1、01FTM10Design Technologies of High SpeedGear Transmissionby: J. Wang, Nuttall GearTECHNICAL PAPERAmerican Gear ManufacturersAssociationDesign Technologies of High Speed Gear TransmissionJeff Wang, Nuttall GearThestatementsandopinionscontainedhereinarethoseoftheauthorandshouldnotbeconstruedasanoffici
2、alactionoropinion of the American Gear Manufacturers Association.AbstractCompetition drives productivity. Productivity drives gear speed. Gear transmissions for steel wire mill drives, turbinedrives and for many other applications are operating at pitch line velocities of 25,000 feet per minute or h
3、igher. Manyfactors are not issues for low speed gears are serious issues for high speed gears. The presented article discusses a fewcritical factors and their effectson high speed gear transmissions. Thefirst factor is centrifugalforce, and its effectsontoothrootstrength,toothexpansionandbacklash,an
4、dtheinterferencefitbetweengearandshaft.Thesecondissystemdynamics, including critical speed, dynamic balancing, and the torsional effects of flexible couplings. The third is thewindagelosswithdifferentcombinationsofhelixandrotationdirection,lubricantflowrate,flowdistributionandtheireffects on tooth b
5、ulk temperature field and tooth thermal expansion.CopyrightGe32001American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 2001ISBN: 1-55589-789-4Design Technologies of High Speed Gear Transmission Jeff Wang, Ph.D. Nuttall Gear Corporation 2221 Niagara Fa
6、lls Blvd. Niagara Falls, New York 14304 Competition drives productivity and gear speed. Gear transmissions for many applications are operating at pitch line velocities in excess of 25,000 feet per minute. Many factors that are not issues for low speed gears could be serious issues for high-speed gea
7、rs. The presented article discusses centrifugal force, system dynamics, windage loss and their effects on high speed gear transmissions. Introduction: Numerous studies have been done on high speed gearing. Each study focused on different aspects of high speed gear transmission. Rotor dynamics, vibra
8、tion and noise have been major subjects 123456. Temperature distribution and its thermal effects are another focus area 78910. However, the application of the research results has not been very productive. There are two major reasons for that. One reason is that most research was conducted in univer
9、sities and laboratories, where either models were created a bit different from that in reality or too many factors were included. The other reason is that most research produces massive and complicated results that are not truly feasible to be directly applied to the real design. This paper discusse
10、s a few critical factors in designing of high speed gears and provides some simple yet effective solutions to corresponding problems. The following table gives basic parameters and working conditions of the studied gear system as our design example throughout this paper. The pinion and gear are carb
11、urized and ground to AGMA class 13. Fine pitch is selected for a higher contact ratio. Oil film bearings are used for this system. Parameters Pinion Gear Number of teeth 32 120 Speed (RPM) 45000 12000 Center Distance (inch) 6 NDP 14 Net Face width (inch) 3 Pressure Angle (degree) 20 Helical Angle (d
12、egree) 35.208 Power Input (HP) 1800 Service Factor 2.0 Figure 1 Prototype of a Double Helical High Speed Gear Pair Centrifugal force Centrifugal force has not received significant attention in designing of high speed gearing. Though a few companies may have considered radial expansion in specificati
13、on of backlash and center distance, most gear design engineers do not even consider this factor. In this article, distributions of stresses resulting from the centrifugal force are investigated. Simplify the gear as a disk, with h R/2, the following equations apply: 0h2r2th)r(rh drd=+0rE)tr)(1)Er(dr
14、d)Et(drd=+Where, r radius at any location E elasticity of the material h thickness of the disk - Poisons ratio t Tangential stress r radial stress - Angular speed R Outside diameter of the disk With given geometry, material and appropriate boundary conditions, the following results at corresponding
15、loading conditions were obtained. Figure 2 Stresses Distribution Due to Centrifugal Force The results show radial stress on a disk due to centrifugal force is a radiation field. The radial stress is zero at the cylindrical surface of the disk. The maximum radial stress is at the very center point of
16、 the disk. Tangential stress, and Von Mises stress for this example, around the location of tooth root is about 1200 PSI, which is about 2-5% of allowable bending stress, depending on heat treatment of the gear material. It may not seem significant in this case, but consideration in the design analy
17、sis would be beneficial. Modeling of the pinion is slightly different from that of the gear. It is more reasonable to be simplified as segments of cylinders rather than a disk. With this assumption, solutions can be found: )2r2(R82-12-3r =)2r23212(R82-12-3t+=)22r2(R42-1a =Where, r radial stress t ta
18、ngential stress a axial stress R Outside radius - Density of the material - Angular speed - Poisons ratio Different from a gear, axial stress 11 caused by centrifugal force for the pinion shaft may be considered due to the significant ratio of length over diameter. The following chart shows the resu
19、lts. Figure 3 Stresses Distribution on Pinion Shaft Due Centrifugal Force Similar to the gear, stresses on the pinion may not be a direct threat to the system, but consideration of these stresses caused by centrifugal force may be more appropriate for certain applications. The effects of radial expa
