1、90 FTM 6IDynamic Responses of Aircraft Gearsby: Kayaalp Buyukataman, General Electric Aircraft Engines Division, III III+,IAmerican Gear Manufacturers AssociationI II I III ITECHNICAL PAPERDynamic Responses of Aircraft GearsKayaalp Buyukataman,General Electric Aircraft Engines DivisionThe Statements
2、 and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.ABSTRACT:Rapid and destructive failures of high quality, high speed, lightweight and highly loaded aircraft gears areindication of the vib
3、rational energy (associated with neutral frequency modes) exceeding the fatigueendurance limit of advance gear materials.This paper reviews:A) Experimental and analytical methods to identify and define resonant mode.B) Affects of gear design and manufacturing variables to the generation and damping
4、of vibrationalenergy.AUTHOR S DEDICATION:This paper has been dedicated to Mr. Joseph S. Alford. Mr. Alford is with the GE Company as engineersince 1934. Mr. Alfords teachings had a major impact on the traditions of GE Aircraft Enginescommunity and the values we have today.Copyright 1990American Gear
5、 Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 1990ISBN: 1-55589-558-1DYNAMIC RESPONSES OF AIRCRAFT GEARSKAYAALP BUYUKATAMANGE AIRCRAFT ENGINES DIVISIONCINCINNATI, OHIOINTRODUCTION One of the earliestsuccessfuleffortsto applythe elementary equations to reson
6、ance phenomena ofThe study of natural frequencies is a three dimensional solids were developed by Love.well-developed branch of engineering. Many The treatment takes into account the inertia ofpublications have been released on the subject of lateral contraction. The Loves assumption arethe applicat
7、ion for practical engineering retained that plane sections remain plane andanalysis, such as; Theory of Elasticity by longitudinal vibrations or stresses over a crossTimeshenko and Goodier (1), Mathematical Theory of section is uniform.Elasticity by Sokolnikoff (2), MechanicalVibrations by Den Horto
8、g (3), A Treatise on the The aircraft gears, in concern, generally consistsMathematical Theory of Elasticity by Love (4), or of two geometrical shapes, the disk and the ring.publications of Peterson (5), Afford (6), Drago Exiting frequency of solid pinions are usually(7),Tuplin(8),and AGMA (9). beyo
9、nd the range of their operation. 3owever,bevel and beveloid gears which are basically disksThe governing equations of natural Frequency, as and the bull or internal gears which are basicallyused in gear resonance analysis, dates back to ring or finite cylinder coupled with thin disksdefinition of co
10、ncepts of elasticity and stress are often being exited, at frequencies close towave propagation by Pythagorean and Aristotlean. one of their resonant frequencies and during theEquations and theories later on refined by Galileo operation163B, RobeFt Hooke 167B, Isac Newton 16B6, ThomasYoung 1807, Cau
11、chy 1822, Novier 1821, Green 1827, We must be alert, during examination of highLordRayleigh1877. frequencywaves, it is importantto distinguishbetween the higher order harmonics of the simpleAs the 20th Century approached, theories of modes and lower order harmonics of higher modes.vibrations began t
12、o advance rapidly and separated Both consists of high frequency vibrations, butfrom the dynamics of elastic materials, the resultant motions are physically veryBasically, the equations we now use and call the different.edge of technology were mathematically derived andverified by large scale tests a
13、t the beginning of With respect to the understanding of the higherthiscentury, modes of vibration,we can examinethe nodalplanes passing through the axis of the cylinder.A circular cylinder vibrating in the nth mode maybe considered as consisting of n pairs ofBACKGROUND diametrically opposite sectors
