1、 - STD.AGMA 97FTMLL-ENGL L997 I b87575 5Lb7 922 97FTM11 Non-Dimensional Characterization of Gear Geometry, Mesh Loss and Windage by: J. Philip Barnes, Northrop Grumman Corporation I I TECHNICAL PAPER COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services STD
2、-AGHA 77FTflLL-ENGL 1777 b87575 O51b b7 Non-Dimensional Characterization of Gear Geometry, Mesh Loss and Windage J. Philip Barnes, Northrop Grumman Corporation nie statements and opinions contained herein arc those of the author and should not be construed as an official action or opinion of the Ame
3、rican Gear ManufacNrers Association. Abstract New relationships for involute spur gear geometry are introduced and integrated with new methods of correlating lubricant traction and windage test data. Compact math models for lubricant density and viscoSity under contact prtssun are proposed. A modem
4、approach to dimensional anaiysis is introduced to characterize lubricant traction data and gear windage data with dimensionless terms which apply to gear systems which may have aconfiguration and/or size different from those tested. Finally, system considerations for optimal gearbox efficiency arc p
5、roposed. Copyright O 1997 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 November. 1997 ISBN: 1-55589-705-3 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSTD-ALMA 77FTMLL-ENGL 1777 b87575 U5Lb7 7T5 W N
6、on-Dimensional Characterization of Gear Geometry, Mesh Loss and Windage J. Philip Barnes Sr. Technid Specialid Northrop Grum Corporation Abstract New relationships for involute spur gear geometry are introduced and integrated with new methods of correlating lubricant traction and windage test data.
7、Compact math models for lubricant density and Viscosity under contact pressure are proposed. A modem approach to dimensional analysis is introduced to characterize lubricant traction data angle (+p). The companion Figure (1.2) describes involute geometry, showing the contact radius (r) which can be
8、“normalized“ as a ratio with the base radius (ib). Similarly, rolling and sliding speeds can be normalized with (V,), gear traction force (J can benormalized with 0, and the mesh power loss can be normalized with the mesh power (FV,). and gear windage data with dimensionless terms which apply to gea
9、r systems which may have a configuration and/or size different from those tested. Finally, system considerations for optimal gearbox efficiency are proposed. Introduction An impomnt link in the chain that ties the prime mover to the products of man is the GEAR. ln the past. gears were only wheels wi
10、th teeth chiseled in stone, carved ln wood . as the gears ground on through rher daily tasks, rhe teeth took on a curvature ar the points of we3r . running a little smoother than they did when they were brand new. T.W Khiraiia In his book (l), Khiralla traces the history of the ubiquitous spur gear.
11、 We learn that Phillipe de La Hire proposed an involute shape for spur gears near the end of me 17th century. However, the mathematics of the invdute spur gear were first developed by Euler in the middle of the 18th century. In todays technology, the rde of the spur gear takes on increasing importan
12、ce. Consequently. we should continually strive to improve our characterizations of its geometry, motion, and power loss For a high-speed gearbox, such characterizations must include me potentially dominant effedts of windage in ie presence of airborne oil. These topics are addressed heran 1 .O Spur
13、Gear Geometry and Motion In this section, we review the well-known equauons for involute geometry and introduce addittonal geometry and velocity characterizations which. although fundamental, appear to have been previously unpublished. These will be integrated, in later sections, with new methods fo
14、r the calculation of mesh loss and windage. Figure 1.1 Base Velocity and Radii Reerences for Non-henaomliziOn Figure 1.2 Involute Geometry Euler Angle (E) Vanes iineafiy with bme (t) Polar Angle 0 1.1 Involute Geometry Figure 1.1 shows the familiar representabon of pinion and gear base circles conne
15、cted by an imaginary string The gear and pinion, at rotational speeds (o, and o2 ), transmit the normal force (F) at constant velocity (Vb=o,r,=m2r2) along a line of action which is inclined at the pressure 9/12/97 1 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Hand
16、ling ServicesSTD-AGMA 77FTflLL-ENGL 1777 b87575 0005L70 4L7 W -1 o Equations (1.1) and (1.2), corresponding to Figure 1.2, are given by Dudley (2). We propose the name “Euler Angle“ for that angle which varies linearly with time throughout the mesh. We also designate a dimensionless contact radius a
17、s (p) and relate it to the contact angle (4) with EQs (1.3) and (1.3a). Note that the contact and pressure angles are equal only at the pitch point. -5 I r 0.01 o1 1 10 100 Euler Angle - Elliptical contact P=(3/2) (F/A) ;i To reduce the number of variables required for lubricant traction and gear me
