1、 STD-AGMA 9bFTM7-ENGL 199b m Ob87575 0004901 3TT m 96FTM7 Dynamic Distribution of Load and Stress on External Involute Gearings by: Dr. Jrg Brner, Dresden University of Technology, and Dr. Donald R. Houser, Ohio State University I I TECHNICAL PAPER STDmAGMA SbFTM7-ENGL L97b Ob87575 OOOi702 23b D Dyn
2、amic Distribution of Load and Stress on External Involute Gearings Dr. Jrg Brner, Dresden University of Technology, and Dr. Donald R Houser, Ohio State University The statements and opinions contained herein are those of the author and should not be construed as an officiai action or opinion of the
3、American Gear Manufacturers Association. Abstract The influence of additional dynamic loads on the disiiutions of load, flask pressure and tooth root stress are shown. The additional dynamic loads excited by the variations of mesh stiffness and infiuenced by tooth deviations and modifications are ca
4、lculated and used as input for the calculation of the distriiutions of load and stresses in the plane of action. The programs used for these calculations were developed at the Institute of Machine Elements at the Dresden University of Technology. Very efficient calculations are obtained by using spe
5、cially developed caiculation algorithms. An extensive graphic presentation of the calculation results is included. The same connection of the caiculation of load distribution with the calculation of dynamic loads is also possiOle in principle on the basis of very similar programs (LDP . iteration st
6、ep for a. The number of harmonics used i: ik solution statement for g, increases greatly in every iteiatic; step caused by the product of the Fourier series of g, and r(r Bur many new excitation harmonics are very low arid thc: can be neglected. The intensity of excitation is analyzed ii the program
7、 GDA and ail harmonics with amplitudes less C .i 8 of the static tooth load are neglected. An unlimited 3 increase of the number of the harmonics is thus avoided in the analysis. The calculation can be simplified if the values of off- diagonal positions on the left-hand side caused only by the dampi
8、ng matrix are much less than the diagonal values resultant from E, 4 and 4. In this case only the diagonals have to be considered and the calculation goes faster. The damping can be considered in GDA as absolute damping on every degree of freedom and as relative damping between the degrees of freedo
9、m. But there is often a great uncertainty regarding the damping values. More data on damping contributions are needed in order to properly consider its effects. Therefore, for the sake of simplicity, modal damping ratios are also possible in GDA. One obtains the Fourier series of the vibration of th
10、e severai degrees of freedom as a resuit of the GDA calculation. The Fourier series of the resultant tooth force and shaft twist are additionally obtained. The time domain vibration traces may be calculated with these series terms provided phase is properly accounted for 4. The performance of LVR an
11、d GDA The distribution of load, Hertzian pressure, contact temperanire and root stresses can be calculated on involute external gears with LVR. Gear units with up to 4 reductions can be analyzed. The loads resultant from neighboring gears on the shafts are considered in the calculation. Fig. 2 shows
12、 the model for the calcuiation on the first reduction of a two reduction unit. The additional loads resulting from the pinion of the second reduction can be seen on the gear shaft. The influencis of the deformations of the surroundings - the bearings and shafts - due to the current load distribution
13、 are included. The calculation is iteratively repeated 4 times for every mesh position to take into account the Hertzian deformarion and the deformation of the mundings due to the calculated load distribution in every iteration step. A maximum of 30 mesh positions can be calculated in a meshing peri
14、od with a minimum ailowable distance between the single loads of 0.2 times module. I ScWdiOtme1Slnduam 1 Fig. 2 Calculation model used in LVR with additional loads on the gear shaft due to the pinion of the 2nd reduction k ml i2 Fig.3 Example .of the calculation model used in program GDA the The vib
15、rations caused by both the tooth mesh inside the geear unit and external somes can be calculated with GDA. One gets also the internal dynamic tooth forces caused by these vibrations. Up to 5 additional masses can be added on the input shaft as well as on the output shaft to consider influences of th
16、e whole driving system. The gear can be modeled as a simple torsional system or as a coupled torsional-bending system. The influence of laterai vibrations of the gears excited by the gear meshing can be considered with this coupled system. An example of the calculation model used in GDA can be seen
17、in Fig. 3. The combination of LVR and GDA makes the calculation of dynamic load distribution possible. The starting point is the calculation of the load distribution with the program LVR for the nominal load. The curves of the tooth stiffness and transmission error are also obtained. The tooth stiff
18、ness k(q) acting in one mesh position in the transverse plane results from the section stiffnesses ki in the nod plane. These section stiffnesses are detemined by the total transmission error xg(p) and the force Fi and the deviation f; on the i-th section of the contact line: Influences of the defor
