1、C O I 98FMl A Method for Predicting the Dynamic Root Stresses of Helical Gear Teeth by: D.R. Houser and J. Harianto, Department of Mechanical Engineering, Ohio State University American Gear Manufacturers Association TECHNICAL PAPER COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by
2、Information Handling ServicesA Method for Predicting the Dynamic Root Stresses of Helical Gear Teeth Dr. Donald R. Houser and Jonny Harianto, Department of Mechanical Engineering, Ohio State University The statements and opinions contained herein are those of the author and should not be construed a
3、s an official action or opinion of the American Gear Manufacturers Association. Abstract The AGMA dynamic factor has traditionally been treated as a dynamic “load” factor where tooth load is the sum of all of the tooth forces that are applied along the plane of action at any instant in time. Knowing
4、 the total load, however, says little about the degree of load sharing or the load position on a tooth when the dynamic load is a maximum. Hence, the dynamic load factor does not directly address the value of either dynamic contact stress or dynamic root stress; quantities that a gear designer shoul
5、d be more interested in. This papers main focus is the prediction of dynamic root stresses. The approach presented in this paper is a relatively simple methodology that does not use finite elements. The method merges the capabilities of a sophisticated load distribution and dynamic excitation predic
6、tion method with a time domain gear dynamics simulation. Because a time integration type of simulation is used, the method can handle both steady state and transient inputs. Inputs that are possible include tooth profile and lead modifications, misalignments, and spacing errors. These can take on av
7、erage values or can have discrete changes applied from tooth to tooth. Examples of the use of some of these errors are provided in this paper. Copyright O 1998 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 October, 1998 ISBN: 1-55589-719-3 COPYRIGHT
8、American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesA Method for Predicting the Dynamic Root Stresses of Helical Gear Teeth Dr. Donald R. Houser Professor, Department of Mechanical Engineering The Ohio State University Columbus, OH 4321 O Jonny Harianto Research Eng
9、ineer, Department of Mechanical Engineering The Ohio State University Columbus, OH 4321 O INTRODUCTION The dynamic factor used in gear design has been the subject of numerous studies, most of them analytic in nature. Papers by Ozguven and Houser l and Harianto and Houser 2 document many of these ana
10、lytical studies. In the paper by Harianto and Houser, three types of dynamic factors are used for spur gears. These are the dynamic load factor used by AGMA, a dynamic load intensity factor that is analogous to a dynamic contact stress factor, and a dynamic bending moment factor that is analogous wi
11、th a dynamic root stress factor. This paper further extends the analyses, such that a dynamic root stress factor may be predicted for both spur and helical gearing. This factor accounts for not only dynamic effects. but also load distribution effects in both the transverse and axial directions. The
12、original method proposed by Harianto and Houser 2 is only appropriate for spur gears while the procedure presented here is also appropriate for helical gears. METHODOLOGY Previous work in predicting dynamic root stresses of helical gears is very sparse. Today it is possible to create a dynamic finit
13、e element model that has both the resolution to predict load distribution and root stresses and the ability to function in a time domain dynamic solution. However, this model would still be extensively time consuming, both in creating the model and in executing it. The only approach of this type tha
14、t the authors are aware of uses Vijayakars finite quasi- prism methodology, but this method has only been applied to two-dimensional dynamic problems 3. Another approach that uses vibrational normal modes has been applied by Boerner 4. Boerners method uses a Fourier series analysis coupled with know
15、ledge of natural frequencies and mode shapes of the system to solve for steady state responses. Sainsot, Velex, and Berthe 5 also have performed similar analyses using normal mode techniques. The time domain approach presented by Harianto and Houser 2 uses a load distribution prediction program to o
16、btain the transmission error and mesh stiffness variation as functions of gear rotation. These quantities are then used as excitations to a multi- degree of freedom time domain dynamics. simulation. This method works well for spur gears but has limitations for helical gears because it is extremely d
17、ifficult to carry along all modeled points of contact in the dynamic simulation. The approach presented in this paper allows for the prediction of time domain -1- COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicesvalues of both root and contact stresses
