LOAD SHARING OF SPUR GEARS IN MESH AN ANALYTICAL AND EXPERIMENTAL STUDY

Transcription

1 NATIONAL TECHNICAL UNIVERSITY OF ATHENS (NTUA) Department of Mechanical Engineering Laboratory of Machine Elements LOAD SHARING OF SPUR GEARS IN MESH AN ANALYTICAL AND EXPERIMENTAL STUDY G. K. Sfantos V. A. Spitas T. N. Costopoulos G. A. Papadopoulos TECHNICAL REPORT No. TR-SM ATHENS March 003

2 ABSTRACT In the present technical report a new analytical approach for the determination of the load-sharing factor in gear pairs is presented. This novel method is used to calculate the load carried by each tooth pair and is verified experimentally using stress-optical methods such as photoelasticity and caustics. This report begins with an introduction to the theoretical calculation of the tooth compliance and the load-sharing factor, and proceeds with a brief presentation of the mathematical background of the method of caustics in contact mechanics and in particular to gear meshing (Sfantos et al. [17]). A spur gear pair of z 1 =z =18 teeth and module m=0mm was studied. Test specimens from plexiglas (PMMA) were cut and loaded in a special device simulating actual meshing conditions. The experimental determination of the load carried by each tooth pair was performed for various angles of meshing. For the same angles of meshing the theoretical load-sharing factor was also calculated for comparison. Charts illustrating both the experimental and the theoretical load-sharing factor were plotted. Comparing the experimental results with the theoretical predictions a significant difference was observed at the ends of the path of contact, due to edge contact and the deformation of engaging / disengaging tooth due to the load applied on the adjacent one, facts that the theoretical model does not take into account. The experimental stress distribution of the loaded gear pair was obtained by photoelasticity. KEYWORDS: Spur gears, Tooth pair, Compliance, Load-sharing factor, Caustics, Photoelasticity 1

3 NOMENCLATURE Symbol Description Units m Module mm z Number of teeth C Compliance t F Tooth thickness on the root circle mm δ Deformation of the tooth along the path of contact mm l Semi-length of contact line mm λ m Magnification factor z o Distance between specimen-screen mm z i Distance between specimen-focus mm b Specimen thickness mm D x Diameter of caustic mm r Radius of the initial curve of the caustic mm c t Stress optical constant for the transmitted rays MPa -1 E Modulus of elasticity MPa G Shear modulus MPa v Poisson ratio P Load per unit width N/mm P t Total load per unit width N/mm R Radius of the body in contact mm C s Tooth thickness coefficient C f Dedendum coefficient C c Rack tip radius coefficient a o Pressure angle rad r c Radius of the rack tip mm LSF Load-sharing factor ε Contact ratio α Angle of load application on a single tooth deg θ Tooth meshing angle deg Suffixes 1 Refers to the pinion Refers to the wheel B Refers to the bending compliance F Refers to the root (foundation) compliance H Refers to the Hertz compliance

4 1. INTRODUCTION In every gear pair under load, gear tooth deformation takes place as the tooth (or teeth) of the pinion compresses the tooth (or teeth) of the wheel for the delivery of the torque [1]. This deformation influences the meshing of the gears, as manufacturing errors had occurred on their flanks. Therefore as teeth bend and deform under load motion errors change and vibrations and noise are generated []. A method to lower the produced noise from a gear pair is to increase the number of simultaneous tooth pairs in contact. As more tooth pairs are in simultaneous contact then the overall mesh stiffness is increased and each tooth pair carries a considerably reduced load. Thus the highly changes of load sharing is reduced and therefore the deformation of each tooth is reduced too. Consequently the produced noise is reduced. Another method to normalize the load sharing is to modify the working tooth flank [3], in order gears teeth mesh in a way that the load sharing did not change very suddenly and therefore lower noise will be produced. The method of modification of the tooth flank in order to reduce the load sharing is composed of two basic parts. The first part of that method is that modifying the working tooth flank, the new path of contact must be known in order to determine the new load sharing. Up to now many theories have been suggested to solve the problem of the calculation of the path of contact when the gear tooth flank is known ([Costopoulos [4], Spitas [5], Buckingham [6]), including the widely used Theory of Gearing by Litvin (Litvin [7]) and the novel theory of Involutization by V.A. Spitas (Spitas et al. [8], Sfantos et al. [9], [10]). The second part of that method is the determination of the load sharing. To determine the load carried by each pair of tooth the deformation of that pair on the line of contact (in the case of involute teeth) must be known. In order to determine the amount of that deformation, a coefficient called Coefficient of Compliance (CoC) is used [Costopoulos [11]). Knowing the CoC of each tooth pair the deformation and the carried load can be calculated whereas in advance the Load Sharing Factor (LSF) can be determined. Few works have been carried out since now for the solution of the determination of the LSF due to the difficultness of the problem. New algorithms are introduced for the determination of the LSF in order to investigate the influence of the deformation of the contact point of the flanks, during the meshing (C. Gosselin, et al. [1], X. Shu [13], 3

