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1、01FTM11 Kinematic and Force Analysis of A Spur Gear System with Separation of Sliding and Rolling Between Meshing Profiles by: D. E. Tananko, Wayne State University TECHNICAL PAPER American Gear Manufacturers Association Copyright American Gear Manufacturers Association Provided by IHS under license
2、 with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 04:03:46 MDTNo reproduction or networking permitted without license from IHS -,-,- Kinematic and Force Analysis of A Spur Gear System with Separation of Sliding and Rolling Between Meshing Profiles D. E. Tanan
3、ko, Wayne State University Thestatementsandopinionscontainedhereinarethoseoftheauthorandshouldnotbeconstruedasanofficialactionor opinion of the American Gear Manufacturers Association. Abstract Thepaperdescribesacomprehensivestudyofanovelexternalspurgeardesignwithphysicalseparationbetweensliding and
4、rollingmotionsinthecontactpointofmeshinggears.Theslidingmotionisaccommodatedbysheardeformationofa thin- -layered rubber- -metal laminate allowing very high compression loads. Kinematic conditions of such “composite” gearsystemwerestudiedanalytically.Themathematicalconceptandkinematicsofthenovelexter
5、nalspurgearwasfully developed and optimized for better suitability of the concept for engineering application of the gear in the power transmission. Closed form solutions were obtained for two different shapes of the composite tooth core, and were optimized for a gear pair used in the final stage of
6、 a helicopter rotor transmission. Static FE stress analyses was also performed, using the finite element approach for complex meshing conditions involving interaction of metal and elastomeric (rubber) materials. The results obtained for the composite gear system compare beneficially to the conventio
7、nalinvolutegears.Thedisplacementofthetoothcorecanbereducedby25%,becauseoftheloaddistribution byrubber- -metallaminatewhichleadstothereducedtransmissionerrorandsequentiallydecreasesnoiseandvibrationof thepowertransmission.Thecontactforcesinthetoothcorecanbereducedby60%,whichrelaxestherequirementsfor
8、the contact strength and costly annealing of the gear. A working prototype was built and tested for the analyzed model, and has shown a good correlation of the strain data as well as kinematics. Copyright ? 2001 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia
9、, 22314 October, 2001 ISBN: 1- -55589- -790- -8 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 04:03:46 MDTNo reproduction or networking permitted without license from IHS -,-,-
10、 1 Kinematic and Force Analysis of A Spur Gear System with Separation of Sliding and Rolling Between Meshing Profiles D. E. Tananko Wayne State University, Detroit, MI Background Power transmission gears play an important role in the modern industry. The overwhelming majority of gears have involute
11、teeth profiles. Although the involute gears have been used for many years and their designs have been significantly improved, they still have several serious shortcomings. The most important problems with conventional involute gears are: 1. Wear of the teeth due to simultaneous rolling and sliding b
12、etween the meshing tooth profiles; 2. Intense dynamic loads in the mesh, which result in objectionable vibration and noise; 3. Different material requirements for tooth surface and tooth core; 4. Clearance in the gear pair, which results in impacts and rattling. While there is a continuous and very
13、substantial effort to solve these problems and many successes have been achieved, it is becoming more difficult to achieve further improvements for conventional designs of involute gears. Even very costly improvements in the gear accuracy, and the use of advanced methods of heat treatment, combined
14、with better materials (steel alloys), bring diminishing returns. The state-of-the-art gears are always made of steel. While special alloying and high metal purity standards contribute to higher performance characteristics of gears, the greatest progress is due to the development of special heat trea
15、tments which, in combination with special bulk and/or surface alloying, provide differing properties of a tooth core (high bending strength) and its surface (high hardness and contact durability). To satisfy conflicting core and surface requirements, new advanced materials with superior specific str
16、ength (high bending strength/weight ratios) cannot be used for industrial gearing. For example, 1, 2, teeth of fiber-reinforced plastic gears show substantial advantages in bending strength but have very poor wear (scoring) resistance. The same is true for high-strength aluminum and titanium alloys
17、3, metal matrix composites, etc. The load-carrying capacity of power transmission gears deteriorates at high rpm due to intense dynamic loads caused by deviations from the ideal geometry. These deviations include pitch errors as well as profile and helix deviations, which can be reduced by accurate
