaceat any measured point in the tooth surface Given a point by(119909lowast 119911lowast) the following system of nonlinear equations can beused to solve for the surface parameters

radic1199092

119894+ 1199102

119894= 119909lowast

119911119894= 119911lowast

(17)

where the coordinates (119909119894 119910119894 and 119911

119894) are three coordinate

components of tooth surface position vector r(119894)1 119894 = 1 2

45The tooth surface of topological modification is no longer

the involute tooth surface and becomes complex because ofthe existence of machining error So the whole tooth surfaceneeds to be digitally processed to get more accurate toothsurface geometric shape namely dividing grid on the toothsurface and computing the coordinate data of the grid pointFigure 5 shows the partition of the grid point inside gear shaftsection after rotation projection in which the grid points aredivided into 5 rows by 9 columns

In Figure 5 1198601 1198602 1198603 and 1198604 are the four boundarypoints of the tooth surface 1198601015840

1 11986010158402 11986010158403 and 119860

1015840

4are the

four new boundary points corresponding to tooth surfacecontraction grid area Here Δ119897

1is the top contraction Δ119897

2is

the front contraction Δ1198973is the root contraction and Δ119897

4is

the back contraction respectively The coordinate of a node119875119894119895(119894 = 1 sim 5 119895 = 1 sim 9) in the tooth surface grid is (119885

119894119895 119877119894119895)

X

O

1

5

9

A2

A9984001

A1

A9984002 A998400

3

A9984004

A4

ZZij

Rij

Pij

A3

Figure 5 Positions of grid points on one tooth

The coordinates of the tooth surface boundary points1198601

1198602 1198603 and 119860

4in the XOZ plane are respectively shown as

follows

1198601

1198851198601= 0

1198831198601= 119903119891

1198602

1198851198602 = 0

1198831198602 = 119903119886

1198603

1198851198603= 119887

1198831198603= 119903119886

1198604

1198851198604= 119887

1198831198604= 119903119891

(18)

here 119903119891is the tooth root radius 119903

119886is the tooth top radius and

119887 is the tooth width respectively

6 Mathematical Problems in Engineering

The new coordinates of tooth surface boundary points1198601015840

11198601015840211986010158403 and1198601015840

4in the XOZ plane are respectively shown

as follows

1198601015840

1

1198851198601 = Δ1198974

1198831198601 = 119903119891 + Δ1198973

1198601015840

2

1198851198602= Δ1198974

1198831198602= 119903119886minus Δ1198971

1198601015840

3

1198851198603= 119887 minus Δ119897

2

1198831198603= 119903119886minus Δ1198971

1198601015840

4

1198851198604 = 119887 minus Δ1198972

1198831198604 = 119903119891 + Δ1198973

(19)

The coordinates (1198851119895 1198831119895) (119895 = 1 sim 9) of 9 equal-division

points 1198751119895 between119860

1015840

1and1198601015840

4in the XOZ plane are shown as

follows

1198851119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

1198831119895= 119903119891+ Δ1198973

(20)

The coordinates (1198855119895 1198835119895) (119895 = 1 sim 9) of 9 equal-division

points 1198755119895between1198601015840

2and1198601015840

3in the XOZ plane are shown as

follows

1198855119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

1198835119895= 119903119886minus Δ1198971

(21)

The coordinates (1198851198941 1198831198941) (119894 = 1 sim 5) of 9 equal-division

points 1198751198941between1198601015840

1and1198601015840

2in the XOZ plane are shown as

follows1198851198941= Δ1198974

1198831198941 = 119903119891 + Δ1198973 +

119894 minus 1

4(119903119886 minus 119903119891 minus Δ1198971 minus Δ1198973)

(22)

The coordinates (1198851198949 1198831198949) (119894 = 1 sim 5) of 9 equal-division

points 1198751198949between1198601015840

4and1198601015840

3in the XOZ plane are shown as

follows1198851198949 = 119887 minus Δ1198972

1198831198949= 119903119891+ Δ1198973+119894 minus 1

4(119903119886minus 119903119891minus Δ1198971minus Δ1198973)

(23)

From the above derivational process the coordinates(119885119894119895 119883119894119895) (119894 = 1 sim 5 119895 = 1 sim 9) of any point 119875

119894119895in the

tooth surface grid in the XOZ plane can be calculated by theformulae that are shown as follows

119885119894119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

119883119894119895= 119903119891+ Δ1198973+119894 minus 1

4(119903119886minus 119903119891minus Δ1198971minus Δ1198973)

(24)

Gear

Theory profile

Trace of probe

Sensitive direction

Y

XΔEg

R

Mi

M998400i

120601

Ci

ni

re(u1 1205791)

r1(u1 1205791)

Figure 6 Relationships of the probe geometry and theory profile

Solving nonlinear equations to get the tooth surfaceparameters (119906

1 1205791) for eachmeasured point which substitute

into (2) can get the theoretical coordinates (119909(119894)1 119910(119894)1 and 119911(119894)

1)

and the unit normal vector n119894 Hence

r(119894)1= [119909(119894)

1119910(119894)

1119911(119894)

1]119879

n(119894)1= [119899(119894)

1119909119899(119894)

1119910119899(119894)

1119911]119879

(119894 = 1 2 2 times 45)

(25)

25Measurement Error Compensation When a point ismea-sured on the modification gear tooth surface the inspectiondatum of an on-machine measurement is the centre of theprobe To improve the measurement accuracy the proberadius can be considered in calculating the profile errorsFigure 6 schematically shows the effect of the probe sizeon the measurement As the probe always has a certaindimension an errorΔ119864119892 is unavoidable Due to the size effectof the probe the actual contact point of the probe is at point119872119894 instead of1198721015840

119894

According to the envelope surface characteristics thenormal vector is perpendicular to the probe sphere andpassed through the centre of the probe 119862119894 The theoreticalmotion trajectory of probe centre is shown as follows

r119890 (1199061 1205791) = r

1 (1199061 1205791) + 119877 sdot n1 (1199061 1205791) (26)

here r119890(1199061 1205791) is the trace of the probe centre and 119877 is theprobe radius

Strictly speaking the measuring tooth surface deviationis the deviation of the actual contact point 119872119894 in thesensitive direction of the probe so an important factor of themeasurement error is the actual contacting position of theprobe In the measurement the probe is also very close tothe contact pointmicroscopically the nearness of the contactpoint119872

119894can approximately be regarded as a small planeThe

Mathematical Problems in Engineering 7

Table 1 Parameters of the gears in the trials

Items Symbol Unit DataDriving gear 119911

1mdash 30

Driven gear 1199112

mdash 30Normal model 119898

119899mm 65

Pressure angle 120572119899

deg 20Helix angle 120573 deg 13Face width 119887 mm 53Top profile crowning 119886

119898119901(119888119891)1mm 00011

Bottom profile crowning 119886119898119901(119889119890)

1mm 00014Top limit angle 119906

119888rad 0496

Bottom limit angle 119906119889

rad 0196Front longitudinal crowning 119886

119898119897(ℎ119886)1mm 00012

Back longitudinal crowning 119886119898119897(119887119895)

1mm 00012Front limit angle 120579

119886rad 0018

Back limit angle 120579119887

rad 0104

measurement error caused by probe radius is Δ119864119892 and its

geometric relationships with probe radius can be expressedas follows

Δ119864119892= 119877(

1

cos120601minus 1) (27)

According to (27) the larger the probe radius 119877 thegreater the measurement error so the probe radius shouldbe small here the radius 119877 is 2mm The normal direction ofthe measuring points and the sensitivity of the probe are alsoincluded in the XOY plane and experimental results showthat the angle 120601 is very small

From Figure 6 it can be seen that the measured headshould be in contact with the theoretical point 1198721015840

119894 and the

119883 119884 direction of the coordinates of the value of a certainamount of compensation can be guaranteed Assuming that119872119894 coordinates are (119909119894 119910119894)119872

1015840

119894coordinates are (1199091015840

119894 1199101015840

119894) so the

relationship between the two expressions is

1199091015840

119894= 119909119894minus 119877119905119892120601

1199101015840

119894= 119909119894 + 119877119905119892120601 sin120601

(28)

The contact points are in agreement with the theoreticalcontact points after compensationThe contact position of themeasuring head does not affect the measurement results

3 Numerical Example

The parameters of the gears are listed in Table 1 The drivinggear is modified in the profile and longitudinal directionsat the same time On the contrary the tooth surface of thedriven gear is a conventional screw involute surface Thencoordinates on the tooth surface are measured and calculatedusing a MATLAB code

The measurement data of tooth surface can be obtainedby measuring the grid points in the order with the probeSelecting the radius of the probe 119877 = 2mm the probecentre trajectory curve is the envelope surface of measured

90

95

100

105

110

01234

0

20

40

60 Rotated topological pointTrace of the centre

of the probe

minus1minus2

y (mm)

x (mm)

z(m

m)

Figure 7The contrast position of rotated topological point and theprobe centre

tooth surface When the tooth surface is detected using polarcoordinate method the trajectory coordinates of the probecentre and the gear rotating angle 120601 are calculated using thecontrol unitThe relationship of relative position between thetopology points in the rotated tooth surface and the trajectoryof probe centre is shown in Figure 7

4 Tooth Flank Errors Separation

According to the predetermined spacing the tooth surfacedeviation of each measured tooth surface detection pointis point-to-point measured by probe along the profile andlongitudinal directions of the gear r0(1199061 1205791) is the standardinvolute tooth surface vector r1(1199061 1205791) is the theoreticaltopological modification tooth surface vector and r1119904(1199061 1205791)is the actual tooth surface vector after grinding respec-tively Due to the error of the machine tool adjustmentand movement the actual tooth surface often deviates fromthe theoretical tooth surface The tooth surface error Δ119864is usually measured along the direction of the unit normalvector n

1and is presented as

Δ119864 = (r1119904(1199061 1205791) minus r1(1199061 1205791)) sdot n1(1199061 1205791) (29)

According to the standard involute tooth surface vectorr0(1199061 1205791) and themodified tooth surface vector r

1(1199061 1205791) the

topological modification amount 120575 can be obtained which isshown as follows

120575 = (r1(1199061 1205791) minus r0(1199061 1205791)) sdot n0(1199061 1205791) (30)

If the tooth surface topology deviation is expressed asΔ120588the tooth surface error Δ119864 can be expressed by Δ120588 and 120575 asfollows

Δ119864 = Δ120588 minus 120575 (31)

Equation (31) shows that the tooth surface error can beobtained by removing the modification amount out of thetopology deviation of tooth surface

8 Mathematical Problems in Engineering

Table 2 The modification amount and tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00564 minus00436 minus00314 minus00257 minus00239 minus00285 minus00291 minus00385 minus004972 minus00449 minus00387 minus00215 minus00159 minus00145 minus00147 minus00214 minus00289 minus004363 minus00406 minus00302 minus00197 minus00115 minus00027 minus0012 minus00173 minus00237 minus003914 minus00479 minus00421 minus00236 minus00214 minus00136 minus00234 minus00257 minus00356 minus004125 minus00535 minus00509 minus00352 minus00298 minus00172 minus00287 minus00327 minus00492 minus00518

Left flankmm

1 2 3 4 5 6 7 8 95 minus00524 minus00487 minus00301 minus00272 minus00196 minus00264 minus00314 minus00501 minus005444 minus00498 minus00410 minus00198 minus00189 minus00035 minus00197 minus00241 minus00417 minus004633 minus00436 minus00375 minus00221 minus00035 minus00021 minus00046 minus00210 minus00324 minus004522 minus00510 minus00417 minus00183 minus00167 minus00138 minus00153 minus00189 minus00394 minus005311 minus00571 minus00462 minus00294 minus00285 minus00227 minus00249 minus00267 minus00472 minus00592

Table 3 The tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00044 minus00047 minus00046 minus00046 minus00062 minus00062 minus00066 minus00061 minus000612 minus00061 minus00064 minus00063 minus00064 minus00079 minus00079 minus00083 minus00077 minus000793 minus00058 minus00061 minus00062 minus00063 minus00075 minus00076 minus00084 minus00074 minus000754 minus00056 minus00059 minus00058 minus00059 minus00074 minus00074 minus00078 minus00072 minus000745 minus00048 minus00051 minus00050 minus00051 minus00066 minus00066 minus00071 minus00064 minus00066

Left flankmm

1 2 3 4 5 6 7 8 95 minus00053 minus00057 minus00055 minus00056 minus00057 minus00058 minus00087 minus00054 minus000654 minus00062 minus00062 minus00068 minus00072 minus00084 minus00067 minus00086 minus00039 minus000743 minus00049 minus00074 minus00072 minus00058 minus00054 minus00069 minus00094 minus00075 minus000872 minus00072 minus00049 minus00085 minus00063 minus00072 minus00069 minus00064 minus00058 minus000761 minus00088 minus00076 minus00067 minus00053 minus00069 minus00072 minus00073 minus00067 minus00069

The tooth surface topology deviation Δ120588 is shown inTable 2 and the tooth surface error Δ119864 is shown in Table 3respectively

The deviation distribution (error and modificationamount) of the tooth surface obtained by an on-machinemeasurement is shown in Figure 8 The dashed parts (thesuperposition of error and modification) in regions 2 46 and 8 of the tooth surface are obviously presented as aparabolic shape in the profile and longitudinal directionsrespectivelyThemaximumvalue of deviation at the tooth topis minus00571mm and the error in the centre of tooth surfacearea 1 is minus00049mm The trend is gradually increasedtowards both sides presenting the shape where the middle ishigh and both sides are low

The measurement using the topological modificationtooth surface equation is shown in Figure 9 The measuredresult reflects the actual tooth surface error the distributionof which has unobvious trend The maximum error in thetooth top is minus00088mm and the error in the centre of righttooth surface area 1 is minus00054mm

To separate flank form deviations from modificationamount and to improve the perception of deviation dia-grams all flank deviations are referenced by the modificationamount The characterization of the tooth surface error andmodification amount are shown in Figures 10 and 11 Themaximum error is about minus88 um in the upper and lower

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus564

minus535

minus524

minus571

minus497

minus518

minus544

minus592

Top land

Figure 8 Topography diagram of modification amount and error(unit um)

surface but the minimum error is about minus54 um which isconcentrated in the central area of the profile and longitudinaldirections The normal error of any intersection of profiledirection and longitudinal direction is obtained by the errorof the contour map

Mathematical Problems in Engineering 9

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus44

minus48

minus56

minus88

minus61

minus66

minus65

minus69

Top land

Figure 9 Topography diagram of error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0minus002minus004minus006minus008

Mod

ifica

tion

amou

nt

and

erro

r (m

m)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)Longitudinal (mm)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

