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Development of a gear measuring device using DFRP method
Precision Engineering 45 (2016) 153–159
Contents lists available at ScienceDirect
Precision Engineering journal homepage: www.elsevier.com/locate/precision
Development of a gear measuring device using DFRP method Jie Tang ∗ , Jianqiao Jia, Zhiqiang Fang, Zhaoyao Shi College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, No. 100 Pingleyuan, Chaoyang District, 100124 Beijing, China
a r t i c l e
i n f o
Article history: Received 25 June 2015 Received in revised form 24 January 2016 Accepted 12 February 2016 Available online 2 March 2016 Keywords: Gear measurement DFRP Rack probe Profile Pitch
a b s t r a c t This paper presents the double-flank rack probe (DFRP) method to measure the accuracy relevant to flanks of gear teeth. In DFRP method, both left and right flanks of the gear to be measured are in mesh with a rack probe to measure pitch and profile deviations on the left and right flanks simultaneously. A gear measuring device (GMD) using DFRP method is developed. The evaluation method for pitch and profile deviations, the structure of the measuring system, and the key mechanical parts are introduced. Repeatability experimental results of the device is less than 2 m, and the results compared to the gear measuring machine (GMM) Klingelnberg P26 shows that this device have consistency with GMM and can be used for the measurement of the accuracy relevant to flanks of gear teeth. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Generally, the deviations relevant to flanks of gear teeth, such as pitch, profile and helix deviations, are measured by gear measuring machine (GMM) and coordinate measuring machine (CMM) in metrology room [1–9]. However, the measurement of the accuracy relevant to flanks of gear teeth at industrial scale manufacturing workshop is still an issue. The double-flank gear rolling test is widely used in gear manufacturing, especially for large-volume inspections in workshop, because of its easy measurement principle and less requirements concerning the environment. Normally, composite deviations are evaluated in traditional double-flank gear rolling test [10,11]. Master gears should be specially designed and manufactured for double-flank gear inspection [12–14]. Lately double-flank gear rolling test with many degrees of freedoms is also used for gears process inspection at shop floor [15–17], for measuring composite, runout, nicks, size, tooth action, lead and taper [18]. The inertia of the mechanical part with multi-freedoms will affect measuring speed. In 1994 & 1995, Zhang proposed to measure gear profile and pitch deviations on left and right flank simultaneously [19,20], but no more authors’ development on this issue had been found. A. Guenther presented the approach for the evaluation of runout deviation at bevel gears based on pitch measurement in 2006 [6],
∗ Corresponding author. Tel.: +86 10 6739 2894; fax: +86 10 6739 0989. E-mail address: [email protected] (J. Tang). http://dx.doi.org/10.1016/j.precisioneng.2016.02.006 0141-6359/© 2016 Elsevier Inc. All rights reserved.
which is based on a simultaneous probing of both flanks of one gap by an appropriate tip at a well-defined pitch diameter near the reference cone. The test was carried out with a pinion bevel gear on 5 different CMMs using probe ball, while rack probe is used for measuring the both flanks of a cylindrical gears simultaneously in this paper. For the purpose of measuring deviations relevant to flanks of gear teeth (pitch, profile and helix deviations) in mass-produced gear manufacturing shop floor, the detailed measurement principle of profile and pitch deviations using double-flank rack probe are described in [21,22]. A gear measuring device based on this measurement principle will be introduced briefly in this paper. 2. DFRP method Gear error calculation system is established to analyse gear error in the DFRP method [23]. The increment on the line of action symbolized extra displacement of the conjugating gear flanks. In Fig. 1, the gear to be inspected is in mesh with a rack probe. The gear error calculation is establishedbydrawing two lines of action on system L and on right flank n R through the contacting point. left flank n All gear errors will be converted to these two lines. Gear error is the function of rotating angle. The gear error on each flank is calculated on each meshing line separately. When gear rotates at a constant speed, displacement increment of the master rack generates along the meshing line. In Fig. 2, the rack probe is driven by the work-piece, left and right flanks of the gear to be inspected are in mesh with a rack probe. The plus material deviation is defined as positive, while the
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J. Tang et al. / Precision Engineering 45 (2016) 153–159
Fig. 1. Gear error calculation system.
minus material deviation is defined as negative. The relationship among the displacement increment along left and right meshing line (FL , FR ), the radial displacement (R) and the tangential displacement (T ) is as follows:
T () = FL () cos ˛ − FR () cos ˛ R() = FL () sin ˛ + FR () sin ˛
(1)
where ˛ is the pressure angle of rack probe (degree). The displacement of rack probe in radial & tangential direction is measured while the work piece is meshing with the rack probe in double-flank with no backlash, then the profile or pitch deviations of the left & right gear flank will be evaluated. 2.1. Profile measurement 2.1.1. Meshing procedure Meshing procedure during profile measurement is shown in Fig. 3, in which the gear to be measured is rotate clockwise as an example, and the pressure angle of the gear to be measured and the rack is both 20 degree. Fig. 3a is the starting measurement position, point a and a is the contact point between the rack probe and the left and right gear flanks, respectively. When gear rotates to the position in Fig. 3b, the tip of the rack contact with the left flank of the gear to be measured on point b , while the right flank of the gear meshes with the rack on point b. The gear keeps on rotating till the position in Fig. 3c, right flank of the gear meshes with the rack on point c, while the left flank comes in contact with the rack at root edge meshing. In Fig. 3d, the rack probe and the left and right gear flanks contact on point d and d, and the right gear flank comes in contact at root edge meshing. The edge meshing will be finished when the measured gear rotating to the position in Fig. 3e.