18、machining. They also include teeth deformations under load as well as shaft misalignments caused by deformations in the housings; especially housings made of light metals, such as helicopter gearboxes. Compensating these deformations is difficult due to their torque dependency; it requires costly te
19、eth profile modifications, as well as derating of the gears. The same deviations also result in high noise levels, which frequently become a critical factor in both civilian and military applications 4, 5. There is an opinion that the required machining accuracy is limited by deformations of the tee
20、th under load. Reducing machining errors beyond these deformations is very expensive, and not very effective. The bulk of geometry-related research is in the domain of involute gears. Profile modifications during machining allow a beneficial redistribution of bulk (bending) stresses between the gear
21、 and the pinion. One- and two-dimensional crowning/flanking allow reducing gear sensitivity to misalignments and to changing deformations caused by variable loading, etc. (e.g., 6, 7, 8). However, these approaches are also nearing their saturation levels, where incremental improvements require incre
22、asing investments in new, sophisticated equipment and tooling. The “reinvention” of conformal gears by Novikov in the late fifties (a slightly different embodiment having been invented by Wildhaber in 1920 9, 10) raised hopes for a dramatic breakthrough in the gear technology due to a theoretically
23、higher strength of the conformal Wildhaber/Novikov (W/N) gears. However, these hopes faded after it was discovered that these gears high noise levels and high sensitivity to center distance deviations are very difficult to Copyright American Gear Manufacturers Association Provided by IHS under licen
24、se with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 04:03:46 MDTNo reproduction or networking permitted without license from IHS -,-,- 2 abate. Still, modifications of conformal gears were successfully used in the Lynx helicopters made by Westland Helicopter
25、Co. 11, 12. New designs of W/N gears were proposed using a different geometrical envelope. Conformal “Symmarc” gears are used in many high- power/medium-speed applications in Japan (Hasegawa Gear Works, Ltd.) 13. W/N gears also were used in general-purpose reducers in the former USSR. Improved W/N g
26、ear designs partially resolve the problems of the involute gears, but they lack such benefits of involute gears as tolerance to center distance variations and have higher noise levels. These properties are essential in many applications of power transmission gears. In 1990, a new type of quasi-invol
27、ute gear (Logi X) has been developed 14 in which the tooth profiles are composed of small involute segments with different parameters. Although sliding in these gears is significantly reduced and Hertzian stresses are also reduced (due to a reduced relative curvature in the contact), it is not clear
28、 how sensitive these gears are to center distance variations. A generic problem for all types of power transmission gears is noise, which becomes the most determining factor for assigning the machining/assembly tolerances for gears and gearboxes and, thus, for their costs. In some cases, noisy gears
29、 require additional very costly treatments, even when the gears are produced with a high degree of accuracy (e.g., in submarines and “low noise” helicopters). Numerous effective techniques for noise abatement, such as the use of plastic or metal- polymer gears are usually associated with a substanti
30、al derating. Noise and dynamic load reduction can be achieved in gears whose rims are insulated from the hubs 15. However, in designs described in 15, the torsional stiffness of the connection is correlated with its radial stiffness; thus for heavy-duty gears a compliant torsional connection results
31、 in unacceptably low radial stiffness, and thus requires unreliable metal-to-metal frictional connections between the hub and the rim. Friction in these connections may negate the effects of torsional compliance. This design is free from these shortcomings, but is not implemented yet. As noted above
32、, after about fifty years of continuous improvements in the gear state of the art, a “saturation” period is now approaching when larger R hence, an increased thickness and strength of each tooth. To highlight the above discussion, we can define two major objectives of this study as follows: 1. Find
33、a solution for the kinematics of the proposed physical separation of rolling and sliding in the general form (case). 2. Investigate the usage of the rubber-metal laminated material for simultaneous attachment of the slider to the composite gear and sliding along the composite tooth core. To perform
