1、I 98FTM5 Low Vibration Design on a Helical Gear Pair by: K. Umezawa, Tokyo Institute of Technology American Gear TECHNICAL PAPER COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesCopyright O 1998 American Gear Manufacturers Association 1500 King Street,
2、Suite 201 Alexandria: Virginia, 27314 * October. 1998 ISBN: 1-55589-723- 1 Low Vibration Design on a Helical Gear Pair Professor Dr.Eng. Kiyohiko Umezawa, Tokyo Institute of Technology The statements and opinionscontained herein are those of the author and should not be construed as an official acti
3、on or opinion of the American Gear Manufacturers Association. Abstract This paper shows a detailed knowledge of how to design the tooth-surface-modification to realize a quiet helical gear pair where face width is finite in length. The performance of gears with bias-in and bias-out modification are
4、discussed considering the effect of the shaft parallelism deviation. To clarify the effect of the tooth deviation types on the vibration behavior of helical gear pairs, performance diagrams on vibration are introduced and created, in which the acceleration level of gear pairs are shown by contour li
5、nes on the contact ratio domain. COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesLOW VIBRATION DESIGN ON A HELICAL GEAR PAIR Professor Dr.Eng. Kiyohiko UMEZAWA 1 . Introduction Precision and Intelligence Laboratory Tokyo Institute of Technology Nagatsu
6、ta, Midori-ku, Yokohama.226-8503, Japan 1.1 The short history of the researchs on helical gear pairs : The first paper titled as “Helical Gears“ was reported by Harry Walker at 1946 July 12 2. In the paper, he wrote “new aspect of gear technique had been seen in the air craft industry, where spur re
7、duction gears were now carring loads that were not long ago unthought of. The author (H. Walker) believes that a similar technique was desirable for the more difficult subject of helical gears.“ He concluded that there was an increase in pressure at each end of contact corresponding to about three t
8、imes the mean rate of loading. H. Walkers paper at 1946 was the first report discussing on the load distribution along the contact line of a helical gear. Then the history of the researchs on the helical gear pairs is very short as only 50 years. A. W. Davise 4 1950, reported on the marine gear that
9、 the load distribution along the contact line were plus 60 per cent and minus 20 per cent of average load, totalling 80 per cent to the mean value at the both ends with the precise experiments by the tooth whose DP was 8/10, pressure angle was 16 , helix angle was 30. On the helical gear with narrow
10、 face width, E. Mnch and A. K. Roy 5 1957, succeeded in observing directly and precisely the load distribution along the contact lines of the gear module was 1 O, gear teeth numbers were 25/25, pressure angle was 20“, helix angle was 27.5, face wideth was 81 mm, with the use of 3 dimensional photo-e
11、lasticity method. These measured results showed that the load distribution along a contact line shaped as like the back of an Arabian camel, that is, the distribution were gently convex and did not inclese at the both ends. Also In 1957, M. D. Trobojevic 6 reported same results as Roys results on th
12、e load distributions with the use of the O tubber tooth whose module was 15mm, face width was 300mm. a In 1962, K. Hayashi 8 showed theoretically the load distribution along the contact line of a helical gear tooth results in the second kind integral equation of Fredholm-type. However the kernel of
13、the integral equation means the deflections due to a concentrated load on a tooth with finite width, which should have been obtained as the solutions of the differential equation on bending deformation of a rack shaped cantilever plates shown by R.G.Olsson (11 in 1934 . For marine gears, in 1950, T.
