1、 STD-AGHA 94FTM8-ENGL 1594 M b87575 OO4bO4 3bb M 94FTMt Reference Point, Mesh Stiffness and Dynamic Behavior of Solid, Semi-Solid and Thin Spur Gears by: J. Brousseau, C. Gosselin and L. Cloutier Laval University, Quebec, Canada Rimmed American Gear TECHNICAL PAPER COPYRIGHT American Gear Manufactur
2、ers Association, Inc.Licensed by Information Handling ServicesSTD.AGMA SVFTMB-ENGL 1994 m b87575 0004b05 2T2 m Reference Point, Mesh Stiffness and Dynamic Behavior of Solid, SemiColid and Thin-Rimmed Spur Gears Jean Brousseau, Claude Gosselin and Louis Clotier, Laval University, Quebec, Canada nie s
3、tatements and opinions contained herein are those of the author and should not be construed as an officiai action or opinion of the American Gear Manufacturers Association. ABSTRACT Most of the models proposed to predict the dynamic behavior of gears do not take into account the blank flexibility an
4、d its naturai modes of vibration. In those models, a gears pair is reduced essentially to two rigid disks coupled by flexible teeth, which we caii Rigid Disks - Flexible Teeth“ models (RD-FT). Contacting teeth pairs are replaced by a spring attached to the disks and acting in the normal ktion at the
5、 contact points. The mesh stiffness, deduced from the stiffness value of each contacting tooth, depends on the position of the contact point and the number of teeth pairs in contact. Regardiess of the inhinsic principle of the method and the boundary conditions applied, the displacement of the loade
6、d point is measured relative to areferencepoint. The compliance of the tooth is largely afected by the choice of the reference point and so is the prediction of the dynamic behavior of the gearset. This paper presents an investigation of the reference point in relation to meshstiffness calculation a
7、nd RD-F dynamic models. The influence of the gear body flexibility is considered and the paper presents results for solid, semi-solid and thin rimmed spur gearsets. The analysis is made on the basis of the natura frequencies, when finite element models of meshing spur gears are taken as reference. R
8、esults show that a reference point inside the gear blank yields excellent correationbetweenthenaturalfrequenciesextractedfromtheRD-lTandFE.A. models. Thatconclusion ismeaslong as no coupling effect between rigid body and gear body modes occurred. The reference point is located at the same place whet
9、her the gear body is solid or thin rimmed Copyright O 1994 American Gear Manufachirers Association 1500 King Street, Suite 201 Alexandna, Vuginia, 223 14 October, 1994 ISBN 1-55589443-X COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services STD-AGUA 94FTUB-E
10、NGL 1994 b87575 0004bb 139 REFERENCE POINT, MESH STIFFNESS AND DYNAMIC BERAVIOR OF SOLID, SEMI-SOLID AND TBIN- SPOR GEAS. Jean Brousseau, Ph. D. Candidate Dr. Claude Gossellin, professor Dr. Louis Cloutier, professor Department of Mechanical Engineering Laval University, Quebec, QC, Canada, GlK-7P4
11、Half of Weber contact width Face width Normal load between teeth pairs Polar inertia of the pinion and the gear Mesh stiffness Stiffness of teeth pairs O and -1 Torsional stiffness of pinion and gear shafts Bending stiffness of pinion and gear shafts Gear body stiffness Mass of the pinion and the ge
12、ar Base pitch radius (pinion and gear) Normal deformation at the contact . point Inertial coordinate (pinion and gear) Distance from the line of action Axial coordinate Rotation of the pinion and the gear 1.0 IXCRODUCTION Gears have been recognized for a long time as an internal excitation source Z
13、of vibration and as a radiating medium 1131. Gears cause vibration mainly because of mass unbalance, transmission error, variable mesh stiffness, and backlash. In running condition, the pinion and gear mass centers move around a static equilibrium position because of the variable loads between teeth
14、 pairs in contact. In addition, elastic waves are generated and propagate through the gear bodies; resonance phenomena may occur if the gear blank is excited at certain frequencies 4, 81 . Ozgven and Houser 12 have reviewed more than 200 hundred mathematical models used in gear dynamics. Most of the
15、 models proposed to predict the dynamic behavior of gears do not take into account the blank flexibility and its natural modes of vibration. In those models, a gear is reduced essentially to two rigid disks coupled by flexible teeth, which we call “Rigid Disks - Flexible Teeth“ models (RD-FT). Conta
16、cting teeth are replaced by a spring attached to two rigid disks and acting in the normal direction at the contact points. The mesh stiffness, deduced from the stiffness of each contacting tooth, depends on the position of the contact point and the number of teeth in contact. Elementary tooth stiffn
17、ess is obtained from the ratio of the applied load to the displacement of the contact point. For spur gears, the problem is normally considered two-dimensional rather than three-dimensional as it is the case for more complex gear types. Several methods are available to calculate the compliance of ge
