1、01FTMS1Optical Technique for Gear Contouringby: F. Sciammarella, Illinois Institute of TechnologyTECHNICAL PAPERAmerican Gear ManufacturersAssociationOptical Technique for Gear ContouringFederico Sciammarella, Illinois Institute of TechnologyThestatementsandopinionscontainedhereinarethoseoftheauthor
2、andshouldnotbeconstruedasanofficialactionoropinion of the American Gear Manufacturers Association.AbstractThree-dimensional contouring of gears is a necessary tool for quality control of high performance gears. Currentmethods involve very large and expensive mechanical devices the carry out point-by
3、-point measurements. This paperpresents an optical technique (projection moir) that is compact and can provide a quick full field analysis of highprecision gears. Comparisons are made between mechanical and optical profiles obtained of a gear tooth.CopyrightGe32001American Gear Manufacturers Associa
4、tion1500 King Street, Suite 201Alexandria, Virginia, 22314October, 2001ISBN: 1-55589-791-61Optical Technique for Gear Contouring F. Sciammarella Illinois Institute of Technology Introduction Gears have existed since the invention of rotating machinery. With the industrial revolution, the eighteenth
5、century saw an explosion in the use of metal gearing. A science of gear design and manufacture rapidly developed through the nineteenth century. These days, the most significant new gear developments are in the area of materials. Modern metallurgy has greatly increased the useful life of industrial,
6、 automotive, and aeronautical gears. With the ever-increasing demand for high performance gears the field of quality control and inspection of gears has also grown tremendously in the past decades. In the aerospace industry very high tolerances are needed in order for helicopters and planes to funct
7、ion at optimal conditions and avoid any major catastrophes. Current methods of measuring gears use a coordinate measurement machine (CMM) of some type. This is a slow point-by-point process, which also requires very large and expensive machinery. It is in this spirit that this paper was written, to
8、introduce an optical technique that performs full field high precision contouring of gears in a simple, fast, and inexpensive manner. Background of Optical Technique The optical technique used in this paper is known as Projection Moir. In Projection Moir an incoherent light source is used to project
9、 a grating onto the surface of observation. When a telecentric lens system is used for projection of the grating and the observer is placed at a large distance the following equation is used to obtain the depth information at any given point: n pz = tan Where n is given as fringe order, p is the pit
10、ch of the grating and is the illumination angle (see Fig. 3). It should be noted that this equation is used to the first order of approximation (this does not satisfy the requirement of micron range accuracy). Later on we introduce corrections in order to obtain depth values within the desired accur
11、acy. These corrections are based on the differences between the mathematical model of the system and the actual physical model. Introduction of any particular model to derive correction expressions was not attempted due to the complexity of the optical system. Our approach to derive depth informatio
12、n extraction algorithms was to use experimental measurements to correct for depth of focus effect, and variation of the sensitivity within the volume of observation. One algorithm is used to correct for the depth of focus effect, while another algorithm is used to correct for the change of sensitivi
13、ty within the volume of observation. Optical Set up An electro optical system was assembled to obtain dimension information from a sample gear provided by Illinois Institute of Technology Research Institute (IITRI). The set up consists of an optical system and a P.C. based computer system (Holo-Moir
14、 Strain Analyzer) for data processing. The optical system consists of two components: 1) Projection system (this projects a grating onto gear surface) 2) Recording system that consists of a microscope and a CCD camera attached to it. Figure 1 shows a schematic representation of the entire set up. Fi
15、gure 2 shows the actual set up used for the experiment. Table I contains the technical data of the projection and recording systems. The gear that is inspected is supported onto a system that has four degrees of freedom (x, y, z, and rotation around gear axis). The accuracy of the x, y, z motion is
