Meta Modeling of Transmission Error for Spur, Helical and Planetary Gears for Wind Turbine ... PDF
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Research report 2013:06 Meta Modeling of Transmission Error for Spur, Helical and Planetary Gears for Wind Turbine Application MUHAMMAD IRFAN Department of Applied Mechanics CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2013 Research report 2006:08 Meta Modeling of Transmission Error for Spur, Helical and Planetary Gears for Wind Turbine Application by MUHAMMAD IRFAN Department of Applied Mechanics CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden, 2013 Meta Modeling of Transmission Error for Spur, Helical and Planetary Gears for Wind Turbine Application MUHAMMAD IRFAN © MUHAMMAD IRFAN, 2013 Research report 2013:06 ISSN 1652-8549 Department of Applied Mechanics Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone +46 (0)31 772 1000 Abstract: Detailed analysis of drive train dynamics requires accounting for the transmission error that arises in gears. However, the direct computation of the transmission error requires a 3-dimensional contact analysis with correct gear geometry, which is impractically computationally intense. Therefore, a simplified representation of the transmission error is desired, a so-called meta-model, is developed. The model is based on response surface method, and the coefficients of the angle- dependent transmission error are dependent on shaft eccentricity (parallel misalignment), shaft bending (angular misalignment), torque and speed. Parallel spur gear, helical gear and planetary gear are studied, with parameters for wind turbine applications is considered and Abaqus 6.12-1 is used for the development of the meta-models. Upon evaluating the results, it is concluded that meta-modeling technique can be an efficient way of predicting the transmission error. Keywords: Transmission error, Wind turbine, Meta-models, Spur gear, Helical gear, Planetary gear, Regression analysis. CHALMERS, Applied Mechanics, Research Report 2013:06 1 Preface This work is carried out at the Division of Dynamics the Department of Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden, under the supervision of Professor Viktor Berbyuk and Lecturer Håkan Johansson. The work is part of the ongoing research into wind turbine drive train system dynamics supported by the Swedish Wind Power Technology Centre (SWPTC) [http://www.chalmers.se/ee/swptc-en]. The work is also reported as final thesis for M.Sc. degree in Mechanical Engineering with Emphasis on Structural Engineering at Blekinge University, Sweden. Formal examiner and subervisor from Blekinge was Ansel Berghuvud, Senior Lecturer at the Department of Mechanical Engineering, Karlskrona, Sweden. CHALMERS, Applied Mechanics, Research Report 2013:06 2 Acknowledgement First and foremost, I would like to thank the Almighty ALLAH who blessed me the ability to complete this work. I would also like to thank my parents and other family members to provide me the support and love. This work is carried out at the Division of Dynamics the Department of Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden, under the supervision of Professor Viktor Berbyuk and Lecturer Håkan Johansson. The work is part of the ongoing research into wind turbine drive train system dynamics supported by the Swedish Wind Power Technology Centre (SWPTC) [http://www.chalmers.se/ee/swptc-en]. The work is also supervised by Ansel Berghuvud, Senior Lecturer at the Department of Mechanical Engineering, Karlskrona, Sweden. It is with immense gratitude that the work would not have been possible without the valuable ideas and comments of Håkan Johansson throughout the work period. I also wish my sincere thanks to Ph.D. students Gaël Le Gigan, Shahab Teimourimanesh, and Xin Li at the Department of Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden, for extending the help. Muhammad Irfan CHALMERS, Applied Mechanics, Research Report 2013:06 3 Contents 1 Notations ......................................................................................................................................... 6 2 Introduction .................................................................................................................................... 8 3 Gear Modeling in Abaqus.............................................................................................................. 11 3.1 Gear Geometry and Part Module ........................................................................................................ 11 3.2 Material Module ................................................................................................................................. 12 3.3 Assembly Module ............................................................................................................................... 12 3.4 Steps Module ...................................................................................................................................... 12 3.5 Interaction Module ............................................................................................................................. 13 3.6 Load Module ...................................................................................................................................... 13 3.7 Boundary Module ............................................................................................................................... 13 3.8 Mesh Module ...................................................................................................................................... 13 4 Meta-Modeling .............................................................................................................................. 16 4.1 Regression Analysis ........................................................................................................................... 16 4.1.1 Random Discrete Variables ......................................................................................................... 16 4.1.2 The Analysis of Variance ............................................................................................................ 16 5 Results ........................................................................................................................................... 