Fig. 3. Procedure of mesh during profile measurement.
2.1.2. Profile deviations The gear to be measured rotate for an angle“”, as shown in Fig. 4, two contacting points a and a moves to points f and f. The relationship between the radial displacement (ypfi ) and the tangential displacement (xpfi ) and the profile deviations on right and
R and F L left gear flank F˛i are as follows: ˛i
⎧ ypfi cos ˛ + xpfi sin ˛ ⎪ R ⎪ ⎨ F˛i = sin 2˛
⎪ ⎪ ypfi cos ˛ − xpfi sin ˛ ⎩ L F˛i =
(2)
sin 2˛
where ˛ is the pressure angle of rack probe (degree). 2.1.3. Requirement of rack probe The requirement of rack probe is that, in profile measurement, the pressure angle of the rack should be designed to fulfil the condition that the tip edge meshing in right flank and the root edge meshing in left flank occurs simultaneously.
Fig. 2. DFRP method.
Fig. 4. Profile deviations evaluation.
J. Tang et al. / Precision Engineering 45 (2016) 153–159
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Fig. 5. Procedure of pitch measurement. Fig. 6. Structure of the device.
2.2. Pitch measurement 2.2.1. Meshing procedure In pitch measurement, rack probe enters into each tooth space to contact with the both left and right flank of the measured gear. The starting tooth of the gear to be measured is identified as 1st tooth. As shown in Fig. 5, rack probe enter into 1st tooth space in Fig. 5a, gear rotates for a pitch angle, and rack probe returns back in Fig. 5b, rack probe enters into 2nd tooth space in Fig. 5c. 2.2.2. Pitch deviations The relationship between the radial displacement ypti and
the tangential displacement xpti and the cumulative pitch deviations, which is the ith tooth compared to 1st tooth, on right and left gear flank
R and F L Fpi pi
are as follows:
⎧ ypti cos ˛ + xpti sin ˛ ⎪ R ⎪ ⎨ Fpi = sin 2˛ cos ˛t
⎪ ⎪ ⎩ F L = ypti cos ˛ − xpti sin ˛ pi
(3)
sin 2˛ cos ˛t
the test and control hardware and the data processing software. The strong points of this method for profile and pitch deviations measurement compared to the traditional method is that, the deviations of the left & right gear flank is evaluated simultaneously. 3.1. Procedure for profile measurement Procedure for profile measurement is as follows. (1) After initialization of the hardware has been executed successfully, the gear to be measured is controlled to rotate to a proper position (2) The rack probe for profile deviations measurement goes ahead to contact with the gear in double-flank with no back lash. (3) Then the rack probe is driven by the gear. (4) Data acquisition of the rotation angle of the gear is executed, which is the radial & tangential displacement increment of the rack.
where ˛t is the transverse pressure angle on reference diameter of the gear to be measured (degree). 2.2.3. Requirement of rack probe The requirement of rack probe is that, in pitch measurement, the contact point is nearby reference diameter. 3. Measuring system The structure of the measuring device is shown in Fig. 6. Rotation of the work-piece is driven by a servomotor. Output from the round grating is 236,800 pulses per revolution after the signal is subdivided. Rack probe can move along the tangential and horizontal direction. The 2D platform contains a linear grating in X-axis for measuring the radial displacement, the resolution of which is 0.2 m, and a micro-displacement sensor used for measuring the tangential displacement in Y-axis, the resolution of which is 0.5 m. 2D platform can move along Z-axis vertical part. Vertical part can move along horizontal direction manually. The developed measuring device using double-flank rack probe method is shown in Fig. 7, which includes the mechanical base unit,
Fig. 7. GMD (DFRP) (Gear measuring device using DFRP method).
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Fig. 9. 2D platform. Fig. 8. Base unit.
(5) After measurement for one set of tooth flanks, the rack probe draws back, the gear rotates an angle to inspect another one set of flanks. (6) When Measurement is accomplished, the deviations are evaluated through data processing software. The measurement results are processed to generate measurement report. Fig. 10. 3D drawing of probe component.
3.2. Procedure for pitch measurement Procedure for pitch measurement is as follows. (1) Work-piece is installed with precision graduated circle coaxially. (2) Rack probe enters to contact both left & right flank on the position of reference circle. (3) The displacement of rack probe in radial & tangential direction is measured. (4) After that, the rack probe draws back, the gear will rotate an angular pitch to begin the next iteration of pitch measurement until all the teeth are inspected. (5) Then the pitch deviations of the left & right gear flank will be evaluated. The measurement results are processed to generate measurement report. 4. Mechanical parts 4.1. Base unit The base unit is shown in Fig. 8. Component for gear to be inspected and the horizontal part is fixed on a granite base. In component for gear to be inspected, the work-piece is supported by a spindle with round grating installed at the lower end. Rack probe is supported by a 2D platform jointed along vertical part which is installed on the horizontal part. 4.2. 2D platform The 2D platform is shown in Fig. 9 (1—supporting plate, 2—linear grating, 3—cross roller, 4—micro-displacement sensor, 5—rack probe and 6—probe component). Probe component (Fig. 10) is fixed with the sliding part of the cross roller, the static part of which is installed on the supporting plate. The linear grating is fixed with the sliding part of the cross roller to output the signal of the radial displacement. A micro-displacement sensor is installed in the probe component, to output the signal of the tangential misalignment guided by a couple of leaf spring.