Mod

ifica

tion

amou

nt

and

erro

r (m

m) 0

minus002minus004minus006minus008

(b) Left flank

Figure 10 The contour map of modification amount and error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

times10minus3

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

Longitudinal (mm)

minus4

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

minus9

times10minus3

(b) Left flank

Figure 11 The contour map of error (unit um)

5 Conclusions

In this study an on-machine profile measurement systemalong with a five-axis CNC gear form grinding machineis developed to improve accuracy of grinding This system

includes the following steps (1) calculate trajectory of thecentre of spherical probe (2) define gear flanks by a grid ofpoints (3) obtain the coordinate values of topology measure-ment points Grinding experiments are performed to verifythe accuracy and efficiency of the topographymeasurements

10 Mathematical Problems in Engineering

With contour map the profile and longitudinal directionsof the error changes are easily seen Using statistical processcontrol techniques to monitor the grinding process cantimely attain the detection of changes in the product errorexceptions and take the necessary measures to prevent theoccurrence of waste As the numerical examples show on-machine measurement method can inspect tooth qualityAn additional advantage of the approach is to characterizemodification amount and tooth surface error respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the National Natural ScienceFoundation of China for its financial support Part of thiswork was performed under Contracts no 51375144 and no51405135 and the Priority Project of Research in Universitiesin Henan Province (Grant no 15A460021)

References

[1] H L Wang X Z Deng and J J Yang ldquoForm grinding andexperiment on segment topographicmodification gearrdquo Journalof Aerospace Power vol 29 no 12 pp 3000ndash3008 2014

[2] H-Y You P-Q Ye J-S Wang and X-Y Deng ldquoDesign andapplication of CBN shape grinding wheel for gearsrdquo Interna-tional Journal of Machine Tools and Manufacture vol 43 no12 pp 1269ndash1277 2003

[3] Q Fan R SDafoe and JW Swanger ldquoHigher-order tooth flankform error correction for face-milled spiral bevel and hypoidgearsrdquo Journal ofMechanical DesignmdashTransactions of theASMEvol 130 no 7 Article ID 072601 2008

[4] J Argyris M De Donno and F L Litvin ldquoComputer programin visual basic language for simulation of meshing and contactof gear drives and its application for design of worm gear driverdquoComputer Methods in Applied Mechanics and Engineering vol189 no 2 pp 595ndash612 2000

[5] Z L Zhang Y Fu Q R Yin and Y Zeng ldquoStudy on the polarmethod of measuring of gear profile errorsrdquo Chinese Journal ofMechanical Engineering vol 37 no 4 pp 70ndash72 2001

[6] C H Gao K Cheng and D Webb ldquoInvestigation on samplingsize optimisation in gear tooth surface measurement using aCMMrdquo The International Journal of Advanced ManufacturingTechnology vol 24 no 7-8 pp 599ndash606 2004

[7] Z L Zhang Y Fu and Y Zeng ldquoExpressing gear involuteerror by polar angle amp generating angle in polar coordinatemeasuring methodrdquo Tool Engineering vol 34 no 4 pp 39ndash402000

[8] F Gao B H Zhao and Y Li ldquoNovel pre-travel calibrationmethod of touch trigger probe based on error separationrdquoChinese Journal of Scientific Instrument vol 34 no 7 pp 1581ndash1587 2013

[9] F Gao Y Li S Tian Y Huang L Hao and J Wang ldquoStudy onthe on-machine measurement method of NC wheel gear formgrinding machinerdquo Chinese Journal of Scientific Instrument vol29 no 3 pp 540ndash544 2008

[10] A Nafi J R R Mayer and A Wozniak ldquoNovel CMM-basedimplementation of the multi-step method for the separation ofmachine and probe errorsrdquo Precision Engineering vol 35 no 2pp 318ndash328 2011

[11] A Nafi J R RMayer andAWozniak ldquoReduced configurationset for the multi-step method applied to machine and probeerror separation on a CMMrdquo Measurement vol 45 no 10 pp2321ndash2329 2012

[12] Y-P Shih and S-D Chen ldquoFree-formflank correction in helicalgear grinding using a five-axis computer numerical control gearprofile grindingmachinerdquo Journal ofManufacturing Science andEngineering vol 134 no 4 Article ID 041006 2012

[13] Y Kobayashi N Nishida Y Ougiya and H Nagata ldquoToothtrace modification processing of helix gear by form grindingmethodrdquo Transactions of the Japan Society of Mechanical Engi-neers Part C vol 61 no 590 pp 4088ndash4093 1995

[14] Y Kobayashi N Nishida and Y Ougiya ldquoEstimation ofgrinding wheel setting error in helical gear processing byform grindingrdquo Transactions of the Japan Society of MechanicalEngineers Part C vol 63 no 612 pp 2852ndash2858 1997

[15] C-K Lee ldquoManufacturing process for a cylindrical crown geardrive with a controllable fourth order polynomial function oftransmission errorrdquo Journal of Materials Processing Technologyvol 209 no 1 pp 3ndash13 2009

[16] C K Lee and C K Chen ldquoMathematical models meshinganalysis and transmission design for robust cylindrical gear setgenerated by double blade-disks with parabolic cutting edgesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 218 no 12 pp1539ndash1553 2004

[17] Z Y Shi and Y Ye ldquoResearch on the generalized polar-coordinate method for measuring involute profile deviationsrdquoChinese Journal of Scientific Instrument vol 22 no 2 pp 140ndash142 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

gyc

za1c

rbxa1

xcR1

u1

M

N

h

C

C0

120572

1205901

ya1

120601

Oa1c

Figure 3 Coordinate systems applied to solve the centre of sphericalprobe coordinates

wheel and longitudinal segment modification is completedby changing the grinding path of the wheel relatively to thegear

As shown in Figure 3 given an involute curve ℎ119892 in thecross section of tooth surfaces of gear if you want to measuretooth profile error of any point such as119872 point you may seta position in normal direction of the involute curve at point119872 That is to say the centre of spherical probe is located inthe extension cord of tangent119873119872 on the base circle

Coordinate systems 1198781(1199091 1199101 1199111) and 119878

119886(119909119886 119910119886 119911119886) are

rigidly connected to the workpiece and rotary table respec-tively So coordinate system 119878

1(1199091 1199101 1199111) is namedworkpiece

coordinate system Coordinate system 119878119888(119909119888 119910119888 119911119888) is the

measurement coordinate system in which the probe rotatesthe workpiece axis The coordinate system 119878

119888initially coin-

cides with 119878119886 Due to no displacement in 119884

1axial direction as

the probe in the measurement the centre of spherical probeonly always can be moved along the119883

1axial direction

Any point such as point 119872(1199091119872 1199101119872 1199111119872) on the tooth

surface measured using on-machine measurement has coor-dinates in the workpiece coordinate system 119878

1(1199091 1199101 1199111)

which can be expressed as

1199091119872

= 119903119887cos (120590

1+ 1199061) + 1199031198871199061sin (120590

1+ 1199061)

1199101119872

= 119903119887sin (120590

1+ 1199061) minus 1199031198871199061cos (120590

1+ 1199061)

1199111119872

= 0

(9)

where 1199061 is the spread angle of an involute 1205901 is half of theangular width of the space on the base circle and 119903119887 is theradius of the base cylinder

Point 119862(1199091119862 1199101119862 1199111119862) the centre of spherical probe maybe expressed in coordinate system 1198781 by

1199091119862 = 1199091119872 + 119877 cos120573119887 sin (1205901 + 1199061)

1199101119862= 1199101119872

minus 119877 cos120573119887cos (120590

1+ 1199061)

1199111119862 = 119877 sin120573119887

(10)

where 120573119887is the helix angle on the base circle

To keep the centre of spherical probe in the coordinateplane 119909

111990011199111 the measurement coordinate system 119878

119888 the

workpiece and the probe should be together rotated by anangle 120601 around the 119885

1axis The transformation matrix 119878

119888to

1198781is expressed as follows

M1119862=

[[[[[

[

cos120601 minus sin120601 0 0

sin120601 cos120601 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(11)

For the rotated point119862 its coordinate point119862(11990911198620

11991011198620

11991111198620

) the centre of spherical probe is expressed in coordinatesystem 119878

1by

[[[[[

[

cos120601 minus sin120601 0 0

sin120601 cos120601 0 0

0 0 1 0

0 0 0 1

]]]]]

]

[[[[[

[

1199091119862

1199101119862

1199111119862

1

]]]]]

]

=

[[[[[

[

11990911198620

11991011198620

11991111198620

1

]]]]]

]

(12)

Because of the centre of spherical probe in the coordinateplane 119909

111990011199111from beginning to end point 119862 is equal to zero

in 1198841direction which can be expressed as

11991011198620

= 0 (13)

Therefore comparing (12) with (13) satisfies the followingrelation

1199091119862

sin120601 + 1199101119862

cos120601 = 0 (14)

So the rotated angle 120601 can be determined as follows

120601 = minus arctan1199101119862

1199091119862

= arctanminus1199101119872

+ 119877 cos120573119887cos (120590

1+ 1199061)

1199091119872

+ 119877 cos120573119887sin (120590

1+ 1199061)

(15)

Consequently the coordinates (11990911198620 11991011198620 and 11991111198620) can

be obtained in coordinate system 1198781 Coordinate system 119878

1is

introduced by the tooth surface generationmodeling and it isrigidly fixed to the gear with 119885

1axis coinciding with the axis

of the gear Hence

11990911198620 = [1199091119872 + 119877 cos120573119887 sin (1205901 + 1199061)] cos120601

minus [1199101119872

minus 119877 cos120573119887cos (120590

1+ 1199061)] sin120601

11991011198620 = 0

11991111198620

= 1199111119872

+ 119877 sin120573119887

(16)

24 Generation of Grid Point of Tooth Surface Formeasuringtheoretical data the tooth surface equation of measured gearmust be known Assuming the grinding parameters of thegear pair are known a series of coordinate transformationand derivation processes are needed to get the tooth surfaceequation Here the topological modification is carried out

Mathematical Problems in Engineering 5

123

451

II

I

2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

1

2

3

4

5j

j

i

i

Figure 4 Definition of tooth flank grid

on the driving gear on the other hand the driven gear is astandard involute gear For the topologicalmodification toothsurface tooth surface measurement areas need to be plannedto get the whole tooth surface error information Accordingto the relevant standards of American gear measurement thedivision of tooth surface grid in the longitudinal directionshould be less than 10 of the tooth width in the profiledirection it should be less than 5 of the working depth butmust be less than or equal to 06mm The density of toothsurfacemeasurement point is generally taken to be 9 columnslong in the longitudinal direction and taken to be 5 rows inthe profile direction a total of 45 points The tooth surfacemeasurement path planning is shown in Figure 4

The plane coordinate system XOZ is an axis of rotationprojection section of the tooth surface where 119874 is centrepoint of base circle and119872lowast is a point in the projection surfaceat any measured point in the tooth surface Given a point by(119909lowast 119911lowast) the following system of nonlinear equations can beused to solve for the surface parameters

radic1199092

119894+ 1199102

119894= 119909lowast

119911119894= 119911lowast

(17)

where the coordinates (119909119894 119910119894 and 119911

119894) are three coordinate

components of tooth surface position vector r(119894)1 119894 = 1 2

45The tooth surface of topological modification is no longer

the involute tooth surface and becomes complex because ofthe existence of machining error So the whole tooth surfaceneeds to be digitally processed to get more accurate toothsurface geometric shape namely dividing grid on the toothsurface and computing the coordinate data of the grid pointFigure 5 shows the partition of the grid point inside gear shaftsection after rotation projection in which the grid points aredivided into 5 rows by 9 columns

In Figure 5 1198601 1198602 1198603 and 1198604 are the four boundarypoints of the tooth surface 1198601015840

1 11986010158402 11986010158403 and 119860

1015840

4are the

four new boundary points corresponding to tooth surfacecontraction grid area Here Δ119897

1is the top contraction Δ119897

2is

the front contraction Δ1198973is the root contraction and Δ119897

4is

the back contraction respectively The coordinate of a node119875119894119895(119894 = 1 sim 5 119895 = 1 sim 9) in the tooth surface grid is (119885

119894119895 119877119894119895)

X

O

1

5

9

A2

A9984001

A1

A9984002 A998400

3

A9984004

A4

ZZij

Rij

Pij

A3

Figure 5 Positions of grid points on one tooth

The coordinates of the tooth surface boundary points1198601

1198602 1198603 and 119860

4in the XOZ plane are respectively shown as

follows

1198601

1198851198601= 0

1198831198601= 119903119891

1198602

1198851198602 = 0

1198831198602 = 119903119886

1198603

1198851198603= 119887

1198831198603= 119903119886

1198604

1198851198604= 119887

1198831198604= 119903119891

(18)

here 119903119891is the tooth root radius 119903

119886is the tooth top radius and

119887 is the tooth width respectively

6 Mathematical Problems in Engineering

The new coordinates of tooth surface boundary points1198601015840

11198601015840211986010158403 and1198601015840

4in the XOZ plane are respectively shown

as follows

1198601015840

1

1198851198601 = Δ1198974

1198831198601 = 119903119891 + Δ1198973

1198601015840

2

1198851198602= Δ1198974

1198831198602= 119903119886minus Δ1198971

1198601015840

3

1198851198603= 119887 minus Δ119897

2

1198831198603= 119903119886minus Δ1198971

1198601015840

4

1198851198604 = 119887 minus Δ1198972

1198831198604 = 119903119891 + Δ1198973

(19)

The coordinates (1198851119895 1198831119895) (119895 = 1 sim 9) of 9 equal-division

points 1198751119895 between119860

1015840

1and1198601015840

4in the XOZ plane are shown as

follows

1198851119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

1198831119895= 119903119891+ Δ1198973

(20)

The coordinates (1198855119895 1198835119895) (119895 = 1 sim 9) of 9 equal-division

points 1198755119895between1198601015840

2and1198601015840

3in the XOZ plane are shown as

follows

1198855119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

1198835119895= 119903119886minus Δ1198971

(21)

The coordinates (1198851198941 1198831198941) (119894 = 1 sim 5) of 9 equal-division

points 1198751198941between1198601015840

1and1198601015840

2in the XOZ plane are shown as

follows1198851198941= Δ1198974

1198831198941 = 119903119891 + Δ1198973 +

119894 minus 1

4(119903119886 minus 119903119891 minus Δ1198971 minus Δ1198973)

(22)

The coordinates (1198851198949 1198831198949) (119894 = 1 sim 5) of 9 equal-division

points 1198751198949between1198601015840

4and1198601015840

3in the XOZ plane are shown as

follows1198851198949 = 119887 minus Δ1198972

1198831198949= 119903119891+ Δ1198973+119894 minus 1

4(119903119886minus 119903119891minus Δ1198971minus Δ1198973)

(23)

From the above derivational process the coordinates(119885119894119895 119883119894119895) (119894 = 1 sim 5 119895 = 1 sim 9) of any point 119875