20、nsion on center distance and backlash resulted from centrifugal force should also be taken into consideration. Theory of elasticity gives an easy solution for the radial expansion. Calculation shows the radial expansion is about 0.19 thousandth of an inch for this specific gear. Radial expansion for
21、 the pinion is slightly different from that of gear, because of the axial stress component. The results show the radial expansion is about 0.05 thousandth of an inch. For the pair of gears, the total radial expansion due to the centrifugal force is about 0.24 thousandth of an inch. When designing pr
22、ecision high speed gear transmissions, the expansion results from centrifugal force, in combination with thermal expansion, which is presented in the paper later, should be considered in specifying of the center distance and backlash. Rotor Dynamics Vibration of high speed gear transmission has alwa
23、ys been a big concern. Quite often the pinion shaft has to be designed into a flexible regime, which means the specified operating speed is higher than first or even second order critical speed. Knowing the critical speeds of the pinion shaft and gear is very important after preliminary design. Bear
24、ing stiffness is always a major factor for critical speeds. Hence, bearing design with optimizing bearing support stiffness is the most effective way to manipulate lower order of system critical speeds. This avoids possibilities of resonance, such as operating speed with any critical speeds, gear me
25、shing frequency to any critical speeds, and other driving or driven mechanisms, like Stress Distribution Due to Centrifugal Force on gear0500100015002000250030000% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%Percentage RadiusStress PSIRadial StressTangential StressVon Mises StressStress Distribution due
26、 to Centrifugal Force on Pinion Shaft-1000-5000500100015002000250030000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percentage RadiusStress, PSIRadial Stresstangential StressVon Mises StressAxial Stressmotor torque ripple frequency or an impeller operating frequency to critical speeds. Of course, in high s
27、peed gear system design, the possibility of mutual excitation between driving or driven mechanism with gear mesh frequency can be avoided by manipulating number of teeth, diametral pitch and even center distance if possible. With the same design example as above stated, the following charts show rot
28、or dynamic analytical results for un-damped critical speeds corresponding with the pinion and the gear. In these charts, operating speed with stabilized bearing stiffness is marked to see where the system is operated on the critical speed map. Tooth meshing frequency and driven propeller operating f
29、requency are also marked to avoid other imaginable excitation sources. Figure 4 Pinion Shaft Critical Speed Map For detail information, please refer to Figure 4 and Figure 5. Figure 4 is the critical speed map of the pinion shaft. The pinion shaft normal operating speed is 45000 RPM, which is 750 Hz
30、. If supporting bearing stiffness can be design between 300K lb/in and 1.05 Million lb/in, then the operating speed is in-between the first and the second order critical speed. Though it is a flexible rotor system, the pinion shaft will not be operating close to any critical speeds. This will assure
31、 no resonance due to slight imbalance. If an impeller with 12 blades is attached to the pinion shaft, the impeller operating frequency, is 9000 Hz. The propeller operating frequency, falling between the fourth and fifth order of critical speeds, which is also shown on the critical speed map. It indi
32、cates no resonance expected. Gear mesh frequency, with rotating speed of 45000 RPM and pinion number of teeth 32, is 24000 Hz. Falling between sixth and seventh order of critical speeds, gear mesh frequency is not likely to be excited. Neither a mutual excitation between driven mechanism operating f
33、requency and gear mesh frequency should be seen. Based on the above analysis, we conclude that the pinion shaft is well designed for its rotor dynamic properties. Refer to Figure 4 for detail information. Figure 5, is a similar critical speed map for the meshing gear and its shaft. Figure 5 Gear and
34、 the Shaft Critical Speed Map Temperature distribution and thermal expansion of gear teeth Quite a few publications are available about windage loss, temperature distribution and its thermal effects on high speed gearing. But how much and at what degree of these research results have been applied in
35、 the high speed transmission design is not certain. With todays computation technology, analyzing the steady state temperature distribution and the thermal expansion of the pinion and gear teeth is not a tough task any more. Uneven teeth expansion can be controlled in different ways, one of the simp
36、le ways is to control the thermal expansion through the combination of rotating direction and helix directions. Special spray nozzles can be designed to get ideal lubricant distribution on face width for an optimized thermal expansion and a preferred load distribution along the face width. With the
37、same set of gears and at the same operating conditions, A Finite Element Model is used to calculate steady state temperature distribution and the corresponding thermal expansion. With application of convection heat transfer theory on the boundaries Un-damped critical speed map - pinion shaft1.E+011.