Table 1 Parameters of the measured gear. Module (mm)
No. of teeth
Pressure angle (◦ )
Helix angle (◦ )
Modification factor
2.5
45
20
0
+0.26
5. Experimental results Parameters of the measured gear and the rack probe are listed in Tables 1 and 2. 5.1. Repeatability test Under repeatability condition of measurement [24], that includes the same measurement procedure, same operators, same measuring system, same operating conditions and same location, and replicate measurements on the same or similar objects over a short period of time, the work-piece and the rack in Tables 1 and 2 mesh on the gear measuring device as shown in Fig. 7 to measure the profile and pitch deviations. Measurements are done on the same object, and 10 sets of measurement results are shown in Tables 3 and 4. Tests under repeatability condition shows that the difference between the maximum and minimum value of the profile deviations is 1.9 m, and the variation of pitch deviations is 1.6 m. Results of profile deviations of 10 tests on left flank and right flank are listed in Table 3, including the total profile deviation F˛ , profile form deviation ff˛ and profile slope deviation fH˛ . The graph for 10 tests of profile deviations is shown in Fig. 11.
Table 2 Parameters of the rack. Pressure angle (◦ )
Helix angle (◦ )
Tooth depth (mm)
Tooth thickness (mm)
23(profile)/21(pitch)
0
10
1
J. Tang et al. / Precision Engineering 45 (2016) 153–159
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Table 3 Results of profile deviations. SN
F␣ (m)
ff␣ (m)
fH␣ (m)
F␣ (m)
LF (Left flank)
ff␣ (m)
fH␣ (m)
RF (Right flank)
1 2 3 4 5 6 7 8 9 10
9.0 9.3 9.9 9.6 10.3 9.8 9.4 10.2 10.2 9.2
10.1 10.4 9.8 9.3 10.6 9.9 10.9 10.4 9.3 9.4
3.1 3.9 4.3 4.0 3.3 3.7 3.3 3.6 2.9 3.5
8.7 9.0 9.6 9.3 10.1 9.5 9.1 9.9 9.9 8.9
5.8 3.9 4.9 5.4 4.8 5.7 4.1 5.1 5.1 4.8
8.1 7.2 8.3 8.4 7.8 7.7 8.5 8.2 7.6 7.8
1.3
1.6
1.4
1.4
1.9
1.2
Fig. 12. Repeatability test of total cumulative pitch deviations.
Table 4 Results of pitch deviations. SN
Fp (m)
fpt (m)
Fp (m)
1 2 3 4 5 6 7 8 9 10
44.3 45.1 45.5 45.2 44.5 44.9 44.5 44.8 44.1 44.7
5.3 4.2 4.8 4.5 5.2 4.7 4.3 5.1 5.1 4.6
44.8 43.9 45.0 45.1 44.5 44.4 45.2 44.9 44.3 44.5
4.9 4.6 4.9 4.3 5.1 4.4 5.4 4.9 3.8 3.9
1.4
1.1
1.3
1.6
LF (Left flank)
fpt (m)
RF (Right flank)
Fig. 13. Repeatability test of single pitch deviation.
fpt . The graph for 10 tests of pitch deviations is shown in Figs. 12 and 13. 5.2. Comparison between GMD (DFRP) and GMM Klingelnberg P26
Fig. 11. Repeatability test of profile deviations.
In the repeatability of the pitch deviation measurement using the developed system, results of pitch deviations of 10 tests on left flank and right flank are listed in Table 4, including the total cumulative pitch deviations Fp and single pitch deviation
To compare the difference of the measurement results between the GMD using DFRP method and the GMM (gear measuring machine) Klingelnberg P26, many tests were implemented separately in the two kinds of machine. The measurement by P26 are done on the same object for three times repeated on one work piece. The results by GMD (DFRP) are the same as those of SN 1, 2 and 3 in Tables 3 and 4, which are the part of the repeatability test by GMD (DFRP). The measurement results for profile deviations were listed in Table 5. For profile deviations measurement, the difference between the mean value of 3 tests on GMD (DFRP) and the mean value of 3 tests on GMM P26 is within 1 m, including the total profile deviation F˛ , profile form deviation ff˛ and profile slope deviation fH˛ . The curve for profile measurement results compared with P26 is shown in Fig. 14, through which the consistency of profile measurement between these two kinds of machines can be found.
Table 5 Comparison between the GMD with P26 (profile deviations). Items
GMD (DFRP) (m) 1
2
(m)
P26 (m) 1
2
3
Avg(Grade)
F␣
LF RF
9.0 8.7
9.3 9.0
3 9.9 9.6
Avg(Grade) 9.4(6) 9.1(6)
10.4 8.1
10.2 8.2
10.3 8.2
10.3(6) 8.2(6)
−0.9 0.9
ff␣
LF RF
10.1 5.8
10.4 3.9
9.8 4.9
10.1(7) 4.9(4)
9.1 4.3
9.0 4.3
9.2 4.4
9.1(7) 4.3(4)
1.0 0.6
fH␣
LF RF
3.1 8.1
3.9 7.2
4.3 8.3
3.8(5) 7.9(7)
4.7 8.4
4.9 8.3
4.9 8.3
4.8(5) 8.3(7)
−1.0 −0.4
Note: The data “9.4(6)” in Table 5 means that the average of three results of total profile deviation 9.0 m, 9.3 m, 9.9 m, measured by GMD (DFRP), is 9.4 m, and the accuracy grade is 6 according to ISO1328-1:1997.