119894119895in the

tooth surface grid in the XOZ plane can be calculated by theformulae that are shown as follows

119885119894119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

119883119894119895= 119903119891+ Δ1198973+119894 minus 1

4(119903119886minus 119903119891minus Δ1198971minus Δ1198973)

(24)

Gear

Theory profile

Trace of probe

Sensitive direction

Y

XΔEg

R

Mi

M998400i

120601

Ci

ni

re(u1 1205791)

r1(u1 1205791)

Figure 6 Relationships of the probe geometry and theory profile

Solving nonlinear equations to get the tooth surfaceparameters (119906

1 1205791) for eachmeasured point which substitute

into (2) can get the theoretical coordinates (119909(119894)1 119910(119894)1 and 119911(119894)

1)

and the unit normal vector n119894 Hence

r(119894)1= [119909(119894)

1119910(119894)

1119911(119894)

1]119879

n(119894)1= [119899(119894)

1119909119899(119894)

1119910119899(119894)

1119911]119879

(119894 = 1 2 2 times 45)

(25)

25Measurement Error Compensation When a point ismea-sured on the modification gear tooth surface the inspectiondatum of an on-machine measurement is the centre of theprobe To improve the measurement accuracy the proberadius can be considered in calculating the profile errorsFigure 6 schematically shows the effect of the probe sizeon the measurement As the probe always has a certaindimension an errorΔ119864119892 is unavoidable Due to the size effectof the probe the actual contact point of the probe is at point119872119894 instead of1198721015840

119894

According to the envelope surface characteristics thenormal vector is perpendicular to the probe sphere andpassed through the centre of the probe 119862119894 The theoreticalmotion trajectory of probe centre is shown as follows

r119890 (1199061 1205791) = r

1 (1199061 1205791) + 119877 sdot n1 (1199061 1205791) (26)

here r119890(1199061 1205791) is the trace of the probe centre and 119877 is theprobe radius

Strictly speaking the measuring tooth surface deviationis the deviation of the actual contact point 119872119894 in thesensitive direction of the probe so an important factor of themeasurement error is the actual contacting position of theprobe In the measurement the probe is also very close tothe contact pointmicroscopically the nearness of the contactpoint119872

119894can approximately be regarded as a small planeThe

Mathematical Problems in Engineering 7

Table 1 Parameters of the gears in the trials

Items Symbol Unit DataDriving gear 119911

1mdash 30

Driven gear 1199112

mdash 30Normal model 119898

119899mm 65

Pressure angle 120572119899

deg 20Helix angle 120573 deg 13Face width 119887 mm 53Top profile crowning 119886

119898119901(119888119891)1mm 00011

Bottom profile crowning 119886119898119901(119889119890)

1mm 00014Top limit angle 119906

119888rad 0496

Bottom limit angle 119906119889

rad 0196Front longitudinal crowning 119886

119898119897(ℎ119886)1mm 00012

Back longitudinal crowning 119886119898119897(119887119895)

1mm 00012Front limit angle 120579

119886rad 0018

Back limit angle 120579119887

rad 0104

measurement error caused by probe radius is Δ119864119892 and its

geometric relationships with probe radius can be expressedas follows

Δ119864119892= 119877(

1

cos120601minus 1) (27)

According to (27) the larger the probe radius 119877 thegreater the measurement error so the probe radius shouldbe small here the radius 119877 is 2mm The normal direction ofthe measuring points and the sensitivity of the probe are alsoincluded in the XOY plane and experimental results showthat the angle 120601 is very small

From Figure 6 it can be seen that the measured headshould be in contact with the theoretical point 1198721015840

119894 and the

119883 119884 direction of the coordinates of the value of a certainamount of compensation can be guaranteed Assuming that119872119894 coordinates are (119909119894 119910119894)119872

1015840

119894coordinates are (1199091015840

119894 1199101015840

119894) so the

relationship between the two expressions is

1199091015840

119894= 119909119894minus 119877119905119892120601

1199101015840

119894= 119909119894 + 119877119905119892120601 sin120601

(28)

The contact points are in agreement with the theoreticalcontact points after compensationThe contact position of themeasuring head does not affect the measurement results

3 Numerical Example

The parameters of the gears are listed in Table 1 The drivinggear is modified in the profile and longitudinal directionsat the same time On the contrary the tooth surface of thedriven gear is a conventional screw involute surface Thencoordinates on the tooth surface are measured and calculatedusing a MATLAB code

The measurement data of tooth surface can be obtainedby measuring the grid points in the order with the probeSelecting the radius of the probe 119877 = 2mm the probecentre trajectory curve is the envelope surface of measured

90

95

100

105

110

01234

0

20

40

60 Rotated topological pointTrace of the centre

of the probe

minus1minus2

y (mm)

x (mm)

z(m

m)

Figure 7The contrast position of rotated topological point and theprobe centre

tooth surface When the tooth surface is detected using polarcoordinate method the trajectory coordinates of the probecentre and the gear rotating angle 120601 are calculated using thecontrol unitThe relationship of relative position between thetopology points in the rotated tooth surface and the trajectoryof probe centre is shown in Figure 7

4 Tooth Flank Errors Separation

According to the predetermined spacing the tooth surfacedeviation of each measured tooth surface detection pointis point-to-point measured by probe along the profile andlongitudinal directions of the gear r0(1199061 1205791) is the standardinvolute tooth surface vector r1(1199061 1205791) is the theoreticaltopological modification tooth surface vector and r1119904(1199061 1205791)is the actual tooth surface vector after grinding respec-tively Due to the error of the machine tool adjustmentand movement the actual tooth surface often deviates fromthe theoretical tooth surface The tooth surface error Δ119864is usually measured along the direction of the unit normalvector n

1and is presented as

Δ119864 = (r1119904(1199061 1205791) minus r1(1199061 1205791)) sdot n1(1199061 1205791) (29)

According to the standard involute tooth surface vectorr0(1199061 1205791) and themodified tooth surface vector r

1(1199061 1205791) the

topological modification amount 120575 can be obtained which isshown as follows

120575 = (r1(1199061 1205791) minus r0(1199061 1205791)) sdot n0(1199061 1205791) (30)

If the tooth surface topology deviation is expressed asΔ120588the tooth surface error Δ119864 can be expressed by Δ120588 and 120575 asfollows

Δ119864 = Δ120588 minus 120575 (31)

Equation (31) shows that the tooth surface error can beobtained by removing the modification amount out of thetopology deviation of tooth surface

8 Mathematical Problems in Engineering

Table 2 The modification amount and tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00564 minus00436 minus00314 minus00257 minus00239 minus00285 minus00291 minus00385 minus004972 minus00449 minus00387 minus00215 minus00159 minus00145 minus00147 minus00214 minus00289 minus004363 minus00406 minus00302 minus00197 minus00115 minus00027 minus0012 minus00173 minus00237 minus003914 minus00479 minus00421 minus00236 minus00214 minus00136 minus00234 minus00257 minus00356 minus004125 minus00535 minus00509 minus00352 minus00298 minus00172 minus00287 minus00327 minus00492 minus00518

Left flankmm

1 2 3 4 5 6 7 8 95 minus00524 minus00487 minus00301 minus00272 minus00196 minus00264 minus00314 minus00501 minus005444 minus00498 minus00410 minus00198 minus00189 minus00035 minus00197 minus00241 minus00417 minus004633 minus00436 minus00375 minus00221 minus00035 minus00021 minus00046 minus00210 minus00324 minus004522 minus00510 minus00417 minus00183 minus00167 minus00138 minus00153 minus00189 minus00394 minus005311 minus00571 minus00462 minus00294 minus00285 minus00227 minus00249 minus00267 minus00472 minus00592

Table 3 The tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00044 minus00047 minus00046 minus00046 minus00062 minus00062 minus00066 minus00061 minus000612 minus00061 minus00064 minus00063 minus00064 minus00079 minus00079 minus00083 minus00077 minus000793 minus00058 minus00061 minus00062 minus00063 minus00075 minus00076 minus00084 minus00074 minus000754 minus00056 minus00059 minus00058 minus00059 minus00074 minus00074 minus00078 minus00072 minus000745 minus00048 minus00051 minus00050 minus00051 minus00066 minus00066 minus00071 minus00064 minus00066

Left flankmm

1 2 3 4 5 6 7 8 95 minus00053 minus00057 minus00055 minus00056 minus00057 minus00058 minus00087 minus00054 minus000654 minus00062 minus00062 minus00068 minus00072 minus00084 minus00067 minus00086 minus00039 minus000743 minus00049 minus00074 minus00072 minus00058 minus00054 minus00069 minus00094 minus00075 minus000872 minus00072 minus00049 minus00085 minus00063 minus00072 minus00069 minus00064 minus00058 minus000761 minus00088 minus00076 minus00067 minus00053 minus00069 minus00072 minus00073 minus00067 minus00069

The tooth surface topology deviation Δ120588 is shown inTable 2 and the tooth surface error Δ119864 is shown in Table 3respectively

The deviation distribution (error and modificationamount) of the tooth surface obtained by an on-machinemeasurement is shown in Figure 8 The dashed parts (thesuperposition of error and modification) in regions 2 46 and 8 of the tooth surface are obviously presented as aparabolic shape in the profile and longitudinal directionsrespectivelyThemaximumvalue of deviation at the tooth topis minus00571mm and the error in the centre of tooth surfacearea 1 is minus00049mm The trend is gradually increasedtowards both sides presenting the shape where the middle ishigh and both sides are low

The measurement using the topological modificationtooth surface equation is shown in Figure 9 The measuredresult reflects the actual tooth surface error the distributionof which has unobvious trend The maximum error in thetooth top is minus00088mm and the error in the centre of righttooth surface area 1 is minus00054mm

To separate flank form deviations from modificationamount and to improve the perception of deviation dia-grams all flank deviations are referenced by the modificationamount The characterization of the tooth surface error andmodification amount are shown in Figures 10 and 11 Themaximum error is about minus88 um in the upper and lower

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus564

minus535

minus524

minus571

minus497

minus518

minus544

minus592

Top land

Figure 8 Topography diagram of modification amount and error(unit um)

surface but the minimum error is about minus54 um which isconcentrated in the central area of the profile and longitudinaldirections The normal error of any intersection of profiledirection and longitudinal direction is obtained by the errorof the contour map

Mathematical Problems in Engineering 9

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus44

minus48

minus56

minus88

minus61

minus66

minus65

minus69

Top land

Figure 9 Topography diagram of error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0minus002minus004minus006minus008

Mod

ifica

tion

amou

nt

and

erro

r (m

m)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)Longitudinal (mm)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

Mod

ifica

tion

amou

nt

and

erro

r (m

m) 0

minus002minus004minus006minus008

(b) Left flank

Figure 10 The contour map of modification amount and error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

times10minus3

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

Longitudinal (mm)

minus4

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

minus9

times10minus3

(b) Left flank

Figure 11 The contour map of error (unit um)

5 Conclusions

In this study an on-machine profile measurement systemalong with a five-axis CNC gear form grinding machineis developed to improve accuracy of grinding This system

includes the following steps (1) calculate trajectory of thecentre of spherical probe (2) define gear flanks by a grid ofpoints (3) obtain the coordinate values of topology measure-ment points Grinding experiments are performed to verifythe accuracy and efficiency of the topographymeasurements

10 Mathematical Problems in Engineering

With contour map the profile and longitudinal directionsof the error changes are easily seen Using statistical processcontrol techniques to monitor the grinding process cantimely attain the detection of changes in the product errorexceptions and take the necessary measures to prevent theoccurrence of waste As the numerical examples show on-machine measurement method can inspect tooth qualityAn additional advantage of the approach is to characterizemodification amount and tooth surface error respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the National Natural ScienceFoundation of China for its financial support Part of thiswork was performed under Contracts no 51375144 and no51405135 and the Priority Project of Research in Universitiesin Henan Province (Grant no 15A460021)

References

[1] H L Wang X Z Deng and J J Yang ldquoForm grinding andexperiment on segment topographicmodification gearrdquo Journalof Aerospace Power vol 29 no 12 pp 3000ndash3008 2014

[2] H-Y You P-Q Ye J-S Wang and X-Y Deng ldquoDesign andapplication of CBN shape grinding wheel for gearsrdquo Interna-tional Journal of Machine Tools and Manufacture vol 43 no12 pp 1269ndash1277 2003

[3] Q Fan R SDafoe and JW Swanger ldquoHigher-order tooth flankform error correction for face-milled spiral bevel and hypoidgearsrdquo Journal ofMechanical DesignmdashTransactions of theASMEvol 130 no 7 Article ID 072601 2008

[4] J Argyris M De Donno and F L Litvin ldquoComputer programin visual basic language for simulation of meshing and contactof gear drives and its application for design of worm gear driverdquoComputer Methods in Applied Mechanics and Engineering vol189 no 2 pp 595ndash612 2000

[5] Z L Zhang Y Fu Q R Yin and Y Zeng ldquoStudy on the polarmethod of measuring of gear profile errorsrdquo Chinese Journal ofMechanical Engineering vol 37 no 4 pp 70ndash72 2001

[6] C H Gao K Cheng and D Webb ldquoInvestigation on samplingsize optimisation in gear tooth surface measurement using aCMMrdquo The International Journal of Advanced ManufacturingTechnology vol 24 no 7-8 pp 599ndash606 2004

[7] Z L Zhang Y Fu and Y Zeng ldquoExpressing gear involuteerror by polar angle amp generating angle in polar coordinatemeasuring methodrdquo Tool Engineering vol 34 no 4 pp 39ndash402000

[8] F Gao B H Zhao and Y Li ldquoNovel pre-travel calibrationmethod of touch trigger probe based on error separationrdquoChinese Journal of Scientific Instrument vol 34 no 7 pp 1581ndash1587 2013

[9] F Gao Y Li S Tian Y Huang L Hao and J Wang ldquoStudy onthe on-machine measurement method of NC wheel gear formgrinding machinerdquo Chinese Journal of Scientific Instrument vol29 no 3 pp 540ndash544 2008

[10] A Nafi J R R Mayer and A Wozniak ldquoNovel CMM-basedimplementation of the multi-step method for the separation ofmachine and probe errorsrdquo Precision Engineering vol 35 no 2pp 318ndash328 2011

[11] A Nafi J R RMayer andAWozniak ldquoReduced configurationset for the multi-step method applied to machine and probeerror separation on a CMMrdquo Measurement vol 45 no 10 pp2321ndash2329 2012

[12] Y-P Shih and S-D Chen ldquoFree-formflank correction in helicalgear grinding using a five-axis computer numerical control gearprofile grindingmachinerdquo Journal ofManufacturing Science andEngineering vol 134 no 4 Article ID 041006 2012

