1、04FTM8Generalized Excitation of TravelingWave Vibration in Gearsby: P.B. Talbert, Honeywell Engines, Systems & ServicesTECHNICAL PAPERAmerican Gear ManufacturersAssociationGeneralized Excitation of Traveling Wave Vibration inGearsPaul B. Talbert, Honeywell Engines, Systems & ServicesThe statements a
2、nd opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractDynamic loading at tooth mesh frequency and its harmonics excites traveling wave vibration in gears. Theassociatedalternatingstressc
3、anbelargeenoughtocausehighcyclefatiguefailureofthegears. Inthispapera generalized excitation of traveling wave vibration in gears is developed for both forward and backwardwaves. Although they do not predict the actual dynamic response of the gear, the developed expressionsquantify the relative exci
4、tation as a function of nodal diameter allowing the analyst to place modes withconstructive excitation outside the gears operating range. These generalized equations allow the followingeffects to be addressed: any combination of tooth counts, any number of gears in mesh with the center gear,symmetri
5、c or non-symmetric spacing of these surrounding gears, and non-equal power extraction from thesurroundinggears. Theequationspresentedarethusapplicableforconfigurationsrangingfromasinglegearset to a sequential, non-symmetric planetary system. Dynamic response from a finite element model ispresented t
6、o verify the generalized excitation for the various configurations.Copyright 2004American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2004ISBN: 1-55589-831-91 Copyright 2004 by AGMA GENERALIZED EXCITATION OF TRAVELING WAVE VIBRATION IN GEARS Paul
7、 B. Talbert Manager, System Dynamics & Shafting Honeywell Engines, Systems & Services Introduction As a gear pair rotates through mesh the number of teeth in contact changes. For a spur gear with a contact ratio between one and two, the number of teeth in contact changes from a single pair, to two p
8、airs, back to a single pair, etc. The angular stiffness between the gears is roughly proportional to the number of teeth in contact. This step change in stiffness contributes to the total transmission error and creates dynamic load at mesh frequency and harmonics of mesh frequency. Flexural vibratio
9、n of a gear occurs when the alternating load associated with tooth mesh is properly timed to constructively reinforce a traveling wave mode shape. The conditions for constructive reinforcement of the wave are not complicated for a single gear set (a gear meshing with just one other gear). The condit
10、ions for constructive reinforcement of the wave become more complicated when the gear meshes with two or more other gears. Stockton (1) identified these conditions for sun gears in both sequential and non-sequential planetary gearboxes, but his work was limited to configurations with symmetric spaci
11、ng of the planet gears. This excellent work laid the foundation to expand knowledge of traveling wave excitation to more general gearbox configurations. This paper presents generalized equations for constructive reinforcement of traveling waves for any gearbox configuration. Nomenclature A Traveling
12、 wave amplitude, inch DAPitch diameter of gear A , inch DiPitch diameter of gear i , inch EiRelative excitation of wave from gear i FmNatural frequency of gear A for mode with m waves (in rotating reference frame) , Hz LiTooth mesh lag fraction of gear i PiRelative power transmitted to gear i Xi , Y
13、iCoordinates of center of gear i , inch f Natural frequency in fixed frame, Hz k Mesh frequency harmonic number m Harmonic waves around circumference nANumber of teeth on gear A niNumber of teeth on gear I iAngular location of center of gear i , rad waveAngular position of traveling wave, rad AAngul
14、ar speed of gear A , rad/sec iAngular speed of gear i , rad/sec waveAngular speed of traveling wave in fixed reference frame, rad/sec Traveling Waves Figure 1 illustrates the difference between standing and traveling waves. A harmonic excitation at a fixed location on a circular part (i.e. a piezoel
15、ectric crystal driver glued a gear) will produce a standing wave with fixed “nodal” locations of zero displacement. Locations between the nodal lines experience harmonic motion undergoing one half a cycle between the blue and red shapes labeled 1 and 3. For harmonic excitation that is not at a fixed
16、 location on the part, the mode shape will travel around the part. Traveling wave vibration occurs in gears because the dynamic load occurs at a fixed angular location (the point of mesh) instead of a fixed location on the gear. time = 0 after 1/4 cycle after 1/2 cycleStanding Wave Traveling Wave113
17、232waveFigure 1. Example of standing and traveling waves 2 Copyright 2004 by AGMA Wave Speed and the Campbell Diagram The angular speed of the traveling wave is not arbitrary because the natural frequency is the same whether the wave is standing or traveling. For a particle on a non-rotating part to