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Fig. 14. Comparison of profile measurement.
Table 6 Comparison between the DFRP GMD with P26 (pitch deviations). Items
Fp fpt
GMD (DFRP) (m)
LF RF LF RF
(m)
P26 (m)
1
2
3
Avg(Grade)
1
2
3
Avg(Grade)
44.3 44.8 5.3 4.9
45.1 43.9 4.2 4.6
45.5 45.0 4.8 4.9
45.0(8) 44.6(8) 4.8(5) 4.8(5)
43.2 43.4 4.1 4.2
43.4 43.3 4.1 4.0
43.2 43.3 4.0 4.2
43.3(8) 43.3(8) 4.1(5) 4.1(5)
1.7 1.3 0.7 0.7
Note: The data “43.3(8)” in the table means that the average of three results of cumulative pitch deviation 43.2 m, 43.4 m, 43.2 m, measured by P26, is 43.3 m, and the accuracy grade is 8 according to ISO1328-1:1997.
The measurement results for pitch deviations were listed in Table 6. For pitch deviations measurement, the difference of the total cumulative pitch deviations Fp of left flank between the mean value of 3 tests on GMD (DFRP) and the mean value of 3 tests on GMM P26 is 1.7 m. The difference for the single pitch deviation fpt is 0.7 m. The difference between GMD (DFRP) and the P26 is that, the DFRP method is more function orientated, and the right flank and left flank can be measured at the same time. When the same work-piece is measured under the same rotation speed, the measurement time using GMD (DFRP) is approximately the half of the P26. As [25,26] proposes a model for the expression and an evaluation method of the correlation uncertainty based on the Axiomatic Design matrix and the Mote Carlo Simulation, there is still much work to do about the conformity assessment between DFRP method and others. 6. Conclusions and outlook In order to meet the demands of the measurement of the accuracy relevant to flanks of gear teeth in mass-produced gear manufacturing shop floor, a gear measuring device (GMD) is developed using double-flank rack probe (DFRP), in which both left and right flanks of the gear to be inspected are in mesh with a rack probe to measure pitch and profile deviations on the left and right flanks simultaneously. The evaluation method for pitch and profile deviations, the structure of the measuring system, and the key mechanical parts are described in this paper. Repeatability experimental results of the device is less than 2 m, and the results compared to the gear measuring machine
(GMM) Klingelnberg P26 shows that this device can be used for fast measurement of the accuracy relevant to flanks of gear teeth. Acknowledgment This research is supported by the National Natural Science Foundation of China (NSFC), grant No. 51105007. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.precisioneng. 2016.02.006. References [1] ISO. ISO 1328-1:2013(en): cylindrical gears-ISO system of flank tolerance classification—Part 1: Definitions and allowable values of deviations relevant to flanks of gear teeth. Geneva, Switzerland: International Organization for Standardization; 2013. http://www.iso.org/iso/home.htm. [2] Yuzaki M. Gear measuring machine by ‘NDG method’ for gears ranging from miniature to super-large. Gear Technol 2011;(March/April):55–60. [3] http://www.klingelnberg.com/en (access in Jan. 17, 2015). [4] http://www.hexagonmetrology.com (access in Jan. 17, 2015). [5] Goch G. Gear metrology. CIRP Ann: Manuf Technol 2003;52(2):659–95. [6] Guenther A. Evaluation of runout deviation at bevel gears based on pitch measurement. Ann CIRP 2006;55(1):539–42. [7] Guenther A. Interpretation of bevel gear topography measurements. Ann CIRP 2011;60(1):551–4. [8] Haertig F, Kniel K. High accurate gear measuring machine. VDI-Berichte 2004;1860:409–15. [9] Frazer RC. An international comparison of involute gear profile and helix measurement. Metrologia 2004;41:12–6. [10] Reiter E, Eberle F. Practical considerations for the use of double-flank testing for the manufacturing control of gearing—Part I. Gear Technol 2014;(January/February):44–51.
J. Tang et al. / Precision Engineering 45 (2016) 153–159 [11] Reiter E, Eberle F. Practical considerations for the use of double-flank testing for the manufacturing control of gearing—Part II. Gear Technol 2014;(March/April):60–9. [12] http://www.frenco.de (access in Jan. 17, 2015). [13] http://www.osk-corp.co.jp (access in Jan.17, 2015). [14] http://www.technomax-j.com (access in Jan.17, 2015). [15] Kawasaki Y, Shinoda M, et al. Development of a new automatic gear selecting machine for automobile transaxle manufacturing. Ann CIRP 1984;33(1): 363–8. [16] Shi ZhY, Tang J, Wei HL, et al. Gear in-line measuring machine based on doubleflank gear rolling test with many degrees of freedom. Chin J of Sci Instrum 2009;30(2):303–7 (Chinese). [17] http://www.marposs.com/product.php/eng/gears inspection (access in Mar.17, 2015). [18] http://www.directindustry.com/prod/itw-heartland/gear-inspectionmachines-65741-513423.html (access in Mar.17, 2015). [19] Zhang ZhL. Measuring gear profile deviations on left and right flank simultaneously. Metrol Meas Tech 1994;1:4–7 (Chinese).
