1、04FTM12Improved Tooth Load Distribution in anInvolute Spline Joint Using Lead ModificationsBased on Finite Element Analysisby: F.W. Brown, J.D. Hayes and G.K. Roddis,The Boeing CompanyTECHNICAL PAPERAmerican Gear ManufacturersAssociationImproved Tooth Load Distribution in an InvoluteSpline Joint Usi
2、ng Lead Modifications Based on FiniteElement AnalysisFrederick W. Brown, Jeffrey D. Hayes and G. Keith Roddis, The BoeingCompanyThe statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Associat
3、ion.AbstractInvolute splines in torque transmitting joints are prone to non-uniform contact loading along their lengthespecially in lightweight, relatively flexible applications such as a helicopter main rotor shaft-to-rotor hubjoint. Thestructuralstiffnessandinternalloadpathsofthetwomembersinthejoi
4、ntaffectsplinetoothcontactpressure distribution. In such applications, in the absence of lead corrections, the torque is transferrednon-uniformlyalongthelengthofthesplineresultinginaconcentrationorpeakingofthetoothcontactloadatone end of the spline.Asignificantlyimprovedtoothloaddistributionwasachie
5、vedforsplinesfortheLowMaintenanceRotor(LMR)version of the CH-47 Chinook helicopter main rotor shaft-to-rotor hub joint by applying, to the internallysplined member, complex lead corrections which varied continuously along the length of the spline. Therequired lead corrections were determined analyti
6、cally using finite element methods (FEM). Rotor hubsplines with the analytically determined lead corrections were manufactured and tested under design loadconditions. A standard CH-47 rotor shaft-to-hub joint, which uses a step lead correction between splines,waspreviouslytested. Straingageswereused
7、toinfercontactloaddistributionalongthelengthofthesplines.Test data indicated that the complex lead corrections resulted in a nearly uniform contact load distributionalongthelengthofthesplineatthedesigntorqueload. Thedataalsoshowedthattheloaddistributionforthesplineswiththecomplex leadcorrections was
8、significantly improvedrelativetothecontactloaddistributionofthe baseline splines. This work was performed under the U.S. Army Aviation and Missile Command(AMCOM) Low Maintenance Rotor (LMR) hub development contract DAAH01-99-3-R001.Copyright 2004American Gear Manufacturers Association500 Montgomery
9、Street, Suite 350Alexandria, Virginia, 22314October, 2004ISBN: 1-55589-835-11 Improved Tooth Load Distribution in an Involute Spline Joint Using Lead Modifications Based on Finite Element Analysis Frederick W. Brown, Jeffrey D. Hayes and G. Keith Roddis The Boeing Company, Rotorcraft Division, Phila
10、delphia, PA Introduction Involute splines offer a compact and weight-efficient means of transferring torque from one shaft to another, or between a shaft and a hub. Involute spline tooth dimensions and tolerances have been standardized by the Society of Automotive Engineers, American Society of Mech
11、anical Engineers and others, and are published in reference 1. The basic equations for involute spline tooth stress calculations assume that spline tooth loading is uniformly distributed along the length of the spline tooth. Non-uniform tooth loading is addressed in some spline load rating calculati
12、ons by applying a “load distribution” factor as in 2 and 3. Load distribution factors are used to account for misalignment (slope) between the internal and external spline members. The load distribution factor is influenced by the magnitude of the misalignment (slope) between the members, and by cro
13、wning of the spline teeth which reduces end-loading of the spline teeth thereby accommodating the misalignment. There is another mechanism that can cause non-uniform contact loading of spline teeth. It has been reported in 4, that as the length of a spline increases relative to its diameter, the tor
14、sional stiffness of the members in the joint exert a stronger influence on the lengthwise contact load distribution. This non-uniform contact load distribution can occur in perfectly aligned spline joints. The mechanism is not dependent on angular misalignment of the members axes, but rather by the
15、relative torsional stiffnesses and deflections (wind-up) of the internal and external members. In longer splines, the tooth contact load peaks near the start of the joint then falls away toward the end of the joint 4. The start of the joint is considered to be where torque first begins to be transfe
16、rred from the inner member to the outer member, as in a shaft to a hub. In splined joints where the inner and outer members have complex geometries (rather than simple cylindrical geometries) torsional stiffness can vary non-uniformly along the length of the joint leading to further non-uniformity o
17、f the tooth contact load distribution. Indeed, in some splined members with complex geometries, very stiff “hard points” may exist that resist torsional deflection and result in high contact loads over relatively short tooth lengths. Predicting tooth contact load distribution for these situations ca
