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1、01FTM6 Performance- -Based Gear- -Error Inspection, Specification, and Manufacturing- -Source Diagnostics by: W.D. Mark and C.P. Reagor, Penn State University TECHNICAL PAPER American Gear Manufacturers Association Copyright American Gear Manufacturers Association Provided by IHS under license with
2、AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:35:20 MDTNo reproduction or networking permitted without license from IHS -,-,- Performance- -Based Gear- -Error Inspection, Specification, and Manufacturing- -Source Diagnostics William D. Mark and Cameron P. Re
3、agor, Penn State University Thestatementsandopinionscontainedhereinarethoseoftheauthorandshouldnotbeconstruedasanofficialactionor opinion of the American Gear Manufacturers Association. Abstract Performance- -relevant imperfections in gear manufacturing machines, cutting tools, and operations are ex
4、hibited as manufacturingerrorsingear- -toothworkingsurfaces. Detailedmeasurementsofsuchgear- -tootherrorscanbeobtained utilizing present- -day dedicated CNC gear measurement machines. The effects of such gear- -tooth errors on gear performance are usefully described in the frequency domain by their
5、rotational harmonic contributions to the transmission error. Using such a frequency- -domain representation, the rotational- -harmonic tooth- -error contributions canbeseparatedfromtheattenuatingeffectsonsucherrorscausedbythesimultaneousmultipletoothcontactofmeshing gear teeth, which can greatly att
6、enuate the contribution of such errors to the transmission error of helical gears. It is shownthatthisfrequency- -domainapproachallowsspecificationoflimitsonthefeaturesofgear- -tootherrorsthatrelate to gear performance as described by the transmission error. In addition, it is shown that one can com
7、pute, from such detailed tooth measurements, the specific error contributions on the teeth that cause any particularly troublesome rotational harmonic contributions to the transmission error, thereby permitting manufacturing- -source identification of such troublesome harmonics. Examples are given i
8、llustrating the above- -described approach to gear- -tooth error measurement, specification, and manufacturing source diagnostics. Copyright ? 2001 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 October, 2001 ISBN: 1- -55589- -785- -1 Copyright Americ
9、an Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:35:20 MDTNo reproduction or networking permitted without license from IHS -,-,- 1 Performance-Based Gear-Error Inspection, Specification, and M
10、anufacturing-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 descri
11、bes 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 are po
12、tentially 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 sources:
13、 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 trans
14、mission 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 stati
15、c transmission error contributions are periodic with a period of one revolution of the gear. Let Rb denote the base-cylinder radius of the gear, and its rotational position in radians. Then b Rx = is an obvious choice 4-7 for an independent variable in which to describe transmission error contributi
16、ons in the “time” domain. (The base pitch ? of both gears of a meshing pair is the same when measured in units of x; it is ,/2NRb= 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 error
17、contributions from a single gear is 2? 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 transmission
18、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-revolution, et
19、c.), 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 its supp
20、orting 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 deviations
21、 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 axial locat
22、ion 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 average
23、deviation surface are provided only to the tooth- (1) (2) (3) (4) Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:35:20 MDTNo reproduction or networking permitted without lice
24、nse from IHS -,-,- 2 meshing 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 working surface modifications from perfect involut
25、e 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 errors. (From this juncture onward, we shall r
26、efer 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 rotational harmonics of the static transmission erro
27、r 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 unique error patterns on the teeth causing any pa
28、rticularly 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 the tooth working surfaces, and as mentioned a
29、bove, 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 considerations of mass (inertia) and force, we shall
30、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 contributions can be computed from tooth-deviation measu
31、rements 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 outlined below. 2. Method of Kinematic Transmi
32、ssion Error Computation Figure 1 illustrates a pair of meshing helical (or spur) gears. Let ?(1) and ?(2) denote deviations of the angular positions of the upper and lower gears in the figure, respectively, from the positions of their perfect counterparts possessing equispaced, rigid, perfect involu
33、te teeth (which would transmit an exactly constant angular velocity ratio). Clearly, ?(1) and ?(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 kinematic transmission error, we assume that the
34、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 helical teeth traversing across the tooth surface. In
35、ertial 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 pair to be the amount the mating teeth come together
36、 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 contributions from each of the two meshing gears )()()(
37、 )2()2()1 () 1( xRxRx bb = where )1( b Rand )2( b Rare 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 contribution of each of the two meshing gears as a
38、 separate entity. (5) Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:35:20 MDTNo reproduction or networking permitted without license from IHS -,-,- 3 The basic assumptions i
39、n 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, defined by radial depth D and axial facewidth F (figure 2). (Tooth deviations are m
40、easured 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.) 3. The stiffness of mating tooth pairs per unit length of line of tooth contact is a
41、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 contribution from each gear, it is necessary to perform detailed measurements on each t
42、ooth 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 multiple line-scanning measurements in a radial direction (profile measurements). In order
43、 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 errors (and intentional modifications of tooth working surfaces) from perfect involute
44、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 illustrate the power of this method, we show in (the lower) figure 3a, 32 samples of a pure
45、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 the Legendre polynomial reconstruction of this sine wave, which utilized only the va
46、lues 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, the reconstruction using the samples would not have been as good as that shown in fig
47、ure 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 ?yk (y) and ?zl (z) denote normalized Legendre polynomials 5,6 of degrees k and l, respectively, in the axial (y-d
48、irection) and radial (z-direction) defined on the tooth surface illustrated in figure 2. Radial coordinate z is defined 6,11 by cRz b +=sin? where Rb is base cylinder radius, ? is roll angle (in radians), ? is pressure angle, and c is a constant determined such that the origin of radial coordinate z
49、 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 2). Then, any geometric deviation on the working surface of tooth j can be represented, exactly, by the doubly infinite sum, , )()(),( , 00 zyczy zlykklj l