1、01FTM6Performance-Based Gear-ErrorInspection, Specification, andManufacturing-Source Diagnosticsby: W.D. Mark and C.P. Reagor, Penn State UniversityTECHNICAL PAPERAmerican Gear ManufacturersAssociationPerformance-Based Gear-Error Inspection,Specification, and Manufacturing-Source DiagnosticsWilliam
2、D. Mark and Cameron P. Reagor, Penn State UniversityThestatementsandopinionscontainedhereinarethoseoftheauthorandshouldnotbeconstruedasanofficialactionoropinion of the American Gear Manufacturers Association.AbstractPerformance-relevant imperfections in gear manufacturing machines, cutting tools, an
3、d operations are exhibited asmanufacturingerrorsingear-toothworkingsurfaces. Detailedmeasurementsofsuchgear-tootherrorscanbeobtainedutilizing present-day dedicated CNC gear measurement machines. The effects of such gear-tooth errors on gearperformance are usefully described in the frequency domain b
4、y their rotational harmonic contributions to thetransmissionerror. Using sucha frequency-domainrepresentation, therotational-harmonic tooth-errorcontributionscanbeseparatedfromtheattenuatingeffectsonsucherrorscausedbythesimultaneousmultipletoothcontactofmeshinggear teeth, which can greatly attenuate
5、 the contribution of such errors to the transmission error of helical gears. It isshownthatthisfrequency-domainapproachallowsspecificationoflimitsonthefeaturesofgear-tootherrorsthatrelateto gear performance as described by the transmission error. In addition, it is shown that one can compute, from s
6、uchdetailed tooth measurements, the specific error contributions on the teeth that cause any particularly troublesomerotationalharmonic contributionsto thetransmission error,thereby permitting manufacturing-source identification ofsuch troublesome harmonics. Examples are given illustrating the above
7、-described approach to gear-tooth errormeasurement, specification, and manufacturing source diagnostics.CopyrightGe32001American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 2001ISBN: 1-55589-785-11Performance-Based Gear-Error Inspection, Specification
8、, and Manufacturing-Source Diagnostics William D. Mark* and Cameron P. Reagor#*Drivetrain Technology Center, Applied Research Laboratory, and Graduate Program in Acoustics, The Pennsylvania State University #Graduate Program in Acoustics, The Pennsylvania State University 1. Introduction This paper
9、describes an approach to measuring, characterizing, specifying, and diagnosing manufacturing errors on spur and helical gears with nominally equispaced involute teeth in a manner that can be related to gear performance as described by the “static” transmission error 1-7. The methods to be described
10、are potentially useful for characterizing and diagnosing errors generated by gear manufacturing machines and their operation, gear cutting and finishing tools, and possibly for spot checking individual gears. The static transmission error contributions from an individual gear have two fundamental so
11、urces: elastic deformations of the teeth and gearbodies, and geometric deviations of the tooth working surfaces from equispaced perfect involute surfaces. (From this juncture onward, we shall refer to such deviations only as geometric deviations of the teeth or, simply, deviations.) Since the static
12、 transmission error is the principal source of vibrations and noise generated by the meshing action of gear pairs 1-3, it is both useful and convenient to utilize its representation in the frequency domain. 1.1 Transmission error representation in the frequency domain Because a gear is circular, its
13、 static transmission error contributions are periodic with a period of one revolution of the gear. Let Rbdenote the base-cylinder radius of the gear, and its rotational position in radians. Then bRx =is an obvious choice 4-7 for an independent variable in which to describe transmission error contrib
14、utions in the “time” domain. (The base pitch G29 of both gears of a meshing pair is the same when measured in units of x; it is ,/2 NRb= where N is the number of teeth on each gear of base-cylinder radius Rb.) It follows from equation (1) that the fundamental period in x of the static transmission e
15、rror contributions from a single gear is 2G42 Rb, the base-cylinder circumference. In the following discussion, it is assumed that the geometric deviations of the working surfaces of all teeth on a gear may differ from one another. It follows 4-7 that rotational harmonic contributions to the transmi
16、ssion error can occur at all integer multiples n of the fundamental n=1 rotational harmonic with period in x of .2 = NRb Typically, the strongest rotational harmonics, other than the tooth-meshing harmonics, ,.,2,1, = ppNn are the first few rotational harmonics (once-per-revolution, twice-per-revolu
17、tion, etc.), the so-called sidebands around the tooth-meshing harmonics, and “ghost tone” harmonics 3,7,8 when they are present. Besides vibration and noise considerations, there are other reasons why transmission error analysis in the frequency domain is useful. If the stiffness of every tooth and
18、its supporting structure is the same as that of every other tooth on a gear, then the contributions to the transmission error from elastic tooth deformations are provided only to the tooth-meshing harmonics, equation (4). Furthermore, let us conceptually measure, in complete detail, the geometric de
