COMPUTATIONAL MODELLING OF SPUR GEAR USING FEM

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1 Conference of Diploma Thesis 007 Institute of Machine and Industrial Design, Institute of Solid Mechanics, Mechatronics and Biomechanics aculty of Mechanical Engineering, Brno University of Technology June 5-6, 007, Brno, Czech Republic COMPUTATIONAL MODELLING O SPUR GEAR USING EM Martin Ševík - martinstroj@seznam.cz Institute of Machine and Industrial Design, aculty of Mechanical Engineering, Brno University of Technology, Technická 896/, Brno ABSTRACT A computational model of spur gear was made to study distribution of contact pressures between teeth in mesh and stresses in tooth root. The results obtained by inite Element Method (EM) are compared with analytical solutions. After tuning of the computational models, the optimal method for gear calculation is chosen. On the basis of four models is demonstrated the dependence of stress on the rim thickness. The analysis includes studying the stresses in the tooth root for the rim thicknesses that are lesser than the standard permission. INTRODUCTION The standard SN [1], used for considering the ultimate states of spur gearing, compares bending stress in tooth foot and contact pressure with some specific allowable value. owever, this standard has some limitation e.g. overlap ratio ε β has to be less than ; thickness of gear rim should be more than 3.5 module. The first condition relates to CR (igh Contact Ratio) gears that are special type of gears usually used in automobile industry. The second condition applies for CR as well for any other types of gears. This second limitation is verified on standard uncorrected spur gearing in this paper. AIM O TE WOR Aim of this work is to develop computational model of spur gearing which performs stress strain analysis using EM focused on distribution of contact pressure and bending stress in tooth root. Results obtained by numerical solution will be compared with analytical solution by standard SN with analytical solution of contact pressure obtained by ertzian theory of two cylinders in contact and with analytical solution of bending stresses of tooth substituted as fixed beam. Another aim is to tune the computational model and create other suitable models for assessment of stresses in the gear root and study the influence of gear rim thickness on stress in tooth root. Martin Vrbka - vrbka.m@fme.vutbr.cz Institute of Machine and Industrial Design, aculty of Mechanical Engineering, Brno University of Technology, Technická 896/, Brno ANALYTICAL SOLUTION Contact pressure obtained by standard SN ertzian pressure in pitch point of gear is defined in standard: - calculated contact stress, E - elasticity factor, zone factor, ε - contact ratio factor, β - helix angle factor, t - nominal tangential load, the transverse load tangential to the reference cylinder, b w - face width, d 1 - reference diameter of pinion, u - gear ratio, A - application factor, v - inner dynamic load factor, β - face load factor, α - transverse load factor. Stress in tooth root fillet obtained by standard SN Tensile stress in tooth foot is defined in standard as follows: or the determination of load capacity of tooth the local bending stress is used in critical area of root fillet. Local stress is defined by: = = t b m w E n ε Y β S t b d Y Y β w ε 1 u + 1 u = β α - tooth-root stress, t - nominal tangential load, the transverse load tangential to the reference cylinder, b w - face width, m n - normal module, - supplementary load factor, A - application factor, v - inner dynamic load factor, β - face load factor, α - transverse load factor, Y S - tip factor, Y b - helix angle factor, Y e - contact ratio factor. A A v v β α Analytical solution of contact pressure The contact pressure in mesh of the gear can be simply calculate using ertzian contact theory of two cylinders that have the same radius as osculatory circles of involutes in