[13] Y Kobayashi N Nishida Y Ougiya and H Nagata ldquoToothtrace modification processing of helix gear by form grindingmethodrdquo Transactions of the Japan Society of Mechanical Engi-neers Part C vol 61 no 590 pp 4088ndash4093 1995

[14] Y Kobayashi N Nishida and Y Ougiya ldquoEstimation ofgrinding wheel setting error in helical gear processing byform grindingrdquo Transactions of the Japan Society of MechanicalEngineers Part C vol 63 no 612 pp 2852ndash2858 1997

[15] C-K Lee ldquoManufacturing process for a cylindrical crown geardrive with a controllable fourth order polynomial function oftransmission errorrdquo Journal of Materials Processing Technologyvol 209 no 1 pp 3ndash13 2009

[16] C K Lee and C K Chen ldquoMathematical models meshinganalysis and transmission design for robust cylindrical gear setgenerated by double blade-disks with parabolic cutting edgesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 218 no 12 pp1539ndash1553 2004

[17] Z Y Shi and Y Ye ldquoResearch on the generalized polar-coordinate method for measuring involute profile deviationsrdquoChinese Journal of Scientific Instrument vol 22 no 2 pp 140ndash142 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

123

451

II

I

2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

1

2

3

4

5j

j

i

i

Figure 4 Definition of tooth flank grid

on the driving gear on the other hand the driven gear is astandard involute gear For the topologicalmodification toothsurface tooth surface measurement areas need to be plannedto get the whole tooth surface error information Accordingto the relevant standards of American gear measurement thedivision of tooth surface grid in the longitudinal directionshould be less than 10 of the tooth width in the profiledirection it should be less than 5 of the working depth butmust be less than or equal to 06mm The density of toothsurfacemeasurement point is generally taken to be 9 columnslong in the longitudinal direction and taken to be 5 rows inthe profile direction a total of 45 points The tooth surfacemeasurement path planning is shown in Figure 4

The plane coordinate system XOZ is an axis of rotationprojection section of the tooth surface where 119874 is centrepoint of base circle and119872lowast is a point in the projection surfaceat any measured point in the tooth surface Given a point by(119909lowast 119911lowast) the following system of nonlinear equations can beused to solve for the surface parameters

radic1199092

119894+ 1199102

119894= 119909lowast

119911119894= 119911lowast

(17)

where the coordinates (119909119894 119910119894 and 119911

119894) are three coordinate

components of tooth surface position vector r(119894)1 119894 = 1 2

45The tooth surface of topological modification is no longer

the involute tooth surface and becomes complex because ofthe existence of machining error So the whole tooth surfaceneeds to be digitally processed to get more accurate toothsurface geometric shape namely dividing grid on the toothsurface and computing the coordinate data of the grid pointFigure 5 shows the partition of the grid point inside gear shaftsection after rotation projection in which the grid points aredivided into 5 rows by 9 columns

In Figure 5 1198601 1198602 1198603 and 1198604 are the four boundarypoints of the tooth surface 1198601015840

1 11986010158402 11986010158403 and 119860

1015840

4are the

four new boundary points corresponding to tooth surfacecontraction grid area Here Δ119897

1is the top contraction Δ119897

2is

the front contraction Δ1198973is the root contraction and Δ119897

4is

the back contraction respectively The coordinate of a node119875119894119895(119894 = 1 sim 5 119895 = 1 sim 9) in the tooth surface grid is (119885

119894119895 119877119894119895)

X

O

1

5

9

A2

A9984001

A1

A9984002 A998400

3

A9984004

A4

ZZij

Rij

Pij

A3

Figure 5 Positions of grid points on one tooth

The coordinates of the tooth surface boundary points1198601

1198602 1198603 and 119860

4in the XOZ plane are respectively shown as

follows

1198601

1198851198601= 0

1198831198601= 119903119891

1198602

1198851198602 = 0

1198831198602 = 119903119886

1198603

1198851198603= 119887

1198831198603= 119903119886

1198604

1198851198604= 119887

1198831198604= 119903119891

(18)

here 119903119891is the tooth root radius 119903

119886is the tooth top radius and

119887 is the tooth width respectively

6 Mathematical Problems in Engineering

The new coordinates of tooth surface boundary points1198601015840

11198601015840211986010158403 and1198601015840

4in the XOZ plane are respectively shown

as follows

1198601015840

1

1198851198601 = Δ1198974

1198831198601 = 119903119891 + Δ1198973

1198601015840

2

1198851198602= Δ1198974

1198831198602= 119903119886minus Δ1198971

1198601015840

3

1198851198603= 119887 minus Δ119897

2

1198831198603= 119903119886minus Δ1198971

1198601015840

4

1198851198604 = 119887 minus Δ1198972

1198831198604 = 119903119891 + Δ1198973

(19)

The coordinates (1198851119895 1198831119895) (119895 = 1 sim 9) of 9 equal-division

points 1198751119895 between119860

1015840

1and1198601015840

4in the XOZ plane are shown as

follows

1198851119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

1198831119895= 119903119891+ Δ1198973

(20)

The coordinates (1198855119895 1198835119895) (119895 = 1 sim 9) of 9 equal-division

points 1198755119895between1198601015840

2and1198601015840

3in the XOZ plane are shown as

follows

1198855119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

1198835119895= 119903119886minus Δ1198971

(21)

The coordinates (1198851198941 1198831198941) (119894 = 1 sim 5) of 9 equal-division

points 1198751198941between1198601015840

1and1198601015840

2in the XOZ plane are shown as

follows1198851198941= Δ1198974

1198831198941 = 119903119891 + Δ1198973 +

119894 minus 1

4(119903119886 minus 119903119891 minus Δ1198971 minus Δ1198973)

(22)

The coordinates (1198851198949 1198831198949) (119894 = 1 sim 5) of 9 equal-division

points 1198751198949between1198601015840

4and1198601015840

3in the XOZ plane are shown as

follows1198851198949 = 119887 minus Δ1198972

1198831198949= 119903119891+ Δ1198973+119894 minus 1

4(119903119886minus 119903119891minus Δ1198971minus Δ1198973)

(23)

From the above derivational process the coordinates(119885119894119895 119883119894119895) (119894 = 1 sim 5 119895 = 1 sim 9) of any point 119875

119894119895in the

tooth surface grid in the XOZ plane can be calculated by theformulae that are shown as follows

119885119894119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

119883119894119895= 119903119891+ Δ1198973+119894 minus 1

4(119903119886minus 119903119891minus Δ1198971minus Δ1198973)

(24)

Gear

Theory profile

Trace of probe

Sensitive direction

Y

XΔEg

R

Mi

M998400i

120601

Ci

ni

re(u1 1205791)

r1(u1 1205791)

Figure 6 Relationships of the probe geometry and theory profile

Solving nonlinear equations to get the tooth surfaceparameters (119906

1 1205791) for eachmeasured point which substitute

into (2) can get the theoretical coordinates (119909(119894)1 119910(119894)1 and 119911(119894)

1)

and the unit normal vector n119894 Hence

r(119894)1= [119909(119894)

1119910(119894)

1119911(119894)

1]119879

n(119894)1= [119899(119894)

1119909119899(119894)

1119910119899(119894)

1119911]119879

(119894 = 1 2 2 times 45)

(25)

25Measurement Error Compensation When a point ismea-sured on the modification gear tooth surface the inspectiondatum of an on-machine measurement is the centre of theprobe To improve the measurement accuracy the proberadius can be considered in calculating the profile errorsFigure 6 schematically shows the effect of the probe sizeon the measurement As the probe always has a certaindimension an errorΔ119864119892 is unavoidable Due to the size effectof the probe the actual contact point of the probe is at point119872119894 instead of1198721015840

119894

According to the envelope surface characteristics thenormal vector is perpendicular to the probe sphere andpassed through the centre of the probe 119862119894 The theoreticalmotion trajectory of probe centre is shown as follows

r119890 (1199061 1205791) = r

1 (1199061 1205791) + 119877 sdot n1 (1199061 1205791) (26)

here r119890(1199061 1205791) is the trace of the probe centre and 119877 is theprobe radius

Strictly speaking the measuring tooth surface deviationis the deviation of the actual contact point 119872119894 in thesensitive direction of the probe so an important factor of themeasurement error is the actual contacting position of theprobe In the measurement the probe is also very close tothe contact pointmicroscopically the nearness of the contactpoint119872

119894can approximately be regarded as a small planeThe

Mathematical Problems in Engineering 7

Table 1 Parameters of the gears in the trials

Items Symbol Unit DataDriving gear 119911

1mdash 30

Driven gear 1199112

mdash 30Normal model 119898

119899mm 65

Pressure angle 120572119899

deg 20Helix angle 120573 deg 13Face width 119887 mm 53Top profile crowning 119886

119898119901(119888119891)1mm 00011

Bottom profile crowning 119886119898119901(119889119890)

1mm 00014Top limit angle 119906

119888rad 0496

Bottom limit angle 119906119889

rad 0196Front longitudinal crowning 119886

119898119897(ℎ119886)1mm 00012

Back longitudinal crowning 119886119898119897(119887119895)

1mm 00012Front limit angle 120579

119886rad 0018

Back limit angle 120579119887

rad 0104

measurement error caused by probe radius is Δ119864119892 and its

geometric relationships with probe radius can be expressedas follows

Δ119864119892= 119877(

1

cos120601minus 1) (27)

According to (27) the larger the probe radius 119877 thegreater the measurement error so the probe radius shouldbe small here the radius 119877 is 2mm The normal direction ofthe measuring points and the sensitivity of the probe are alsoincluded in the XOY plane and experimental results showthat the angle 120601 is very small

From Figure 6 it can be seen that the measured headshould be in contact with the theoretical point 1198721015840

119894 and the

119883 119884 direction of the coordinates of the value of a certainamount of compensation can be guaranteed Assuming that119872119894 coordinates are (119909119894 119910119894)119872

1015840

119894coordinates are (1199091015840

119894 1199101015840

119894) so the

relationship between the two expressions is

1199091015840

119894= 119909119894minus 119877119905119892120601

1199101015840

119894= 119909119894 + 119877119905119892120601 sin120601

(28)

The contact points are in agreement with the theoreticalcontact points after compensationThe contact position of themeasuring head does not affect the measurement results

3 Numerical Example

The parameters of the gears are listed in Table 1 The drivinggear is modified in the profile and longitudinal directionsat the same time On the contrary the tooth surface of thedriven gear is a conventional screw involute surface Thencoordinates on the tooth surface are measured and calculatedusing a MATLAB code

The measurement data of tooth surface can be obtainedby measuring the grid points in the order with the probeSelecting the radius of the probe 119877 = 2mm the probecentre trajectory curve is the envelope surface of measured

90

95

100

105

110

01234

0

20

40

60 Rotated topological pointTrace of the centre

of the probe

minus1minus2

y (mm)

x (mm)

z(m

m)

Figure 7The contrast position of rotated topological point and theprobe centre

tooth surface When the tooth surface is detected using polarcoordinate method the trajectory coordinates of the probecentre and the gear rotating angle 120601 are calculated using thecontrol unitThe relationship of relative position between thetopology points in the rotated tooth surface and the trajectoryof probe centre is shown in Figure 7

4 Tooth Flank Errors Separation

According to the predetermined spacing the tooth surfacedeviation of each measured tooth surface detection pointis point-to-point measured by probe along the profile andlongitudinal directions of the gear r0(1199061 1205791) is the standardinvolute tooth surface vector r1(1199061 1205791) is the theoreticaltopological modification tooth surface vector and r1119904(1199061 1205791)is the actual tooth surface vector after grinding respec-tively Due to the error of the machine tool adjustmentand movement the actual tooth surface often deviates fromthe theoretical tooth surface The tooth surface error Δ119864is usually measured along the direction of the unit normalvector n

1and is presented as

Δ119864 = (r1119904(1199061 1205791) minus r1(1199061 1205791)) sdot n1(1199061 1205791) (29)

According to the standard involute tooth surface vectorr0(1199061 1205791) and themodified tooth surface vector r

1(1199061 1205791) the

topological modification amount 120575 can be obtained which isshown as follows

120575 = (r1(1199061 1205791) minus r0(1199061 1205791)) sdot n0(1199061 1205791) (30)

If the tooth surface topology deviation is expressed asΔ120588the tooth surface error Δ119864 can be expressed by Δ120588 and 120575 asfollows

Δ119864 = Δ120588 minus 120575 (31)

Equation (31) shows that the tooth surface error can beobtained by removing the modification amount out of thetopology deviation of tooth surface

8 Mathematical Problems in Engineering

Table 2 The modification amount and tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00564 minus00436 minus00314 minus00257 minus00239 minus00285 minus00291 minus00385 minus004972 minus00449 minus00387 minus00215 minus00159 minus00145 minus00147 minus00214 minus00289 minus004363 minus00406 minus00302 minus00197 minus00115 minus00027 minus0012 minus00173 minus00237 minus003914 minus00479 minus00421 minus00236 minus00214 minus00136 minus00234 minus00257 minus00356 minus004125 minus00535 minus00509 minus00352 minus00298 minus00172 minus00287 minus00327 minus00492 minus00518

Left flankmm

1 2 3 4 5 6 7 8 95 minus00524 minus00487 minus00301 minus00272 minus00196 minus00264 minus00314 minus00501 minus005444 minus00498 minus00410 minus00198 minus00189 minus00035 minus00197 minus00241 minus00417 minus004633 minus00436 minus00375 minus00221 minus00035 minus00021 minus00046 minus00210 minus00324 minus004522 minus00510 minus00417 minus00183 minus00167 minus00138 minus00153 minus00189 minus00394 minus005311 minus00571 minus00462 minus00294 minus00285 minus00227 minus00249 minus00267 minus00472 minus00592

Table 3 The tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00044 minus00047 minus00046 minus00046 minus00062 minus00062 minus00066 minus00061 minus000612 minus00061 minus00064 minus00063 minus00064 minus00079 minus00079 minus00083 minus00077 minus000793 minus00058 minus00061 minus00062 minus00063 minus00075 minus00076 minus00084 minus00074 minus000754 minus00056 minus00059 minus00058 minus00059 minus00074 minus00074 minus00078 minus00072 minus000745 minus00048 minus00051 minus00050 minus00051 minus00066 minus00066 minus00071 minus00064 minus00066

Left flankmm

1 2 3 4 5 6 7 8 95 minus00053 minus00057 minus00055 minus00056 minus00057 minus00058 minus00087 minus00054 minus000654 minus00062 minus00062 minus00068 minus00072 minus00084 minus00067 minus00086 minus00039 minus000743 minus00049 minus00074 minus00072 minus00058 minus00054 minus00069 minus00094 minus00075 minus000872 minus00072 minus00049 minus00085 minus00063 minus00072 minus00069 minus00064 minus00058 minus000761 minus00088 minus00076 minus00067 minus00053 minus00069 minus00072 minus00073 minus00067 minus00069