18、 have the same frequency as a standing wave, the angular speed of a forward or backward traveling wave is: mFmwave2= (1) For a part rotating at speed , the angular speed of a traveling wave is: mFmmwave=22(2) The frequency of the traveling wave as observed from the fixed reference frame is: 2mFfm= (
19、3) Campbell (2) identified the phenomena of traveling waves in his investigation of steam turbine disk vibration failures. He graphically presented traveling waves in his classic frequency-speed diagram, now commonly known as a Campbell Diagram (Figure 2). Gear speed A, rpm Frequency,HzTooth Mesh Ex
20、citationGear natural frequencyFrequency of forward waveFrequency of backward waveActual interferenceApparent interference60AmmF+60AmmFmFmnA+mnAnAFigure 2. Campbell Diagram for single gear set Single Mesh Excitation For a single gear set, traveling wave resonance will occur when the tooth mesh freque
21、ncy, nAA, coincides with the frequency of either a forward or backward traveling wave. =22AmAAmFn (4) The Campbell Diagram shown in Figure 2 illustrates the speeds at which the tooth mesh excitation frequency coincides with the frequency of either the forward or backward traveling wave. As noted by
22、Stockton (1), the “actual interference” between the tooth mesh excitation and the traveling wave occurs in the fixed reference frame. A strain gage on the gear observes the “apparent interference” at the natural frequency. Note that the abscissa in Figure 2 has units of rpm, thus Eq. 3 has a differe
23、nt conversion factor to achieve cycles per second. For a single gear set, mesh excitation is perfectly timed at some speed to reinforce traveling wave vibration for any value of m. All modes of the gear, both forward and backward traveling, will be excited provided that the resonance defined by Eq.
24、4 is within the operating range and the rim displacement of the mode shape creates a non-zero modal force. Multiple Mesh Excitation Traveling wave excitation becomes more complex when a gear meshes with more than one other gear. Figure 3 illustrates the geometry of a general, multiple mesh gear set.
25、 Gear A is the center gear experiencing traveling wave excitation from the surrounding gears B, C, D, E, etc (gear i in general). The center of gear A is located at coordinates (0,0). Gear B is located at 12 oclock above gear A . gear Agear Bgear iABiiXiYiFigure 3. Multiple mesh gear set geometry de
26、finition3 Copyright 2004 by AGMA The diameter and angular speed of gear i can be determined from the diameter and speed of gear A and the ratio of their tooth counts. AiAinnDD = (5) iAAinn = (6) The center of gear i is located at Xi, Yi. iiAiDDX sin2+= (7)iiAiDDY cos2+= (8) The number of teeth on ge
27、ar A and the angular spacing of gear i relative to gear B determines the timing of gear i relative to gear B. If gear i is located an integer number of teeth from gear B, its alternating force is exactly in phase with the force from gear B . The fractional part of the number of gear teeth where gear
28、 i is located relative to gear B determines its tooth mesh lag fraction. =ABiinFracL/2(9) Forces Acting On The Center Gear: Assume that gear i applies a radial force acting inward on gear A. The mesh force is actually along the line of action, but this does not change the conclusions that follow. Th
29、e relative magnitude of the force at gear i at time t is: ()iAAiiLtnPF 2cos = (10) The term nAAis the tooth mesh frequency, and the term 2Liis the phase lag of gear i relative to gear B. On a relative basis the force is proportional to the power transmitted, Pi. For graphical illustration, the coord
30、inates of the end point of the force are: ()iAiAAiDLtnPX sin22cos11 += (11) ()iAiAAiDLtnPY cos22cos11 += (12) Figure 4 shows these forces at time t for a symmetrical five planet, fixed carrier system with 65 teeth on gear A . For this non-sequential gear system, the forces from all five planets have
31、 equal magnitude at all times. The tooth mesh lag fraction of each planet gear is zero because the number of teeth on gear A is an integer multiple of the number of planet gears. The straight lines in Figure 4 originating at the center of each gear show their angular position at that instant. All th
32、ese clocking lines start in the vertical direction at time t = 0. gear Agear Bgear Cgear Dgear Egear FFigure 4. Forces for example non-sequential system Figure 5 shows these forces at time t for a symmetrical five planet, fixed carrier system with 63 teeth on gear A . For this example sequential gea
33、r system, the forces from all five planets do not all have equal magnitude at all times. The tooth mesh lag fractions of planet gears C, D, E, and F are 0.60, 0.20, 0.80, and 0.40, respectively. The time shown in Figure 5 is when the force at gear B is at its maximum value. The forces at gears C and