18、ar teeth: the complex potential method SI, the finite element analysis (F.E.A. ) 14, boundary element method 91, and other methods based on the strength of material and elasticity theory using or not the energy principle 6, 161. Whatever the intrinsic principle of the method and the boundary conditi
19、ons applied, the displacement of the loaded point is measured relative to a reference point. The compliance of the tooth is largely affected by the choice of the reference point and so is the prediction of the dynamic behaviour of the gear pair. If the reference point is chosen at the root radius, t
20、he compliance is smaller than if the reference point is situated at the bore radius. The determination of tooth stiffness is also an essential step in the calculation of load sharing. For static load sharing predictions, every displacement that makes teeth move relative to each other must be taken i
21、nto consideration 14, 151. If the question of the reference point can be answered for static calculations, it is not yet 1 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services STD-AGHA SqFTHB-ENGL L97Li E resolved for dynamic prediction 17, 151. In our kno
22、wledge, no study has investigated the question of the reference point in relation to the difficult problem of the dynamic behaviour of gears. This paper presents an investigation of the reference point in relation to mesh stiffness calculation and RD-FT models. The influence of the gear body flexibi
23、lity is also considered. Globally, the analysis is made on the basis of the natural frequencies. F.E.A. models of meshing gears are taken as reference. The F.E.A models are used to study the static behavior of meshing gears, to evaluate the mesh stiffness relative to different reference points and t
24、o calculate the natural frequencies of the gearsets. Gears are not in rotation but fixed in a given contact position. The stiffness values evaluated with the F.E.A. models are used in RD-ET models and, after the homogeneous dynamic equations of RD-FT models have been established, the natural frequen
25、cies are extracted and compared to those obtained with the F.E.A. models. This paper presents results for solid, semi-solid and thin rimed spur gearsets. 2.0 STATIC BEHAVIOUR OF SPUR U 2.1 Finite element model of solid, semi-solid and thin rimmed spur gears We assume that F.E.A. adequately represent
26、s the elastic behaviour of solid bodies. We also assume that the gears are mounted on shafts and for this reason, boundary conditions are imposed on bore surfaces. Table 1 summarizes the geometry of the three gearsets studied. The pinion body is solid in every case. The mating gears are meshed using
27、 solid quadratic elements for the bodies and the teeth, and inter-9 interface elements for the contact zones 11 . The interface elements provide coupled displacements in the direction normal to the meshing teeth surfaces. Figure 1 presents half of the finite element models used for the thin rimmed g
28、ear. In order to correctly predict the contact phenomena, the contact zones are finely meshed. Figure 2 shows the contact position studied in this paper. According to that figure, adjacent teeth are referred to as follows: tooth -1 is going out of mesh, tooth O is the main tooth in contact and tooth
29、 1 is coming into mesh; a median line divides a tooth in two equal parts from the bore to the tip of the tooth: the median contact point is at the intersection between the line of action and the median line. 2.2 Contact pressure and deformation Let us assume the bore surface of the gear is completel
30、y fixed and a displacement equivalent to a rotation is applied to the bore surface of the pinion. Let us also consider that there is no interface element between teeth pair O, such that teeth pair -1 is the only one in contact Figures 3 and 4 present the contact pressure and deformation at mid face
31、width of teeth pair -1 obtained with finite elumnt models of mating gear using interface elements between teeth pair -1. Results are compared with those obtained with Hertz theory and Weber formulae. On figure 3, the y coordinate is the perpendicular distance between the line of action and the point
32、 where the contact pressure is measured (see figure 2), and b is half of the Weber contact width. As the results in figure 4 show, contact deformations calculated with Weber formulae 1161 are just between those obtained for solid and semi-solid gears. Ob87575 ODOLib07 075 E Figure 5 shows how the lo
33、ad is distributed across the face width. For the solid gear, the load is almost uniformly distributed while for the senu- solid and for the thin rimmed gears, the load decreases from the center to the side of the tooth. Table 1: Geometry of the gearsets I D Pipion I ku l Module (nun) 3.175 Pressure
