1、01FTM11Kinematic and Force Analysis of A SpurGear System with Separation of Slidingand Rolling Between Meshing Profilesby: D. E. Tananko, Wayne State UniversityTECHNICAL PAPERAmerican Gear ManufacturersAssociationKinematic and Force Analysis of A Spur Gear Systemwith Separation of Sliding and Rollin
2、g BetweenMeshing ProfilesD. E. Tananko, Wayne State UniversityThestatementsandopinionscontainedhereinarethoseoftheauthorandshouldnotbeconstruedasanofficialactionoropinion of the American Gear Manufacturers Association.AbstractThepaperdescribesacomprehensivestudyofanovelexternalspurgeardesignwithphys
3、icalseparationbetweenslidingandrollingmotionsinthecontactpointofmeshinggears.Theslidingmotionisaccommodatedbysheardeformationofathin-layered rubber-metallaminateallowing very high compression loads.Kinematicconditionsof such “composite”gearsystemwerestudiedanalytically.Themathematicalconceptandkinem
4、aticsofthenovelexternalspurgearwasfullydeveloped and optimized for better suitability of the concept for engineering application of the gear in the powertransmission. Closed form solutions were obtained for two different shapes of the composite tooth core, and wereoptimized for a gear pair used in t
5、he final stage of a helicopter rotor transmission. Static FE stress analyses was alsoperformed, using the finite element approach for complex meshing conditions involving interaction of metal andelastomeric (rubber) materials. The results obtained for the composite gear system compare beneficially t
6、o theconventionalinvolutegears.Thedisplacementofthetoothcorecanbereducedby25%,becauseoftheloaddistributionbyrubber-metallaminatewhichleadstothereducedtransmissionerrorandsequentiallydecreasesnoiseandvibrationofthepowertransmission.Thecontactforcesinthetoothcorecanbereducedby60%,whichrelaxestherequir
7、ementsforthecontactstrengthandcostlyannealingofthegear.Aworkingprototype wasbuilt andtested forthe analyzedmodel,and has shown a good correlation of the strain data as well as kinematics.CopyrightGe32001American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314Octo
8、ber, 2001ISBN: 1-55589-790-81Kinematic 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 majori
9、ty of gears have involute 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 simultan
10、eous rolling and sliding between 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
11、 is a continuous and very 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
12、 heat treatment, combined 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 develo
13、pment of special heat treatments 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
14、with superior specific strength (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 alu
15、minum and titanium alloys 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 c
16、an be reduced by accurate 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 depende
17、ncy; it requires costly teeth 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
18、by deformations of the teeth 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)
19、 stresses between the gear 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
20、improvements require increasing 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 technolo
21、gy due to a theoretically 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 2abate. Still, modifications of conformal gears wer
22、e successfully used in the Lynx helicopters made by Westland Helicopter 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 w
23、ere used in general-purpose reducers in the former USSR. Improved W/N gear 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 appli
24、cations of power transmission gears. In 1990, a new type of quasi-involute 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 reduce
25、d (due to a reduced relative curvature in the contact), it is not clear 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 gea
26、rs and gearboxes and, thus, for their costs. In some cases, noisy gears 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
27、plastic or metal-polymer gears are usually associated with a substantial 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 stiffne
28、ss; thus for heavy-duty gears a compliant torsional connection results 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 fr
29、ee from these shortcomings, but is not implemented yet. As noted above, 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,
30、 we can define two major objectives of this study as follows: 1. Find 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
31、composite gear and sliding along the composite tooth core. To perform 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
32、meshing process; 2. FEA of the composite and conventional involute spur 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 an
33、d load characteristics, calculated by FE method. These steps are tightly 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, de
34、scribed above, defines mathematical formulation of the composite gear 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 solu
35、tion. Also, the analytical solution for the external profile of the slider 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
36、 the power transmission and in FE Analyses as well as in the building 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 compos
37、ite gear design. Further work, described in Future Work section, must 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 comp
38、osite gear must behave as the pinion meshing with the involute gear. The 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
39、the composite gear design preserve all the benefits of an involute transmission 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.
40、The sliders motion along the composite tooth core, which accommodates 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 calculat
41、ions of the external profile of the slider can be divided into two cases: 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. 7Figure 2 - Logical Block Diagram of the Design and
42、Verification steps To analyze the kinematics of the composite gear, the problem was formulated as follows: how the geometrical 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 t
43、o realize (or, at least, approximate) the conjugate engagement with the counterpart involute profile, while the slider 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 involut
44、e pinion. FLImplement Engineering SolutionOptimize Verify FEA Model Mathematical RequirementsCheck KinematicsCalculate 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
45、in coordinate frame xOy; Apply the second statement above and solve for the profile L and function (t). The first step gives the parametric form of the external profile L” of the slider in coordinate frame x”O”y”: ()() cos() sin()( ) sin( ) cos( )xt R t t tbyt R t t tb =+ =(1) The external profile o
46、f 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 the meshing teeth contact, in the coordinate system of the composite gear, moves along an involute curve, i.e., the requirements of co
47、nstant speed and smoothness of transmission are automatically fulfilled due to the geometrical properties of the involute. On the other hand, due to the ability of the slider to move, any traveling of the contact point along the sliders external profile must be equal to the travel of the contact poi
48、nt along the profile of the involute tooth, which is meshing with the composite tooth. Thus friction in the contact point is avoided. Now we must describe the transformation of the “virtual” involute curve of the external profile of the slider in the composite gear coordinate system into the actual
49、shape of that profile in the coordinate system connected to the slider. As stated above, this can be done by assuming one of the possible shapes of the external profile of the tooth core. Thus further discussion can be divided into two cases: the case with an arc-shape external tooth core profile and the case with a straight profile. Arc-shape tooth core Coordinate frame xOy associated with the slider can be derived form coordinate frame x”O”y” rigidly connected with the composite gear by rotating x”O”y” around the center of the slider rotat