18、n be quite difficult. Figure 1 LMR Rotor Hub and Shaft (Section of Hub Removed for Clarity) One application that utilizes a relatively long splined joint, between a shaft and a hub with complex geometries, occurs on the CH-47 helicopter. The splined joint in question transfers torque and rotary moti
19、on from the rotor shaft to the rotor hub. The rotor hub provides the attachment (via lugs) and load transfer to the helicopter rotor blades. A sectioned view of the rotor hub with the Start of Joint End of Joint Rotor Hub Rotor Shaft 2 mating shaft is shown in Figure 1. The shaft spline teeth that m
20、ate with the hub are split into two lengths, an upper and lower spline, with an un-splined cylindrical section between them as shown in Figure 2. Figure 2 CH-47 Rotor Shaft A section-view through the members is shown in Figure 3. The splined joint in this application is fixed and preloaded. The bend
21、ing loads generated in the rotor hub are transferred primarily by the “clamp-up” between the hub and a shoulder on the rotor shaft, rather than through bearing on the spline teeth. The spline teeth act to transfer torque between the rotor shaft and hub, in turn, driving the main rotor blades. Figure
22、 3 Cross-Section Through Rotor Shaft/Hub Splined Joint In this application, the splined joint has functioned well for many years. However, it has been shown in fatigue testing to be one of the critical sections of the rotor shaft (inner member) and is an impediment to further torque growth. During f
23、atigue testing, torsional fatigue loads are increased (overloaded) until fatigue cracking occurs. A torsional fatigue crack in the splined area of the rotor shaft, from fatigue overload testing, is shown in Figure 4. Figure 4 Shaft fatigue test specimen showing a crack at the spline Of course it too
24、k loads much greater in magnitude than actual aircraft loads to generate this failure. There is evidence, such as spline wear patterns, that indicate the pressure distribution along the splines is highly non-uniform. The current splined joint configuration has a simple “step” modification to reduce
25、load peaking at the lower end of the lower spline. This “step” modification is applied to the hub (internal spline) member. The “step” modification was designed to cause the upper spline to carry a larger portion of the total torque transmitted by the joint. The “step” modification effectively index
26、es the lower spline teeth relative to the upper spline teeth and was accomplished by thinning the lower spline teeth. No lead modifications are applied to the upper and lower spline teeth themselves, but the upper spline is indexed relative to the lower spline. The indexing results in a 0.005-0.006
27、inch tangential position difference between drive flanks of the upper and lower spline teeth. 3 The upper spline teeth precede the lower spline teeth into contact. The “step” modification increases the contact loading on the upper spline while decreasing the contact pressure on the lower spline. Thi
28、s approach results in better load sharing between the upper and lower spline, but load peaking on the splines was still observed. Further improvement is needed to extend the service life of the rotor shaft and to support anticipated torque growth. As part of the Low Maintenance Rotor project funded
29、by the U. S. Army, a new rotor hub for the CH-47 helicopter was designed. This new hub design presented an opportunity to improve the splined joint tooth load distribution using tooth lead modifications as well as “step” indexing. A ground rule for our design investigation stated that the shaft desi
30、gn was to remain unchanged, therefore lead modifications to the hub spline teeth (internal splines) only, were evaluated. Technical Approach The overall technical objective was to increase the service lives of the forward and aft rotor shafts in CH-47 aircraft configured with LMR rotor hubs. Ground
31、rules for the LMR program dictated that the new hub had to mate with the existing externally splined rotor shafts. This forced improvements to be obtained from modifying the hub design. Two basic features of the splined joint were contemplated for improvement: 1) Incorporate lead modifications on th
32、e internal hub splines to improve spline load distribution 2) Utilize the full length of the lower rotor shaft spline for torque transmission to the hub, to obtain increased load carrying capability. In the present CH-47 configuration a portion of the rotor shaft lower spline is used to engage a swa
33、shplate drive collar and is not available to transmit main rotor torque to the hub. In the LMR design, the collar is driven by a secondary spline on the hub. This allows the entire length of the lower rotor shaft spline to be utilized for torque transmittal to the rotor hub. The general approach to