34、angl e Addendum factor Dedendum factor Cutter end radius (mm) Face width (mm) 88.9 Bore radius (mi 12.7 Figure 1: Half of the finite element model of the thin-rimed gear. Medien line Median contact pant Contact point i Figure 2: Meshing teeth pairs. 2 COPYRIGHT American Gear Manufacturers Associatio
35、n, Inc.Licensed by Information Handling Servicesa Contecf pressure a FIZ P I PmaxHertz 1.2 4 -Thin nrnme 0.8 i 0.6 , -Herb- i 04 i t I o2 i I t / b -1.2 -1 4.8 4.6 4.4 -0.2 O 02 0.4 0.6 0.1) 1 12 yh Figure 3: Contact pressure between teeth -1. Contact deformation at F12 (pm) 10 , . i 9 -fbl O 100 Mo
36、 300 400 500 Load per unit of face width (Nlmm) Figure 4: Contact deformation of teeth pair -1. W I Wmoy 14 Load distnbution along the contact line I - Center of the tooth (Z I F=O) O8 07 O o1 02 03 04 05 06 ZIF Figure 5: Load distribution along the contact line of teeth pair -1. 2.3 Teeth and gear
37、body flexibility Again, let us assume that the teeth pair -1 is the only one in contact, the bore surface of the gear is completely fixed and a displacement equivalent to a rotation is applied to the bore surface of the pinion. For the solid gearset, the normal displacements (projected in the direct
38、ion of the line of action) of the median lines situated at mid face width (Z = O), from bore to tip, appear in figure 7 for the pinion and figure 6 for the gear. Results for the median lines situated at 2 = F: (side of the gears) are not shown because they are almost identical to the curves appearin
39、g ir. fique 6 and 7. Noml displacement (pn) 30 25 20 15 10 5 O Pinion of fite soli gearset Normal displacement of median hnes Teeth pair 4 m contact, 400 Nimm Median contact point, tooth -1 - Median contact point, tooth O - Root radius -. 10 15 20 25 Y) 35 40 45 50 Radial position (mm) Figure 6: Pro
40、jected displacement of pinion median lines (solid gearset). Normal displacement (pm) Gear of the solid gearset Nom1 displacement of median lines Teeth pair -1 in contact, 4oON/mm ,-I Median contact point tooth O-, Median contact point, tooth -1- , Tooth o - Tooth 1- Departure point - of gaiil body 0
41、1 - 3050 70 90 110 130 150 170 190 Radial position (mm) Figure 7: Projected displacement of gear median lines (solid gearset). From the results in figure 6, two phenomena are readily apparent: i) the body of the blank deforms linearly from ,the bore up to a point called depazture point“; ii) from th
42、is point on, the loaded tooth deflects in the direction of the applied load and the displacement curves for the other two teeth do not follow a straight line, but rather deviate in a direction opposite to that of the loaded tooth. Similar behaviour is clearly visible in the results for the gear in f
43、igure 7. In this case the unloaded teeth deviate from a straight line in the direction of the loaded tooth. For the thin rimmed gearset, the displacements projected in the direction of the line of action of the teeth median lines situated at three different axial position (2 = O, F/4 and F/2), from
44、bore to tip, appear in figure B for the solid pinion and in figures 9 and 10 for the thin rimmed gear. As we have seen, load is not uniformly distributed along the face width, and therefore, the median lines behave differently depending on their axial position. As figures 9 and 10 show, the thin rim
45、ed body is much more flexible than the solid one but still deforms linearly from the bore up to the departure point. 3 STD-AGMA 74FTMB-ENGL L794 D Ob87575 0004b08 TO1 D COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesNormal displacement (pm) 30 Sdid pi
46、nion in mesh with thin rimmed gear To the results are shown in figure 12 for tooth -1. From that figure two phenomena are readily apparent: i) the deformation curves of the solid and semi-solid gears are very close to each other; ii) the radial position of the departure point is at the same location
47、 regardless of the gear considered. Noml dbplacsmanlpm) Sdid, semiGdid and thin rimmed gears Teeth pair -1 in contact, 400 Nimm Rawlls atz4 ,Tm -1 100 ao 60 40 20 O 30 50 70 90 110 130 150 170 190 210 Radial position (mm) Figure 11: Projected displacement of the solid, semi- solid and thin rimmed ge
48、ars median lines. Normal displacement (pm) 12 -. Solid, semi.oolid and thin-rimmed geats Teeth wir -1 in contact, 100 Nimm Straight line defomration of gear body subtracted 14 1 10 Resu ii the deformation of the body relative to a chosen reference point is the displacement measured between the media
49、n contact point and the reference point, on the straight line deformation of the body. Nonnal displacement I 1 1 Radial position Pigurc 13: Tooth bending and gear body displacement. 4 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services. STD-AGMA 74FTMB-ENGL 2774 b87575 0004bL bbT Let us assume that adjacent and unloaded teeth follow, from bore up to tooth. tip, the straight line deformation of the body. The following mesh stiffness model for two meshing teeth pairs in contact is propose