14、 of a circle. The npairs of sectors may be chosen in two ways, eitherResonance phenomena in gears is complex in bounded by nodal planes of radial motion or nodalunderstanding. However, elementary equations planes of angular motion. These planes arewhich guide the design of high power density separat
15、ed by an angle (_/2n). Fig. (1)aerospace gearing can be derived directly from the illustrates schematically the motion across crossequations of motion and by using certain sections of cylinders vibrating in the modes n =simplifying assumptions. O, O, l, 2, and 3 (the fundamental is consideredfor eac
16、h mode). It is noted that n = O, O, and l It is interesting to note that when the gear blankare ;special,“i.e., the cylinder vibrates as a is effected at a frequency sufficiently close towhole, and these modes do not conveniently lend one of its resonant frequencies. Deflections andthemselves to gen
17、eralizations; they are included stresses can increase to levels where failure canfor completeness, occur within a very short time. Fig. (2) showsvariation of stress levels during investigation ofWith respect to the understanding of the higher a bevel gear. Strain gages in this case wereharmonics of
18、vibration, we examine the number of located on the flange of the gear.amplitude nodal cylinders concentric with the axisof the cylinder. The first harmonic is the _RESS_I:_Psl ii_i_i_ _i_.-T“_-_-_-:_:-_._:i_i_._;_;_.ii _simplest vibration for which reflection occurs, PP _ _J-“_“ -“ _:_ a-;_i : _-_-
19、_ ;the reflected waves combining to produce one nodal 30,000_=_ .:_ _Jcylinder. Higher harmonics would be of increasingcomplexities, the waves combining to give two,three, etc., nodal cylinders. 20,000.In addition to the infinity of roots of a given zO,ooOdispersion relation for real propagationcons
20、tants, there also exists an infinite number of oroots for complex propagation constants. These 2 ,ooo _ 35,_complexdispersioncurvesare extensionsof the RPMhigher order harmonics below their cut-Off Figure2frequencies. VariationofStressLevelswith_ Change in Speed at Constant Torque.1. GEARDISK VIBRAT
21、IONS,(RADIAL)Natural frequency of gear disks were so_vedAnalytical by Kirchoff in 1850 and since then(a) “_-_/ (b) basic equations remained same. During early1900s Peterson expressed Kirchoff equations ine_j _e=_“r_= _ the following format.o.,o,.o.=neutral ax;s in _ r T“_Stm;ned “neutm_ ax_s_ ;n_._,
22、_H_=_-f p is frequency X 2_. /P-C,fl.,y E is Youngs modulusM is half-thickness, “_-_(=) a is radiusa_. _Noa=I_I_o_ eI_-“ _ is numerical coefficient depending on mode ofvibration and on Poisons ratio._ |,_;,_r_o. _“_L_ y is density_%/_,_ _X_J-. /_/A/_ _._/_ The mode of vibration having the lowest fre
23、quencyhas two nodal diameters, for which _ = ID.61when a value of 0,3 is used for Poisons ratio.Using a value of 30 X 106 lb. per sq. in. for Eand a density of 490 lb. per cu. ft., thefollowing formula will give the lowest frequency(_t _/ _,_o,_o_ _= (.) of a thin steel disk:Figure I f = 208,400 h/d
24、2 (2)Motion across cross sections of cylindersvibrating in the fundamentals in the first five _heremodes (schematic).f is frequency of vibration in cycles per second.(a) Longitudinal mode; n = O;uo = O;ur = O;uz = O. h is thickness of disk in inches.(b) Torsional mode; n = O;ur = uz = O;uo = O. d is
25、 diameter of disk in inches.(c) Transverse mode; n = I.(d) Screw mode; n = 2. Inasmuch as the foregoing formula applies only to(e) No designation; n = 3. thin disks it was necessary to determine frequencyvalues experimentally for relatively thick disks.Peterson (5) used three different diameters and