34、increasing service life of the rotor shaft was to use lead modification to improve load distribution. Fundamentally, fatigue failures of spline joints typically originate at areas of peak stresses. If the contact load distribution along the lengths of the splines can be made more uniform, tooth bend
35、ing stress peaking would be attenuated. The analytical approach to the problem was to impose a uniform tooth loading condition on the splines, then work backwards from this state to determine the lead modifications necessary to achieve this condition. The plan of action was to develop lead modificat
36、ion using finite element models of the hub and rotor shaft components. Then verify the calculated lead modification by experimentally measuring spline tooth bending stresses at various points along the length of the rotor shaft splines. Another indicator of effective tooth loading is the portion of
37、the total torsional load transmitted at points along the rotor shaft. Torque measurements were performed at points along the shaft to determine the portion of the total torque transmitted by the upper spline. Finite element analysis of the rotor shaft/hub spline joint was used to: 1) Define lead mod
38、ifications for the hub spline teeth 2) Predict, by analysis, the uniformity of the resulting tooth loading The results of two linear static finite element models, one of the rotor shaft and one of the hub, were used to define tooth lead modification. A non-linear finite element model of the rotor sh
39、aft and rotor hub would be used to verify the proposed tooth lead modification. A strain survey test program was used to experimentally verify the effectiveness of the lead modification in minimizing load peaking along the length of the spline under the design rotor torque. Tooth bending stresses in
40、 the fillets at points along the spline teeth and rotor torque at sections below the splines were chosen as metrics to define success. 4 Analytical Approach The current state of engineering computing hardware and software allowed the use of detailed finite element models for the analysis of the comp
41、lex rotor hub and rotor shaft geometry. Two linear static finite element models, one of the rotor shaft and one of the rotor hub, were created. Each of these models included their spline teeth. A uniform spline tooth contact load distribution was applied to the finite element model of the rotor shaf
42、t. The resulting tangential deflections along the rotor shaft spline teeth at the pitch circle were computed. Tangential deflections were calculated at the rotor hub spline teeth in the same manner using the rotor hub finite element model. The numerical difference between the calculated tangential d
43、eflections of the hub spline teeth and the shaft spline teeth, at a number of points along the tooth length, defined the proposed tooth lead modifications. In other words, the lead modification required to produce a uniform tooth load distribution is the difference between shaft tooth and hub tooth
44、tangential deflections, calculated under uniform load conditions. This approach yields an optimal lead modification for the applied torque load. For the LMR lead modification calculations, the torque load selected was 100% of the rotor design torque. A third finite element model, consisting of the r
45、otor hub installed on the rotor shaft, was created. This non-linear model utilized “gap” elements to simulate the contact between mating spline teeth. The initial magnitude of the “gap” elements, at various points along the tooth length, described the tooth lead modification determined from the prev
46、ious finite element analyses. With the lead modifications thus inputted, the shaft and hub were loaded and the contact load distribution between the spline teeth were calculated. This model analytically verified that the lead modification determined from the previous analyses resulted in a uniform c
47、ontact load distribution along the spline teeth. To generate the finite element models described above, detailed solid geometry models of the rotor shaft and rotor hub were created using the CATIA program. The solid models were imported into the SDRC I-Deas Master Series analysis preprocessor softwa
48、re. In the preprocessor package, the solid model was partitioned into volumes to create useful features for applying loads and boundary conditions and to aid in meshing. The resulting volumes were meshed with linear tetrahedron solid elements. The linear static hub finite element model consisted of
49、147,466 solid elements and 36,399 nodes. The linear static models were run using the NASTRAN 101 solution sequence. Uniform tangential line loads totaling the design rotor torque were applied to nodes along the pitch diameter of the splines on the drive side of the teeth. The hub was restrained from translating by applying normal constraints to the surface which contacts the drive collar end face and normal constraints to the outer portion of the hub lugs inner bores. These two constraints resulted in a fully constrained model. Figure 5 Rotor Hub (green) and Sh
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