16、2.5 10 4in., the rotation is accurate to six seconds of arc. The set up contains a reference plane (see Fig. 3) this plane is used to align the optical system with the recording system and also provides the reference plane that is necessary for obtaining the shape information of the gear. Data Acqui
17、sition It is important to establish the following calibrations in the experiment prior to data acquisition to ensure the most accurate measurements possible. 2Focal length of projection lens 381 mm (15”) Angular aperture of projection system 0.00787 Microscope Angular aperture 0.00624 Magnification
18、1.01612 Distance of reference plane to the objective lens of microscope 483 mm (19”) CCD camera pixels 768 (h) 494 (v) CCD camera format 1/2” HMSA pixel 512 (h) 480 (v) x/y in reference plane 10.281/9.958 (/pixel) Grating pitch 211.51 Table 1 - Basic parameters of optical systemEstablish the proper
19、geometry seen by the CCD camera on the reference plane. 1) Determination of the grating pitch to five significant figures. 2) Determination of the projected grating pitch onto reference plane. 3) Correction function for depth of focus effect. 4) Establish sensitivity function. Once these calibration
20、s are determined the images must be recorded. A four-phase step technique is used to record the phases of the projected grating on the reference plane (Fig. 4) and on the gear (Fig. 5). These two images are subtracted from each other and produce a final phase output (Fig. 6) that will be used for th
21、e data processing. Data Processing The Holo-Moir Strain Analyzer (H.M.S.A) is an in house developed electro-optical device that contains dedicated hardware and specialized software for the analysis of fringe information with a high degree of accuracy. The output of the system is the phase of the pro
22、jected grating as a function of the coordinates of the CCD camera. The phase is transformed into depth information by an equation of the form: () () (), ,cZ xy Z xy f xy=+ where the correction for sensitivity is given by the following equation, () () ( )121,2cZxy pfxyxfxyzGe9Gf9=GebGfbThe first term
23、 represents the phase of the reference grating on the reference plane. f1(x, y) is the sensitivity of the reference plane and contains the effect of both the projection system and the recording camera. The pitch of the reference grating is given by p. The second term contains the function f2that is
24、a function of the optical system the recording system and the shape of the contoured body. The first term can be removed by subtracting the reference plane phase from the actual phase of the gear tooth being analyzed. After the first correction was done a second correction was made for depth of focu
25、s G44f (x, y). The necessary algorithms to correct for the second term were developed and enabled us to get the contour of the gear into a universal coordinate system. Results The optical profiles obtained in this experiment are plotted against the mechanical data that was obtained experimentally an
26、d are shown in Figures 7-9. Mechanical data was measured using the Mitutoyo Contracer CBH-400. According to the manufacturer and the range of measured values the accuracy in the coordinate location x = + 2 G6dm, and the z coordinate is 0.025% of the range measured. Our measurements are made in a ran
27、ge of 6000 G6dm, and therefore the expected accuracy is + 15G6dm. Figures 7-9 also show the differences in between the two profiles obtained in the x and y direction. As it can be seen in these figures the differences between the mean values of the two profiles are well within the accuracy of the me
28、chanical measurements. Figure 10 shows the 3-D view of a half tooth that was obtained from the H.M.S.A. From this profile one can extract information concerning the cross sections of the gear (Fig. 12) or horizontal sections providing a gear profile plot (Fig. 13). All this information is numericall
29、y stored into data files in which data can be extracted to obtain the tooth geometry information of interest. Table II shows the average of these differences between the two profiles (optical, mechanical) as well as their corresponding standard deviations as seen in Figs 7-9. 3Profile Average differ
30、ence () Standard deviation () x (thickness) z (depth) x (thickness) z (depth) 0.1 in 2.400 5.139 4.991 8.808 0.2 in 1.338 3.635 4.343 7.798 0.3 in 3.699 7.367 5.553 9.826 Table 2 Difference in thickness and depth of the optical and mechanical profiles Discussion of Results and Conclusions As seen in
31、 Table II there is excellent agreement between mechanical and optical measurements. In spite of the fact that the current optical set up was limited to 10.281 microns/pixel in the x-direction the average agreement between the two profiles is within 2 microns in a total length of about 5000 microns (