5 Zhang and Fang [14]). All of these works use numerical tools to calculate the tooth bending stiffness and linearized Hertzian theory for the calculation of the contact deformation. Moreover they did not take into account the effectiveness of the displacements on the tooth surface at the contact point, the edge contact phenomenon and the movement of one tooth due to the load applied on an adjacent tooth ([1], [13]), and their controversial results are not verified against an experimental method ([1], [14]). In this technical report a new way for determining the load-sharing factor experimentally is introduced. The proposed way takes into account the deformation near the contact region of the tooth pair and using the optical method of caustics (Papadopoulos [15], Sfantos et al. [16]) the carried load by each tooth pair is determined (Sfantos et al. [17]). The carried load by each tooth pair can be calculated from the geometry of the caustic using the solution of the application of caustics on contact problems, proposed by G. K. Sfantos [17]. At the beginning of this technical report an overview on the theoretical way of calculation of the load-sharing factor is discussed. Then a brief description of the application of the caustics on contact problems and the final relations that characterize the specific problem are presented. Afterwards a presentation of the experimental set-up follows. Finally charts, illustrating the variation of the experimental and the theoretical LSF against with the contact ratio are plotted. Moreover experimental stress distribution of the meshing of the pinion with the wheel by photoelasticity is presented, for a more qualitative analysis of the phenomenon of carried load and stress developed inside a tooth pair.. DETERMINATION OF THE LOAD SHARING FACTOR a) Theoretical investigation To calculate the load-sharing factor in a gear train the deformation on the path of contact of each tooth pair in mesh must be known. In order to determine that deformation the dimensionless coefficient of compliance (CoC) is introduced as follows: When a load (P) is applied on a tooth of width (b), the CoC (C) determines the 4

6 deformation (δ) of the contact point of the flanks, on the path of contact (PoC) (Costopoulos [11]). E b δ C = (1) P where E is the modulus of elasticity of the gear material. The above coefficient consists of the Bending Compliance (C B ), the Hertz Compliance (C H ) and the Root Compliance (C F ). The total compliance of the tooth pair depends on the point of contact (on the PoC) and is calculated by the following relation: C + = CB1 + CB + CF1 + CF CH () where 1 refers to the pinion and to the wheel. For the calculation of the bending compliance the tooth is considered as a constrained beam, loaded normal to its flank, as Fig. 1 illustrates. The bending compliance is calculated using the following equation (Costopoulos [11]): tan C = 1 cos + 0. ( 1+ ) + ϕ B ϕ I I1 ν (3) 1 where the integrals I 1 = Y dy 0 t ( y Y ) I = Y dy (4) 0 3 t are calculated numerically. The root compliance is calculated from the following relation [11]: Y 4.8 tan ϕ C F = + 1+ ( ) (5) 3π t F 1 ν tf π.4 1+ ν 50 Y ( ) ( 1 ν ) 1 ν cos ϕ + where t F is the width of the tooth at the root circle, which is calculated from the theory of the fillet as follows: t x F = x = r 1 w = C z B = m C r c o c o = m C cos w s f c + r ( r B) ( C C ) c o f c sin w tan a o Cc + cosa o (6) where C s is the tooth thickness coefficient, C f is the dedendum coefficient, C c is the rack tip radius coefficient and a o is the pressure angle. 5

