1、91 FTM 8vMethods of Statistical Dynamics for theCalculation of Gear Stress Distribution and itsEffect on Gear Failure Probabilityby: M. Haykin, Wayne State UniversityAmerican Gear Manufacturers AssociationI IITECHNICAL PAPERMethods of Statistical Daynamics for the Calculation of Gear Stress Distribu
2、tion and itsEffecton GearFailureProbability vM. Haykin, Wayne State UniversityTheStatementsandopinions containedhereinare thoseof the authorandshouldnotbe construed asan officialaction oropinion of the American Gear ManufacturersAssociation.ABSTRACT:The process of most machine loading has a random c
3、haracterwhich is determined by external varialion and dynamicqualifiesof the system. Such an approachwas usedto obtainthe load spectrum for gears and the probabilityof its failure.Analysis and experiments showed that gain factor for gear is distinguishedby the similar parameter of the entire drivesy
4、stem. This is explained by the uniform distribution of gear stress even for cases of smile loading. Method of gearstrength calculationwith the st_ati_licalparameter is discussed.Copyright 1991American Gear ManufacturersAssociation1500 King Street, Suite 201Alexandria, Virginia, 22314October, 1991ISB
5、N: 1-55589-613-8AMETHODS OF STATISTICAL DYNAMICS FOR THECALCULATION OF GEAR STRESS DISTRIBUTION ANDITS EFFECT ON GEAR FAILURE PROBABILITYMikhai! HaykinWayne State University, Detroit, MichiganI. INTRODUCTION machines and using the same materials. On the otherhand, its application reduces the importa
6、nce ofIn I Ing. Theodor Hosel compared the determination of the real picture of the stressaccepted standards of AGMA, ISO, DIN, and Comecon. distribution for the gear tooth as it allows one toOne of the conclusions of this research is that use only nominal load.“Different interpretations of geometri
7、cal data and C.C.Wang, offering the economic approach fordifferent calculating methods within the standards the use of an application factor, noted that-thecan lead to considerable differences of surface “Empirical data is a good reasonable approximation,stress, root stress, and transmissible torque
8、, but, it is an over-simplified shortcut for dynamiccalculation for one and the same gear pair“ i; systems“ 3; page i.page 13. This shows that such differences are In fact, there is a real scheme of stressdetermined because of different values of some distribution and the possibility of simulating t
9、hisfactors that were applied for stress analysis, scheme is dependent on our ability to reflect theFor example, the author calculated the Y real physical processes by analytical methods.factor for the same gear (AGMA-Y=O.917, DIN- Among the various factors which are taken intoY=0.799) and explained
10、that a 15% difference in consideration for determination of maximum stress,this factor can be understood if we take into the value of the load distribution factor is oneconsideration the different AGMAs and DINs of the most important.definitions of the critical points and critical There are many con
11、ditions which determinesections at the tooth root. Another example of his nonuniform load intensity, and each of them has anresearch is when the use of the same helical gear influence on the length of contact line between( _ =I_) yielded a difference in increasing the pair of gears.load capacity ben
12、efit. The author showed that, the The real contact length of a pair of teethtorque ratio between the helical gear and the spur compared with the whole of the face width dependsgear is : on two main factors which are reflected in everyone of the standards.Tm/To =1.87 (AGIMA); T_/To =1.45 (DIN); - pos
13、ition of gears relative to the bearingsupports.Tm/T o =1.3 (Comecon); - ratio of the whole of the face width of thegear to the diameter of the pinion.But, it can be noted that, as a rule, despitedifferences in the value of various factors of Besides, there are researches for evaluationstrength calcu
14、lations, the results can be smoothed of the influence of another special characteristicout due to the application or a “safety factor“, on the length of contact line and consequently, onDespite the fact that AGMAs fundamental stress distribution. In 4, T. Tobe and K. Inoueformulae do not contain a d
15、enominated Safety analyzed the influence of element of manufacturingj Factor, the sense of AGMAs formulae is the same errors to estimate the maximum load intensity. Inand the standard explains that, “The application 5, W. Mark described the transmission error byfactors make allowance for any externa
16、lly applied three scalar components and showed that even aloads in excess of the nominal tangential load W“ factor such as bearing misalignments may beincluded in the calculation.2; page 14. It is clear that the practice ofapplying the safety factor reflects our knowledge Because of this, some of th
17、e authors gavebased on the experience of studying similar integral coefficients (K _ which are dependent onsuch known parameters as face width to diameter one of the reasons which requires to apply safetyratio (_=_), or hardness of gears , as well as on or application factor.design _ _ear case and e