The tooth surface topology deviation Δ120588 is shown inTable 2 and the tooth surface error Δ119864 is shown in Table 3respectively

The deviation distribution (error and modificationamount) of the tooth surface obtained by an on-machinemeasurement is shown in Figure 8 The dashed parts (thesuperposition of error and modification) in regions 2 46 and 8 of the tooth surface are obviously presented as aparabolic shape in the profile and longitudinal directionsrespectivelyThemaximumvalue of deviation at the tooth topis minus00571mm and the error in the centre of tooth surfacearea 1 is minus00049mm The trend is gradually increasedtowards both sides presenting the shape where the middle ishigh and both sides are low

The measurement using the topological modificationtooth surface equation is shown in Figure 9 The measuredresult reflects the actual tooth surface error the distributionof which has unobvious trend The maximum error in thetooth top is minus00088mm and the error in the centre of righttooth surface area 1 is minus00054mm

To separate flank form deviations from modificationamount and to improve the perception of deviation dia-grams all flank deviations are referenced by the modificationamount The characterization of the tooth surface error andmodification amount are shown in Figures 10 and 11 Themaximum error is about minus88 um in the upper and lower

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus564

minus535

minus524

minus571

minus497

minus518

minus544

minus592

Top land

Figure 8 Topography diagram of modification amount and error(unit um)

surface but the minimum error is about minus54 um which isconcentrated in the central area of the profile and longitudinaldirections The normal error of any intersection of profiledirection and longitudinal direction is obtained by the errorof the contour map

Mathematical Problems in Engineering 9

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus44

minus48

minus56

minus88

minus61

minus66

minus65

minus69

Top land

Figure 9 Topography diagram of error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0minus002minus004minus006minus008

Mod

ifica

tion

amou

nt

and

erro

r (m

m)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)Longitudinal (mm)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

Mod

ifica

tion

amou

nt

and

erro

r (m

m) 0

minus002minus004minus006minus008

(b) Left flank

Figure 10 The contour map of modification amount and error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

times10minus3

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

Longitudinal (mm)

minus4

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

minus9

times10minus3

(b) Left flank

Figure 11 The contour map of error (unit um)

5 Conclusions

In this study an on-machine profile measurement systemalong with a five-axis CNC gear form grinding machineis developed to improve accuracy of grinding This system

includes the following steps (1) calculate trajectory of thecentre of spherical probe (2) define gear flanks by a grid ofpoints (3) obtain the coordinate values of topology measure-ment points Grinding experiments are performed to verifythe accuracy and efficiency of the topographymeasurements

10 Mathematical Problems in Engineering

With contour map the profile and longitudinal directionsof the error changes are easily seen Using statistical processcontrol techniques to monitor the grinding process cantimely attain the detection of changes in the product errorexceptions and take the necessary measures to prevent theoccurrence of waste As the numerical examples show on-machine measurement method can inspect tooth qualityAn additional advantage of the approach is to characterizemodification amount and tooth surface error respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the National Natural ScienceFoundation of China for its financial support Part of thiswork was performed under Contracts no 51375144 and no51405135 and the Priority Project of Research in Universitiesin Henan Province (Grant no 15A460021)

References

[1] H L Wang X Z Deng and J J Yang ldquoForm grinding andexperiment on segment topographicmodification gearrdquo Journalof Aerospace Power vol 29 no 12 pp 3000ndash3008 2014

[2] H-Y You P-Q Ye J-S Wang and X-Y Deng ldquoDesign andapplication of CBN shape grinding wheel for gearsrdquo Interna-tional Journal of Machine Tools and Manufacture vol 43 no12 pp 1269ndash1277 2003

[3] Q Fan R SDafoe and JW Swanger ldquoHigher-order tooth flankform error correction for face-milled spiral bevel and hypoidgearsrdquo Journal ofMechanical DesignmdashTransactions of theASMEvol 130 no 7 Article ID 072601 2008

[4] J Argyris M De Donno and F L Litvin ldquoComputer programin visual basic language for simulation of meshing and contactof gear drives and its application for design of worm gear driverdquoComputer Methods in Applied Mechanics and Engineering vol189 no 2 pp 595ndash612 2000

[5] Z L Zhang Y Fu Q R Yin and Y Zeng ldquoStudy on the polarmethod of measuring of gear profile errorsrdquo Chinese Journal ofMechanical Engineering vol 37 no 4 pp 70ndash72 2001

[6] C H Gao K Cheng and D Webb ldquoInvestigation on samplingsize optimisation in gear tooth surface measurement using aCMMrdquo The International Journal of Advanced ManufacturingTechnology vol 24 no 7-8 pp 599ndash606 2004

[7] Z L Zhang Y Fu and Y Zeng ldquoExpressing gear involuteerror by polar angle amp generating angle in polar coordinatemeasuring methodrdquo Tool Engineering vol 34 no 4 pp 39ndash402000

[8] F Gao B H Zhao and Y Li ldquoNovel pre-travel calibrationmethod of touch trigger probe based on error separationrdquoChinese Journal of Scientific Instrument vol 34 no 7 pp 1581ndash1587 2013

[9] F Gao Y Li S Tian Y Huang L Hao and J Wang ldquoStudy onthe on-machine measurement method of NC wheel gear formgrinding machinerdquo Chinese Journal of Scientific Instrument vol29 no 3 pp 540ndash544 2008

[10] A Nafi J R R Mayer and A Wozniak ldquoNovel CMM-basedimplementation of the multi-step method for the separation ofmachine and probe errorsrdquo Precision Engineering vol 35 no 2pp 318ndash328 2011

[11] A Nafi J R RMayer andAWozniak ldquoReduced configurationset for the multi-step method applied to machine and probeerror separation on a CMMrdquo Measurement vol 45 no 10 pp2321ndash2329 2012

[12] Y-P Shih and S-D Chen ldquoFree-formflank correction in helicalgear grinding using a five-axis computer numerical control gearprofile grindingmachinerdquo Journal ofManufacturing Science andEngineering vol 134 no 4 Article ID 041006 2012

[13] Y Kobayashi N Nishida Y Ougiya and H Nagata ldquoToothtrace modification processing of helix gear by form grindingmethodrdquo Transactions of the Japan Society of Mechanical Engi-neers Part C vol 61 no 590 pp 4088ndash4093 1995

[14] Y Kobayashi N Nishida and Y Ougiya ldquoEstimation ofgrinding wheel setting error in helical gear processing byform grindingrdquo Transactions of the Japan Society of MechanicalEngineers Part C vol 63 no 612 pp 2852ndash2858 1997

[15] C-K Lee ldquoManufacturing process for a cylindrical crown geardrive with a controllable fourth order polynomial function oftransmission errorrdquo Journal of Materials Processing Technologyvol 209 no 1 pp 3ndash13 2009

[16] C K Lee and C K Chen ldquoMathematical models meshinganalysis and transmission design for robust cylindrical gear setgenerated by double blade-disks with parabolic cutting edgesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 218 no 12 pp1539ndash1553 2004

[17] Z Y Shi and Y Ye ldquoResearch on the generalized polar-coordinate method for measuring involute profile deviationsrdquoChinese Journal of Scientific Instrument vol 22 no 2 pp 140ndash142 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

The new coordinates of tooth surface boundary points1198601015840

11198601015840211986010158403 and1198601015840

4in the XOZ plane are respectively shown

as follows

1198601015840

1

1198851198601 = Δ1198974

1198831198601 = 119903119891 + Δ1198973

1198601015840

2

1198851198602= Δ1198974

1198831198602= 119903119886minus Δ1198971

1198601015840

3

1198851198603= 119887 minus Δ119897

2

1198831198603= 119903119886minus Δ1198971

1198601015840

4

1198851198604 = 119887 minus Δ1198972

1198831198604 = 119903119891 + Δ1198973

(19)

The coordinates (1198851119895 1198831119895) (119895 = 1 sim 9) of 9 equal-division

points 1198751119895 between119860

1015840

1and1198601015840

4in the XOZ plane are shown as

follows

1198851119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

1198831119895= 119903119891+ Δ1198973

(20)

The coordinates (1198855119895 1198835119895) (119895 = 1 sim 9) of 9 equal-division

points 1198755119895between1198601015840

2and1198601015840

3in the XOZ plane are shown as

follows

1198855119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

1198835119895= 119903119886minus Δ1198971

(21)

The coordinates (1198851198941 1198831198941) (119894 = 1 sim 5) of 9 equal-division

points 1198751198941between1198601015840

1and1198601015840

2in the XOZ plane are shown as

follows1198851198941= Δ1198974

1198831198941 = 119903119891 + Δ1198973 +

119894 minus 1

4(119903119886 minus 119903119891 minus Δ1198971 minus Δ1198973)

(22)

The coordinates (1198851198949 1198831198949) (119894 = 1 sim 5) of 9 equal-division

points 1198751198949between1198601015840

4and1198601015840

3in the XOZ plane are shown as

follows1198851198949 = 119887 minus Δ1198972

1198831198949= 119903119891+ Δ1198973+119894 minus 1

4(119903119886minus 119903119891minus Δ1198971minus Δ1198973)

(23)

From the above derivational process the coordinates(119885119894119895 119883119894119895) (119894 = 1 sim 5 119895 = 1 sim 9) of any point 119875

119894119895in the

tooth surface grid in the XOZ plane can be calculated by theformulae that are shown as follows

119885119894119895= Δ1198974+119895 minus 1

8(119887 minus Δ119897

2minus Δ1198974)

119883119894119895= 119903119891+ Δ1198973+119894 minus 1

4(119903119886minus 119903119891minus Δ1198971minus Δ1198973)

(24)

Gear

Theory profile

Trace of probe

Sensitive direction

Y

XΔEg

R

Mi

M998400i

120601

Ci

ni

re(u1 1205791)

r1(u1 1205791)

Figure 6 Relationships of the probe geometry and theory profile

Solving nonlinear equations to get the tooth surfaceparameters (119906

1 1205791) for eachmeasured point which substitute

into (2) can get the theoretical coordinates (119909(119894)1 119910(119894)1 and 119911(119894)

1)

and the unit normal vector n119894 Hence

r(119894)1= [119909(119894)

1119910(119894)

1119911(119894)

1]119879

n(119894)1= [119899(119894)

1119909119899(119894)

1119910119899(119894)

1119911]119879

(119894 = 1 2 2 times 45)

(25)

25Measurement Error Compensation When a point ismea-sured on the modification gear tooth surface the inspectiondatum of an on-machine measurement is the centre of theprobe To improve the measurement accuracy the proberadius can be considered in calculating the profile errorsFigure 6 schematically shows the effect of the probe sizeon the measurement As the probe always has a certaindimension an errorΔ119864119892 is unavoidable Due to the size effectof the probe the actual contact point of the probe is at point119872119894 instead of1198721015840

119894

According to the envelope surface characteristics thenormal vector is perpendicular to the probe sphere andpassed through the centre of the probe 119862119894 The theoreticalmotion trajectory of probe centre is shown as follows

r119890 (1199061 1205791) = r

1 (1199061 1205791) + 119877 sdot n1 (1199061 1205791) (26)

here r119890(1199061 1205791) is the trace of the probe centre and 119877 is theprobe radius

Strictly speaking the measuring tooth surface deviationis the deviation of the actual contact point 119872119894 in thesensitive direction of the probe so an important factor of themeasurement error is the actual contacting position of theprobe In the measurement the probe is also very close tothe contact pointmicroscopically the nearness of the contactpoint119872

119894can approximately be regarded as a small planeThe

Mathematical Problems in Engineering 7

Table 1 Parameters of the gears in the trials

Items Symbol Unit DataDriving gear 119911

1mdash 30

Driven gear 1199112

mdash 30Normal model 119898

119899mm 65

Pressure angle 120572119899

deg 20Helix angle 120573 deg 13Face width 119887 mm 53Top profile crowning 119886

119898119901(119888119891)1mm 00011

Bottom profile crowning 119886119898119901(119889119890)

1mm 00014Top limit angle 119906

119888rad 0496

Bottom limit angle 119906119889

rad 0196Front longitudinal crowning 119886

119898119897(ℎ119886)1mm 00012

Back longitudinal crowning 119886119898119897(119887119895)

1mm 00012Front limit angle 120579

119886rad 0018

Back limit angle 120579119887

rad 0104

measurement error caused by probe radius is Δ119864119892 and its

geometric relationships with probe radius can be expressedas follows

Δ119864119892= 119877(

1

cos120601minus 1) (27)

According to (27) the larger the probe radius 119877 thegreater the measurement error so the probe radius shouldbe small here the radius 119877 is 2mm The normal direction ofthe measuring points and the sensitivity of the probe are alsoincluded in the XOY plane and experimental results showthat the angle 120601 is very small

From Figure 6 it can be seen that the measured headshould be in contact with the theoretical point 1198721015840

119894 and the

119883 119884 direction of the coordinates of the value of a certainamount of compensation can be guaranteed Assuming that119872119894 coordinates are (119909119894 119910119894)119872

1015840

119894coordinates are (1199091015840

119894 1199101015840

119894) so the

relationship between the two expressions is

1199091015840

119894= 119909119894minus 119877119905119892120601

1199101015840

119894= 119909119894 + 119877119905119892120601 sin120601

(28)

The contact points are in agreement with the theoreticalcontact points after compensationThe contact position of themeasuring head does not affect the measurement results

3 Numerical Example

The parameters of the gears are listed in Table 1 The drivinggear is modified in the profile and longitudinal directionsat the same time On the contrary the tooth surface of thedriven gear is a conventional screw involute surface Thencoordinates on the tooth surface are measured and calculatedusing a MATLAB code

The measurement data of tooth surface can be obtainedby measuring the grid points in the order with the probeSelecting the radius of the probe 119877 = 2mm the probecentre trajectory curve is the envelope surface of measured

90

95

100

105

110

01234

0

20

40

60 Rotated topological pointTrace of the centre

of the probe

minus1minus2

y (mm)

x (mm)

z(m

m)

Figure 7The contrast position of rotated topological point and theprobe centre

tooth surface When the tooth surface is detected using polarcoordinate method the trajectory coordinates of the probecentre and the gear rotating angle 120601 are calculated using thecontrol unitThe relationship of relative position between thetopology points in the rotated tooth surface and the trajectoryof probe centre is shown in Figure 7

4 Tooth Flank Errors Separation

According to the predetermined spacing the tooth surfacedeviation of each measured tooth surface detection pointis point-to-point measured by probe along the profile andlongitudinal directions of the gear r0(1199061 1205791) is the standardinvolute tooth surface vector r1(1199061 1205791) is the theoreticaltopological modification tooth surface vector and r1119904(1199061 1205791)is the actual tooth surface vector after grinding respec-tively Due to the error of the machine tool adjustmentand movement the actual tooth surface often deviates fromthe theoretical tooth surface The tooth surface error Δ119864is usually measured along the direction of the unit normalvector n