38、E+021.E+031.E+041.E+051.E+04 1.E+05 1.E+06 1.E+07Bearing stiffness (Lb/in)Criticalspeed(Hz)Mode -1Mode -2Mode -3Mode -4mode -5Mode -6Mode -7Operating speedDriven mechanism operating frequencyGear mesh frequencyUn-damped critical speed map-gear and the shaft1.E+001.E+011.E+021.E+031.E+041.E+051.E+04
39、1.E+05 1.E+06 1.E+07Bearing StiffnessCriticalspeedHzMode-1Mode-2Mode-3Mode-4Mode-5Mode-6Mode-7Mode-8Operati ng Speeddriven mechanismoperating frequencyGear mesh frequencyand conduction heat transfer in the tooth bulk volume, temperature distribution at nodes of pitch circle in the middle of the toot
40、h can be found at different operating speeds. The most important factor is the convection heat transfer coefficient on the meshing tooth flank. In this model, the convection heat transfer coefficient developed by J. Wang 10 is used. Figure 6 shows calculated temperature distributions at different pi
41、tch line velocities, with evenly distributed lubricant flow. One combination of the rotating direction and helix direction will result in mesh in engagement, by which we mean the contact line on gear mesh would move from both edges towards the middle of the face width. In this article, this kind of
42、mesh is called mesh in for convenience. Studies show this combination creates very unfavorable temperature distribution and thermal expansion on gear teeth. The high speed motion of the contact line in the axial direction creates strong pump effects. This motion forces the lubricant and air mixture
43、fluid to flow at extremely high speeds along the channel between adjacent gear teeth flanks. This stream of the fluid flows from edge to the middle of the face width, until it is pushed out into the middle gap of the double helical gear. At such a high speed, (pitch line velocity divided by tangenti
44、al of helix angle), the fluid actually heats the tooth surface rather than cooling it, due to the friction between lubricant particles and metal tooth surface. Both fluid temperature and tooth flank temperature increase rapidly along the path of the fluid. When the fluid approaches the other end of
45、the mesh, temperature decreases slightly, where forced convection heat transfer is stronger due to increased surface area. In addition, in the bulk material of the gear body, both ends of the face width have more surface area for the convection heat transfer between the lubricant and the gear body.
46、The temperature at both ends of face width are apparently lower then that in the middle of the face width. Opposite to mesh in, if the gear mesh started from the other end, at the gap of the double helical gear and meshes out towards two edges of the face width, lets call it mesh out for convenience
47、, the temperature distribution is significantly different from the above situation. The theory is the same, but applied to a different meshing direction. Figure 7 shows the analytical results of temperature distribution at different pitch line velocities. Comparing Figure 6 and Figure 7, the later h
48、as a much more favorable temperature distribution. Not only is the temperature difference along the face width much less, but the distribution profile is also much more favorable. In the next section, analytical results of thermal expansion on the tooth flanks, which affects load distribution will c
49、onfirm this point. Temperature and thermal expansion distributions will also be further discussed later on with lubricant flow rate controlled along the face width. Figure 6 Temperature Distribution along the Face Width withMesh In Working Condition Figure 7 Temperature Distribution along the Face Width with Mesh Out Working Condition With temperature distribution results available, thermal stress and strain follow the principle of elasticity. Thermal expansion on the