14、 J. Jaramillo 3 successed in solving the deflections due to a transverse concentrated load acting an arbitrary point of an infinitely long cantilever plate of constant width and thickness. G. Niemann and T. Hsel 9 examined precisely and reported the relation between noise level and, overlap and tran
15、sverse contact ration, in 1966 . In 1970 G. Niemann and J. Baethge lo reported that the behaviors of the driven gear under loading could be substituted with the changing of the total length of contact lines of a helical gear pair in 1970. D. R. Houser and A. G. Seireg ll, 12 examined the dynamic loa
16、d formula for spur and helical gears with the use of four sets of specially designed gears, mounted with forty-one strain gages each, in 1970 . T. F. Conry and A. Seireg 1131 presented a mathematical programming method for design of elastic bodies in contact in 1971. And they 14 reported a mathemati
17、cal programming technique for the evaluation of load distribution and optimal modifications for parallel gear systems in 1973. However, in this study, they used Jaramillos solution of an infinitely long cantilever plate and the mirror image law proposed by E.J. Wellauer and A. G. Seireg m 1960, to d
18、etermine the deflections due to bending of a gear tooth being finite in length. In the early 1970% main frame computer and computer aided technology became available to all research field, of course to the researchs on helical gears, and lead the studies more precise. 1 COPYRIGHT American Gear Manuf
19、acturers Association, Inc.Licensed by Information Handling Services1.2 Introduction of this paper : It was fortunate for the author that the author could start and continue studying a helical gear pair in the same age when main frame computers were developed and evolved explosively. From the viewpoi
20、nt on the vibration performance, power transmission helical gear pairs with narrow face width, are classified theoretically into three categories over the contact ratio domain. And it has been verified with experiments. From the experimental results, at the same time the relation between the vibrati
21、on magnitude and the shaft parallelism deviation have been investigated. To clarify the effect of the tooth deviation types on the vibration behavior of helical gear pairs, performance diagrams on vibration are introduced, in which the acceleration level of gear pairs are shown by contour lines on t
22、he contact ratio domain. Finally, the performance of gears with bias-in and bias- out modification are discussed considering the effect of the shaft parallelism deviation with use of the developed simulator on a helical gear unit. It becomes clear that there is an asymmetrical feature on the relatio
23、n between the vibration magnitude of a gear pair and the direction of each deviation. 2. Three Categories of a Parallel Pair : It has been confirmed theoretically and experimentally from the viewpoint on the vibration performance that power transmission helical gear pairs with comparatively narrow f
24、ace- width can be classified into three categories over the proposed contact ratio domain whose abscissa is the transverse contact ratio and whose ordinate is the overlap contact ratio. - 10-9 I t Prn “-0 II (i) Spur gear E= 1.48 2.1 lhe Helix Angle and the Transmission Behaviors of a Driven Gear 19
25、 : The author solved numerically the deflections of a thi under a concentrated load shown by R.G.Olssonl by usin i) plate with finite width (151 and a rack shaped cantilever the finite difference method. Furthermore, the load distribution along the line of contact and the compliance of a helical gea
26、r tooth pair from the start of meshing to the end of meshing have been revealed 17-19. Under the condaion that the width of gear face is constant, .e. three times the whole depth, the relation between the helix angle and the calculated behaviors of the driven gear under loading is shown in Fig.1. Th
27、ese results are analyzed under the unique condition that the normal pitch e, normalyzed with the whole depth, is 0.6. Then the whole contact ratio is calculated for each helix angle. And the overlap ratio is calculated from this whole contact ratio and the assumption that the transverse contact rati
28、o is E, = 1.4. This contact ratio was calculated for a spur gear pair when the normal pitch P, is 0.6. When the helix angle is 14 in Fig.1, the sum of the transverse and overlap contact ratio is smaller than 2. So this pair of gears transmits load alternately with one pair and with two pairs of mati
29、ng teeth. It is observed that the load sharing ratio for this pair of gears varies more smoothly than that of spur gears. But sometimes there occur some knobs in this curve when the meshing condition transits from one pair meshing to two pairs meshing. 1-(iii) or (iv) , the total contact ratio is ov
30、er 2. So the gears transmit load alternately with two and three mating pairs of gear teeth. Under these conditions the load sharing ratio varies i, When the helix angle is 20 or 30 as shown in Fi (ii) /I =14 (iii) /I =20 O =1.48, ea =1.48, E =0.44 cS =0.66 Fig.1 The behaviours of the driven gear and
31、 the helix angle with b/h=3.0 2 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicessmoothly. The behavior of the driven gear, or the transmission error, also varies smoothly . Especially when the helix angle is 30, the overlap contact ratio is over 1.0, t
