Finite Element Analysis of High Contact Ratio Gear

Transcription

1 10FTM06 AGMA Technical Paper Finite Element Analysis of High Contact Ratio Gear By M. Rameshkumar, G. Venkatesan and P. Sivakumar, DRDO, Ministry of Defence

2 Finite Element Analysis of High Contact Ratio Gear M. Rameshkumar, G. Venkatesan and P. Sivakumar, DRDO, Ministry of Defence [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 Modern dayvehiclesdemand higher load carrying capacity with less installed volume and weight. The gears used in the vehicles should also have lesser noise and vibration. Even though helical gears will meet the requirement,they are prone for additional axial thrust problem. High contact ratio (HCR) is one such gearing concept used for achieving high load carrying capacity with less volume and weight. Contact ratio greater than 2.0 in HCR gearing results in lower bending and contact stresses. Previously published literature deal with studies on various parameters affecting performance of HCR gears but a comparison of HCR and normal contact ratio (NCR) gears with same module and center distance has not been carried out so for. This paper deals with finite element analysis of HCR, NCR gears with same module, center distance and the comparison of bending, contact stress for both HCR, NCR gears. A two dimensional deformable body contact model of HCR and NCR gears is analyzed in ANSYS software. ANSYS Parametric Design language (APDL) is used for studying the bending and contact stress variation on complete mesh cycle of the gear pair for identical load conditions. The study involves design, modeling, meshing and post processing of HCR and NCR gears using single window modeling concept to avoid contact convergence and related numerical problems. Copyright 2010 American Gear Manufacturers Association 500 Montgomery Street, Suite 350 Alexandria, Virginia, October 2010 ISBN:

3 Finite Element Analysis of High Contact Ratio Gear M. Rameshkumar, G. Venkatesan and P. Sivakumar, DRDO, Ministry of Defence Introduction A majority of the heavily loaded transmissions used in military applications use gears with a contact ratio less than 2.0. The contact ratios of these transmissions are in the range of 1.3 to 1.8. So, the number of teeth in engagement at any instant is either one or two. Many gear designs use increased pressure angle for increasing the load carrying capacity of gears with fixed module and center distance, but the contact ratio decreases. Tooth dynamic loads and noise increase due to decreased pressure angle. Hence increasing the load carrying capacity of gears for the above conditions can be done by the design of gears with a contact ratio greater than 2.0. High contact ratio gears having a contact ratio greater than 2.0 have load sharing between two or three teeth during engagement and less load per tooth [1]. The high contact ratio (HCR) gears guarantees that a minimum of two teeth always share the load. The variation of gear mesh stiffness for HCR gears is less than the normal contact ratio (NCR) gears; the transmission error for HCR gears is minimum compared to NCR gears. The literature survey indicated that HCR gearing was designed [2] and successfully used in helicopter transmissions [3], to improve power to weight ratio of the gear trains. This study deals with estimation and comparison of tooth root bending stress and contact stress over the path of contact for high contact ratio gears [1, 4] and normal contact ratio gears designed with identical module, center distance, gear ratio and face width using Finite Element Analysis. In order to overcome the numerical and convergence difficulties [5] involved a new single window modeling approach [6] using ANSYS Parametric Design Language (APDL). The contact stress and bending stress are compared and plotted for identical load conditions. Design for HCR gear pair Contact ratio of a gear pair is defined as the average number of teeth in contact during the course of engagement. The contact ratio of the gear pair plays an important role in increasing the load carrying capacity of gears. The contact ratio (CR) for any gear pair is given by equation 1. CR = where + r1 + a 2 r 2 1 cos2 α π m cos α r2 + a 2 r 2 2 cos2 α r1 + r 2 sin α π m cos α (1) r 1, r 2 are the operating pitch radius of the pinion and gear, α is the operating pressure angle; m is the module and a is the addendum (based on the operating pitch radius) which is equal to one module for standard gears. High contact ratio can be achieved by different ways namely: S Increasing the number of teeth; S Lowering the pressure angle; S Increasing the addendum factor. Figure 1, Figure 2 and Figure 3 show the variations of contact ratio with respect to above parameters. In order to achieve a contact ratio more than 2.0 for a gear pair with identical module, center distance, gear ratio and pressure angle the addendum factor of the gears pair is increased from a standard 1.0 m to 1.25 m. The entire tooth parameters of a HCR gear pair are calculated using a Matlab code and tabulated in Table 1. 3