7 To calculate the Hertzian compliance the tooth pair is considered as two cylinders in contact (Fig. ) of equal radius with the radius of the involute curve at the point of contact and the pressure distribution on the contact region is considered to be Hertzian. Thus the deformation of the tooth pair, due to contact stresses, is given by the following relation [11]: δ l H KPR = π ( 1 ν ) P = π b E 1 1 R = + R1 R 1 4h1h ln l ν 1 ν and considering eq. (1), the total compliance of the tooth pair is calculated as follows: ( ν ) 1 4h1h ν C H = ln (8) π l 1 ν (7) By observing all the equations for the calculation of each compliance (eqs. 3, 5, and8) it is clear that both the bending and the root compliance are independent of the carried load by the tooth pair, while the Hertz compliance depends on the load as it is a function of the semi-length (l) of contact. Therefore for calculating the Hertz compliance of a tooth pair the procedure below is followed. When two tooth pairs are in meshing, both the tooth pairs carry the total load from the delivery of the torque. The load carried by the first (A) and the second (B) tooth pair is calculated by the following equations (Costopoulos [11]): P P A B Pt CB = C + C A PC t A = C + C A B B (9) where P t is the total load, and C A, C B the total compliances of each tooth pair. To calculate the total compliances of each tooth pair (eq. ), the Hertz compliance of each tooth pair must be known. Thus using eqs. (7) and (8) the Hertz compliances are calculated by: 6

8 C C HA HB ( ν ) 1 = π ( ν ) 1 = π h 1hπ ν ln KPA RA 1 ν h1hπ ν ln KPB RB 1 ν (10) Hence using eqs. (), (9) and (10) a non-linear 4X4 equation system is appeared, which its solution gives the Hertz compliance of each tooth pair. At this point the concept of the Covered Contact Ratio ε C ( θ ) is introduced. This magnitude depends on the meshing angle θ and expresses the ratio of the length of contact already covered by the tooth pair in mesh to the base pitch. Therefore it is evident that its minimum value is 0 at the engagement of the tooth pair and its maximum value is equal to ε at the disengagement of the pair. When all the compliances are known the total compliance is calculated using eq. (), and the load-sharing factor (LSF) for a tooth pair against the angle of meshing is calculated using the following equations (Costopoulos [11]): LSF P P A B = = when two tooth pairs are in meshing, for 0 ε C <ε-1 t C C + C A B Pt LSF = = 1 when one tooth pair is in meshing, for ε-1 ε C 1 P t (11) LSF P P B A = = when two tooth pairs are in meshing, for 1<ε C ε t C C + C where ε is the contact ratio. A B b) Experimental investigation The experimental procedure that is followed to determine the LSF in a gear train is based on the deformation near the contact region of every tooth pair. In every contact between two elastic solids a stress singularity appears because of the high strain gradients at the contact region. The optical method of caustics is able to transform the stress singularity into an optical singularity, using the reflection laws of geometric optics (Papadopoulos [15], Sfantos et al. [16]) thus providing all the information needed for the evaluation of the stress singularity. 7