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[20] Zhang ZhL. Measuring gear pitch deviations on left and right flank simultaneously. Metrol Meas Tech 1995;2:10–2 (Chinese). [21] Tang J, Shi ZhY, Fang ZhQ. Novel measuring principle of pitch deviation evaluated through double-flank gear rolling test with rack probe. Key Eng Mater 2012;523–524:832–5. [22] Tang J, Shi ZhY, Fang ZhQ. Novel measurement principle of profile deviations evaluated through double-flank rolling test with rack probe. Proc SPIE 2013;8759, 87591P-1-87591P-5. [23] Bai YX. Gear accuracy and inspection. Shanghai: Shanghai Sci & Technol Press; 1986. [24] BIPM, IEC, IFCC, ILAC, ISO, IUPAC et al., International vocabulary of metrology—basic and general concepts and associated terms (VIM) JCGM 200:2012, Bureau International des Poids et Mesures, 2012. http://www.bipm. org/en/publications/guides/vim.html (access in Jan.17, 2015). [25] Vincent JP, Dantan JY, Bigot R. Virtual meshing simulation for gear conformity verification. CIRP J Manuf Sci Technol 2009;2(1):35–46. [26] Dantan JY, Vincent JP, Goch G, Mathieu L. Correlation uncertainty—application to gear conformity. Ann CIRP 2010;59(1):509–12.
Contents lists available at ScienceDirect
Precision Engineering journal homepage: www.elsevier.com/locate/precision
Development of a gear measuring device using DFRP method Jie Tang ∗ , Jianqiao Jia, Zhiqiang Fang, Zhaoyao Shi College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, No. 100 Pingleyuan, Chaoyang District, 100124 Beijing, China
a r t i c l e
i n f o
Article history: Received 25 June 2015 Received in revised form 24 January 2016 Accepted 12 February 2016 Available online 2 March 2016 Keywords: Gear measurement DFRP Rack probe Profile Pitch
a b s t r a c t This paper presents the double-flank rack probe (DFRP) method to measure the accuracy relevant to flanks of gear teeth. In DFRP method, both left and right flanks of the gear to be measured are in mesh with a rack probe to measure pitch and profile deviations on the left and right flanks simultaneously. A gear measuring device (GMD) using DFRP method is developed. The evaluation method for pitch and profile deviations, the structure of the measuring system, and the key mechanical parts are introduced. Repeatability experimental results of the device is less than 2 m, and the results compared to the gear measuring machine (GMM) Klingelnberg P26 shows that this device have consistency with GMM and can be used for the measurement of the accuracy relevant to flanks of gear teeth. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Generally, the deviations relevant to flanks of gear teeth, such as pitch, profile and helix deviations, are measured by gear measuring machine (GMM) and coordinate measuring machine (CMM) in metrology room [1–9]. However, the measurement of the accuracy relevant to flanks of gear teeth at industrial scale manufacturing workshop is still an issue. The double-flank gear rolling test is widely used in gear manufacturing, especially for large-volume inspections in workshop, because of its easy measurement principle and less requirements concerning the environment. Normally, composite deviations are evaluated in traditional double-flank gear rolling test [10,11]. Master gears should be specially designed and manufactured for double-flank gear inspection [12–14]. Lately double-flank gear rolling test with many degrees of freedoms is also used for gears process inspection at shop floor [15–17], for measuring composite, runout, nicks, size, tooth action, lead and taper [18]. The inertia of the mechanical part with multi-freedoms will affect measuring speed. In 1994 & 1995, Zhang proposed to measure gear profile and pitch deviations on left and right flank simultaneously [19,20], but no more authors’ development on this issue had been found. A. Guenther presented the approach for the evaluation of runout deviation at bevel gears based on pitch measurement in 2006 [6],
∗ Corresponding author. Tel.: +86 10 6739 2894; fax: +86 10 6739 0989. E-mail address: [email protected] (J. Tang). http://dx.doi.org/10.1016/j.precisioneng.2016.02.006 0141-6359/© 2016 Elsevier Inc. All rights reserved.
which is based on a simultaneous probing of both flanks of one gap by an appropriate tip at a well-defined pitch diameter near the reference cone. The test was carried out with a pinion bevel gear on 5 different CMMs using probe ball, while rack probe is used for measuring the both flanks of a cylindrical gears simultaneously in this paper. For the purpose of measuring deviations relevant to flanks of gear teeth (pitch, profile and helix deviations) in mass-produced gear manufacturing shop floor, the detailed measurement principle of profile and pitch deviations using double-flank rack probe are described in [21,22]. A gear measuring device based on this measurement principle will be introduced briefly in this paper. 2. DFRP method Gear error calculation system is established to analyse gear error in the DFRP method [23]. The increment on the line of action symbolized extra displacement of the conjugating gear flanks. In Fig. 1, the gear to be inspected is in mesh with a rack probe. The gear error calculation is establishedbydrawing two lines of action on system L and on right flank n R through the contacting point. left flank n All gear errors will be converted to these two lines. Gear error is the function of rotating angle. The gear error on each flank is calculated on each meshing line separately. When gear rotates at a constant speed, displacement increment of the master rack generates along the meshing line. In Fig. 2, the rack probe is driven by the work-piece, left and right flanks of the gear to be inspected are in mesh with a rack probe. The plus material deviation is defined as positive, while the
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J. Tang et al. / Precision Engineering 45 (2016) 153–159
Fig. 1. Gear error calculation system.