1and is presented as

Δ119864 = (r1119904(1199061 1205791) minus r1(1199061 1205791)) sdot n1(1199061 1205791) (29)

According to the standard involute tooth surface vectorr0(1199061 1205791) and themodified tooth surface vector r

1(1199061 1205791) the

topological modification amount 120575 can be obtained which isshown as follows

120575 = (r1(1199061 1205791) minus r0(1199061 1205791)) sdot n0(1199061 1205791) (30)

If the tooth surface topology deviation is expressed asΔ120588the tooth surface error Δ119864 can be expressed by Δ120588 and 120575 asfollows

Δ119864 = Δ120588 minus 120575 (31)

Equation (31) shows that the tooth surface error can beobtained by removing the modification amount out of thetopology deviation of tooth surface

8 Mathematical Problems in Engineering

Table 2 The modification amount and tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00564 minus00436 minus00314 minus00257 minus00239 minus00285 minus00291 minus00385 minus004972 minus00449 minus00387 minus00215 minus00159 minus00145 minus00147 minus00214 minus00289 minus004363 minus00406 minus00302 minus00197 minus00115 minus00027 minus0012 minus00173 minus00237 minus003914 minus00479 minus00421 minus00236 minus00214 minus00136 minus00234 minus00257 minus00356 minus004125 minus00535 minus00509 minus00352 minus00298 minus00172 minus00287 minus00327 minus00492 minus00518

Left flankmm

1 2 3 4 5 6 7 8 95 minus00524 minus00487 minus00301 minus00272 minus00196 minus00264 minus00314 minus00501 minus005444 minus00498 minus00410 minus00198 minus00189 minus00035 minus00197 minus00241 minus00417 minus004633 minus00436 minus00375 minus00221 minus00035 minus00021 minus00046 minus00210 minus00324 minus004522 minus00510 minus00417 minus00183 minus00167 minus00138 minus00153 minus00189 minus00394 minus005311 minus00571 minus00462 minus00294 minus00285 minus00227 minus00249 minus00267 minus00472 minus00592

Table 3 The tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00044 minus00047 minus00046 minus00046 minus00062 minus00062 minus00066 minus00061 minus000612 minus00061 minus00064 minus00063 minus00064 minus00079 minus00079 minus00083 minus00077 minus000793 minus00058 minus00061 minus00062 minus00063 minus00075 minus00076 minus00084 minus00074 minus000754 minus00056 minus00059 minus00058 minus00059 minus00074 minus00074 minus00078 minus00072 minus000745 minus00048 minus00051 minus00050 minus00051 minus00066 minus00066 minus00071 minus00064 minus00066

Left flankmm

1 2 3 4 5 6 7 8 95 minus00053 minus00057 minus00055 minus00056 minus00057 minus00058 minus00087 minus00054 minus000654 minus00062 minus00062 minus00068 minus00072 minus00084 minus00067 minus00086 minus00039 minus000743 minus00049 minus00074 minus00072 minus00058 minus00054 minus00069 minus00094 minus00075 minus000872 minus00072 minus00049 minus00085 minus00063 minus00072 minus00069 minus00064 minus00058 minus000761 minus00088 minus00076 minus00067 minus00053 minus00069 minus00072 minus00073 minus00067 minus00069

The tooth surface topology deviation Δ120588 is shown inTable 2 and the tooth surface error Δ119864 is shown in Table 3respectively

The deviation distribution (error and modificationamount) of the tooth surface obtained by an on-machinemeasurement is shown in Figure 8 The dashed parts (thesuperposition of error and modification) in regions 2 46 and 8 of the tooth surface are obviously presented as aparabolic shape in the profile and longitudinal directionsrespectivelyThemaximumvalue of deviation at the tooth topis minus00571mm and the error in the centre of tooth surfacearea 1 is minus00049mm The trend is gradually increasedtowards both sides presenting the shape where the middle ishigh and both sides are low

The measurement using the topological modificationtooth surface equation is shown in Figure 9 The measuredresult reflects the actual tooth surface error the distributionof which has unobvious trend The maximum error in thetooth top is minus00088mm and the error in the centre of righttooth surface area 1 is minus00054mm

To separate flank form deviations from modificationamount and to improve the perception of deviation dia-grams all flank deviations are referenced by the modificationamount The characterization of the tooth surface error andmodification amount are shown in Figures 10 and 11 Themaximum error is about minus88 um in the upper and lower

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus564

minus535

minus524

minus571

minus497

minus518

minus544

minus592

Top land

Figure 8 Topography diagram of modification amount and error(unit um)

surface but the minimum error is about minus54 um which isconcentrated in the central area of the profile and longitudinaldirections The normal error of any intersection of profiledirection and longitudinal direction is obtained by the errorof the contour map

Mathematical Problems in Engineering 9

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus44

minus48

minus56

minus88

minus61

minus66

minus65

minus69

Top land

Figure 9 Topography diagram of error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0minus002minus004minus006minus008

Mod

ifica

tion

amou

nt

and

erro

r (m

m)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)Longitudinal (mm)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

Mod

ifica

tion

amou

nt

and

erro

r (m

m) 0

minus002minus004minus006minus008

(b) Left flank

Figure 10 The contour map of modification amount and error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

times10minus3

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

Longitudinal (mm)

minus4

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

minus9

times10minus3

(b) Left flank

Figure 11 The contour map of error (unit um)

5 Conclusions

In this study an on-machine profile measurement systemalong with a five-axis CNC gear form grinding machineis developed to improve accuracy of grinding This system

includes the following steps (1) calculate trajectory of thecentre of spherical probe (2) define gear flanks by a grid ofpoints (3) obtain the coordinate values of topology measure-ment points Grinding experiments are performed to verifythe accuracy and efficiency of the topographymeasurements

10 Mathematical Problems in Engineering

With contour map the profile and longitudinal directionsof the error changes are easily seen Using statistical processcontrol techniques to monitor the grinding process cantimely attain the detection of changes in the product errorexceptions and take the necessary measures to prevent theoccurrence of waste As the numerical examples show on-machine measurement method can inspect tooth qualityAn additional advantage of the approach is to characterizemodification amount and tooth surface error respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the National Natural ScienceFoundation of China for its financial support Part of thiswork was performed under Contracts no 51375144 and no51405135 and the Priority Project of Research in Universitiesin Henan Province (Grant no 15A460021)

References

[1] H L Wang X Z Deng and J J Yang ldquoForm grinding andexperiment on segment topographicmodification gearrdquo Journalof Aerospace Power vol 29 no 12 pp 3000ndash3008 2014

[2] H-Y You P-Q Ye J-S Wang and X-Y Deng ldquoDesign andapplication of CBN shape grinding wheel for gearsrdquo Interna-tional Journal of Machine Tools and Manufacture vol 43 no12 pp 1269ndash1277 2003

[3] Q Fan R SDafoe and JW Swanger ldquoHigher-order tooth flankform error correction for face-milled spiral bevel and hypoidgearsrdquo Journal ofMechanical DesignmdashTransactions of theASMEvol 130 no 7 Article ID 072601 2008

[4] J Argyris M De Donno and F L Litvin ldquoComputer programin visual basic language for simulation of meshing and contactof gear drives and its application for design of worm gear driverdquoComputer Methods in Applied Mechanics and Engineering vol189 no 2 pp 595ndash612 2000

[5] Z L Zhang Y Fu Q R Yin and Y Zeng ldquoStudy on the polarmethod of measuring of gear profile errorsrdquo Chinese Journal ofMechanical Engineering vol 37 no 4 pp 70ndash72 2001

[6] C H Gao K Cheng and D Webb ldquoInvestigation on samplingsize optimisation in gear tooth surface measurement using aCMMrdquo The International Journal of Advanced ManufacturingTechnology vol 24 no 7-8 pp 599ndash606 2004

[7] Z L Zhang Y Fu and Y Zeng ldquoExpressing gear involuteerror by polar angle amp generating angle in polar coordinatemeasuring methodrdquo Tool Engineering vol 34 no 4 pp 39ndash402000

[8] F Gao B H Zhao and Y Li ldquoNovel pre-travel calibrationmethod of touch trigger probe based on error separationrdquoChinese Journal of Scientific Instrument vol 34 no 7 pp 1581ndash1587 2013

[9] F Gao Y Li S Tian Y Huang L Hao and J Wang ldquoStudy onthe on-machine measurement method of NC wheel gear formgrinding machinerdquo Chinese Journal of Scientific Instrument vol29 no 3 pp 540ndash544 2008

[10] A Nafi J R R Mayer and A Wozniak ldquoNovel CMM-basedimplementation of the multi-step method for the separation ofmachine and probe errorsrdquo Precision Engineering vol 35 no 2pp 318ndash328 2011

[11] A Nafi J R RMayer andAWozniak ldquoReduced configurationset for the multi-step method applied to machine and probeerror separation on a CMMrdquo Measurement vol 45 no 10 pp2321ndash2329 2012

[12] Y-P Shih and S-D Chen ldquoFree-formflank correction in helicalgear grinding using a five-axis computer numerical control gearprofile grindingmachinerdquo Journal ofManufacturing Science andEngineering vol 134 no 4 Article ID 041006 2012

[13] Y Kobayashi N Nishida Y Ougiya and H Nagata ldquoToothtrace modification processing of helix gear by form grindingmethodrdquo Transactions of the Japan Society of Mechanical Engi-neers Part C vol 61 no 590 pp 4088ndash4093 1995

[14] Y Kobayashi N Nishida and Y Ougiya ldquoEstimation ofgrinding wheel setting error in helical gear processing byform grindingrdquo Transactions of the Japan Society of MechanicalEngineers Part C vol 63 no 612 pp 2852ndash2858 1997

[15] C-K Lee ldquoManufacturing process for a cylindrical crown geardrive with a controllable fourth order polynomial function oftransmission errorrdquo Journal of Materials Processing Technologyvol 209 no 1 pp 3ndash13 2009

[16] C K Lee and C K Chen ldquoMathematical models meshinganalysis and transmission design for robust cylindrical gear setgenerated by double blade-disks with parabolic cutting edgesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 218 no 12 pp1539ndash1553 2004

[17] Z Y Shi and Y Ye ldquoResearch on the generalized polar-coordinate method for measuring involute profile deviationsrdquoChinese Journal of Scientific Instrument vol 22 no 2 pp 140ndash142 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1 Parameters of the gears in the trials

Items Symbol Unit DataDriving gear 119911

1mdash 30

Driven gear 1199112

mdash 30Normal model 119898

119899mm 65

Pressure angle 120572119899

deg 20Helix angle 120573 deg 13Face width 119887 mm 53Top profile crowning 119886

119898119901(119888119891)1mm 00011

Bottom profile crowning 119886119898119901(119889119890)

1mm 00014Top limit angle 119906

119888rad 0496

Bottom limit angle 119906119889

rad 0196Front longitudinal crowning 119886

119898119897(ℎ119886)1mm 00012

Back longitudinal crowning 119886119898119897(119887119895)

1mm 00012Front limit angle 120579

119886rad 0018

Back limit angle 120579119887

rad 0104

measurement error caused by probe radius is Δ119864119892 and its

geometric relationships with probe radius can be expressedas follows

Δ119864119892= 119877(

1

cos120601minus 1) (27)

According to (27) the larger the probe radius 119877 thegreater the measurement error so the probe radius shouldbe small here the radius 119877 is 2mm The normal direction ofthe measuring points and the sensitivity of the probe are alsoincluded in the XOY plane and experimental results showthat the angle 120601 is very small

From Figure 6 it can be seen that the measured headshould be in contact with the theoretical point 1198721015840

119894 and the

119883 119884 direction of the coordinates of the value of a certainamount of compensation can be guaranteed Assuming that119872119894 coordinates are (119909119894 119910119894)119872

1015840

119894coordinates are (1199091015840

119894 1199101015840

119894) so the

relationship between the two expressions is

1199091015840

119894= 119909119894minus 119877119905119892120601

1199101015840

119894= 119909119894 + 119877119905119892120601 sin120601

(28)

The contact points are in agreement with the theoreticalcontact points after compensationThe contact position of themeasuring head does not affect the measurement results

3 Numerical Example

The parameters of the gears are listed in Table 1 The drivinggear is modified in the profile and longitudinal directionsat the same time On the contrary the tooth surface of thedriven gear is a conventional screw involute surface Thencoordinates on the tooth surface are measured and calculatedusing a MATLAB code

The measurement data of tooth surface can be obtainedby measuring the grid points in the order with the probeSelecting the radius of the probe 119877 = 2mm the probecentre trajectory curve is the envelope surface of measured

90

95

100

105

110

01234

0

20

40

60 Rotated topological pointTrace of the centre

of the probe

minus1minus2

y (mm)

x (mm)

z(m

m)

Figure 7The contrast position of rotated topological point and theprobe centre

tooth surface When the tooth surface is detected using polarcoordinate method the trajectory coordinates of the probecentre and the gear rotating angle 120601 are calculated using thecontrol unitThe relationship of relative position between thetopology points in the rotated tooth surface and the trajectoryof probe centre is shown in Figure 7

4 Tooth Flank Errors Separation

According to the predetermined spacing the tooth surfacedeviation of each measured tooth surface detection pointis point-to-point measured by probe along the profile andlongitudinal directions of the gear r0(1199061 1205791) is the standardinvolute tooth surface vector r1(1199061 1205791) is the theoreticaltopological modification tooth surface vector and r1119904(1199061 1205791)is the actual tooth surface vector after grinding respec-tively Due to the error of the machine tool adjustmentand movement the actual tooth surface often deviates fromthe theoretical tooth surface The tooth surface error Δ119864is usually measured along the direction of the unit normalvector n

1and is presented as

Δ119864 = (r1119904(1199061 1205791) minus r1(1199061 1205791)) sdot n1(1199061 1205791) (29)

According to the standard involute tooth surface vectorr0(1199061 1205791) and themodified tooth surface vector r

1(1199061 1205791) the

topological modification amount 120575 can be obtained which isshown as follows

120575 = (r1(1199061 1205791) minus r0(1199061 1205791)) sdot n0(1199061 1205791) (30)

If the tooth surface topology deviation is expressed asΔ120588the tooth surface error Δ119864 can be expressed by Δ120588 and 120575 asfollows

Δ119864 = Δ120588 minus 120575 (31)

Equation (31) shows that the tooth surface error can beobtained by removing the modification amount out of thetopology deviation of tooth surface