18、sh loss characterizations. we can use the well-known equivalent elastic modulus of EQ (3.3) and the equivalent curvature radius for rectangular contact, given by EQ (3.4). These are then used to calculate rectangular contad pressure with EQ (3.5). Again, we are using (F) for normal force and (w) for
19、 width. Equivalent Modulus = 3572 GPa Steel gear and pinion 2 E, = x E, 2 / 1 ( l-vp-) + E2 ( 1-v:) 3.3 Equiv. Curvature Radius, Rect. Contact Ra = Rj b / (Ri+: . . _. . - - . . _ _ . - . - - -. - - - . _& A - - -.-*-_ 10 100 1wO Combted Wndage. Wab 9112197 9 COPYRIGHT American Gear Manufacturers As
20、sociation, Inc.Licensed by Information Handling ServicesSTDSAGMA 97FTMLL-ENGL 1997 = Ob87575 0005378 708 4.1 Sample Windage Calculation Here, we calculate the windage of a 5O-twth gear with a pitch diameter (D=O.lO m), facewidth (w=O.Cn m), and rotational speed 25000 RPM (o = 2618 radls) at sea leve
21、l. From EQ (7.9), the non-dimensional tooth height is (&=h/o=O.O45). Also, the non-dimensional facewidth is (G,=w/D=0.2 ). The air density is ( p=1.2256 kam3) and air viscosity is p=1.783 (107 kglm-s 1. The Reynolds number is then %=1.8 (lo6 )J. From EQs (4.4-4.9, the nondimensional windage power is
22、 (G,=0.0037). Finally, multiplying by (Po%? we obtain (zo=814 W=0.814 rcw) for the windage power loss in dry air at sea level. .Before leaving our windage correlation, We point out that an additional dimensionless group, representing the pitch Mach numbty, would be required to characterize gear wind
23、age if the pitch speed approaches or exceeds the speed of sound. In this case, the windage can be expected to increase significantiy. 5.0 Design for High-speed Gearbox Efficiency Let us assume now that we can accurately calculate the power losses due to gear traction, dry-air windage, and other well
24、-known factors such as bearings, lube pumps, and seals. We now need to verify these methods by testing a geared system. At this point, however, there is a good chance that the measured power loss can exceed our prediction by 50 % if we have a high-speed gearbox. V. Cunningham (8) points out the pote
25、ntially large power losses due to gear churning in a wet sump. In his paper, he strongly recommends a dry sump for high-speed gearboxes and provides guidelines for dry-sump system design. However, given a dry sump, we should recognize that airborne oil can easily dominate the total power loss. In a
26、high-speed gearbox, a significant portion of the oil which may be thought to reside in a quiescent tank is likely to be airborne, playing “volleyball” with the gears. Repeated impact with high-speed gears leads to oil atomization, and ultimately, to gasification. That portion of the airborne oil whi
27、ch remains a liquid strikes the walls of the gearcase and may inhibit drainage of “solid” oil, thereby promoting localized churning. More importantly, that portion which has become gasified increases the effective density and viscosity of the gearcase atmosphere to the point where the windage may be
28、 perhaps five times its dry-air level. To reduce the effects of airborne oil, all oil flows should be minimized. This includes bearing lubrication, since that oil usually finds its way into the gearcase. Reduced bearing lubrication frequently leads to reduced bearing temperatures together With reduc
29、ed bearing losses. Must of the bearing power loss is due to fluid mechanisms which are similar to those causing gear mesh losses. c If reduced lubrication flow is not smcient to met gearbox efficiency requirements, shrouds can be instailed around some, or all, of the gears. The basic intent of shrou
30、ds is to provide a *rain shelter“ for the gears, thereby reducing repeated gear/dl impact and the attendant oil gasification. Also, shrouds can promote rapid drainage of sdid oil to the storage tank In some cases, shrouds have reduced total gearbox power loss by 30 %. 6.0 Summary and Recommendations
31、 We have introduced and integrated the non-dimensional characterization of gear geometry and motion with dimen- sional analyses of lubricant traction and windage. Also, we have described phenomena which can make windage the dominant power loss mode in a high-speed gearbox It is hoped that other rese
32、archers will apply and enhance the methods herein to better characterize lubricant traction, film thickness, and bearing power loss. In particular, those interested may wish to apply the methods of Sections 2 and 3 to characterize the density, viscosity, and traction of the widely-used MIL-L-23699 l
33、ubricant. . For traction testing, higher sliding speeds should be added to the database used herein. For gear windage, the effects of various shroud configurations, oil mist, and subambient gearcase air pressure should be measured, particularly at high speed. For future tests of any type, researcher