9 In the case of a gear train under load, the appeared caustic from the illumination of the contact region of a tooth pair, can give all the information needed for the determination of the load carried by the specific tooth pair (Sfantos et al. [17]). The experimental set-up for obtaining a caustic curve from a transparent test specimen with stress singularity is illustrated in Fig. 3. The light beam of a He-Ne gas laser passes through a special filter and two convergent lenses, to become a divergent beam. In front and behind the specimen, screens are placed at distances z o and parallel to the mid-plane of the specimen. On these screens the caustics, from reflected and transmitted (in case of transparent specimen) light rays, are formed. The reason for the use of the specific set-up is that the produced caustics are magnified by a specific factor λ m given by the relation: z + z o i λ m = (1) zi where z o is the distance between the mid-plane of the specimen and the screen and z i is the distance between the focus of the bundle and the mid-plane of the specimen (Fig. 3). The distance z i takes positive values when the focus is located in front of the specimen, and negative when the focus is located behind the specimen. In contact applications the focus lies usually behind the specimen. If a parallel light beam impinges directly on the specimen then the magnification factor is λ m =1. The appeared caustic on the screen behind the specimen is illustrated in Fig. 4. If one of the two teeth, in contact, is transparent the caustic appears as in Fig. 4a, where D x is the diameter of the caustic, while if both the two teeth are transparent the caustic appears as in Fig. 4b, because both the two teeth are loaded by the same force, and the appeared caustic from each tooth is identical to the another after a rotation of 90 o due to the fact that the faces of the tooth flanks are opposite. The relation of the diameter of the caustic in respect with the semi-length of a contact of two solids is given by the following equations (Sfantos et al. [17]): g Dx = rλmcosθ + sin θ 1 ( + ) 1 1 8g 1 θ = sin 4g ** Ct l g = 3 r (13) 8

10 where r is the radius of the initial curve of the caustic given by: and ** C t is a constant given by: C r = l 1+ l ** t 3 1 (14) C ** t zotc = KR λ m t (15) K ( k + 1) ( k + 1) 1 = +, 4G 1 4G 3 ν 1, k 1, =, 1+ ν 1, 1 R 1 1 = + R1 R (16) Thus measuring the diameter of the caustic (D x ) and solving numerically eqs. (13) the semi-length (l) of the contact line can be calculated. From the knowledge of the semi-length of the contact line, the applied load can be calculated by the following equation (Sfantos et al. [17]): 1 1 l + R1 R π P = (17) K Knowing the load carried by each tooth pair at any time of mesh, the load-sharing factor can be calculated as follows: P LSF = (18) P t where P t is the total load the gear train is subjected from the delivery of the torque. 3. EXPERIMENTAL SET-UP The experimental set-up consists of an arrangement of a He-Ne laser with a system of optical filters and lenses, the specimens loading device and the test specimens themselves in meshing (Fig. 5). The specimens used were gear segments from gear with module m=0mm and z=18 teeth, consisting of four (4) teeth each. The geometry of the gear segments including the tooth profiles was calculated using the equations of gearing (Buckingham [6], Costopoulos [4], Spitas et al. [5], Sfantos et al. [9], [10]). Then the geometrical data 9

11 were translated into a CAM code on a CNC machining center where the test specimens were actually manufactured. The specimens were cut from a 5mm thick PMMA (Plexiglas) sheet, which is an optical isotropic material. Therefore the produced caustic curves were developed exclusive due to the deformation around the contact line and were independent of optical anisotropy (in case of optical anisotropic materials, the caustic curves appear double and it is more difficult to measure their diameters [17]). In order to determine the load-sharing factor during the whole meshing, from the beginning until the end, of a tooth pair, thirteen different angles of meshing were considered. Table 1 illustrates the angles of meshing that were subjected to LSF determination against the covered contact ratio of the specific gear train of z 1 =z =18 teeth and module m=0mm. The angle of meshing starts counting from the first time a new pair of tooth comes in mesh (angle a=0 o ), relative to the rolling circle. Angle of Meshing Covered Contact ratio 0 o o o o o o o o o o o o o 1.53 Table 1: Angle of meshing vs. the Covered Contact ratio The gear segments were clamped on the fixing support of the loading device and were statically loaded using the same horizontal force per unit width P t.x =55.1N/mm. Using the arrangement of Fig. 5 the light rays of the He-Ne laser pass through the optical 10