4 modeled exactly using the procedure suggested by Buckingham [7]. An APDL computer language code in ANSYS was developed for generating an exact tooth profile with a trochoidal fillet. The trochoidal fillet form is generated from the dedendum circle up to the limiting circle, where it meets the involute profile at the common point of tangency and the involute profile extends up to the addendum circle. Figure 1. Contact ratio versus number of teeth Figure 2. Contact ratio versus pressure angle Table 1. Gear parameters SI No. Parameters NCR HCR 1. Profile Involute Involute 2. DIN accuracy class Module, m 2.5 mm 2.5 mm 4. Number of teeth in gear, Z 1 5. Number of teeth in pinion, Z 2 7. Profile correction in gear, X 1 8. Profile correction in pinion, X Center distance, C d 122 mm 122 mm 11. Reduction ratio, G r Addendum factor, Y a 13. Contact ratio, CR Face width, F 18 mm 18 mm Figure 3. Contact ratio versus addendum factor Figure 4 shows generation of a trochoidal fillet by the tip of the basic rack with a = 0 (sharp cutter). Coordinates of the trochoid are calculated using the APDL program using equation 2 through 8 as noted in [7]. The type of Trochoidal fillet changes with parameters of the cutter like type of cutter (pinion or rack), edge radius (a), addendum (b), pressure angle and profile correction required in the gear. Generation of gear pair model Spur gear geometry The profile of an involute spur gear tooth is comprised of two curves. The working portion is the involute and the fillet portion is the trochoid. The trochoid tooth fillet as generated by a rack cutter is R 2 (R b)2 R t 2 t θ t = tan 1 (R b) tan ψ t = R R b2 R 2 t R R 2 t R b2 2 (R b)2 R (2) (3) 4

5 Finite element analysis of NCR and HCR gear pair A two dimensional deformable body symmetric contact model of HCR and NCR gear pairs was modeled using a ANSYS APDL looping program and a quasi static Finite element analysis of the gear pair was carried out. The various parameters such as load sharing ratio, bending stress, and contact stress are estimated over the path of contact for both NCR and HCR gearing. Figure 4. Trochoid profile generation The co-ordinates of the actual fillet are determined from: R f = R 2 t + A 2 2 AR t sin ψ t (4) θ f = θ t + cos 1 R t A sin ψ t R f (5) x i = R f sin θ f (6) y i = R f cos θ f (7) θ f = δ t + θ f (8) where A b R is the cutter tip edge radius; is the distance between the pitch line of the cutter and the center of the outer edge; is the operating pitch circle radius; R t, R f is the radius vector of the trochoid and root fillet; ψ is the angle between the radius vector and tangent to trochoid; θ t is the angle between the radius vector and centerline of trochoid; δ is the angle between the center of the gear tooth and the center of the trochoid; θ f is the vectorial angle of fillet in reference to selected center line; θ f is the original vectorial angle of fillet; x i is the abscissa of the profile co-ordinate; is the ordinate of the profile co-ordinate. y i Assumptions for finite element models and meshing A list of the assumptions adopted in the present work is given below: S The gear material is assumed to be homogeneous, isotropic and elastic according to Hooke s law, and the material properties required for the analysis are Young s modulus of elasticity (2.1e5 N/mm 2 ) and Poisson s ratio (0.3). S The load distribution along the face width is assumed uniform and plane strain method is adopted. [8] S The effect of case hardness, case depth and the oil film thickness is neglected. S The surface asperities and waviness is neglected. S Root fillet curves are assumed to be circular [9], here assumed actual trochoid. S S Sliding friction between the mating gear teeth is neglected, since its effect on deflection is small. [10] All the manufacturing errors and geometrical errors are neglected. Finite element model of gear and mesh generation Both the NCR and HCR gears are kept in contact by positioning at the stipulated center distance (122 mm) with respect to the global coordinate system and only the plane area models are used for the FEA. Quadratic two dimensional (2D) plane 183 higher order elements as shown in Figure 5 are used with plane a strain option [8]. The element is defined by eight nodes having two degrees of freedom at each node which are nodal translations in x and y directions. For easy convergence of a contact solution, the finite element models are meshed with 5