19、viations of every point on the working surface of every tooth on a gear, as a function of axial location and roll angle. Next, conceptually place these N measured surfaces in a stack and form the arithmetic average of these N surfaces, which creates a deviation surface expressed as a function of axi
20、al location and roll angle. (If the geometric deviations of the working surface of every tooth on the gear were the same, then each such deviation surface would coincide with the above-described average deviation surface.) The contributions to the static transmission error from the above-described a
21、verage deviation surface are provided only to the tooth-(1)(2)(3)(4)2meshing harmonics, equation (4), along with contributions from elastic deformations of the teeth. For high-quality gears, the contributions from the above-described average deviation surface usually are dominated by intentional wor
22、king surface modifications from perfect involute surfaces. Next, conceptually form the difference between the deviation surface measured for each tooth on a gear and the above-described average deviation surface for that gear. The resulting “difference surfaces” clearly are a result of manufacturing
23、 errors. (From this juncture onward, we shall refer to these surfaces only as geometric “difference surfaces” or, simply, “difference surfaces”.) For gear pairs rotating at constant speed and transmitting constant torque, the above-described difference surfaces provide the contributions to the rotat
24、ional harmonics of the static transmission error 4-8, excluding the tooth-meshing harmonics, equation (4). In what follows, it is shown that these rotational harmonic contributions to the transmission error can be computed from detailed gear tooth measurements made on a single gear, and that the uni
25、que error patterns on the teeth causing any particularly troublesome rotational harmonic, or set of harmonics, can be computed from such detailed tooth measurements. 1.2 Kinematic transmission error The transmission error contributions described in this paper arise only from geometric deviations of
26、the tooth working surfaces, and as mentioned above, do not include contributions from the force-dependent elastic deformations of the teeth that contribute only to the tooth-meshing harmonics, equation (4). Since the term “kinematic” 9,10 refers to the study of motions of objects apart from consider
27、ations of mass (inertia) and force, we shall use the term “kinematic transmission error” to describe the contributions to the transmission error that arise only from geometric deviations of the tooth working surfaces, as explained in the following pages. Such kinematic transmission error contributio
28、ns can be computed from tooth-deviation measurements made on only a single gear, and thus characterize the tooth-deviation contributions to the transmission error of the measured gear without regard to the mating gear. The procedure for computing these kinematic transmission error contributions is o
29、utlined below. 2. Method of Kinematic Transmission Error Computation Figure 1 illustrates a pair of meshing helical (or spur) gears. Let G32(1)and G32(2)denote deviations of the angular positions of the upper and lower gears in the figure, respectively, from the positions of their perfect counterpar
30、ts possessing equispaced, rigid, perfect involute teeth (which would transmit an exactly constant angular velocity ratio). Clearly, G32(1)and G32(2)are functions of the nominal rotational positions of the gears designated, here, by the independent variable x, equation (1). In our computation of the
31、kinematic transmission error, we assume that the gears are operating under a constant loading of sufficient magnitude to insure full contact on a rectangular region of the tooth working surfaces. Such a rectangular region is illustrated in figure 2, along with the line of contact between mating heli
32、cal teeth traversing across the tooth surface. Inertial effects are assumed to be negligible. Figure 1. Pair of meshing gears (adapted from reference 4). Figure 2. Line of tooth contact moving across tooth face (adapted from reference 4). We define 4,7,11 the transmission error of a meshing gear pai
33、r to be the amount the mating teeth come together in the plane of contact relative to their above-described perfect involute counterparts. Then, it follows directly from the sign conventions illustrated in figure 1 that the transmission error (x) of the gear pair can be expressed as the sum of contr
34、ibutions from each of the two meshing gears )()()()2()2()1()1(xRxRxbb =where )1(bR and )2(bR are the base cylinder radii of gears (1) and (2), respectively, and (1)and (2)are measured in radians. From equation (5) it follows that we can rigorously consider the kinematic transmission error contributi
35、on of each of the two meshing gears as a separate entity. (5)3The basic assumptions in the kinematic transmission error algorithms and developed computer software are: 1. The geometric zone of tooth contact on all tooth working surfaces of a gear is the same specified rectangular region on all teeth
36、, defined by radial depth D and axial facewidth F (figure 2). (Tooth deviations are measured in this rectangular region.) 2. Tooth measurements are sufficiently dense to adequately characterize deviations on tooth working surfaces. (Methods have been developed to insure this, as illustrated below.)