32、he behavior changes very smoothly with little fluctuation as shown. Gear pair Normalmodule 2.2 Three Categories of a parallel axes gear pair 20 : The theoretical and experimental study on the static meshing behavior under loading has proved that a power transmission parallel gear pair can be classif
33、ied into three categories on the contact ratio domain based on its facility for reducing the vibration as shown Fig.2 (refer to Fig.1). They are categorized as: K(I) Total contact ratio is less than 2.0. Poor performance. H1 I H2 I H3 H4 3.5 4 K(II) Total contact ratio is over 2.0 and overlap ratio
34、is less than 1 .O. The tooth modification is recommended. K(III) Overlap ratio is over 1.0. Good performance without tooth modification. Helix angle degl Pressure angle tdegl Face width mml 3. The Vibration Magnitude and the Shaft Parallelisrn21 : The proposed three classifications of a parallel gea
35、r were verified experimentally by dynamic meshing test under loading. At the same time the relation between the vibration and the parallelism of axes was investigated for three kinds of helical gear pairs classified into three categories. Two kinds of shaft misalignment were implemented in this stud
36、y, i.e the in-plane and the out-of-plane deviation. For realizing the out-of-plane or the in-plane parallelism, the pedestal of the driving gear shaft was tipped in the vertical plane or in the horizontal plane, respectively. The vibration was measured by two accelerometers attached directly to the
37、driven gear blank surface. 30 15 20 10 I 20 I 25 I 28 3.1 Dimensions of test gears and test apparatus : Test gear pairs are designed to belong to each category classified over the contact ratio domain, and are named HI, H2, H3 and H4 as shown in Fig.2. Dimensions of each gear pair are displayed in T
38、able.1. All test gears were hardened about Hrc55, and finished by the MAAG 30-BC Gear Grinder. Tooth profile and tooth traces are made with as little deviations as possible. L .- I -0.17 O Addendum modification coefficient Transverse contact ratio 1.4 1.57 3.2 Shaft misalignment set up : The realiza
39、tion of shaft misalignment in the actual experimental set up is shown in Fig.3. It was carried out by inserting several thickness gages either on the surface of the base plate or on its side surface for the out-of-plane or in-plane deviation, respectively. In this set up the thickness gages used wer
40、e of 0.1 mm-0.4mm and the angular deviations realized were about 0.5 x rad and i. i x rad. The amount of the out-of-plane and the in-plane deviation were measured with two dial indicators. Each gear shaft misalignment was introduced for not only the leading side bearing but also the trailing side be
41、aring. 3.3 Influence of the out-of-plane deviation : The relation between the rotational vibration response vs. speed and the out-of-plane deviation is shown in Fig.4 for the gear pair H3. In the case of the proper alignment condition 1 .O (u“ 2 a 3 6 0.5 O .- u 4 - o 1.0 1.5 2.0 Transverse contact
42、ratio E, Fig.2 Classification of parallel pairs Table 1 Dimensions of test gear pairs I I tNurnber of teeth I 30 I 29 I I I Overlap contact ratio I 0.45 I 0.9 1 I 1.14 I 0.58 in-plane Out-of-plane e Driven Gesr Driving Gesr H+XIO-, IX IO- Fig.3 Shaft misalignment setup 3 COPYRIGHT American Gear Manu
43、facturers Association, Inc.Licensed by Information Handling ServicesGearH3 ffl Out-of-plane n 2000 3000 Speed RPM IO00 L“ o 0.5 1 .o 1.5 Tooth mesh frequency I1.0, where the vibration level is small at the low and middle speed, the vibration becomes strong in the area of es =1.3 and E, =1.5. Moreove
44、r, a strong vibration area extends to around the total contact ratio cy = 2.9 - 3.0. These phenomena indicate that the vibration of a pair having no error is determined only by the behavior of the stiffness of a pair in Eq.(2). Therefore the performance diagram at fz/ f, = 0.91 is very similar to th
45、e equi-amplitude contours of first order component of the Fourier series of stiffness. It is reasonable that the contour lines become dense and high at fz/fn =0.98 because it is near to the resonance speed. 7 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Ser
46、vices4.4.2 Performance diagrams on a pair having each kind of deviations : On the helical gear pairs having kinds of deviations, the produced performance diagrams on vibration are shown at thespeeds fz/f, =0.63 and fJf, =0.98 in Fig.11 and 12 respectively. The influence of each kind of deviations is
47、 clear by Comparison with the case of no error as shown in (a) in each figure. It must be must payed attention that vibration levels in Fig.11 are shown with the different scale of those in Fig.10. Each performance diagram is calculated under the condition that every tooth of the driven gear has the
48、 identical deviation, and that the non dimensional amount of deviation is 1.0, that is, the maximum amount of deviation is set to be the same as the mean deformation produced by the transmitting load. At the middle speed Cf,/f, =0.63) : In the diagram on pitch error (single pitch deviation in ISO),
49、Fig.11-(g), the contour lines become parallel to the direction, which is 45 degrees to each coordinate, at which the total contact ratio is constant. And the vibration does not become CO large in the area of E 1.0, almost regardless of the influence of the deviation. This feature is understood because the pitch deviation is 2.0 1.5 ru- 2 1.0 n O .- - m al - L 6 0.5 1 .O 1.5 2.0 2.3 Transverse contact ratio E, Vibration levels 0.080 Ib Pressure ande error 1 .O9 2.0 1.5 1 .O 0.5 u 1 .O 1.5 2.0 2.3 Transverse contact ra