18、ven on types of bearings But, it is clear that real load on the gear A6. tooth can be changed during operating time because,_a K_# as a rule, on one side there is alternatingexternal load, and on the other side, the gear is_I _ I, ;b _! _ only one of the elements of a complex dynamicy _$ system subj
19、ected to such load._ -“ influence of real dynamic external load on the.! character of maximum stress.A _/ / _ /! As any nonuniform load can be expressed by_ _ #_ i_ , / , f _ the parameters of random processes, such an_ /j_jr / _/ approach required considering some special tasks,. _ _ ;,ti#/_*“ _“ w
20、hich are discussed below :- - analysis of transformation of external. t t_ _- dynamic load for different parts of the mechanicalilll_ll ll_ _ti I.,2 ;o_ llll system in order to determine the gear tooth load asa random function dependent on statisticala b characteristics of the external load and on d
21、ynamicFig. I Integral coefficient of load distribution characteristic of the system.along face width of tooth. - theoretical and experimental determination(a) Type of design of the stress distribution under static and dynamic(b) Value of KH_ for the gear with hardness load.350HB - method of strength
22、 calculation with thestatistical parameters: variance and spectrumIn Fig.i (a) 1.5; 0.75 a b(b/m=5; 3.3; 1.65 ) we can obtain any curves of Fig. 8. Stress _stribution along the faceload distribution for different points of width of the gear tooth.application of the load. a) Scheme of testing arrang_
23、ent.Fig. 7 shows the comparison of cu_es of b) Result of test: 1-7, from concentratedben_ng moments on the fillet of the tooth which load applied at the point of transducer location.was obtained for three _fferent cases of applied 8, resultant cu_e from unit loads. 9, cu_e fromforces: unifo_stribute
24、d load.- concentrated load, applied at the _d-point Loa_ng of the tooth on the hydropulser was(Cu_e-I)created both by concentrated and uniformly- concentrated loads aapplied at two distributed load. It was used as the principle ofsymmetrical points res_ctively to the _d_e line. supe_osition, for ind
25、ependent forces. In Fig.8 the(cu_e-2) stress _stribution is sho_ along the face width- _iformly _stributed load. (cu_e -3) and, as we can see, the resultant cu_es obtainedM due to the summation of curves obtained for_ concentrated forces (8) and for _stributed load_.I (9) are ve_ close. The _fferenc
26、e for the _xim_stress is not bigger than 6%. Experiments proved_ .3 that the maximum stress under a concentrated load_)/ , _ _ is placed on the point of application of force. Ifi the point of application of force will be moving/_ I k respective to the tooth face width, the resultantI I cu_e will _ _
27、splaced practically without changein its shape. When the load is _ifo_ and there_Q _ I,$ 0 1.65 _ b _ are no _salignments of the shaft, then the law ofFig. 7.Change of ben_ng moment on the fillet stress _stribution for the theoretical case orof the tooth, their deflections is ve_ close to parabolic
28、factor1-from concentrated load located at the _d _=1.22.point.2-from two concentrated loads located It is understood that the experimentalsymmetrically from_d-point respectively, character of stress _stribution _ffers from the3-from unifo_ _stributed load. real stress since the hydropulser can not p
29、roducethe real conditions of engagement because ofThe cu_e (3) shows that even a uniformly _saligru_ents connected with the deformations of_stributed load along the face width of the tooth the shafts, bearings, cases, and so on.does not provide a unifo_ bending moment on thefillet. As a result there
30、 is a critical point with In order to compare the real results ofmaximum stress. In this case, the normal ben_ng stress distribution, the s_e gear was checked onstress is approxi_ted well enough by cosine cu_e the shaft of the real Shearer _ning _chine. Theof 3rd power. In this case the _xim_ stress
31、 for tooth loa_ng was created _ a static moment on thethe calculation has to _ taken as: free end of the motor shaft and by fixing of theoutput shaft. The results were obtained by_-_VMA_=v_ _ *_ increasing the moment on the shaft, andWhere _. - average _itude of stress, conse_ently, the ma_itude of
32、the tooth loado_n - coefficient of nonuniform changed the stress _stribution cu_e. When theload _stributi dependent on face width to module moment of the motor shaft was only 55% fromratio (see t_le i) nounal, the cu_e of the stress _stributed acrossthe gear tooth was close to the result obtainedwit
33、h the hydrop_ser, and the peak of the stressbl_ 3.3 5 66 |0 coincided with the _d_e point of the tooth. Then,under increasing load, _x_ stress moves to the8 I.|3 117 1.2 |.25 e_e of the tooth. (see Fig. 9).A nounal (static) moment provides averageTable I. stress on the tooth which is about 310 _a an