34、 F are nearly zero while the forces at gears D and E are at intermediate values. gear Agear Bgear Cgear Dgear Egear FFigure 5. Forces for an example sequential system 4 Copyright 2004 by AGMA Figure 6 shows the forces for this same sequential system at a time 1/5 of a tooth mesh cycle later. The for
35、ce at gear D is now at its maximum value. This pattern repeats with the maximum force in turn moving to gear F, then gear C, then gear E before returning to gear B at exactly one full tooth mesh cycle after the time shown in Figure 5. gear Agear Bgear Cgear Dgear Egear FFigure 6. Forces for an examp
36、le sequential system Traveling Wave Excitation: Figure 7 illustrates a typical traveling wave response for the symmetrical five planet, fixed carrier system with 65 teeth on gear A. The traveling wave shown is a five nodal diameter mode ( m = 5 ). As previously noted, the alternating forces from the
37、 five symmetrically spaced planets are all exactly in phase. The wave is initially aligned with the maximum inward force at gear B. It will be shown that the speed of this wave places the five outward displaced “lobes” in line with the planets at a time of a tooth mesh cycle later when the alternati
38、ng forces are minimum. gear Agear Bgear Cgear Dgear Egear FwaveFigure 7. Non-sequential system with m = 5 wave The excitation of the traveling wave is quantified by integrating the product of the alternating force from each surrounding gear and the traveling wave displacement of gear A at the mesh p
39、oints for one tooth mesh cycle. =01jiiidtFE (13) The alternating force at each surrounding gear is initially timed so that the force at gear B is maximum and the forces at the other gears follow their respective tooth mesh lag fractions. The traveling wave is initially aligned so that maximum inward
40、 displacement occurs at gear B. The traveling wave displacement of gear A at each of the mesh locations is: ()tmwaveii = cos (14) Figure 8 shows the displacement of gear A for a m = 3 wave for the example symmetrical five planet, fixed carrier system with 63 teeth on gear A. The forces shown are at
41、the same instant in time as shown in Figure 6. Note that gears C and E have minimum force at this instant when the outward displacement of the wave is aligned with these planet gears. Constructive re-enforcement of the traveling wave occurs when the maximum force from each gear coincides with maximu
42、m inward displacement of the wave. gear Agear Bgear Cgear Dgear Egear FwaveFigure 8. Constructive excitation of traveling wave Substituting the force and displacement at the mesh locations from Eqs. 10 and 14 into Eq. 13, the relative excitation of the traveling wave is: ()()=01cos2cosjiwaveiiAAidtt
43、mLtnPE(15) 5 Copyright 2004 by AGMA The excitation from each individual gear will super-impose, so calculating each independently: ()()=0cos2cos dttmLtnPEwaveiiAAii(16) Backward Traveling Wave: Focusing first on excitation of the backward traveling wave, resonance occurs when tooth mesh frequency eq
44、uals the frequency of the wave in the fixed reference frame. Eq. 4 for a backward wave is: =22AmAAmFn (17)Recalling the expression for wave speed from Eq 2. waveAAmn = (18) Substituting Eq. 18 into Eq. 16: ()()+=0cos2cos dttnmLtnPEAAiiAAii(19) Utilizing the trigonometric relationships: () sinsincosc
45、oscos += () sinsincoscoscos =+ ()()()iAAiAALtnLtn 2coscos2cos = ()()iAALtn 2sinsin+ ()()()tnmtnmAAiAAi coscoscos =+ ()( )tnmAAi sinsin Multiplying the two cosine terms: ()()=+ tnmLtnAAiiAA cos2cos ()()()()tnmLtnAAiiAA coscos2coscos ()()()()tnmLtnAAiiAA sinsin2coscos ()()()()tnmLtnAAiiAA coscos2sinsi
46、n+ ()()()()tnmLtnAAiiAA sinsin2sinsin Collecting the constant terms from the product together: ()()=+ tnmLtnAAiiAA cos2cos ()tnaAAi2cos ()()tntnbAAAAi sincos ()()tntncAAAAi cossin+ ()tndAAi2sin (20)where: ()()iiimLa cos2cos= ()()iiimLb sin2cos= ()()iiimLc cos2sin= ()()iiimLd sin2sin= Substituting th
47、ese terms into Eq. 19, the excitation from each individual gear becomes: ()=02cos dttnaPEAAiii()()()+0sincos dttntnbcPAAAAiii()02sin dttndPAAii(21) Integrating: ()042sin2+=AAAAiiintntaPE ()()042cos+AAAAiiintncbP()042sin2AAAAiintntdP (22) The time for a one full tooth mesh cycle is the inverse of the
48、 mesh frequency: AAn2= (23) Evaluating the integral from 0 to : ()+=AAAAiiinnaPE44sin()()+AAAAiiinncbP4144cos()AAAAiinndP44sin(24) 6 Copyright 2004 by AGMA The final expression for the relative excitation of the backward traveling wave due to an individual surrounding gear is: ()() ( )()iiiiAAiimLmLnPE sin2sincos2cos =(25) The total excitation on the center gear is the sum from all the surrounding gears. Forward Traveling Wave: Focusing next on excitation of the forward traveling wave, resonance occurs when tooth mesh fr