26、Note = u is the dlrectlonal displacement, the results given are shown graphically inn is number of nodal diameters , or number Fig. (4). It will be noticed that for thin disksof mode.2the agreement between theory and experiment is frequency of the same body with all constraintsclose, but that for th
27、icker disks the slopes of removed. Implying of course the free state hasthe experimental curves decrease with increasing the lowest natural frequency. This statement isthickness. In our case thin disks are the ones 120 years old and stil being verified by eachused in aircraftgears, test we go throug
28、hto understandthe phenomenoninquestion. The equation of u (Eq. (3) as definedH in Lord Rayleighsstudies containssingleT variableparameter.i:_ asThisshownparameterbelow:“S“ is due to Stodola and appearsu= a rS Sinn e Cosfat (3)“iu is axial deflection at radius “r“ (inches)f a ismagnitudeconstant.r is
29、 radius of u (inches).S is the Stodola parameter.n is the number of complete vibration waves in a_ circumference,no.ofnodaldiameters.!_ 1 0 is angular distance around disk fromreferenCeradius(radians)1 .#_ fa isthenaturalfrequencyofaxialvibration: (radians/sec)-daX - Axis : t is the time in secondsF
30、igure3 1 TCross Section of a Spur GearShowing Notations Used. _ A-_#: J r_. hz A- Jlsooo / = +I - -/I Figure 5/ZI _e_/“ # _ Components of Equivalent Disks_z_/J_ # In the case of steel spur gears Rayleighs_ 1 (77)B is Amplitude of ring vibration, in.a is Acceleration, in/sec.f = g/2_ cycles per sec,
31、frequency.The pressure, P, usually arises from centrifugalforces on the damper ring. Most damper rings aresplit or severed in one location, to ensure a high _ _P, but thisis not a requirementfor radialmodes ! S_II_!whenn _2. The pressuremay be obtainedby a _ j_force-fit of the ring by means of heati
32、ng the gear ,._body and cooling the damper ring.A most important parameter that must be chosen bythe designer is Q, the change in the magnificationfactor due to the damper ring. Welded andmachined structure often have an inherent Q ashigh as 3,000 - 5,000, Ref. (27). Based on Ref. _(27), a satisfact
33、ory Q for hollow shafts might bein the order of 200 - 500. %In axial dampers; the axial motion is thevibration amplitude, Bi. Bi is a function ofthe ratio, Eq. (78).pA (2T,f)2 ) (78)where /Figure 26A is Ring area, In2 A Damper Ring Configuration afterfn is natural frequency of n nodal dia. Extensive
34、 Service use. 16theradiodeatoentoconsiderationforstiff rings. Third, the ring will be unstable andtry to overturn, and fourth, a short life mayresult from the small wear area. Fig. (25) shows Gear Xa successfully used damper configuration foraircraft applications. _/_._ Damper / ._,/_Fig. (26)Shows
35、an aircraft damper configuration _._ _ Ring / _ _v_/ afterabout8000 hrs. of life. The particulargear had both dampers on, one on each side, one _ _-_ _2, _ 3_! _wore out, one remained new. This was judged to be ./_ythe result of over turn condition and gear _wtoalignment under load. Once the cause i
36、s knowncorrective action can easily be implemented. The f ,x _ /key point is several comparisons may be made fordampers that dissipate axial motion and those thatdissipate radial motion and some that dissapete “_ I Figure 27both like in Fig. 26. THE AFFECTIVENESSOF DAMPER _ _Axial VelocityDiagram.DO
37、ES LIE IN THE EXPERIENCE OF THE DESIGNER TO _CHANGE THE SHAPE FOR THE MODE INTENDED. Thecapabilities of damper rings are often limited by Fig. (28) defines a typical example on the affectsgeometry, speed, rings natural frequency and the of coefficient of friction to energy dissipationadded stresseso