34、d its_xim_ value is 400 _a. It has increased by 29%.Such a result was confi_ed by tests with Fig.10 shows the comparison between the stressreal gears. Parameters of gears:module, m = 12mm; Number of teeth, N =19; _stributions for the hydropulser curve (cu_e i)Face width, 2b = ll6mm, and the real mac
35、hine test (curve 2), both doneb/m, relative width = 9.7 under an e_al load. The _iformly _stributed loadon the hy_opulser was 156,000 N, and the moment onheat treatment - carburizing 1.2 - 1.6 mm., the gear shaft was 17700 N-m.hardening and tempering HRC 58.Sdesigned to create vibration and to chang
36、e theo._ frequency of the load.2 The same gear with the strain gages was usein order to get the amplitude _-frequencyJ _ I_I I I characteristic between the exit of the mechanicalt$_. _ _ ! J _ J also put on the shaft of this gear, in order to,_J J/_ : J .J 11 _ I obtain the amplitude-frequency chara
37、cteristicbetween the same exit of the system and the shaft._I I i I i I I l d Tests confirmed that, due to the dependence betweenthe average load on the shaft and the maximum value_B i_ _ E _ _ I of the distributed static load on the gear tooth_-_28“-_ 8-t16 tJ (see paragraph above), there is a diff
38、erencebetween the characteristics of the shaft and theFig. 9 Stress distribution on the tooth of gear. The amplitude-frequency characteristicthe gear mounted on the gearbox, obtained for the shaft (Fig. Ii, curve i)1-Curve from minimum static load. corresponds to the average tooth stress.2-Curve fro
39、m maximum static load.zoo/ = 7“/P uloo vzz I r iIZS MZ2Y Y =a-,6 =“=/ Xkk -,Fig. I0. Comparison of stress distribution Qcurve of gear mounted on: ;oO .,7.00 _ _ _ L-“l-Pulsar.2-Real gear-case. Fig. Ii. Amplitude-frequency characteristics:1-According to the average stress of tooth.So, the static load
40、ing of the tooth gave 2-According to the maximum stress of tooth.several important results, as follows : l- there is a parabolic shape of the stress However, the curve of amplitude-frequenc_distribution curve when the gear tooth is characteristic obtained for the values of peak load-subjected to a u
41、niformly distributed on the tooth was different (Fig. ii, curve 2). But,load. The maximum value of such a stress actually such a curve has to be taken for strengthis a function of the face width to calculation of the gear. The comparison of themodule ratio, curves showed that the maximum increase of
42、 the load- the actions of the real factors of for the maximum load is bigger than the sameengagement (mis alignment s of shafts, increase for the average load by 21%. Thedeflection, and deformation of coefficient of gain for the random loads, for atransmission elements) causes, the given mechanical
43、system, which can appear at thenonsymmetrical shape of the stress exit in the wide frequency spectrum is the squaredistribution curve and additionally of the area under the obtained curve. Theincreases the maximum value of stress, comparison of the areas under the curves (I) D_/.E = 0.45 DE ;D_.E =0
44、.15 DZ.In order to study the characters of the geartooth stress distribution under the dynamic loadson the output shaft, a special vibrator was6where, D E -co-on variance of the external where, n - conditional safety factor ofload on the drum strength.A D,_. -the variance of the external F-Loload in
45、 the field of low nfrequency _DH. -the variance of the external _ - expectation of strengthload in the field of high endurance limi_for the gearfrequency _ - expectation of load on the gear-the variance of the periodic _,7 variation coefficient ofcomponent strength enduranh_e limit_/ _ vaa:iatioD_ue
46、effic-ient of loadIn this case, the variance of the load on the on the gearshaft (Ds_.) is :DS_“ = DZ. $_. + DH._+ D_._= DZ.E + D _ + I_. Using the standard methods of calculation of_._$:_ D_._ the variation coefficient of the external load on“ the cutting drum, we can give an example of such a= 0.4
47、 DE. + 0.45 x 1.77 D + 0.15 x 1.15 D calculation. For two different types of cutting= drums (Shearer and Trepanner), such coefficients= 1.39 DE. were determined as :but, the variance of the load for the gear For Shearer drum : v_d_= 0.36(D_) is : For Trepanner drum : vr_._/_.= 0.41D_ + + Then, in ac
48、cordance with the results shown“ - above, the variation coefficients for the gear are= 0.4 DE + 0.45 x 2.39 DE + 1.15 x 0.15 DE given in Table 2, which was obtained from theexpression: v g_A_ = vm_ /j_-7= 1.674 DE.The analysis of our obtained results shows Type0f variation coefficientthat common loa
49、d variance for gear tooth is biggerby 19.3% in comparison with the same parameter for Drum 0rgearusmg for gear uslngthe shaft (transmission element) and by 67% in for (1rum averaqe load rnaxlmum loadcomparison with the load variance on the cuttingdrum. So, we have to conclude that the nonuniform shearer 0.36 0.426 0.47character of stress distribution throughout thetooth face width creates an increased dynamicregime for the tooth in the area of maximum stress Trepanner 0.41 0.485 0.53in comparison with the transmission. Obvio