8 Mathematical Problems in Engineering

Table 2 The modification amount and tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00564 minus00436 minus00314 minus00257 minus00239 minus00285 minus00291 minus00385 minus004972 minus00449 minus00387 minus00215 minus00159 minus00145 minus00147 minus00214 minus00289 minus004363 minus00406 minus00302 minus00197 minus00115 minus00027 minus0012 minus00173 minus00237 minus003914 minus00479 minus00421 minus00236 minus00214 minus00136 minus00234 minus00257 minus00356 minus004125 minus00535 minus00509 minus00352 minus00298 minus00172 minus00287 minus00327 minus00492 minus00518

Left flankmm

1 2 3 4 5 6 7 8 95 minus00524 minus00487 minus00301 minus00272 minus00196 minus00264 minus00314 minus00501 minus005444 minus00498 minus00410 minus00198 minus00189 minus00035 minus00197 minus00241 minus00417 minus004633 minus00436 minus00375 minus00221 minus00035 minus00021 minus00046 minus00210 minus00324 minus004522 minus00510 minus00417 minus00183 minus00167 minus00138 minus00153 minus00189 minus00394 minus005311 minus00571 minus00462 minus00294 minus00285 minus00227 minus00249 minus00267 minus00472 minus00592

Table 3 The tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00044 minus00047 minus00046 minus00046 minus00062 minus00062 minus00066 minus00061 minus000612 minus00061 minus00064 minus00063 minus00064 minus00079 minus00079 minus00083 minus00077 minus000793 minus00058 minus00061 minus00062 minus00063 minus00075 minus00076 minus00084 minus00074 minus000754 minus00056 minus00059 minus00058 minus00059 minus00074 minus00074 minus00078 minus00072 minus000745 minus00048 minus00051 minus00050 minus00051 minus00066 minus00066 minus00071 minus00064 minus00066

Left flankmm

1 2 3 4 5 6 7 8 95 minus00053 minus00057 minus00055 minus00056 minus00057 minus00058 minus00087 minus00054 minus000654 minus00062 minus00062 minus00068 minus00072 minus00084 minus00067 minus00086 minus00039 minus000743 minus00049 minus00074 minus00072 minus00058 minus00054 minus00069 minus00094 minus00075 minus000872 minus00072 minus00049 minus00085 minus00063 minus00072 minus00069 minus00064 minus00058 minus000761 minus00088 minus00076 minus00067 minus00053 minus00069 minus00072 minus00073 minus00067 minus00069

The tooth surface topology deviation Δ120588 is shown inTable 2 and the tooth surface error Δ119864 is shown in Table 3respectively

The deviation distribution (error and modificationamount) of the tooth surface obtained by an on-machinemeasurement is shown in Figure 8 The dashed parts (thesuperposition of error and modification) in regions 2 46 and 8 of the tooth surface are obviously presented as aparabolic shape in the profile and longitudinal directionsrespectivelyThemaximumvalue of deviation at the tooth topis minus00571mm and the error in the centre of tooth surfacearea 1 is minus00049mm The trend is gradually increasedtowards both sides presenting the shape where the middle ishigh and both sides are low

The measurement using the topological modificationtooth surface equation is shown in Figure 9 The measuredresult reflects the actual tooth surface error the distributionof which has unobvious trend The maximum error in thetooth top is minus00088mm and the error in the centre of righttooth surface area 1 is minus00054mm

To separate flank form deviations from modificationamount and to improve the perception of deviation dia-grams all flank deviations are referenced by the modificationamount The characterization of the tooth surface error andmodification amount are shown in Figures 10 and 11 Themaximum error is about minus88 um in the upper and lower

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus564

minus535

minus524

minus571

minus497

minus518

minus544

minus592

Top land

Figure 8 Topography diagram of modification amount and error(unit um)

surface but the minimum error is about minus54 um which isconcentrated in the central area of the profile and longitudinaldirections The normal error of any intersection of profiledirection and longitudinal direction is obtained by the errorof the contour map

Mathematical Problems in Engineering 9

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus44

minus48

minus56

minus88

minus61

minus66

minus65

minus69

Top land

Figure 9 Topography diagram of error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0minus002minus004minus006minus008

Mod

ifica

tion

amou

nt

and

erro

r (m

m)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)Longitudinal (mm)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

Mod

ifica

tion

amou

nt

and

erro

r (m

m) 0

minus002minus004minus006minus008

(b) Left flank

Figure 10 The contour map of modification amount and error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

times10minus3

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

Longitudinal (mm)

minus4

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

minus9

times10minus3

(b) Left flank

Figure 11 The contour map of error (unit um)

5 Conclusions

In this study an on-machine profile measurement systemalong with a five-axis CNC gear form grinding machineis developed to improve accuracy of grinding This system

includes the following steps (1) calculate trajectory of thecentre of spherical probe (2) define gear flanks by a grid ofpoints (3) obtain the coordinate values of topology measure-ment points Grinding experiments are performed to verifythe accuracy and efficiency of the topographymeasurements

10 Mathematical Problems in Engineering

With contour map the profile and longitudinal directionsof the error changes are easily seen Using statistical processcontrol techniques to monitor the grinding process cantimely attain the detection of changes in the product errorexceptions and take the necessary measures to prevent theoccurrence of waste As the numerical examples show on-machine measurement method can inspect tooth qualityAn additional advantage of the approach is to characterizemodification amount and tooth surface error respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the National Natural ScienceFoundation of China for its financial support Part of thiswork was performed under Contracts no 51375144 and no51405135 and the Priority Project of Research in Universitiesin Henan Province (Grant no 15A460021)

References

[1] H L Wang X Z Deng and J J Yang ldquoForm grinding andexperiment on segment topographicmodification gearrdquo Journalof Aerospace Power vol 29 no 12 pp 3000ndash3008 2014

[2] H-Y You P-Q Ye J-S Wang and X-Y Deng ldquoDesign andapplication of CBN shape grinding wheel for gearsrdquo Interna-tional Journal of Machine Tools and Manufacture vol 43 no12 pp 1269ndash1277 2003

[3] Q Fan R SDafoe and JW Swanger ldquoHigher-order tooth flankform error correction for face-milled spiral bevel and hypoidgearsrdquo Journal ofMechanical DesignmdashTransactions of theASMEvol 130 no 7 Article ID 072601 2008

[4] J Argyris M De Donno and F L Litvin ldquoComputer programin visual basic language for simulation of meshing and contactof gear drives and its application for design of worm gear driverdquoComputer Methods in Applied Mechanics and Engineering vol189 no 2 pp 595ndash612 2000

[5] Z L Zhang Y Fu Q R Yin and Y Zeng ldquoStudy on the polarmethod of measuring of gear profile errorsrdquo Chinese Journal ofMechanical Engineering vol 37 no 4 pp 70ndash72 2001

[6] C H Gao K Cheng and D Webb ldquoInvestigation on samplingsize optimisation in gear tooth surface measurement using aCMMrdquo The International Journal of Advanced ManufacturingTechnology vol 24 no 7-8 pp 599ndash606 2004

[7] Z L Zhang Y Fu and Y Zeng ldquoExpressing gear involuteerror by polar angle amp generating angle in polar coordinatemeasuring methodrdquo Tool Engineering vol 34 no 4 pp 39ndash402000

[8] F Gao B H Zhao and Y Li ldquoNovel pre-travel calibrationmethod of touch trigger probe based on error separationrdquoChinese Journal of Scientific Instrument vol 34 no 7 pp 1581ndash1587 2013

[9] F Gao Y Li S Tian Y Huang L Hao and J Wang ldquoStudy onthe on-machine measurement method of NC wheel gear formgrinding machinerdquo Chinese Journal of Scientific Instrument vol29 no 3 pp 540ndash544 2008

[10] A Nafi J R R Mayer and A Wozniak ldquoNovel CMM-basedimplementation of the multi-step method for the separation ofmachine and probe errorsrdquo Precision Engineering vol 35 no 2pp 318ndash328 2011

[11] A Nafi J R RMayer andAWozniak ldquoReduced configurationset for the multi-step method applied to machine and probeerror separation on a CMMrdquo Measurement vol 45 no 10 pp2321ndash2329 2012

[12] Y-P Shih and S-D Chen ldquoFree-formflank correction in helicalgear grinding using a five-axis computer numerical control gearprofile grindingmachinerdquo Journal ofManufacturing Science andEngineering vol 134 no 4 Article ID 041006 2012

[13] Y Kobayashi N Nishida Y Ougiya and H Nagata ldquoToothtrace modification processing of helix gear by form grindingmethodrdquo Transactions of the Japan Society of Mechanical Engi-neers Part C vol 61 no 590 pp 4088ndash4093 1995

[14] Y Kobayashi N Nishida and Y Ougiya ldquoEstimation ofgrinding wheel setting error in helical gear processing byform grindingrdquo Transactions of the Japan Society of MechanicalEngineers Part C vol 63 no 612 pp 2852ndash2858 1997

[15] C-K Lee ldquoManufacturing process for a cylindrical crown geardrive with a controllable fourth order polynomial function oftransmission errorrdquo Journal of Materials Processing Technologyvol 209 no 1 pp 3ndash13 2009

[16] C K Lee and C K Chen ldquoMathematical models meshinganalysis and transmission design for robust cylindrical gear setgenerated by double blade-disks with parabolic cutting edgesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 218 no 12 pp1539ndash1553 2004

[17] Z Y Shi and Y Ye ldquoResearch on the generalized polar-coordinate method for measuring involute profile deviationsrdquoChinese Journal of Scientific Instrument vol 22 no 2 pp 140ndash142 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 2 The modification amount and tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00564 minus00436 minus00314 minus00257 minus00239 minus00285 minus00291 minus00385 minus004972 minus00449 minus00387 minus00215 minus00159 minus00145 minus00147 minus00214 minus00289 minus004363 minus00406 minus00302 minus00197 minus00115 minus00027 minus0012 minus00173 minus00237 minus003914 minus00479 minus00421 minus00236 minus00214 minus00136 minus00234 minus00257 minus00356 minus004125 minus00535 minus00509 minus00352 minus00298 minus00172 minus00287 minus00327 minus00492 minus00518

Left flankmm

1 2 3 4 5 6 7 8 95 minus00524 minus00487 minus00301 minus00272 minus00196 minus00264 minus00314 minus00501 minus005444 minus00498 minus00410 minus00198 minus00189 minus00035 minus00197 minus00241 minus00417 minus004633 minus00436 minus00375 minus00221 minus00035 minus00021 minus00046 minus00210 minus00324 minus004522 minus00510 minus00417 minus00183 minus00167 minus00138 minus00153 minus00189 minus00394 minus005311 minus00571 minus00462 minus00294 minus00285 minus00227 minus00249 minus00267 minus00472 minus00592

Table 3 The tooth surface error data (unit mm)

Location 1 2 3 4 5 6 7 8 9

Right flankmm

1 minus00044 minus00047 minus00046 minus00046 minus00062 minus00062 minus00066 minus00061 minus000612 minus00061 minus00064 minus00063 minus00064 minus00079 minus00079 minus00083 minus00077 minus000793 minus00058 minus00061 minus00062 minus00063 minus00075 minus00076 minus00084 minus00074 minus000754 minus00056 minus00059 minus00058 minus00059 minus00074 minus00074 minus00078 minus00072 minus000745 minus00048 minus00051 minus00050 minus00051 minus00066 minus00066 minus00071 minus00064 minus00066

Left flankmm

1 2 3 4 5 6 7 8 95 minus00053 minus00057 minus00055 minus00056 minus00057 minus00058 minus00087 minus00054 minus000654 minus00062 minus00062 minus00068 minus00072 minus00084 minus00067 minus00086 minus00039 minus000743 minus00049 minus00074 minus00072 minus00058 minus00054 minus00069 minus00094 minus00075 minus000872 minus00072 minus00049 minus00085 minus00063 minus00072 minus00069 minus00064 minus00058 minus000761 minus00088 minus00076 minus00067 minus00053 minus00069 minus00072 minus00073 minus00067 minus00069

The tooth surface topology deviation Δ120588 is shown inTable 2 and the tooth surface error Δ119864 is shown in Table 3respectively

The deviation distribution (error and modificationamount) of the tooth surface obtained by an on-machinemeasurement is shown in Figure 8 The dashed parts (thesuperposition of error and modification) in regions 2 46 and 8 of the tooth surface are obviously presented as aparabolic shape in the profile and longitudinal directionsrespectivelyThemaximumvalue of deviation at the tooth topis minus00571mm and the error in the centre of tooth surfacearea 1 is minus00049mm The trend is gradually increasedtowards both sides presenting the shape where the middle ishigh and both sides are low

The measurement using the topological modificationtooth surface equation is shown in Figure 9 The measuredresult reflects the actual tooth surface error the distributionof which has unobvious trend The maximum error in thetooth top is minus00088mm and the error in the centre of righttooth surface area 1 is minus00054mm

To separate flank form deviations from modificationamount and to improve the perception of deviation dia-grams all flank deviations are referenced by the modificationamount The characterization of the tooth surface error andmodification amount are shown in Figures 10 and 11 Themaximum error is about minus88 um in the upper and lower

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus564

minus535

minus524

minus571

minus497

minus518

minus544

minus592

Top land

Figure 8 Topography diagram of modification amount and error(unit um)

surface but the minimum error is about minus54 um which isconcentrated in the central area of the profile and longitudinaldirections The normal error of any intersection of profiledirection and longitudinal direction is obtained by the errorof the contour map

Mathematical Problems in Engineering 9

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus44

minus48

minus56

minus88

minus61

minus66

minus65

minus69

Top land

Figure 9 Topography diagram of error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0minus002minus004minus006minus008

Mod

ifica

tion

amou

nt

and

erro

r (m

m)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)Longitudinal (mm)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

Mod

ifica

tion

amou

nt

and

erro

r (m

m) 0

minus002minus004minus006minus008

(b) Left flank

Figure 10 The contour map of modification amount and error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

times10minus3

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

Longitudinal (mm)

minus4

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

minus9

times10minus3

(b) Left flank

Figure 11 The contour map of error (unit um)

5 Conclusions

In this study an on-machine profile measurement systemalong with a five-axis CNC gear form grinding machineis developed to improve accuracy of grinding This system

includes the following steps (1) calculate trajectory of thecentre of spherical probe (2) define gear flanks by a grid ofpoints (3) obtain the coordinate values of topology measure-ment points Grinding experiments are performed to verifythe accuracy and efficiency of the topographymeasurements

10 Mathematical Problems in Engineering

With contour map the profile and longitudinal directionsof the error changes are easily seen Using statistical processcontrol techniques to monitor the grinding process cantimely attain the detection of changes in the product errorexceptions and take the necessary measures to prevent theoccurrence of waste As the numerical examples show on-machine measurement method can inspect tooth qualityAn additional advantage of the approach is to characterizemodification amount and tooth surface error respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the National Natural ScienceFoundation of China for its financial support Part of thiswork was performed under Contracts no 51375144 and no51405135 and the Priority Project of Research in Universitiesin Henan Province (Grant no 15A460021)