minus material deviation is defined as negative. The relationship among the displacement increment along left and right meshing line (FL , FR ), the radial displacement (R) and the tangential displacement (T ) is as follows:
T () = FL () cos ˛ − FR () cos ˛ R() = FL () sin ˛ + FR () sin ˛
(1)
where ˛ is the pressure angle of rack probe (degree). The displacement of rack probe in radial & tangential direction is measured while the work piece is meshing with the rack probe in double-flank with no backlash, then the profile or pitch deviations of the left & right gear flank will be evaluated. 2.1. Profile measurement 2.1.1. Meshing procedure Meshing procedure during profile measurement is shown in Fig. 3, in which the gear to be measured is rotate clockwise as an example, and the pressure angle of the gear to be measured and the rack is both 20 degree. Fig. 3a is the starting measurement position, point a and a is the contact point between the rack probe and the left and right gear flanks, respectively. When gear rotates to the position in Fig. 3b, the tip of the rack contact with the left flank of the gear to be measured on point b , while the right flank of the gear meshes with the rack on point b. The gear keeps on rotating till the position in Fig. 3c, right flank of the gear meshes with the rack on point c, while the left flank comes in contact with the rack at root edge meshing. In Fig. 3d, the rack probe and the left and right gear flanks contact on point d and d, and the right gear flank comes in contact at root edge meshing. The edge meshing will be finished when the measured gear rotating to the position in Fig. 3e.
Fig. 3. Procedure of mesh during profile measurement.
2.1.2. Profile deviations The gear to be measured rotate for an angle“”, as shown in Fig. 4, two contacting points a and a moves to points f and f. The relationship between the radial displacement (ypfi ) and the tangential displacement (xpfi ) and the profile deviations on right and
R and F L left gear flank F˛i are as follows: ˛i
⎧ ypfi cos ˛ + xpfi sin ˛ ⎪ R ⎪ ⎨ F˛i = sin 2˛
⎪ ⎪ ypfi cos ˛ − xpfi sin ˛ ⎩ L F˛i =
(2)
sin 2˛
where ˛ is the pressure angle of rack probe (degree). 2.1.3. Requirement of rack probe The requirement of rack probe is that, in profile measurement, the pressure angle of the rack should be designed to fulfil the condition that the tip edge meshing in right flank and the root edge meshing in left flank occurs simultaneously.
Fig. 2. DFRP method.
Fig. 4. Profile deviations evaluation.
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Fig. 5. Procedure of pitch measurement. Fig. 6. Structure of the device.
2.2. Pitch measurement 2.2.1. Meshing procedure In pitch measurement, rack probe enters into each tooth space to contact with the both left and right flank of the measured gear. The starting tooth of the gear to be measured is identified as 1st tooth. As shown in Fig. 5, rack probe enter into 1st tooth space in Fig. 5a, gear rotates for a pitch angle, and rack probe returns back in Fig. 5b, rack probe enters into 2nd tooth space in Fig. 5c. 2.2.2. Pitch deviations The relationship between the radial displacement ypti and
the tangential displacement xpti and the cumulative pitch deviations, which is the ith tooth compared to 1st tooth, on right and left gear flank
R and F L Fpi pi
are as follows:
⎧ ypti cos ˛ + xpti sin ˛ ⎪ R ⎪ ⎨ Fpi = sin 2˛ cos ˛t
⎪ ⎪ ⎩ F L = ypti cos ˛ − xpti sin ˛ pi
(3)
sin 2˛ cos ˛t
the test and control hardware and the data processing software. The strong points of this method for profile and pitch deviations measurement compared to the traditional method is that, the deviations of the left & right gear flank is evaluated simultaneously. 3.1. Procedure for profile measurement Procedure for profile measurement is as follows. (1) After initialization of the hardware has been executed successfully, the gear to be measured is controlled to rotate to a proper position (2) The rack probe for profile deviations measurement goes ahead to contact with the gear in double-flank with no back lash. (3) Then the rack probe is driven by the gear. (4) Data acquisition of the rotation angle of the gear is executed, which is the radial & tangential displacement increment of the rack.
where ˛t is the transverse pressure angle on reference diameter of the gear to be measured (degree). 2.2.3. Requirement of rack probe The requirement of rack probe is that, in pitch measurement, the contact point is nearby reference diameter. 3. Measuring system The structure of the measuring device is shown in Fig. 6. Rotation of the work-piece is driven by a servomotor. Output from the round grating is 236,800 pulses per revolution after the signal is subdivided. Rack probe can move along the tangential and horizontal direction. The 2D platform contains a linear grating in X-axis for measuring the radial displacement, the resolution of which is 0.2 m, and a micro-displacement sensor used for measuring the tangential displacement in Y-axis, the resolution of which is 0.5 m. 2D platform can move along Z-axis vertical part. Vertical part can move along horizontal direction manually. The developed measuring device using double-flank rack probe method is shown in Fig. 7, which includes the mechanical base unit,
Fig. 7. GMD (DFRP) (Gear measuring device using DFRP method).
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Fig. 9. 2D platform. Fig. 8. Base unit.
(5) After measurement for one set of tooth flanks, the rack probe draws back, the gear rotates an angle to inspect another one set of flanks. (6) When Measurement is accomplished, the deviations are evaluated through data processing software. The measurement results are processed to generate measurement report. Fig. 10. 3D drawing of probe component.