37、3. The stiffness of mating tooth pairs per unit length of line of tooth contact is a constant value. (This constant stiffness value is not required.) Otherwise, the computation method is virtually exact. 2.1 Tooth working surface representation In order to compute the kinematic transmission error co
38、ntribution from each gear, it is necessary to perform detailed measurements on each tooth of the gear. Using current technology, such measurements are performed by multiple line-scanning measurements on the tooth working surfaces in a direction parallel to the gear axis (lead measurements) and multi
39、ple line-scanning measurements in a radial direction (profile measurements). In order to insure smooth, convergent, nonoscillatory polynomial interpolation, these measurements are located at the positions of the zeros of normalized Legendre polynomials 12. Our method of representing manufacturing er
40、rors (and intentional modifications of tooth working surfaces) from perfect involute surfaces utilizes two-dimensional normalized Legendre polynomials. This method requires interpolation across line scanning measurements (which are located at the zeros of normalized Legendre polynomials). To illustr
41、ate the power of this method, we show in (the lower) figure 3a, 32 samples of a pure sine wave having 8 full cycles, where these 32 samples are taken at the locations of the zeros of a normalized Legendre polynomial of degree 32. Using only these 32 discrete samples, we show in (the upper) figure 3b
42、 the Legendre polynomial reconstruction of this sine wave, which utilized only the values of the 32 samples. One can see that this interpolated reconstruction is a virtually perfect replica of the original sampled sine wave shown in figure 3a. (If significantly fewer than 32 samples had been taken,
43、the reconstruction using the samples would not have been as good as that shown in figure 3b.) Figure 3a (lower figure). 32 samples of a sine wave possessing 8 full cycles. Figure 3b (upper figure). Legendre polynomial reconstruction of sine wave from the 32 samples. Let G52yk(y) and G52zl(z) denote
44、normalized Legendre polynomials 5,6 of degrees k and l, respectively, in the axial (y-direction) and radial (z-direction) defined on the tooth surface illustrated in figure 2. Radial coordinate z is defined 6,11 by cRzb+= sinG70where Rbis base cylinder radius, G2c is roll angle (in radians), G4e is
45、pressure angle, and c is a constant determined such that the origin of radial coordinate z is located at the midpoint of the range D of z where tooth contact is assumed to take place; similarly, the origin of the axial coordinate y is located at the midpoint of the axial contact range F (see figure
46、2). Then, any geometric deviation on the working surface of tooth j can be represented, exactly, by the doubly infinite sum, ,)()(),(,00zyczyzlykkljlkjGe5Ge5= where the expansion coefficients in equation (7) are obtained from the deviation surface j(y, z) by .)()(),(12/2/2/2/,dydzzyzyFDczlykjFFDDklj
47、Gf2Gf2=Equation (8) is implemented in our algorithms using the method of Gaussian quadrature 12,13, which yields finite upper limits in the summations in equation (7). 2.2 Fourier series representation of kinematic transmission error contribution from a single gear Once the expansion coefficients cj
48、, klare computed from measurements made on each tooth j=0, 1, . . . ,N-1 of the gear, the next step 4-6,11 is computation of the “finite discrete Fourier transforms” 14 of these expansion (6) (7)(8)4coefficients with respect to tooth number j, where N is the number of teeth on the gear: ,2,1,0,1)(/2
49、,10=Ge5necNnBNnjikljNjklNotice that n is the rotational harmonic number of the gear. From the rotational harmonic number spectral contributions Bkl(n), the Fourier series coefficients G22nof the kinematic transmission error contribution from the gear under consideration then are computed 4-6,11 by ,2,1,0),/()(00=Ge5Ge5=nNnnBklkllknThe functions )/(Nnkl in equation (10) are called “mesh transfer functions” 5,6,11 or “mesh attenuation functions.” These functions quantitatively describe the attenuating e