References

[1] H L Wang X Z Deng and J J Yang ldquoForm grinding andexperiment on segment topographicmodification gearrdquo Journalof Aerospace Power vol 29 no 12 pp 3000ndash3008 2014

[2] H-Y You P-Q Ye J-S Wang and X-Y Deng ldquoDesign andapplication of CBN shape grinding wheel for gearsrdquo Interna-tional Journal of Machine Tools and Manufacture vol 43 no12 pp 1269ndash1277 2003

[3] Q Fan R SDafoe and JW Swanger ldquoHigher-order tooth flankform error correction for face-milled spiral bevel and hypoidgearsrdquo Journal ofMechanical DesignmdashTransactions of theASMEvol 130 no 7 Article ID 072601 2008

[4] J Argyris M De Donno and F L Litvin ldquoComputer programin visual basic language for simulation of meshing and contactof gear drives and its application for design of worm gear driverdquoComputer Methods in Applied Mechanics and Engineering vol189 no 2 pp 595ndash612 2000

[5] Z L Zhang Y Fu Q R Yin and Y Zeng ldquoStudy on the polarmethod of measuring of gear profile errorsrdquo Chinese Journal ofMechanical Engineering vol 37 no 4 pp 70ndash72 2001

[6] C H Gao K Cheng and D Webb ldquoInvestigation on samplingsize optimisation in gear tooth surface measurement using aCMMrdquo The International Journal of Advanced ManufacturingTechnology vol 24 no 7-8 pp 599ndash606 2004

[7] Z L Zhang Y Fu and Y Zeng ldquoExpressing gear involuteerror by polar angle amp generating angle in polar coordinatemeasuring methodrdquo Tool Engineering vol 34 no 4 pp 39ndash402000

[8] F Gao B H Zhao and Y Li ldquoNovel pre-travel calibrationmethod of touch trigger probe based on error separationrdquoChinese Journal of Scientific Instrument vol 34 no 7 pp 1581ndash1587 2013

[9] F Gao Y Li S Tian Y Huang L Hao and J Wang ldquoStudy onthe on-machine measurement method of NC wheel gear formgrinding machinerdquo Chinese Journal of Scientific Instrument vol29 no 3 pp 540ndash544 2008

[10] A Nafi J R R Mayer and A Wozniak ldquoNovel CMM-basedimplementation of the multi-step method for the separation ofmachine and probe errorsrdquo Precision Engineering vol 35 no 2pp 318ndash328 2011

[11] A Nafi J R RMayer andAWozniak ldquoReduced configurationset for the multi-step method applied to machine and probeerror separation on a CMMrdquo Measurement vol 45 no 10 pp2321ndash2329 2012

[12] Y-P Shih and S-D Chen ldquoFree-formflank correction in helicalgear grinding using a five-axis computer numerical control gearprofile grindingmachinerdquo Journal ofManufacturing Science andEngineering vol 134 no 4 Article ID 041006 2012

[13] Y Kobayashi N Nishida Y Ougiya and H Nagata ldquoToothtrace modification processing of helix gear by form grindingmethodrdquo Transactions of the Japan Society of Mechanical Engi-neers Part C vol 61 no 590 pp 4088ndash4093 1995

[14] Y Kobayashi N Nishida and Y Ougiya ldquoEstimation ofgrinding wheel setting error in helical gear processing byform grindingrdquo Transactions of the Japan Society of MechanicalEngineers Part C vol 63 no 612 pp 2852ndash2858 1997

[15] C-K Lee ldquoManufacturing process for a cylindrical crown geardrive with a controllable fourth order polynomial function oftransmission errorrdquo Journal of Materials Processing Technologyvol 209 no 1 pp 3ndash13 2009

[16] C K Lee and C K Chen ldquoMathematical models meshinganalysis and transmission design for robust cylindrical gear setgenerated by double blade-disks with parabolic cutting edgesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 218 no 12 pp1539ndash1553 2004

[17] Z Y Shi and Y Ye ldquoResearch on the generalized polar-coordinate method for measuring involute profile deviationsrdquoChinese Journal of Scientific Instrument vol 22 no 2 pp 140ndash142 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

1

1

2

2

3

3

4

4

5

51 2 3 4 5 6 7 8 9

Right flank

Left flank

minus44

minus48

minus56

minus88

minus61

minus66

minus65

minus69

Top land

Figure 9 Topography diagram of error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0minus002minus004minus006minus008

Mod

ifica

tion

amou

nt

and

erro

r (m

m)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)Longitudinal (mm)

minus0005

minus001

minus0015

minus002

minus0025

minus003

minus0035

minus004

minus0045

minus005

minus0055

Mod

ifica

tion

amou

nt

and

erro

r (m

m) 0

minus002minus004minus006minus008

(b) Left flank

Figure 10 The contour map of modification amount and error (unit um)

9294

9698

100102

104106

1020

3040

50

Profile (mm)

Longitudinal (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

times10minus3

(a) Right flank

9294

9698

100102

104106

1020

3040

50

Profile (mm)

0

minus0002

minus0004

minus0006

minus0008

minus001

Erro

r (m

m)

Longitudinal (mm)

minus4

minus45

minus5

minus55

minus6

minus65

minus7

minus75

minus8

minus85

minus9

times10minus3

(b) Left flank

Figure 11 The contour map of error (unit um)

5 Conclusions

In this study an on-machine profile measurement systemalong with a five-axis CNC gear form grinding machineis developed to improve accuracy of grinding This system

includes the following steps (1) calculate trajectory of thecentre of spherical probe (2) define gear flanks by a grid ofpoints (3) obtain the coordinate values of topology measure-ment points Grinding experiments are performed to verifythe accuracy and efficiency of the topographymeasurements

10 Mathematical Problems in Engineering

With contour map the profile and longitudinal directionsof the error changes are easily seen Using statistical processcontrol techniques to monitor the grinding process cantimely attain the detection of changes in the product errorexceptions and take the necessary measures to prevent theoccurrence of waste As the numerical examples show on-machine measurement method can inspect tooth qualityAn additional advantage of the approach is to characterizemodification amount and tooth surface error respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the National Natural ScienceFoundation of China for its financial support Part of thiswork was performed under Contracts no 51375144 and no51405135 and the Priority Project of Research in Universitiesin Henan Province (Grant no 15A460021)

References

[1] H L Wang X Z Deng and J J Yang ldquoForm grinding andexperiment on segment topographicmodification gearrdquo Journalof Aerospace Power vol 29 no 12 pp 3000ndash3008 2014

[2] H-Y You P-Q Ye J-S Wang and X-Y Deng ldquoDesign andapplication of CBN shape grinding wheel for gearsrdquo Interna-tional Journal of Machine Tools and Manufacture vol 43 no12 pp 1269ndash1277 2003

[3] Q Fan R SDafoe and JW Swanger ldquoHigher-order tooth flankform error correction for face-milled spiral bevel and hypoidgearsrdquo Journal ofMechanical DesignmdashTransactions of theASMEvol 130 no 7 Article ID 072601 2008

[4] J Argyris M De Donno and F L Litvin ldquoComputer programin visual basic language for simulation of meshing and contactof gear drives and its application for design of worm gear driverdquoComputer Methods in Applied Mechanics and Engineering vol189 no 2 pp 595ndash612 2000

[5] Z L Zhang Y Fu Q R Yin and Y Zeng ldquoStudy on the polarmethod of measuring of gear profile errorsrdquo Chinese Journal ofMechanical Engineering vol 37 no 4 pp 70ndash72 2001

[6] C H Gao K Cheng and D Webb ldquoInvestigation on samplingsize optimisation in gear tooth surface measurement using aCMMrdquo The International Journal of Advanced ManufacturingTechnology vol 24 no 7-8 pp 599ndash606 2004

[7] Z L Zhang Y Fu and Y Zeng ldquoExpressing gear involuteerror by polar angle amp generating angle in polar coordinatemeasuring methodrdquo Tool Engineering vol 34 no 4 pp 39ndash402000

[8] F Gao B H Zhao and Y Li ldquoNovel pre-travel calibrationmethod of touch trigger probe based on error separationrdquoChinese Journal of Scientific Instrument vol 34 no 7 pp 1581ndash1587 2013

[9] F Gao Y Li S Tian Y Huang L Hao and J Wang ldquoStudy onthe on-machine measurement method of NC wheel gear formgrinding machinerdquo Chinese Journal of Scientific Instrument vol29 no 3 pp 540ndash544 2008

[10] A Nafi J R R Mayer and A Wozniak ldquoNovel CMM-basedimplementation of the multi-step method for the separation ofmachine and probe errorsrdquo Precision Engineering vol 35 no 2pp 318ndash328 2011

[11] A Nafi J R RMayer andAWozniak ldquoReduced configurationset for the multi-step method applied to machine and probeerror separation on a CMMrdquo Measurement vol 45 no 10 pp2321ndash2329 2012

[12] Y-P Shih and S-D Chen ldquoFree-formflank correction in helicalgear grinding using a five-axis computer numerical control gearprofile grindingmachinerdquo Journal ofManufacturing Science andEngineering vol 134 no 4 Article ID 041006 2012

[13] Y Kobayashi N Nishida Y Ougiya and H Nagata ldquoToothtrace modification processing of helix gear by form grindingmethodrdquo Transactions of the Japan Society of Mechanical Engi-neers Part C vol 61 no 590 pp 4088ndash4093 1995

[14] Y Kobayashi N Nishida and Y Ougiya ldquoEstimation ofgrinding wheel setting error in helical gear processing byform grindingrdquo Transactions of the Japan Society of MechanicalEngineers Part C vol 63 no 612 pp 2852ndash2858 1997

[15] C-K Lee ldquoManufacturing process for a cylindrical crown geardrive with a controllable fourth order polynomial function oftransmission errorrdquo Journal of Materials Processing Technologyvol 209 no 1 pp 3ndash13 2009

[16] C K Lee and C K Chen ldquoMathematical models meshinganalysis and transmission design for robust cylindrical gear setgenerated by double blade-disks with parabolic cutting edgesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 218 no 12 pp1539ndash1553 2004

[17] Z Y Shi and Y Ye ldquoResearch on the generalized polar-coordinate method for measuring involute profile deviationsrdquoChinese Journal of Scientific Instrument vol 22 no 2 pp 140ndash142 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

With contour map the profile and longitudinal directionsof the error changes are easily seen Using statistical processcontrol techniques to monitor the grinding process cantimely attain the detection of changes in the product errorexceptions and take the necessary measures to prevent theoccurrence of waste As the numerical examples show on-machine measurement method can inspect tooth qualityAn additional advantage of the approach is to characterizemodification amount and tooth surface error respectively

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the National Natural ScienceFoundation of China for its financial support Part of thiswork was performed under Contracts no 51375144 and no51405135 and the Priority Project of Research in Universitiesin Henan Province (Grant no 15A460021)

References

[1] H L Wang X Z Deng and J J Yang ldquoForm grinding andexperiment on segment topographicmodification gearrdquo Journalof Aerospace Power vol 29 no 12 pp 3000ndash3008 2014

[2] H-Y You P-Q Ye J-S Wang and X-Y Deng ldquoDesign andapplication of CBN shape grinding wheel for gearsrdquo Interna-tional Journal of Machine Tools and Manufacture vol 43 no12 pp 1269ndash1277 2003

[3] Q Fan R SDafoe and JW Swanger ldquoHigher-order tooth flankform error correction for face-milled spiral bevel and hypoidgearsrdquo Journal ofMechanical DesignmdashTransactions of theASMEvol 130 no 7 Article ID 072601 2008

[4] J Argyris M De Donno and F L Litvin ldquoComputer programin visual basic language for simulation of meshing and contactof gear drives and its application for design of worm gear driverdquoComputer Methods in Applied Mechanics and Engineering vol189 no 2 pp 595ndash612 2000

[5] Z L Zhang Y Fu Q R Yin and Y Zeng ldquoStudy on the polarmethod of measuring of gear profile errorsrdquo Chinese Journal ofMechanical Engineering vol 37 no 4 pp 70ndash72 2001

[6] C H Gao K Cheng and D Webb ldquoInvestigation on samplingsize optimisation in gear tooth surface measurement using aCMMrdquo The International Journal of Advanced ManufacturingTechnology vol 24 no 7-8 pp 599ndash606 2004

[7] Z L Zhang Y Fu and Y Zeng ldquoExpressing gear involuteerror by polar angle amp generating angle in polar coordinatemeasuring methodrdquo Tool Engineering vol 34 no 4 pp 39ndash402000

[8] F Gao B H Zhao and Y Li ldquoNovel pre-travel calibrationmethod of touch trigger probe based on error separationrdquoChinese Journal of Scientific Instrument vol 34 no 7 pp 1581ndash1587 2013

[9] F Gao Y Li S Tian Y Huang L Hao and J Wang ldquoStudy onthe on-machine measurement method of NC wheel gear formgrinding machinerdquo Chinese Journal of Scientific Instrument vol29 no 3 pp 540ndash544 2008

[10] A Nafi J R R Mayer and A Wozniak ldquoNovel CMM-basedimplementation of the multi-step method for the separation ofmachine and probe errorsrdquo Precision Engineering vol 35 no 2pp 318ndash328 2011

[11] A Nafi J R RMayer andAWozniak ldquoReduced configurationset for the multi-step method applied to machine and probeerror separation on a CMMrdquo Measurement vol 45 no 10 pp2321ndash2329 2012

[12] Y-P Shih and S-D Chen ldquoFree-formflank correction in helicalgear grinding using a five-axis computer numerical control gearprofile grindingmachinerdquo Journal ofManufacturing Science andEngineering vol 134 no 4 Article ID 041006 2012

[13] Y Kobayashi N Nishida Y Ougiya and H Nagata ldquoToothtrace modification processing of helix gear by form grindingmethodrdquo Transactions of the Japan Society of Mechanical Engi-neers Part C vol 61 no 590 pp 4088ndash4093 1995

[14] Y Kobayashi N Nishida and Y Ougiya ldquoEstimation ofgrinding wheel setting error in helical gear processing byform grindingrdquo Transactions of the Japan Society of MechanicalEngineers Part C vol 63 no 612 pp 2852ndash2858 1997

[15] C-K Lee ldquoManufacturing process for a cylindrical crown geardrive with a controllable fourth order polynomial function oftransmission errorrdquo Journal of Materials Processing Technologyvol 209 no 1 pp 3ndash13 2009

[16] C K Lee and C K Chen ldquoMathematical models meshinganalysis and transmission design for robust cylindrical gear setgenerated by double blade-disks with parabolic cutting edgesrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 218 no 12 pp1539ndash1553 2004

[17] Z Y Shi and Y Ye ldquoResearch on the generalized polar-coordinate method for measuring involute profile deviationsrdquoChinese Journal of Scientific Instrument vol 22 no 2 pp 140ndash142 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of