34、s are urged to tabulate all significant measurands, no matter how confident they may be in their own characterizations of the test data. Also, as illustrated herein, researchers should accompany any proposed empirical formula with a “scatter plot” which shows the applicable data range and which comp
35、ares the formulas correlations with the test data. Nomenclature Subscripts 1 pinion 2 gear a addendum b base d dedendum e equivalent f form L limit p pitch Symbols A . Hertzian contact area a addendum b dedendum C center distance D pitch diameter E elastic modulus e exponent or coefficient 9/12/97 1
36、0 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesF G h N P R RPM T T t V W Unit M i t T normai force dimensioniess group tooth height number of teeth Hertzian contact pressure radius of curvature rotational speed, rev/min contact radius temperature tr
37、action force tim velocity face width P k L v P W D c, h 1 1 O O 1 1 O o 1 -1 1 1 1 -3 U 1 2 O -1 -3 O -1 O -1 O -2 -3 O -1 O O o o O -1 -1 Greek Symbols Euler angle contact angle pressure angle viscocity Poissons ratio polar angle dimensionless radius density torque dimensionless time rotational spe
38、ed, raus References 1. Khiraila, T.W., On the Geometry of &teml Involute Spur Gears, published by the author, available from UC Santa Barbara Library 2. Dudley, D.W., Handbook of Practical Gear Design, McGraw Hill, 1984 3. Lord Rayieigh, Nature, Vol. 95, p. 66,1915 4. Dawson, P.H., Effect of Metalli
39、c Control on the Pitting of Lubricated Rolling Surfaces, J. Mech. Engr. Sciences, Vol. 4, N0.1,1962 * . 5. Smith, R.L, Research on Elastohydmdymmic Lubrication of High Sped RollingSliding Contacts, AFAPL-TR-72-56, Wright-Patterson AFB, 1972 6. Walowit, J.A., Traction characteristics of a Mil-1-7808
40、Oil, Journal of Lubrication Technology, Oct 1976 7. Dawson, P.H., Wndage loss in Larger Highspeed Gears, Journal article, source unknown, 1983 8. Cunningham, V., Airplane Mounted Accessory Gearbox Design, SE 841605, Oct 1984 Appendix 1 DIMENSIONAL ANALYSIS METHOD Example: Heat Transfer of Hot Gas In
41、jected into a Tank O bjedive: Hot gas is injected with flow (w) and velocity (V) into a tank with hemi-spherical ends of diameter (a) and overall tank length (L). Correlate the heat transfer coefficient (h) with gas injection conditions, tank geometry, and fluid properties - thermal conductivity (4,
42、 constant pressure specific heat (cp) and viscosity (p). STEP 1 Postulate: h = e, pel ke2 Le3 Ve4 pal wa2 Da3 c a4 P I-Primary Variables+ -Secondary- I Here, the coefficient & and primary exponents (el . a) are to be empirically-derived, and the secondary exponents (al . a) are to be calculated by t
43、he solution of (n4) simultaneous equations for (n) applicable units - mass (M), length (I), time (t), and temperature (T). The combination of exponents ( el . a) must obtain units of (h) on the right-hand side of E (1). Our objective is to obtain five dimensionless groups, G, GI, . G4, each consisti
44、ng of one primary vanable (h, u k, L, V), combined with one or more secondary variables (p, w, D, ,). Any variable can be located in either group (primary or secondary, but not both) provided all applicable units (4 here) are represented at least once in the secondary group and provided the secondar
45、y units matM (shaded below) is non-singular. 9/12/97 li COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSTD-AGMA 77FTMLL-ENGL 1777 = b87575 0005180 3bb a, 1 O 1 -2 1-el-e2 O+e4 a2 O O -1 2 e1-ep-e3-e4 l-el-eq-e4 a3 3 1 3 -4 3+el+3e2+e4 -2+el+e2-e3+2e4
46、a4 O O O -1 -1+e2 , ,l-e2 * = - Appendc 1, continued * STEP 3 ( llel+( lIe2 +( O)e3+( Ole4 +( l)al +( l)at+( Ola3 +( Ola4 = 1 (-1lel +( lle2 +( lle3 +( lIe4 +(-31al +( Ola2 +( lla3 +( 2)a4 = O (-1Ie1 +(-3)e2 +( Ole3 +(-l)e4 +( Olal +(-l)aZ +( Ola3 +(-2)a4 = -3 ( Ole1 +(-l)e2 +( Ole3 +( Ole, +( Ola1
47、+( Ola2 +( Ola3 +(-l)a4 = -1 Write Equations for Consistent Units Rewrite: STEP 4 Solve the Equations For short hand, we will rewrite equation series (2) in matrix format, whereby A a = (b. The columnar matrix a represents the secondary exponents, and the columnar matrix b represents the right-hand
48、side of EQ. (2). We invert the matrix A, and we set a = (A- b as follows: el ke2 Le3 Ve, pe4 ,1-e1-e2-e4 D-2+el+e2-e3+2e4 1-e2 P Therefore, h = e, p whereby, (3) The left-hand side of EQ (3) represents a form of the Stanton number. The first group on the right-hand side represents a form of the Reyn
49、olds number, inverted. The second group is the inverted product of the Reynolds and Prandtl numbers. Therefore, we multiply this second group by (cL/cl) in EQ (4). Since the exponents (and their signs) are to be experimentally determined, we are free to re-define el ( = &+e) and to invert any group. The final resuit is EQ (5). wc 9/12/97 12 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services