2 contact point of the gears. The contact pressure for elastic cylinders with the same materials is defined []: = E π (1 µ ) br n eqv = + R eqv R R Η - contact pressure obtained by ertzian theory, E - Young s modulus of elasticity, n - normal force on cylinders, µ - Poisson s ratio, b - cylinders width, R eqv - equivalent radius of cylinders, R 1, R - radii of cylinders. Analytical solution of tensile stress in tooth root Presumption for analytical calculating of the stress in tooth foot is usually fixed beam loaded by force, see ig. 1. Bending stress, shear stress and compression stress are taken into consideration. The contact pressure between teeth is not included for simplification. One of a few equations for analytical solution of maximal tensile stress in tooth root has been developed by Aida and Terauchi [3]: t max 1 0,08 0,66 Nb 0,40 Nb 36τ N 1, 15 r = Nc max maximal tensile stress in tooth root, t tooth thickness in root passage location, r radius of root passage, Nb bending stress produced by tangential force, Nc compression stress produced by radial force, τ N shear stress produced by tangential force. igure 1 Simplified scheme of gear load by [3] COMPUTATIONAL MODELLING USING EM The analytical solution of stresses and deformations in whole volume of gear is impossible because the geometry of the tooth is very difficult. The stress and strain of gear was studied by photoelasticimetry in former times. Progressive approach for studying of gear stress uses numerical solution; especially EM. Suitable software system for EM simulations is e.g. ANSYS which is used in this research. 1 Model of geometry A primary task was to create an accurate model of gear geometry. Software PROILDATA was used for generating of coordinates of tooth profile with an accuracy of 1 decimal places. Additional modeling was made in ANSYS software system in order to create areas (for D problems) and volumes (only for 3D problems). Position of gears was modeled so that the contact point and the pitch point are the same. Parameters of considered gear are in Tab. 1. On base sources [4, 5, 6], models of both gears must have at least three teeth. The notch effect on one side of tooth causes additionally the stress on the other side of the tooth. or this reason, three teeth were always modeled on the gear. The pinion had half of teeth or all teeth; this depends on type of plane problems. In 3D problems the pinion had three teeth as well the gear. The sub-areas were prepared around the contact point and the root passage to make easier finite element (E) mesh creating and interpretation of results. gear parameter value number of teeth on pinion z 1 = 5 number of teeth on gear z = 47 normal module active gear width transmitted power m n = 4 mm b = 30 mm P n =. kw Table 1 Parameters of considered gear Model of materials The material of both gears is homogenous, isotropic, linear elastic, with Young s modulus E = 10GPa and Poisson s ratio µ = 0.3. Creating of E mesh The 8-nodes quadratic plane elements PLANE8 were used for all plane problems. The mesh was mapped in sub regions around contact point and root passage. Rest of areas was meshed by free mesh. The 0-nodes quadratic element SOLID95 and 8-nodes linear elements SOLID45 were used to create mesh for 3D problems. The transitions between quadratic and linear elements were filled by transition pyramids. The table 3 shows approximate numbers of nodes and degrees of freedom (DO) of the individual models. type of problem the smallest element size [mm] number of all nodes Table Characteristics of E mesh number of DO plane (D) solid (3D)

3 The smallest elements were located around the contact point (plane problems) or along contact line (3D problem). The E mesh of 3D problem is shown in ig.. igure E mesh of 3D problem Bindings There were an inner binding - contact between pinion and gear. The contact element CONTA17 is defined for pinion and target element TARGE169 is defined for gear in plane problems. In 3D problem the contact element CONTA174 is defined on pinion and TARGE170 is defined on gear. The value of friction coefficient f = 0.05 was implemented to the models only in models with rim. Outer bindings are shown in ig. 3. A rotation axis of pinion was replaced by zero displacements of nodes whose coordinates are the same one as coordinates of center of pinion Nodes on lines, which delimitating the cut of the gear, were fixed in all directions. The symmetric boundary condition was used only for 3D problem. or that reason the plane of symmetry was one side of the model. Loads The forces were placed into nodes antisymetrically about axis of pinion to induce torsion moment. Stress peaks, which could arise, are far from the tooth root. orce values correspond to nominal power Pn =. kw at 1500 min-1. Computational variants Primary task was to tune the plane model of geometry and compare stresses and pressures with values obtained by analytical solutions. Models for plane strain (full D), plane stress (PS), plane stress with thickness (PS/wT) and 3D problem (full 3D) were studied. The plane strain (full D) option was chosen as suitable model for calculating stresses and strain in this concrete gear. This model of calculation was used to study the pinion with thin rim. our new geometries of pinion were created to study the influence of gear rim thickness on the stress and strain in the tooth root fillet. ollowing geometries of pinion were chosen: a) with rung under the tooth (rung A) see ig. 4 b) with rung back of the tooth (rung B) see ig. 4 c) with axis of the groove in tooth axis (key A) see ig. 5 d) with axis of groove in tooth space axis (key B) see ig. 5 rung under the tooth (rung A) rung back of the tooth (rung B) igure 4 E model of pinion with rung GEAR UX = UY = 0 contact PINION UX = UY = 0 igure 3 Bindings and loads of the E model axis of the groove in tooth axis (ey A) axis of groove in tooth space axis (ey B) igure 5 E model of pinion with groove