18、along each contacting line and at any root positior,. The basis of the dynamic portion of the analysis presented here is the dynamic transmission error method developed by Ozguuen and Houser 6. The computer program DYTEM that is used to apply this methodology uses a Runge-Kutta time integration proc
19、edure to solve the six-degree of freedom dynamic model presented in Fig. 1. Excitations to the system are time varying mesh stiffness, k(t) and displacement transmission error input, e(t), hat are depicted in the first segment of Fig. 1. Nonlinearities such as loss of contact and back tooth contact
20、are predicted using this methodology. However, when heavy nonlinearities such as those occurring in gear rattle are being simulated, chaotic solutions and/or non-convergence may arise such that special care must be taken when applying Runge-Kutta type algorithms (Barlow, Padmanabhan and Singh) 7. Fi
21、g. 2 shows the scheme for predicting the dynamic root stresses of helical gear teeth. First, the static transmission error and mesh stiffness variations are predicted using the Load Distribution Program (LDP) 8. Then these values are used to compute dynamic loads (the values used to compute dynamic
22、load factors in rating procedures) using the time domain simulation called DYIEM 6. Now the load distribution program is re-run with the loads that are predicted with the dynamics program. This procedure works well in most cases, but will have problems if the dynamic loads are so great that there ar
23、e significant load distribution shifts when going from the static to the dynamic situation. Experience has shown, however, that for reasonably aligned helical gears, these situations seldom occur. Brief descriptions of the programs used are provided in the Appendix of this paper. SPUR GEAR EXAMPLE A
24、 spur gear example will be used to present verification that the new method provides similar results for predicting dynamic root stress factors as those obtained with Harianto and Housers 2 method that predicts dynamic tooth bending moments. Here we define dynamic root stress factor (DRSF) and dynam
25、ic bending moment factor (DBMF) as follows: Peak dynamic root stress Peak static root stress DRSF = Peak dynamic bending moment Peak static bending moment DBMF = where bending moment is found by computing the .; product of the tooth normal force and the distance from the force application to the cri
26、tical radius in the root region. The geometry of the spur gear set used in this example is shown in Table 1. This set has the same 1:l ratio geometry as was used by Rebbechi, et al 9. The gears in the analysis were run with no profile modifications so that at the transitions from single to double to
27、oth pair contact there are steps in the loading. This situation is far more severe than most other loading cases that are encountered with gears that have appropriate tip and root relief to reduce the step inputs lo. In reality there is corner contact that must also be taken into account at these st
28、ep locations l 11. Corner contact analysis reduces the steps, but results in impacts that are undesirable. The techniques described here have the ability to deal with corner contact, but for this papers purposes corner contact is not considered. Fig. 3 shows the static transmission error and mesh st
29、iffness values that are sent from LDP to DYIEM. Fig. 4 shows the dynamic torque that is computed by DYTEM and then used by LDP to compute dynamic stresses. In all simulations, the damping ratio value of 0.06 is used for each uncoupled vibration mode. Fig. 5 shows two modeling situations predicts the
30、 dynamic bending moment and the secon one using the original DYTEM procedure that o being the new procedure using a program called DLDP. In each case, the fine line is the predicted result for static loading and the bold line is the predicted dynamic response. The results show that the shapes of the
31、 curves are practically identical, thus showing for these spur gears that the two models give similar results. Note that the peak dynamic stress does not occur at the same position in the meshing cycle as the peak static stress. Although not shown, when computing the dynamic root stress factor with
32、DLDP, we must check the root stresses across the entire face width of the gear prior to taking the ratio between the dynamic and static values. Fig. 6 shows a similar comparison, only at a speed at which one of the system natural frequencies is being excited, thus giving a much larger dynamic factor
33、. Fig. 7 shows the effect of speed on the dynamic factors predicted by the two prediction .methods. These plots show that, except for the region at about 13286 rpm (6200 Hz), the predictions are almost identical, thus giving more credence to the new method. rn Tip relief was added to each of gears s
