1、97FM14 Refinements in Root Stress Predictions for Edges of Helical Gear Teeth by: Aaron Dziech and Donald R. Houser, Ohio State University TECHNICAL PAPER COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesRefinements in Root Stress Predictions or Edges o
2、f Helical Gear Teeth Aaron Dziech and Donald R Houser, Ohio State University The statements and opinions contained herein are those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract Industry demands for higher power de
3、nsity gearboxes to improve product performance require utilization of the load carrying capability of every inch of available face width. As a result, load distributions and the resulting root stresses are of considerable importance. Current analytical methods include three dimensional finite elemen
4、t analysis (FEA) and combinations of Wellauer and Seiregs moment image method with two dimensional boundary element analysis or beam bending formulas. Three dimensional FEA, although reliable, is time consuming and of great expense in comparison to the use of classical techniques and approximations.
5、 In helical gears these approximations give reasonable estimates of root stress distributions along the face width but lack the accuracy to design to engineering limits. Based upon finite element and experimental results, a discrepancy in the approximate methods is the stiffness change in the normal
6、 plane associated with the ends of helical gear teeth. Tooth stiffness is lower at the acute edge, where the normal force associated with tooth contact protrudes beyond the transverse edge of the gearon the back side of the tooth. The opposite result occurs at the obtuse edge, where the normal force
7、 is not producing beyond the transverse edge of the tooth. Analytical and experimental studies for a limited number of cases have been completed. The experimental results are used to verify the simplified three dimensional FEAparametric study on the effect of helix angle. The parametric study result
8、s, when completed, will be used to determine a root stress correction factor. The focus of this paper is on the initial parametric results and experimental studies, with an introduction to possible correction techniques. The correction factors are currently being researched and will be the topic of
9、future publications. Copyright Q 1997 American Gear Manufacturen Awciation 1500 King Street, Suite 201 Alexandria, Virginia, u314 r 1 O November, 1997 ISBN: 1-55589-708-8 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesRefinements in Root Stress Predic
10、tions for Edges of Helical Gear Teeth Aaron M. Dziech Dr. Donald R. Houser The Ohio State Universii) Gear Dynamics and Gear Noise Research Laboratoq Columbus, OH 43210 Introduction Variation in root stresses at the edges of helical gear teeth due to geometric differences is conceptually realistic an
11、d easily grasped. By placing a point load at each edge of a helical gear tooth as in Figure 1, it is easy to see two distinct geometries are involved. At the left end the normai plane of the load protrudes beyond the tooth edge and is referred to as the acute edge. This edge can be thought of as les
12、s stiff than an equivalent spur tooth edge since less material exists directly behind the load. At the opposite end, called the obtuse edge, the load is supported by the additional material that extends beyond the plane of the load. This material can be thought of as a slight buttress, effectively m
13、alung the edge more stiff than an equivalent spur gear tooth. Past research efforts have mentioned this potential effect 1 J 2 and some work has been completed in this area 3. Current approximations are considered to be adequate for practical gear design, especially when safety factors and lead modi
14、fications designed to centralize contact are taken into account. However, these effects may become important in high power density applications and improved predictions are always beneficial to gear design. 0 The driving factor behind the onussion of edge effects in helical gear analysis is computat
15、ional speed. Full three dimensional finite element analysis can accurately predict root stresses and can include the edgereffects but is costly in terms of preparation time, program costs, and computational time. Special purpose gear design programs greatly reduce the computational time involved but
16、 rely more heavily on empirical mathematical formulations which do not easily lend themselves to the inclusion of edge effects. Therefore, the most desirable solution is to find an appropriate modification or adjustment to the current calculation techniques which includes the edge effects with a rea
17、sonable degree of accuracy. The goal of this research is to further develop the root stress prediction capability of the Load Distribution Program (LDP) 4 and the associated post processor, GGR4PH. LDP and- GGRAPH, computer programs for general gear analysis, are the result of research and developme