32、approximately 0.04%). In the z-direction the average agreement between the two profiles is within 3 microns in 6000 microns total depth of the profile (approximately 0.05%). Therefore the optical technique (Projection Moir) presented in this paper proves to be a very useful and practical technique f
33、or the contouring of gears with high accuracy. Due to the selection of the optical system parameters a very simple model can be used to transform phase value of the camera pixels into the 3-D object coordinate space. This model ensures flexibility in the optical set-up without the use of complex adj
34、ustment operations. The observation and illumination units are compact and very easy to handle. The calibration procedures are simple and provide a quick determination of the system parameters. As mentioned above there seems to be good agreement between the mechanical an optical profiles obtained. F
35、urthermore the advantage of having the optical profile is that a one can obtain full field information of a gear tooth whereas the mechanical profile provides information from only a few points. With full field information it allows one to make quality control decisions with greater confidence. Furt
36、her developments can be made to this optical technique so that the entire process would require minimal user interface, and so that one could conceivably obtain quasi real time results. ACKNOWLEDGMENTS First of all I would like to thank Dr. Ali Manesh (IITRI) for his support of this project that oth
37、erwise would not be possible without it. I would also like to thank Dr. C.A. Sciammarella my father and mentor, and Dr. Luciano Lambertti both of whom provided guidance and support during this project and made it possible for me to present this information. Figure 1- Schematic of optical set-up xyzc
38、ollimation lensprojection lensgratingmicroscopeCCD camerageaririscondenser lensillumination sourceH.M.S.Amonitor=30ooptical axis of projection systemoptical axis of CCD camerareference plane 4Figure 2 View of the optical set-up Figure 3 - Model used to design the optical set-up Gear tooth Optical ax
39、is of microscope Reference plane Optical axis of projection system =sinpz x z z (Depth) 5Figure 4 Phase of the reference plane Figure 5 Phase of gear tooth Figure 6 Phase of the gear tooth after subtracting the phase of reference plane 6Figure 7 - Differences in the X and Z directions of mechanical
40、and optical profiles (0.1 in from edge) 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 550001000200030004000500060007000Thickness (X) (microns)Depth (Z)(microns)mech.optical0 1000 2000 3000 4000 5000 6000 7000-10-50510152025Depth (microns)DifferenceDX(microns)500 1000 1500 2000 2500 3000 3500 4000
41、 4500 5000 5500-20-10010203040Thickness (microns)DifferenceDZ(microns)7Figure 8 - Differences in the X and Z directions of mechanical and optical profiles (0.2 in from edge) 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500-30-20-100102030Thickness (X) (microns)Difference DZ(microns)0 1000 2000
42、3000 4000 5000 6000 7000-10-505101520Depth (Z) (microns)Difference DX(microns)500 1000 1500 2000 2500 3000 3500 4000 4500 5000 550001000200030004000500060007000Thickness (X) (microns)Depth (Z) (microns)mechoptical8Figure 9 - Differences in the X and Z directions of mechanical and optical profiles (0
43、.3 in from edge) 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500-20-10010203040Thickness (X) (microns)Difference DZ(microns)0 1000 2000 3000 4000 5000 6000 7000-15-10-50510152025Depth (Z) (microns)Difference DX(microns)500 1000 1500 2000 2500 3000 3500 4000 4500 5000 55000100020003000400050006
44、0007000Thickness (X) (microns)Depth (Z) (microns)mechoptical901000200030004000500002000400060008000100000100020003000400050006000Thickness (X) (microns)Width (Y) (microns)Depth (Z) (microns)01000200030004000500002000400060008000100000100020003000400050006000Width (Y) (microns)Thickness (X) (microns)
45、Depth (Z) (microns)Figure 10 Three-dimensional views of the tooth 1000 1500 2000 2500 3000 3500 4000 4500 50000100020003000400050006000Thickness (X) (microns)Depth (Z) (microns)Figure 11 Ten separate tooth profiles ( 1000 spacing) 10Figure 12 Lead tooth profiles and detail 1000 2000 3000 4000 5000 6
46、000 7000 8000 900038004000420044004600480050005200540056005800Width (Y) (microns)Depth(Z) (microns)4200 4400 4600 4800 5000 5200 54003700372037403760378038003820Width segment (Y) (1200 microns)Depth segment(Z)(125microns)1000 2000 3000 4000 5000 6000 7000 8000 90000100020003000400050006000Width (Y) (microns)Depth (Z) (microns)