18 5.1 Spur Gear ............................................................................................................................................ 18 5.1.1 Radial Misalignment:S1 .............................................................................................................. 18 5.1.2. Angular Misalignment:S2 .......................................................................................................... 23 5.1.3. Radial and Angular Misalignment .............................................................................................. 26 5.1.4. Transmission error for torque:S3 ................................................................................................ 28 5.1.5. Transmission error for radial, angular, and torque:S4 ................................................................ 31 5.1.6. Transmission error for pressure angle:S5 ................................................................................... 32 5.1.7. Transmission error for gear ratio:S6 ........................................................................................... 34 5.1.8. Transmission error for addendum:S7 ......................................................................................... 36 5.2 Helical Gear ........................................................................................................................................ 38 5.2.1 Radial Misalignment:H1 ............................................................................................................. 38 5.2.2. Angular Misalignment:H2 .......................................................................................................... 39 5.2.3. Radial and Angular Misalignment .............................................................................................. 41 5.2.6. Transmission error for torque:H3 ............................................................................................... 42 5.2.5. Transmission error for radial, angular misalignments and torque:H4 ............................. 43 5.2.6. Transmission error for pressure angle:H5 .................................................................................. 44 5.2.7. Transmission error for gear ratio:H6 .......................................................................................... 45 5.2.8. Transmission error for addendum:H7 ........................................................................................ 46 5.2.9. Transmission error for helix angle:H8 ....................................................................................... 47 CHALMERS, Applied Mechanics, Research Report 2013:06 4 5.3 Planetary Gear .................................................................................................................................... 49 5.3.1. Radial and Angular Misalignment .............................................................................................. 49 5.3.2. Transmission error for radial, angular misalignments and Torque:P4 ............................ 51 5.3.3. Transmission error for radial, angular misalignments, torque, pressure angle, gear ratio, and addendum of planetary gears. ................................................................................................................ 51 5.3.4. Comparison of TE from meta-models and simulation ............................................................... 52 6 Evaluation ..................................................................................................................................... 54 6.1. Spur gear............................................................................................................................................ 54 6.2 Helical gear......................................................................................................................................... 57 6.3. Planetary gear .................................................................................................................................... 58 7 Conclusion ..................................................................................................................................... 59 8 Future work .................................................................................................................................. 60 References ......................................................................................................................................... 61 Appendices ........................................................................................................................................ 64 A. Matlab Coding for TE .......................................................................................................................... 64 B. Matlab Coding to Plot TE lines and Mean values ................................................................................ 65 C. Matlab Coding for Meta-model and Polynomial Fit of Coefficients .................................................... 66 D. Matlab Coding for Radial and Angular Misalignments Meta-models ................................................. 68 E. Meshing Technique ............................................................................................................................... 69 CHALMERS, Applied Mechanics, Research Report 2013:06 5 1 Notations mm millimeter MW Megawatt Dependent output variable Independent input variable 𝑦 Coefficient of model 𝑥 𝛽 𝑜 Random error Estimated coefficient 𝜀 Residual 𝑏 Out variable in matrix form 𝑅 Input variable in matrix form 𝑌 Estimated value 𝑋 Variance 𝐸 Identity matrix 𝑉𝑎𝑟 𝐼𝑁 Standard deviation 2 𝜎 Estimated output variable 𝑌� Estimated input variable ′ 𝑥𝑝 Estimated standard deviation 2 Average value 𝑠 Statistic estimation checking 𝑦� Null hypothesis 𝐹 𝐻𝑜 Alternative hypothesis 𝐻𝛼 Coefficient of determination 2 Adjusted statistic estimation 𝑅 2 𝑅 𝐴 Statistic estimation checking C3D8R 8-node linear brick element 𝑡 Indices Number of variables Number of observations 𝑝 Level of statistic estimation 𝑁 A Adjusted statistic 𝛼 CHALMERS, Applied Mechanics, Research Report 2013:06 6
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