4 To prevent the stress concentration in the notch the large fillet was used between rim and rung. Dimensions of the key were assumed from standard SN [7]. In the models with key, the thickness of the rim is distance between bottom of the groove and root diameter. Configuration of solver The Precondition Conjugate Gradient (PCG) solver was used for analysis. The PCG solver is markedly faster than direct solvers. Large deformations were included. The load was divided into 50 steps. ANALYSIS O RESULTS Introductory comparative computations The maximal contact pressure between teeth and von Misses stress in tooth root on the tension side were analyzed using numerical simulations. The maximal contact pressures depending on the variant of models are shown in ig. 6. Misses stress in tooth root obtained by standard SN is higher due to involving additional factors. von Misses stress in the tooth root [MPa] ,414 6,017 6,017 6,175 6,304 10, maximal contact pressure [MPa] full D 184, PS 163,6 163,6 PS/wT full 3D 180,1 analytical 180,8 00,5 standard SN igure 6 Results of introductory comparative computations - maximal contact pressure The suitable approach for calculating the maximal contact stress seems to be the 3D problem (full 3D) option, possibly option with plane strain (full D). The differences between values of contact pressures obtained from numerical solution using EM and analytical solution is less than %. The value of maximal contact stress obtained by standard SN is higher due to involving additional factors. The maximal von Misses stress in the tooth root on the tension side was studied. Results are shown in ig. 7. This simulation showed that all numerical options of solutions are in good agreement with analytical solution. The value of von 0 full D PS PS/wT full 3D analytical standard SN igure 7 Results of introductory comparative computations von Misses stress in the tooth root The difference between stresses in the tooth root obtained by analytical solution and numerical solution using EM is less than 7 %. The suitable analysis for calculating the maximal contact pressure and von Misses stress is the plane strain (full D). The advantage of this option is speed and sufficient accuracy. The difference of values between analytical and numerical solution using EM approach with plane strain (full D) was less than %. Influence of rim thickness on stress in tooth root The objective of the next step was to calculate stress in the tooth root that depends on the rim thickness. There is no possibility to compare these values of stresses with standard SN because the standard was made up provided that the rim thickness is at least 3.5 module. The analysis started for all geometries at 15 mm in thickness of the rim that is 3.75 module and continued to 5mm in thickness of the rim that is 1.5 module where step between the thicknesses was mm. or the geometry with rung the analysis was accomplished to 3 mm in thickness that is 0.75 module. Results of this analysis are shown in ig. 8.

5 von Misses stress in the tooth root [MPa] rung A key A rim thickness [mm] rung B key B 3.5xmodule igure 8 Von Misses stress in the tooth root depending on rim thickness CONCLUSION A computational model of gear set was made in order to perform stress and strain analysis of spur gear. Within tuning of computational model the plane strain, plane stress, plane stress with thickness and 3D model were used to calculate values of stresses. These values were compared with analytical solutions using standard SN , ertzian theory of contact and analytical solution of stress in the tooth root. The option of computational model with plane strain was chosen to analyze geometries of pinion with rim. our geometries were modeled in order to study influence of thickness of the rim on the stress in the tooth root. or geometries with rung the calculation were accomplished up to 3 mm in thickness that is 0.75 module. Geometries with key were calculated up to 5mm in thickness that is 1.5 module. The computational model was made up which makes possible studying the stresses in the tooth root for the rim thicknesses that are lesser than the standard permission. The stresses were numerically calculated also on geometries where no calculation by standard SN was possible because the standard can only be used where rim thickness is grater than 3.5 module. All numerical calculations were performed as a static thus there are not included any dynamic factors or fatigue. The effect of over sizing on the contact pressure and fracture parameters will by studied on spur gears and computational model of helical gears will be developed in future work. ACNOWLEDGMENTS This research was supported by Czech Science oundation (projects: No. 101/06/P035). REERENCES [1] Pevnostní výpoet elních a kuželových ozubených kol SN MDT :539 [] Bolek, A., ochman, J.: ásti stroj -. svazek Technický prvodce, svazek 6; SNTL, Praha, 1990; ISBN [3] Pilkey, Walter D.: Peterson's Stress Concentration actors (nd Edition). (pp ). John Wiley & Sons [online]. URL < [cit ] [4] ORYL, P. - LAVÁOVÁ, M.: 3D modely ozubení. 1. ANSYS User's Meeting, rubá-skála, Sborník pednášek v elektronické form. 004, p.1-7. [5] ORYL, P. - LAVÁOVÁ, M.: Contact and substandatd gearing.in Conference Proceedings 11st CAD-EM Users Meeting 003 Int. Congress on EM Technology. Potsdam: CAD-EM Gmb, 003. p ISBN [6] ORYL, P.: The numerical study of contact pressure for substandard gearing. Sborník vdeckých prací Vysoké školy báské - Technické univerzity Ostrava, ada strojní, 00, vol. XLVII, (. 1), s ISSN [7] LEINVEBER, J. ASA, J. VÁVRA, P.: Strojnické tabulky. Scientia, 1999; ISBN [8] Ansys Inc. Ansys help [9] Moravec, V.: onstrukce stroj a zaízení II elní ozubená kola. Ostrava, Montanex a.s ISBN [10] LEWICI, D. G. BALLARINI, R.: Effect of rim thickness on gear crack propagation path. Prepared for the Seventh International Power Transmission and gearing Conference 6-9, October 1996, San Diego, California. U.S. Army Research Laboratory ARL-TR-1110 [online]. URL < [cit ] [11] Calculation of load capacity of spur and helical gears, ISO [1] ULAGA, S et al.: Synthesis of the gear pair with the help of global optimization approach. In Gearing, Transmissons, and mechanical systems: proceeding of the international conference 3-6, July 000, Nottingham Trent University. U. Professional Engineering Publishing, 000. s ISBN