19、mations of the surroundings and of tooth modifications and deviations are contained in the total transmission error xg(4p). The constant amount of the transmission emr x, does not influence the excitation of vibrations. The transmission error is transformed into a tooth deformation xZ( by subtractio
20、n of xgm and the addition of the mean tooth deformation x,. This mean tooth deformation results from the nominal tooth load Fbt and the mean value k, of the tooth stiffness k(cp): Fbl X, =- km 4 STD-AGUA 4bFTM7-ENGL 177b = b87575 0004907 818 m The following equation is used for the tooth force F,(rp
21、) depends on the angle of rotation rp: The effective tooth deviation fi(rp) resultant from the load distribution is then determined by: The Fourier series of the curves k(p and fdp) are provided by LVR in an input data file for GDA. This file also contains the other parameters (inertias, masses, tor
22、sional stiffnesses, damping) needed for the model used in GDA. These parameters are calculated in the background using the input data of LVR. The time-varying tooth deformation xxq) (which is equal to the time-varying transmission error) is also provided for in GDA. This curve can be approximately u
23、sed in connection with the mean tooth stiffness as a force excitation if the time-varying part of the tooth stiffness is neglected. The influence of the parametric excitation is neglected with this proceure. But this approximation is exact enough if only a small time-varying transmission error occur
24、s in comparison to the mean tooth deformation. This calculation method can be chosen also in the program GDA to increase the speed of calculation because the iteration based on equation (IS) is not necessary. The tooth force curve Calculated in GDA for a given input speed is again provided for use i
25、n LVR. The dynamic load dismbution can be calculated in a repeated run of the program LVR with the use of this tooth load curve instead of the nominal load. The high frequency vibrations caused by the tooth mesh are superimposed on low frequency vibrations caused by external excitations if .such exc
26、itations are additionally used in GDA. The tooth force curve provided by GDA for LVR then contains the curve of that mesh position in which the maximum dynamic tooth load appears. the following analysis is shown in Fig. 4. But the neglectable transmission error amplitude is valid only in the unioade
27、d situation which is of little practical use. An increased contac! ratio occurs as soon as the gears are loaded. This is the result of the off line of action tip edge contact. Only a very small spacing exists between the tooth tip edge and the opposite tooth flank shortly before and after the theore
28、tical contact on the line of action. Fig. 4 Example of the analyzed HCR-gearing Fig. 5 shows in a greatly enlarged view this situation c;1 the HCR-gearing. The small spacing can be fully filled by the deformkion of the loaded tooth pairs thus causing the tip edge contact. The duration of additional
29、tip edge contact increases with increasing load. A larger additional contact duration OCCU in the case of large numbers of teeth than iri the case of low numbers of teeth due to the lower flank curvature. .Because of this, a relative long duration of additional tip edge contact occurs on HCR-gears b
30、ecause they have to be manufactured with larger numbers of teeth to reduce the undercutting problem. Finally, the calculated load distribution considers the influences of deforrnations along the line of action and along the face width as well as additional dynamic loads caused by internal and extern
31、al excitation sources. The method of the combined use of LVR and GDA will be shown in the following example. Fig.5 Spacing on the tip edge short before the begin c contact in the unloaded situation 5. Analysis of high contact ratio spur gearing The use of high contact ratio spur gears is often conne
32、cted with the attempt to minimize the amplitude of the transmission error with the contact ratio E= = 2. Two tooth pairs are always in contact and little change of tooth stiffness occurs. The result is a theoretically neglectable transmission error amplitude. Such a gear set, which will be the subje
33、ct of 5 o2 Oi Oh OBC PMhacamdparhmSepW Fig. 6 Transmission error on the HCR-gear set; without tip relief; withfull design load A short period with three teeth in contact occurs if the gearing is designed with a theoretical contact ratio The variation of the transmission em for the resulting HCR-gear
34、 set under design load is shown in Fig. 6. The negative effect of the additional tip edge contact can be subsequently compensated for with short tip relief. The well balanced transmission error me obtained with those tip reliefs is plotted in Fig. 7. This nearly optimum curve is only valid around th
35、e design load. Gears are often used under conditions with lower loads than the design load. The transmission em shown in Fig. 8 occurs if the analyzed = 2. wing is loaded at one-third of the design load. Q.7 Transmission error on the HCR-gear set; wi relief; underfull design load 2 . .i . # , *. . .