3.2. Procedure for pitch measurement Procedure for pitch measurement is as follows. (1) Work-piece is installed with precision graduated circle coaxially. (2) Rack probe enters to contact both left & right flank on the position of reference circle. (3) The displacement of rack probe in radial & tangential direction is measured. (4) After that, the rack probe draws back, the gear will rotate an angular pitch to begin the next iteration of pitch measurement until all the teeth are inspected. (5) Then the pitch deviations of the left & right gear flank will be evaluated. The measurement results are processed to generate measurement report. 4. Mechanical parts 4.1. Base unit The base unit is shown in Fig. 8. Component for gear to be inspected and the horizontal part is fixed on a granite base. In component for gear to be inspected, the work-piece is supported by a spindle with round grating installed at the lower end. Rack probe is supported by a 2D platform jointed along vertical part which is installed on the horizontal part. 4.2. 2D platform The 2D platform is shown in Fig. 9 (1—supporting plate, 2—linear grating, 3—cross roller, 4—micro-displacement sensor, 5—rack probe and 6—probe component). Probe component (Fig. 10) is fixed with the sliding part of the cross roller, the static part of which is installed on the supporting plate. The linear grating is fixed with the sliding part of the cross roller to output the signal of the radial displacement. A micro-displacement sensor is installed in the probe component, to output the signal of the tangential misalignment guided by a couple of leaf spring.
Table 1 Parameters of the measured gear. Module (mm)
No. of teeth
Pressure angle (◦ )
Helix angle (◦ )
Modification factor
2.5
45
20
0
+0.26
5. Experimental results Parameters of the measured gear and the rack probe are listed in Tables 1 and 2. 5.1. Repeatability test Under repeatability condition of measurement [24], that includes the same measurement procedure, same operators, same measuring system, same operating conditions and same location, and replicate measurements on the same or similar objects over a short period of time, the work-piece and the rack in Tables 1 and 2 mesh on the gear measuring device as shown in Fig. 7 to measure the profile and pitch deviations. Measurements are done on the same object, and 10 sets of measurement results are shown in Tables 3 and 4. Tests under repeatability condition shows that the difference between the maximum and minimum value of the profile deviations is 1.9 m, and the variation of pitch deviations is 1.6 m. Results of profile deviations of 10 tests on left flank and right flank are listed in Table 3, including the total profile deviation F˛ , profile form deviation ff˛ and profile slope deviation fH˛ . The graph for 10 tests of profile deviations is shown in Fig. 11.
Table 2 Parameters of the rack. Pressure angle (◦ )
Helix angle (◦ )
Tooth depth (mm)
Tooth thickness (mm)
23(profile)/21(pitch)
0
10
1
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Table 3 Results of profile deviations. SN
F␣ (m)
ff␣ (m)
fH␣ (m)
F␣ (m)
LF (Left flank)
ff␣ (m)
fH␣ (m)
RF (Right flank)
1 2 3 4 5 6 7 8 9 10
9.0 9.3 9.9 9.6 10.3 9.8 9.4 10.2 10.2 9.2
10.1 10.4 9.8 9.3 10.6 9.9 10.9 10.4 9.3 9.4
3.1 3.9 4.3 4.0 3.3 3.7 3.3 3.6 2.9 3.5
8.7 9.0 9.6 9.3 10.1 9.5 9.1 9.9 9.9 8.9
5.8 3.9 4.9 5.4 4.8 5.7 4.1 5.1 5.1 4.8
8.1 7.2 8.3 8.4 7.8 7.7 8.5 8.2 7.6 7.8
1.3
1.6
1.4
1.4
1.9
1.2
Fig. 12. Repeatability test of total cumulative pitch deviations.
Table 4 Results of pitch deviations. SN
Fp (m)
fpt (m)
Fp (m)
1 2 3 4 5 6 7 8 9 10
44.3 45.1 45.5 45.2 44.5 44.9 44.5 44.8 44.1 44.7
5.3 4.2 4.8 4.5 5.2 4.7 4.3 5.1 5.1 4.6
44.8 43.9 45.0 45.1 44.5 44.4 45.2 44.9 44.3 44.5
4.9 4.6 4.9 4.3 5.1 4.4 5.4 4.9 3.8 3.9
1.4
1.1
1.3
1.6
LF (Left flank)
fpt (m)
RF (Right flank)
Fig. 13. Repeatability test of single pitch deviation.
fpt . The graph for 10 tests of pitch deviations is shown in Figs. 12 and 13. 5.2. Comparison between GMD (DFRP) and GMM Klingelnberg P26
Fig. 11. Repeatability test of profile deviations.
In the repeatability of the pitch deviation measurement using the developed system, results of pitch deviations of 10 tests on left flank and right flank are listed in Table 4, including the total cumulative pitch deviations Fp and single pitch deviation
To compare the difference of the measurement results between the GMD using DFRP method and the GMM (gear measuring machine) Klingelnberg P26, many tests were implemented separately in the two kinds of machine. The measurement by P26 are done on the same object for three times repeated on one work piece. The results by GMD (DFRP) are the same as those of SN 1, 2 and 3 in Tables 3 and 4, which are the part of the repeatability test by GMD (DFRP). The measurement results for profile deviations were listed in Table 5. For profile deviations measurement, the difference between the mean value of 3 tests on GMD (DFRP) and the mean value of 3 tests on GMM P26 is within 1 m, including the total profile deviation F˛ , profile form deviation ff˛ and profile slope deviation fH˛ . The curve for profile measurement results compared with P26 is shown in Fig. 14, through which the consistency of profile measurement between these two kinds of machines can be found.