6 a very fine mesh i.e., 0.01mm element edge length where the tooth will experience contact. The finite element mesh of NCR and HCR gear pairs are shown in Figure 6 and Figure 7, respectively. Figure 5. Plane 183 quad element Figure 6. Finite element meshed model of NCR gear pair Figure 7. Finite element meshed model of HCR gear pair Loading and boundary conditions In order to study the load sharing, identical bending stress and contact stress loads were applied for both NCR and HCR gear pairs. A maximum of 373 Nm is applied at all nodes lying on the circumference of the inner hub diameter of the pinion gear (47 teeth) and the pinion is arrested in the radial direction with respect to the local coordinate system. The gear of the teeth gear pair (ie) 50 teeth gear is fully constrained in all directions. A symmetric contact element [11] was created at the involute portion of the gear pair in contact. Solution and post processing Estimation of percentage load sharing Each gear is rotated as a rigid body according to the gear ratio. The solution is repeated for both NCR and HCR gears rotated with same amount of angular increment according to the gear ratio. Approximately angular increments with 0.5 degree steps are used for this analysis and the analysis is carried out with the help of the customized APDL (ANSYS Parametric Design Language) looping program [11]. Root stress, load sharing ratio and the contact stress are obtained for all the gear mesh positions. The nodal forces at each node of the contact element are captured from the ANSYS post processing for each individual gear tooth. By this methodology, the percentages of load sharing between teeth are estimated for both the gear pairs throughout the path of contact. Accordingly, the maximum percentage of load shared by the individual teeth for the NCR and HCR gears are estimated for the entire path of contact. The individual tooth loads have been determined by comparing the total normal load to the sum of the normal loads contributed by each pair of contact equally. It is observed from the above FEA that for the NCR gearing the maximum load of 100% is taken by the single tooth at the HPSTC point (Highest point of single tooth contact) and at the tip of teeth only 40% load is shared. However, for the HCR gearing, the maximum load of 57% load is taken at the FLPDTC (First lowest point of double tooth contact) which is above the pitch circle and only 20% load is shared by the tip of the teeth during the course of engagement. 6

7 Load sharing comparison NCR and HCR Load sharing ratio in terms of percentage load shared from root to tip of a particular tooth for the NCR gearing 47/50 is shown in Figure 8. The load sharing ratio is plotted with respect to rotation angle for a single tooth of the 47 gear tooth from root to tip which corresponds to a rotation angle of 0.5 deg(root) and 13.5 deg (tip). It can be seen from the graph that the double tooth contact band is from 0.5 deg to 5.5 and from 8.5 deg to 13.5 deg, the maximum and minimum percentage of load shared in these double tooth contact bands are 59% and 40% respectively. The single tooth contact band starts from 5.5 deg and ends at 8.5 deg. In the entire range of the single tooth contact, 100% load is taken by the single tooth. It is observed that the rate of increase of percentage load sharing from 0.5 deg to 5.5 deg is gradual from 40% to 59%, whereas from a rotation angle of 5.5 deg to 6.5 deg, the rate of increase of percentage load sharing is drastic ie from 59% to 100% during the change over from double tooth to single tooth. Again two tooth contact starts at 8.5 and gradually the load sharing reduced to 40% at the tip, corresponding to It can also be observed that the rate of decrease is very rapid from 8 o to 8.5 in view of single to two tooth contact and the load sharing ratio decreases gradually towards tip. This phenomenon is common to any tooth of the 47 tooth gear which is in contact. However, the tooth of the mating gear which comes in to contact carries the balance of the load in both the cases i.e., 47/ 50 teeth gears. Figure 9 shows the load sharing ratio in terms of percentage of load from root to tip for a single tooth of the HCR gearing (47/50 teeth). The load sharing ratio is plotted with respect to rotation angle for a single tooth of the 47 tooth gear from root to tip which corresponds to rotation angle of 0.5 deg (root) and after 18 deg (tip) period. It can be seen from the plot that the triple tooth contact band is from 0.5 deg to 3 deg, 7 deg to 10.5 deg and from 15.5 deg to 18 deg along the path of contact. On a single tooth, the triple tooth contact occurs three times and during this period the maximum percentage of load shared by any single tooth during the three tooth contact band is 46% and minimum percentage load is 20%. Similarly, the double tooth contact is from 3 deg to 7 deg and from 10.5 deg to 15.5 deg, along the path of contact. On a single tooth, the double tooth contact occurs twice and the maximum percentage of load shared by any single tooth during the double tooth contact band is 57% and minimum percentage of load shared is 42.5%. Figure 8. Load sharing of 47/50 teeth 7