34、o that t transmission error was minimized. Fig. 8 shows t profile shapes that provide “optimum“ tip relief and Fig. 9 shows the gears predicted bending moment and root COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services8 i stress. It is interesting to not
35、e that for the speed shown, the dynamic factor is less than one, even though the gear is operating near a natural frequency. This is because the smooth transition in load sharing due to the tip relief greatly reduces the excitation at this frequency. Fig. 10 shows comparisons of the speed effects on
36、 the dynamic factors predicted by the two methods. For the optimized gear, the shapes of the curves for the two methods are extremely close and the peak dynamic factors are always less than 1 .lo. It is interesting to note that the dynamic factor for the gear is less than 1 .O over most of the speed
37、 range. HELICAL GEAR EXAMPLE The helical gear pair whose geometry is given in Table 2 is used to demonstrate the procedure for helical gears. The first set of runs is for an unmodified (no tip relief) pair of gears and the second set of runs is for a set with modifications that minimize transmission
38、 error. These gears have the same geometry as a test set run in a noise study by Oswald, et al 12 and the load used is one of the test loads used by Oswald. Since this load is roughly one-half of the rated load, it is expected that the predicted dynamic factors at this load will be greater than thos
39、e that would occur at the rated load. Fia 11 shows the static root stresses and the dynamic root stresses that are predicted for the unmodified helical pinion. Traces for the gear and pinion are very similar, so the gear root stress traces are not shown. The three traces are for different locations
40、across the face width, starting at the edge of one tooth and then progressing to the other edge in 50/0 increments (OTO, 50%, and 100% of face width). The steps in the stresses at the entering region of the 0% location and at the exit of the 100% location indicate that there is corner contact at the
41、 entering and exiting locations along the face width. As will be shown later, proper tip relief eliminates these steps. Note that the peak stress occurs at one edge of the tooth. Had appropriate lead crowning been applied, the peak stress would have occurred closer to the center of the tooth. Fig. 1
42、2 shows the effect of speed on the dynamic root stress factors for both the pinion and the gear. The resonant peaks are not as pronounced as with the spur gears and the dynamic values for the unmodified helical gears are in general less than those for the unmodified spur gears. When the optimum prof
43、ile modifications of Fig. 13 are applied to the gear pair, the transmission error excitation reduces by about 90%. A comparison of the dynamic root stress factors for the unmodified and modified helical gears is given in Fig. 14. Pinion dynamic factors are virtually the same as the gear factors at a
44、ll operating speeds. It is interesting to note that the modified gears dynamic factor is virtually 1 .O0 indicating that optimum modifications literally eliminate dynamic effects, even at resonance. Another interesting fact is that the static stresses of the modified gear pair increase by about 1.7
45、times from the unmodified gear due to the change in transverse load sharing that occurs when the tip relief is added. PROFILE ERROR EFFECTS Rather than performing a complex matrix of errors like that run by Harianto and Houser 2, profile error effects are studied by simply applying a pressure angle
46、error to the gear pair. In this case a rather severe pressure angle error of 0.001 in. from root to tip was applied (equivalent to an AGMA Quality 8 error) and the dynamic factor analysis was repeated for the unmodified helical gear. Fig. 15 shows the effect of speed on the pinions dynamic root stre
47、ss factor for both the unmodified gear pair and the pair with pressure angle error. The gears dynamic root stress factor is almost identical to the pinion values shown in the figure so it is not presented. As can be seen in the figure, there is very little change in the dynamic factor at low speeds
48、and there is some reduction in the factor at very high speeds. These relatively small changes in the dynamic factor are in spite of the fact that the error is fairly severe. However, when one observes the actual predicted root stresses, one finds a 80% increase that is the result of a different amou
49、nt of transverse load sharing. MISALIGNMENT EFFECTS Misalignment also has a much greater effect on load distribution than it does on dynamic factor. As was done for pressure angle error, 0.001 in. of misalignment was applied to the unmodified gear pair (equivalent to AGMA quality 8). This misalignment was extreme enough to cause the contact to shift heavily to one side of the tooth as is indicated in the plot of load distribution for one mesh position that is shown in Fig. 16. Fig. 17 shows the time traces for both static root stresses and