Table 5 Comparison between the GMD with P26 (profile deviations). Items
GMD (DFRP) (m) 1
2
(m)
P26 (m) 1
2
3
Avg(Grade)
F␣
LF RF
9.0 8.7
9.3 9.0
3 9.9 9.6
Avg(Grade) 9.4(6) 9.1(6)
10.4 8.1
10.2 8.2
10.3 8.2
10.3(6) 8.2(6)
−0.9 0.9
ff␣
LF RF
10.1 5.8
10.4 3.9
9.8 4.9
10.1(7) 4.9(4)
9.1 4.3
9.0 4.3
9.2 4.4
9.1(7) 4.3(4)
1.0 0.6
fH␣
LF RF
3.1 8.1
3.9 7.2
4.3 8.3
3.8(5) 7.9(7)
4.7 8.4
4.9 8.3
4.9 8.3
4.8(5) 8.3(7)
−1.0 −0.4
Note: The data “9.4(6)” in Table 5 means that the average of three results of total profile deviation 9.0 m, 9.3 m, 9.9 m, measured by GMD (DFRP), is 9.4 m, and the accuracy grade is 6 according to ISO1328-1:1997.
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Fig. 14. Comparison of profile measurement.
Table 6 Comparison between the DFRP GMD with P26 (pitch deviations). Items
Fp fpt
GMD (DFRP) (m)
LF RF LF RF
(m)
P26 (m)
1
2
3
Avg(Grade)
1
2
3
Avg(Grade)
44.3 44.8 5.3 4.9
45.1 43.9 4.2 4.6
45.5 45.0 4.8 4.9
45.0(8) 44.6(8) 4.8(5) 4.8(5)
43.2 43.4 4.1 4.2
43.4 43.3 4.1 4.0
43.2 43.3 4.0 4.2
43.3(8) 43.3(8) 4.1(5) 4.1(5)
1.7 1.3 0.7 0.7
Note: The data “43.3(8)” in the table means that the average of three results of cumulative pitch deviation 43.2 m, 43.4 m, 43.2 m, measured by P26, is 43.3 m, and the accuracy grade is 8 according to ISO1328-1:1997.
The measurement results for pitch deviations were listed in Table 6. For pitch deviations measurement, the difference of the total cumulative pitch deviations Fp of left flank between the mean value of 3 tests on GMD (DFRP) and the mean value of 3 tests on GMM P26 is 1.7 m. The difference for the single pitch deviation fpt is 0.7 m. The difference between GMD (DFRP) and the P26 is that, the DFRP method is more function orientated, and the right flank and left flank can be measured at the same time. When the same work-piece is measured under the same rotation speed, the measurement time using GMD (DFRP) is approximately the half of the P26. As [25,26] proposes a model for the expression and an evaluation method of the correlation uncertainty based on the Axiomatic Design matrix and the Mote Carlo Simulation, there is still much work to do about the conformity assessment between DFRP method and others. 6. Conclusions and outlook In order to meet the demands of the measurement of the accuracy relevant to flanks of gear teeth in mass-produced gear manufacturing shop floor, a gear measuring device (GMD) is developed using double-flank rack probe (DFRP), in which both left and right flanks of the gear to be inspected are in mesh with a rack probe to measure pitch and profile deviations on the left and right flanks simultaneously. The evaluation method for pitch and profile deviations, the structure of the measuring system, and the key mechanical parts are described in this paper. Repeatability experimental results of the device is less than 2 m, and the results compared to the gear measuring machine
(GMM) Klingelnberg P26 shows that this device can be used for fast measurement of the accuracy relevant to flanks of gear teeth. Acknowledgment This research is supported by the National Natural Science Foundation of China (NSFC), grant No. 51105007. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.precisioneng. 2016.02.006. References [1] ISO. ISO 1328-1:2013(en): cylindrical gears-ISO system of flank tolerance classification—Part 1: Definitions and allowable values of deviations relevant to flanks of gear teeth. Geneva, Switzerland: International Organization for Standardization; 2013. http://www.iso.org/iso/home.htm. [2] Yuzaki M. Gear measuring machine by ‘NDG method’ for gears ranging from miniature to super-large. Gear Technol 2011;(March/April):55–60. [3] http://www.klingelnberg.com/en (access in Jan. 17, 2015). [4] http://www.hexagonmetrology.com (access in Jan. 17, 2015). [5] Goch G. Gear metrology. CIRP Ann: Manuf Technol 2003;52(2):659–95. [6] Guenther A. Evaluation of runout deviation at bevel gears based on pitch measurement. Ann CIRP 2006;55(1):539–42. [7] Guenther A. Interpretation of bevel gear topography measurements. Ann CIRP 2011;60(1):551–4. [8] Haertig F, Kniel K. High accurate gear measuring machine. VDI-Berichte 2004;1860:409–15. [9] Frazer RC. An international comparison of involute gear profile and helix measurement. Metrologia 2004;41:12–6. [10] Reiter E, Eberle F. Practical considerations for the use of double-flank testing for the manufacturing control of gearing—Part I. Gear Technol 2014;(January/February):44–51.
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