18、nt canied out at the Ohio State Universitys Gear Dynamics and Gear Noise Research Laboratory. Currently these programs account for stiffness reduction at the edges of gear teeth by application of Wellauer and Seiregs 5 moment image method to a two-dimensional boundary element model that analyzes the
19、 normal cross section of the gear tooth. For helical gears the edges of the teeth are assumed to be the spur equivalent, meaning that the tooth edge runs parailel to the nod plane of the tooth. This simplification is a reasonable approximation, especially for small helix angles. However, variations
20、in the stresses due to edge effects will increase with increasing helix angle. The results and information presented focus upon experimental data taken for a 14 degree helix angle gear, three-dimensional finite element analysis for an approximation of the gear tooth, and the helix angle parametric s
21、tudies generated using the finite element model. Based upon a comparison of the values and properties of the experimental and analytical results, the FEA model is believed to be adequate for use in the determination of a correction for LDP. Corrections are currently being researched and one prelimin
22、ary concept is presented. 1 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesBacksound The following is a brief sum of the calculations involved with root stress predictions in the current calculation method. Determination of load distribution for the d
23、esired positions in the mesh cycle is accomplished using the elastic body contact method of Conry and Seireg 6. Jarhllos 7 solution for bending moments in an infinite cantilever plate, in conjunction with the moment image method of Wellauer and Seireg, uses the calculated load distribution to determ
24、ine the moment distribution at the root of the gear tooth. The appropriate stress concentration factor is determined using two dimensional boundary element analysis for the normal plane tooth geometry. A more complete description of the process is given by Clapper i. Several previous studies involvi
25、ng root stress measurement and prediction have shown good correlation between experimental results, finite element analysis, and gear analysis programs. Yoshida SI performed experimental measurements on 33 degree helical gears and found good agreement when proper misalignment was included in the LDP
26、 analysis. The same experimental data also shows good agreement with Borner 9. Clapper lo analyzed fully reversed root base stresses on 33 degree helical gears with the finite element program of Vijayakar 1 i. However. the focus of thesc experiments was not to predict edge effects. The current edge
27、effect srudies are a logical extension to these successful research efforts. Experimental Data Extensive experimental data has been collected for a 14 degree helical gear using the instrumentation and data acquisition system developed by ONeal 121. A series of strain gages have been placed at each e
28、dge of a tooth at the critical radius determined by a two dimensional boundary element analysis 131. The test gearset consists of a 23 tooth pinion and 43 tooth gear with 4.621 nonrial diametral pitch, 25 degree normal pressure angle, 3 inch face width, operating at a 7.5 inch center distance. The o
29、uter diameter and root diameter of the pinion are 5.704 and 4.663 inches respectively. The normal chordal thickness on the pinion is .345 inches at a diameter of 5.227. The corresponding values for the gear are 10.168.9.127 and 35 at a diameter of 9.773 inches. The gear geometry used in this experim
30、ent is quite advantageous for several reasons. The coarse pitch allows enough room in the root of the gear for proper placement of the strips of gages. The 3 inch face width insures that moment reflection from the opposite edge of the tooth will not convolute the test data. Placing the gages on the
31、gear (10.168 OD) minimizes the amount of twisting that occurs in the gage strips. in order to identify the effect of loading at various distances from the edge of the tooth, a piece of shim stock is placed on the tooth to simulate point loading as shown in Figure 2. The selected shim stock is thick
32、enough so that the only point of contact during mesh occurs at the shim location but is less than the backlash setting. The shim is also made wide enough to support the contact load without crushing. The length of the shim is selected such that contact is maintained throughout a large portion of the
33、 mesh cycle without the risk of interfering with or damaging the gages. The selected shim for these experiments is 0.1X0.004X0.4 inches. The shim is placed on the gear in the transverse plane and is held .in place by. a small amount of grease. The shim starts at the edge and is indexed towards the c