34、a comprehensive study of the new “composite” gear design, several design and verification steps have been undertaken: 1. Kinematic analysis, which finds the shape of the composite gear and study the kinematics of the gears in the meshing process; 2. FEA of the composite and conventional involute spu
35、r gears to obtain and compare the stresses in the contact point and in the fillet area; 3. Building a working prototype to validate both the kinematical characteristics of the composite gear calculated in the kinematic analysis and load characteristics, calculated by FE method. These steps are tight
36、ly connected and interact with each other as it is shown in Figure 2. It is obvious, that a comprehensive study of the proposed design requires several iterations in the logical loop described in Figure 2. That general concept, described above, defines mathematical formulation of the composite gear
37、concept. Mathematical statement of the problem has been developed in Kinematical Analysis. The main goal of this chapter is to develop the general form solution for the external profile of the slider, as well as optimize this solution. Also, the analytical solution for the external profile of the sl
38、ider was compared to the conventional involute profile in terms of constant transmission ratio. The generated external profile of the slider and other parameters of the composite gear system were then used in 3-D solid modeling of the power transmission and in FE Analyses as well as in the building
39、Working Prototypes. Several FE models and working prototypes were built while improving, optimizing and enhancing the general concept of the composite gear. However, the present work shows only conceptual base for the novel composite gear design. Further work, described in Future Work section, must
40、be accomplish in order to fully reveal the potential of the proposed design. Kinematic Analysis. Analytical calculation of the kinematics of the composite gear was based on two hypotheses: The involute pinion meshing with the composite gear must behave as the pinion meshing with the involute gear. T
41、he slider motion along the tooth core of the composite gear should accommodate the physical friction with pinion, so pure rolling is achieved in the contact between the slider and the pinion tooth. These two conditions applied to the composite gear design preserve all the benefits of an involute tra
42、nsmission and, at the same time, allow one to separate the simultaneous rolling and sliding (friction), occurred in the point of contact for involute gears between internal and external sides of the slider for the composite gear. The sliders motion along the composite tooth core, which accommodates
43、pure sliding, is supposed to occur the along self-adjacent line. In other words, if the sliding of the slider is the motion of a rigid body, then the internal profile of the slider must be either arc or straight line. All calculations of the external profile of the slider can be divided into two cas
44、es: arc-shaped and straight-line shape internal profile of the slider. Solutions for these two cases were parametrically dependent on the positions of the arc and straight line. Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/111111100
45、1, User=Wing, Bernie Not for Resale, 04/18/2007 04:03:46 MDTNo reproduction or networking permitted without license from IHS -,-,- 7 Figure 2 - Logical Block Diagram of the Design and Verification steps To analyze the kinematics of the composite gear, the problem was formulated as follows: how the g
46、eometrical properties of an involute tooth profile can be simulated while rolling and sliding motions are separated? The external profile of the slider has to be constructed in order to realize (or, at least, approximate) the conjugate engagement with the counterpart involute profile, while the slid
47、er performs a pure rolling motion in the contact between its external profile and the profile of a counterpart involute tooth (Figure 3). Figure 3 - Composite gear in mesh with involute pinion. F L Implement Engineering Solution Optimize Verify FEA Model Mathematical Requirements Check Kinematics Ca
48、lculate Stress Make the transformation of the coordinate frame from x”O”y” to xOy and apply this transformation on the involute profile L”, so that profile L will be in the parametric form in coordinate frame xOy; Apply the second statement above and solve for the profile L and function (t). The fir
49、st step gives the parametric form of the external profile L” of the slider in coordinate frame x”O”y”: () () ( )cos( )sin( ) ( )sin( )cos( ) x tRttt b y tRttt b =+ = (1) The external profile of the slider is an involute in the coordinate system of the composite gear (xOy), which meets all the requirements to a power transmission gear. Indeed, the point of