8 Figure 9. Load sharing of 467/50 teeth HCR gear Bending stress comparison of NCR and HCR gear pair The variation of bending stress from root to tip of any tooth of the 47 tooth NCR gear which is in contact with the 50 tooth gear is shown in Figure 10. It can be seen from the figure that the rate of increase and decrease of bending stress during two teeth contact is the same as the load sharing ratio. However, the stress level is more near the tip region in view of the cantilever effect and the tooth shape. The bending stress at the root is about MPa and it increases up to 500 MPa at a 6 rotation angle corresponding to that of single tooth contact. The stress varies from 500 MPa to 580 MPa throughout single tooth contact when the load sharing ratio is constant i.e. 100%. The variation is again due to the cantilever effect on the tooth. The bending stress again reduces from 580 MPa to 445 MPa at the tip. However, the bending stress is more near the tip compared to the root as explained above. The maximum bending stress occurs at the HPSTC point and the minimum bending stress occurs at the root. The variation of bending stress from root to tip of any tooth of the 47 tooth HCR gear which is in contact with the 50 tooth gear is shown in Figure 11. The load is shared by two teeth in the region BC and DE, while load is shared by three teeth in the region AB, CD and EF. 20% load sharing occurs at the tip and as the load increases, the stress also increases and the maximum bending stress of 478 MPa is observed at the first lowest point of double tooth (FLPDTC) contact. As the load increases, in the three teeth contact zone CD, the bending stress variation is identical to the load sharing behavior. Further, in the two tooth and three tooth regions, DE and EF, respectively, the bending stress decreases as the load decreases. From Figure 10 and Figure 11, it is noted that the maximum bending stress occurs at the highest point of single tooth contact (HPSTC) for the NCR gear and it occurs at the FLPDTC for the HCR gear. The maximum tooth root bending stress is 18% less in the HCR gear compared to the NCR gear. Li [12] has found that the increase in addendum increases the number of teeth in contact thereby decreases the tooth contact stress and the root bending stress. But this increase also makes the tooth depth long and allows the tooth to be deformed easily. Therefore, though, the increment in addendum can reduce the tooth contact stress, there is no guarantee that this increment can certainly reduce the tooth root bending stress. Tooth root stress is increased if the addendum becomes 8

9 longer and the number of contact teeth has no change. Tooth root stress can also be reduced when the number of teeth in contact is increased through increasing the addendum. But there is no guarantee that this increment of the number of teeth can certainly reduce the root stress. This is because the increase of addendum also makes the whole depth of the teeth longer so that a larger moment occurred at the tooth root. Figure 10. Bending stress of 47 teeth NCR gear Figure 11. Bending stress of 47 teeth HCR gear 9

10 Contact stress comparison of NCR and HCR gear pair In this study, the contact stress at the meshes was calculated based on the tooth load distributed on a unit contact area of the tooth surface. Tooth contact stress was analyzed using the Hertz formula but it is not precise enough [13]. Tooth contact stresses are often calculated with the Hertz formula when the tooth loads are known. Since the Hertz formula was deduced from two symmetric elastic cylinders, it is not precise enough for contact stress calculations of gear teeth, a cantilever structure with involute profiles. Especially at engagement positions such as tooth tip and root contacts, the Hertz formula does not give the correct contact stress values [6]. The maximum load applied at the FLPDTC, where two teeth are in contact, is only 57% and hence the maximum tooth contact stress is lesser by 19% in the HCR gear compared to the NCR gear. The reduction in the contact stress is due to the increase of the addendum factor and the number of contact teeth, resulting in a lesser load. Tooth loading condition is a critical parameter especially in HCR gear applications. The tooth contact stress which is critical to wear, pitting and operating temperature was studied by Cornell and Westervelt [14]. Wang and Howard [6] carried out an analysis of HCR spur gears with tooth profile modification. They concluded that the contact stress was significantly reduced compared to the unmodified gears. The contact stresses tended to be smooth with only small stress irregularities at the relief starting point. The variation of contact stress on the 47 tooth NCR gear as shown in Figure 12 resembles its load sharing behavior. It varies from MPa at the root to 1200 MPa at the start of single tooth contact, corresponding to 6 degrees, and the stress is maintained constant till 8 degrees and further decreases to 811 MPa at the tip, similar to the load sharing pattern. The contact stress pattern of the 47 tooth HCR gear is shown in Figure 13. As in the case of the NCR gear, its variation in contact stress of the HCR gear resembles its load sharing behavior. The contact stress increases from MPa at the root to 988 MPa at the start of FLPDTC. The contact stress reduces in the double tooth contact region initially and further the contact stress increases and reaches the maximum value of 988 MPa at the start of FLPDTC equivalent to rotation angle of 10.5 degrees. Thereafter the contact stress decreases with respect to the rotation angle in both the two tooth and three tooth region. Wang and Howard (2008) carried out research on the contact zone of the NCR and the HCR gears with profile modification [6]. For the tip relieved HCR gears, the width of single, double and triple contact zone will change dramatically as the load increases. The HCR gears will have single tooth contact under light loads. Single contact zone in the mesh cycle can quickly disappear when the load is increased to become a triple contact zone that will expand in zone width as the load is further increased. Figure 12. Contact stress of 47 teeth NCR gear 10