34、enter of the tooth by one shim width each test run. A sample of one data run is shown in Figure 3. in this particular case the shim is located at the edge of the tooth. Gage 1 is located closest to the edge of the tooth and each proceeding gage is located farther in on the tooth. As expected, gage 1
35、. located directly below the load point. shows the highest stress while gage 10, located over an inch away from the load point, shows no stress at all. The notch in the data is attributed to friction. The normal load on the tooth results in a tensile bending stress in the strain gage. Due to frictio
36、n, an additional load is seen in the data which changes sign dependent upon the direction of sliding. At the tip contact is sliding towards the strain gage, causing a compressive stress which subtracts from the bending stress. Once contact passes through the pitch point the direction of sliding reve
37、rses, and the friction force produces a tensile stress in the gage which adds to the bending stress. A complete set of data runs for tip loading at the acute and obtuse edges can be plotted together as shown in Figure 4 and Figure 5 respectively. Each data series represents a different point load lo
38、cation with the 2 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicesb distance of the shim centerline from the edge provided in the legend. For example, by examining Figure 5 it is reasonable to conclude that no moment image or edge effects are seen in t
39、he data when the point load is located at least 1.1 to 1.4 inches from the edge of the tooth based upon the fact that the stresses in the fillet . 1 , inches from the edge of the tooth are nearly zero and the peak stress at the load location is not changing for the 1.155 and 1.375 inch point load ca
40、ses. It is important to notice the differences in the characteristics of these two graphs, especially the differences in the shape of the stress distributions at the edge. Figure 6 shows stress distributions for loading at each edge. The acute edge peaks early and appears to have a decreasing slope
41、in the same region that the obtuse edge results are still rapidly increasing. Finite Element Analysis An ANSYS finite element program, incorporating some geometric simplifications, is used for verification of the experimental data and to perform parametric studies. The model is three-dimensional and
42、 uses a trapezoid and fillet to approximate the tooth form. If considerable care is taken the model can be made to match the tooth form quite closely, including the critical stress region of the trochoid as shown in Figure 7. Mesh size and computation time are kept to a minimum by modeling the fille
43、t only on the tensile side and working with only one edge at a time using 2 inches of face width. This 2 inch face width is sufficient for this case based on experimanetal data and analytical studies. By utilizing the ANSYS programming language, a variety of tooth forms and helix angles can be inves
44、tigated without additional mode I preparation ti me. Using the same normal cross section and the appropriate point load, the finite element model will simulate the experimental set up. In addition the helix angle can be removed from the model. thus adding an equivalent spur approximation for compari
45、son of edge effects. Figure 8 shows the finite element model results for the acute edge, obtuse edge, and a spur gear with the same normal cross section with load applied near the edge of the tooth. As expected the acute edge has the highest peak stress, the obtuse edge the lowest, and the spur gear
46、 lies in between. Figure 9 is a similar plot for loading .5 inches from the edge. Although differences are small on a percentage basis it is important to keep in mind that these results are for a rather modest helix angle of 14 degrees. Comparison of FEA and Experimental Data Direct comparisons of t
47、he finite element results to experimental data is a difficult task. Due to the uncertainty in gage location lo, actual experimental loads, and the effect of the geometric approximations it is not surprising to see some discrepancies in magnitude between the analytical and experimental data. Of great
48、er importance than absolute magnitudes in this study are the qualitative traits shared by both the experimental and FEA results. Figure 10 is a comparison of the central loading case at the tip of the tooth. In this case, the load is applied far enough from the edge of the tooth that no edge effects
49、 are present. Although considerably higher in magnitude, the shape of the FEA stress distribution matches fairly well with the experimental data. Figure 11 shows a similar trend in shape for loading on the acute edge. Occasionally the comparisons are not quite as good as expected but still show some relative shape characteristics, as in the data shown in Figure 12 for-loading .5 inches in from . the obtuse edge. Based upon these comparisons, the FEA model is a viable tool for performing parametric studies to generate a correction routine and