36、i I LAbyaL; Oi 0.4 0;6 OBc 1.0 A Pathof eorud pu * Flach The tip reliefs are too large for this case as less tooth defonnation occurs. The transmission error curves are the results of the calculation of load distribution with LVR with the use of the design load as nominal load. These transmission er
37、ror curves were taken to calculate the wes of the dynamic factor versus the input speed with the program GDA. The simple torsional model of the gear pair was used with a damping ratio of 0.06. The resultant curves of the dyhamic factor are shown in Fig. 9. The main resonance is reached at an input s
38、peed of ni,= 6500 min“. b dvnamic faor - LwWdihilhlidd-M -bd.dwhhMdaM o - B s s! E D Fig. 9 Dynamic factor on the HCR-gear set with tip relief; forfull and a thrd of design load A considerably higher dynamic factor occurs in the case of the reduced load due to the higher amplitude of the transmissio
39、n error. The effect of the dynamic forces on the tooth loads will be shown for the example of an input speed of n, = 2150 min- which is at one-third of the main resonance speed- The me of the tooth force within one mesh period is plotted in Fig. 10 for the case of the full design load. As expected,
40、the variation of the tooth force is very low. A much higher amplitude of the tooth force occus in the case of the loading with only a third of the design load as shown in Fig. 1 1. 01) 0.1 02 O3 0.4 OS O 0.7 Ob 0.9 11) -p.ilod Fig.10 Effective tooth force related to the nominal load within one mesh
41、period at ne 2150 min- and full design load Fig.8 Transmission error on the Ha-gear set; wizh tip relief; with a third of desim load 6 STD*AGMA SbFTN7-ENGL 199b = Ob87575 0004909 690 amc lactor OP 0.1 02 03 0.4 015 On 0.7 0:s 0.9 li0 hpmd ig. 11 Effective tooth force related to the nominal load with
42、in one mesh period at n,= 2150 min-; third of design load The curves of the tooth forces are now used as input for the repeated calculation with LVR, the last step in the combined calculation. The resultant dynamic load distribution is shown for the case of full design load at ni,= 2150 min- in Fig.
43、 12. There are only low variations of the load due to the low force variation. The gear has lead crowning to avoid high edge loads due to misalignment. This causes the higher load intensity in the middle of the face width (tooth deviations are neglected). The distribution of the Hertzian pressure, w
44、hich is shown in Fig. 13, can be calculated simultaneously with LVR. . Load diSnwuon Que 10 GXRXlJ N 80 mLlmm bod huu.=34732 Nhm O h-.no L=0.15-1.85 Fig. 12 Load distribution on the Ha-gear set at nh= 2150 min“ andfull load I Fig. 13 Distribution of Hertzian pressure at nin= 2150 min- andfull load T
45、he pressure at the beginning of contact is even larger than the pressure in the range of the operating pitch circle with the maximum load intensities. This is caused by the influence of the flank curvature which is considered in Is0 6336 wirh the factor for the single point of contact. The distribut
46、ion of the resultant maximum local tooth root stress on the critical section (at the 30 -tangent) is shown for the pinion in Fig. 14. The increase from the beginning of contact through the center of the line of action is caused by the increasing bending moment arm of the contact point. This level in
47、creases until the end of contact, but the load decrease leads to a nearly constant root stress in therangeL 1. The load distribution in Fig. 15 is significantly more unbalanced than those from Fig. 12. The dynamic load dismbution in Fig. 15 shows the influence of the larger amplitude of the tooth fo
48、rce variation in the one-third design load case. A better view of the resultant load variations is visible in another perspective in Fig. 16. Analogous variations are obtained for the distributions of Hertzian pressure and root stresses which are not shown. These large Variations are not cnticai in
49、regard to the maximum pressure and root stresses in the case of only a third of design load. These values do not exceed the clearly higher values in the case of full design load. But the load variations in the case of reduced load do have a considerably negative effect on the excitation of gear noise. The noise excitation level is 15 dB higher in the case of a third of design load in comparison to the levei in the case of the much higher fui1 load. Fig. 14 Distribution of the maximum local tooth root stress min-; a third of design load 7 STD*AGMA S