12 filter and the two lenses before illuminating the contact region of the specimens. The focusing point lies behind the specimen in order the caustics appear in the form of a circle since the specimen is transparent and the contact region undergoes compression leading to the local increase of thickness due to the Poisson effect (Sfantos et al. [17]). One screen is been placed behind the specimen in order to record the transmitted caustic. For all the considered mesh angles the same magnification factor is used. At first both the two segments are transparent in order to examine if the two caustics are same from each contact region are the same. Then the segment of the wheel becomes untransparent (placing a sheet of paper in front of it), in order to record the geometry of each caustic and finally measure her diameter. 4. RESULTS AND DISCUSSION The test specimens were loaded, according to the procedure that was presented in the previous chapter and caustics appeared on the screen behind the specimens. To minimize the uncertainty and to validate the repeatability of the test method, each specimen was set and loaded five times, observing and measuring the geometrical data of each caustic. After that, if the repeatability was verified the final measurement of the diameter was made. In addition photos of each caustic were taken for a more rigorous analysis and post-processing. Transmitted caustic curves as they formed on the screen behind the specimen are illustrated in Figs. 6 and 7. In those figures the deviation of the diameters of the caustics is obvious as the contact ratio (or mesh angle) increases. By carefully observing those figures it is clear that when a tooth pair comes in mesh, the carried load at first is little due to the edge contact, and increases as the ratio contact increases. The opposite phenomenon is observed at the end of the meshing of a tooth pair. Fig. 8 illustrates some caustics appeared only from the pinion segment and were used to measure their diameters. Measuring the diameters of each caustic and using eqs (1) (16) the semi-length of the contact line was calculated. Then using eq. (17) and (18) the carried load by each tooth pair and the LSF were calculated respectively. Table illustrates the diameter of 11

13 each caustic, the semi-length of each contact line, the carried load by each tooth pair and the LSF against the angle of meshing and the covered contact ratio. Angle of meshing θ ( o ) Covered Contact Ratio Equivalent Radius of Pinion Curvat. R 1 (mm) Equivalent Radius of Wheel Curvat. R (mm) Diameter of caustic D x (mm) Semilength l (mm) Carried load P (N) LSF Table : The experimental determined LSF where c t = MPa -1, G 1, =1000 MPa, ν 1, =0.34, λ m = 1.5, t = 5mm, z o =3405mm. The experimentally determined load carried by each tooth pair is plotted against the percentage-covered length of contact in Fig. 9. In the same figure it is also plotted the variation of the total load carried by the pair as it is determined experimentally. In Fig. 10 the experimentally determined LSF is illustrated. It is obvious that there is an increase of the carried load until one tooth pair is in mesh and a decrease of the carried load until the tooth pair is disengaged. The theoretical coefficients of compliance are calculated for the same meshing angles using eqs. () (10). The bending and the root compliance components of the pinion and the mating gear and the hertzian compliance of the pair are plotted against the percentage of the covered length of contact in Fig. 11. The total resulting compliance of the pair is also plotted on the same graph. The symmetry of the total compliance curve 1

14 about the 50% vertical line is due to the fact that the pinion and the wheel have the same number of teeth (z=18). The theoretical and the experimental load-sharing factor are plotted versus the percentage of the covered length of contact in Fig. 1. It is obvious that there is some difference at the beginning and the end of meshing of the tooth pair and this can be attributed to the fact that the theoretical model does not take into account edge contact and deflections of the teeth in mesh. On the contrary the experimental method is sensitive to this phenomena particularly as the teeth are made from a flexible material (PMMA). In order to obtain better understanding of the LSF and the stresses developed inside a tooth pair during meshing of a gear train, the stress distribution in respect to the contact ratio was determined by the experimental method of photoelasticity. Figures 13-0 illustrate the isochromatic fringes for different angles of meshing. The change of the stress distribution is obvious as the meshing progresses. 5. CONCLUSION In this technical report a novel approach for the determination of the load-sharing factor in gear pairs was presented. The experimental method used to validate the theoretical findings was based on the stress optical method of caustics suitably modified to fit the specific contact mechanics problem. After analyzing the experimental results obtained for a spur gear pair of z 1 =z =18 teeth and compared with theoretical calculations of the LSF, it evident that there is a significant difference at the ends (engagement and disengagement) of the path of contact. This difference is due to the fact that the theoretical calculation of the LSF does not take into account edge contact phenomena and tooth deflections of the unloaded teeth as they come in mesh. These deflections occur due to the deformation of the adjacent loaded teeth. From the experimental determination of the LSF useful data can be collected in order to model and design new stronger tooth profiles of advanced stiffness and lower noise during their meshing. Moreover this method can be applied on every multiple contact and load-sharing problem as they appeared in splines, bearings, chains, etc. 13