11 Figure 13. Contact stress of 47 teeth HCR gear Conclusions Quasi static finite element analysis was carried out for NCR and HCR gears with fixed module, center distance and gear ratio. Here the increased contact ratio is obtained by increasing the addendum factor from 1.0 to 1.25 m. Hence a contact ratio of more than 2.0 was achieved for the same number of teeth. Two dimensional deformable body contact models for both HCR gear and NCR gears were created using the ANSYS-APDL loop program. Various parameters such as load sharing ratio, bending stress and contact stress were evaluated and compared over the path of contact. The maximum percentage of load sharing occurs at the HPSTC point in the case of a NCR gear and the FLPDTC point in the case of a HCR gear pair. At any point of time, the HCR gear tooth experiences a maximum of 57% of the total load against 100% in the NCR gear pair. At the tip of the tooth, a HCR gear shares 20% of the total load against 40% in a NCR gear pair. The maximum bending stress for a HCR gear is 18% less and contact stress is 19% less as compared to a NCR gear for the specific case of same module and fixed center distance. Hence the load carrying capacity of the HCR gear is 18% more than the NCR gear designed for the same weight and volume for fixed module and same center distance gear pairs. References 1. Elkholy,A.H., Tooth load sharing in high contact ratio spur gears. Trans. ASME. Journal of Mechanisms, Transmission and Automation in Des.107: Rosen, M.K., and Frint, H.K., Design of high contact ratio gears, J. Am. Helicopter Soc., (1982) Leming, J.C., High Contact Ratio (2+) Gears, Gear design manufacturing and inspection manual, SAE, Warrendale PA, 1990 pp Sivakumar, P., Hanumanna, D. and Gopinath, K., High contact ratio gearing concept for military tracked vehicle application, 17 th national convention of production engineers on Recent trends in design and manufacture of gears and bearings, Chennai (2002) pp a. Mohanty, S.C., Tooth load sharing and contact stress analysis of high contact ratio spur gears in Mesh. IE Journal 84 (2003) Coy, J.J., and Chao, C.H., A method of selecting grid size to account for Hertz Deformation in 11

12 Finite Element Analysis of Spur Gears, 1982, ASME J. Mech Des. 103, pp Wang, J., Howard, i., Finite Element Analysis of High Contact Ratio Spur Gears in Mesh, ASME J. of Tribology.,127, pp , Buckingham, E., Analytical mechanics of gears, Dover publication, New York, Cornell,R.W., Compliance and stress sensitivity of spur gear teeth, ASME Trans., J. Mech. Des., April Arafa, M.H., and Megahed, M.M., Evaluation of spur gear mesh compliance using the finite element method, Proc Instn Mech Engrs Vol 213 Part C, pp Walton, D., Tessema, A.A., Hooke, C.J., and Shippen, J. Load sharing in metallic and non metallic gears. Proc. Instn Mech. Engrs, Journal of Mechanical Engineering Science, 1994, 208 (C2), ANSYS Release 11.0, Structural analysis guide, ANSYS Inc., Canonsburg, PA 15317, USA Li, S., Effect of addendum on contact strength, bending strength and basic performance parameters of a pair of spur gears, Mechanism and Machine Theory, (2008). 13. Dolan, T.J., and Broghamer, E.J., A photoelastic study of the stresses in gear tooth fillets,bulletin No.355. University of Illinois, Engineering experiment station, Cornell, R.W., and Westervelt, W.W., Dynamic Tooth Loads and Stressing for High Contact Ratio Spur Gears, American Society of Mechanical Engineers, pp. 77-DET-101,