38、n the structure, with speed and at a given frequency.The following equations may be used in conjunction Similarly by combining Eqs. (81 and 83) lock-upwith previous ones to define operating speed due to centrifugal forces becomescharacteristics of damper rings.W2:2XW2 (85)rThe work done per cycle.=
39、4Fx cos wt (_9) and maximum energy dissipation for the damper ring“ as a function of maximum displacement can bewto is phaseangle.(Fig.27) expressedby combiningaboveequations,asX isaxialdisplacementofgear. - = W2X_ aF frictionforce. Wmax _ax(8Y_ (R+c) (B6)2 (80) X _“ .0_F=_R=_yaRWr _ = .20 = ._0 =is
40、 coefficient of friction. /f_._is radial force. ,oo6is the work done per cycle.,005Definition of basic variables and phase angle areshown in Fig. (27), shaded area of Fig. (27) ,oo_- defines the energy damped by the damper.Lets replace cos wt of Eq. (79) by its equivalent. ,oo3v_l 2 2- ,oo_F (Bl)W =
41、 4FX -_ y2a-_ 001where o , ,0 _0 60 00(Thous_dg_a is cross sectional area. _e_8_-R_Wr is rotational/speed of gear. (2_N/60) Figure 2BW = 2wf, circular natural frequency. Typical Friction Energy Dissipation withVariations in Coefficients of Friction.By equating Eqs. (79 and 81) the value of frictionf
42、orce becomes CONCLUSIONF = (_2/w) _ X W2a (82) The natural frequencies of free vibrations can bepredicted accurately for simple or cor_pIPx,gearand substituting Eq. (82) into Eq. (9) and gearshaft geometries by using the analyticaland experimental methods or by using finite- 4- element methods as ou
43、tlined in this paper.Wmax = _- ya X2 N2 (83)and by combining Eqs. (80 and 83) THE ELEMENTARY EQUATIONS OF FREE VIBRATIONS,RESONANCE FREQUENCIES, EXCITATION FREQUENCIES,SHIFT OF EXCITATION FREQUENCIES, SH _ OFN2 = _2 XW2 (84) RESONATING FREQUENCIES AND DAMPING ARE ALL OERIVEDr _ _T FROMTHE EQUATIONSD
44、F MOTIONANDBY USINGCERTAINgives us the speed for maximum energy dissipation. SIMPLIFYING ASSUMPTIONS.17Referencesgivenwhere possible. II. Campbell, W., “Protection of Steam TurbineDisks from Axial Vibration“, Trans. ASME 1924.It should be re-emphasized that resonance ofaircraft gears during their op
45、erating range does 12. Bradley, W. A., “Sound Gear Quality“,not always imply high energy levels and/or Mechanical Engineering, Oct. 1972.possible failure. However, in order to avoidunnecessary and complicated re-design of these 13. Dale, A. K., “Gear Noise and Side Bandlightweight and complex gears
46、provisions should be Phenomenon“, ASME B4-DET-174, 1989.made to place damper rings in the event ofunpredictable complex frequency modulation 14. Dudley, D. W., “Private Notes“.generated high energy outburst.15. Harris, C. M. and Crede, C. E., ed, “Shock andVibration Handbook“, McGraw-Hill, New York,
47、ACKNOWLEDGMENT 1961.The subject of resonance phenomenon is becoming 16. Plunkett, R., ed., “Mechanical Impedanceincreasingly important as the speed of the Methods for Mechanical Vibrations“, ASME, Newaircraft gears increase and as their weight York, 1958.decrease due to requirements of high systemef
48、ficiencyand low power density. 17. Malvern, E., “Effect of Damping on VibrationFrequencies of Simple Systems“, Proc. ThirdThe efforts made in this paper to define the Midwestern Conf. on Solid Mech., University ofoperating environment has been helped immeasurably Michigan, April 1957, 195-205.by bot
49、h formal and informal contacts and by thesupport provided to complete this documentation by 18. Jacobsen, L. S., “Steady forced vibration asMessrs. P. Bissett, R. N. Kloos, 3. Alford, D.W. influenced by damping“, Trans. ASME,Dudley, R. Drago, W. Bradley, and by the APM-52-15,1930.technical publication of Messrs. W. Campbell, D.H. Landis, R. E. Peterson, H. A. Lowe, and AGMA. 19. Van Bommel P., “A Simple Mass System with DryDamping Subjected to Harmonic Vibrations;The author also thanks GE Aircraft E