15 6. REFERENCES [1] C.A. Spitas, Kinematic and Dynamic Analysis of Spur Gears, NTUA, Dep. of Mech. Engineering, Laboratory of Machine Elements, TR-SM-98, [] C.A. Spitas, T.N. Costopoulos, A New Theory for the Determination of the Actual Characteristics of Loaded Spur Gears in Mesh, Proc. ASME International-Greek Section 1 st National Conference on Recent Advances in Mechanical Engineering, September 17-0, 001, Patras, Greece. [3] Yi Zhang, F.L. Litvin, R.F. Handschuh, Computerized Design of Low-Noise Face- Milled Spiral Bevel Gears, Journal of Mechanism and Machine Theory, Vol. 30, No. 8, 1995, pp [4] T. N. Costopoulos, Generalized theory of gearing and tooth stress, Proceedings of Ninth World Congress on the theory of Machines and Mechanisms, Milan, Italy, Vol. I, 1995, p [5] V.A. Spitas, Modeling and Design of Optimum Gears using Analytical, Numerical and Experimental Methods, Ph.D. Thesis, National Technical University of Athens, Athens 000. (in Greek) [6] E. Buckingham, Analytical mechanics of gears, Dover Publications Inc., New York, [7] F. L. Litvin, Theory of gearing, Nasa Reference Publication 11, AVSCOM Technical Report, 88-C-035, [8] V. A. Spitas, T. N. Costopoulos, Ch. A. Spitas, A quick and efficient algorithm for the calculation of gear profiles based on flank Involutization, 4 th GRACM Congress on Computational Mechanics, Patra, Greece, 00 [9] G. K. Sfantos, V. A. Spitas, T. N. Costopoulos, Application of the theory of Gear Tooth Flank Involutization on external modified involute gear teeth, NTUA, Dep. of Mech. Engineering, Laboratory of Machine Elements, No. TR-SM-003, August 00. [10] G. K. Sfantos, V. A. Spitas, T. N. Costopoulos, Application of the theory of Gear Tooth Flank Involutization on Cycloidal Gearing (Spur Gears Lobe Pumps), NTUA, Dep. of Mech. Engineering, Laboratory of Machine Elements, No. TR-SM- 004, September 00. [11] T. N. Costopoulos, Gearing and Speed Reducers, Symeon, Athens, 1991 (in Greek). 14

16 [1] C. Gosselin, L. Cloutier, Q.D. Nguyen, A general formulation for the calculation of the Load Sharing and Transmission Error under Load of Spiral Bevel and Hypoid Gears, Journal of Mechanism and Machine Theory, Vol. 30, No. 3, 1995, pp [13] S. Xiao-Long, Determination of Load Sharing Factor for Planetary Gearing with small tooth number difference, Journal of Mechanism and Machine Theory, Vol. 30, No., 1995, pp [14] Y. Zhang, Z. Fang, Analysis of Tooth Contact and Load Distribution of Helical Gears with Crossed Axes, Journal of Mechanism and Machine Theory, 34, 1999, pp [15] G.A. Papadopoulos, Fracture Mechanics, The experimental method of Caustics and the Det.-Criterion of Fracture, Springer-Verlag, London, 199. [16] G. K. Sfantos, V. A. Spitas, G.A. Papadopoulos, T. N. Costopoulos, Determination of the Stress Intensity Factors (K I, K II ) in Loaded Cracked Spur Gear Teeth by the optical method of Caustics, NTUA, Dep. of Mech. Engineering, Laboratory of Machine Elements, No. TR-SM-0301, December 003. [17] G. K. Sfantos, V. A. Spitas, T. N. Costopoulos, Analytical and Experimental study of Bearing Stresses developed on Machine Elements in Contact, NTUA, Dep. of Mech. Engineering, Laboratory of Machine Elements, No. TR-SM-030, February

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