MEV Mechatronics, Electrical Power, and Vehicular Technology 03 (2012) 87-94 Mechatronics, Electrical Power, and Vehicular Technology e-ISSN: 2088-6985 p-ISSN: 2087-3379 Accreditation Number: 432/Akred-LIPI/P2MI-LIPI/04/2012 www.mevjournal.com Β© 2012 RCEPM - LIPI All rights reserved doi: 10.14203/j.mev.2012.v3.87-94 ANALYTICAL AND NUMERICAL DEFLECTION STUDY ON THE STRUCTURE OF 10 KW LOW SPEED PERMANENT MAGNET GENERATOR KAJIAN DEFLEKSI ANALITIS DAN NUMERIK PADA STRUKTUR GENERATOR MAGNET PERMANEN KECEPATAN RENDAH KAPASITAS 10 KW Hilman S. Alam a, *, Pudji Irasari b, Dyah Kusuma Dewi c a Technical Implementation Unit for Instrumentation Development, Indonesian Institute of Sciences, Jl. Sangkuriang Komplek LIPI Gedung 30 Bandung, 40135, Indonesia b Research Center for Electrical Power and Mechatronics, Indonesian Institute of Sciences, Jl. Sangkuriang Komplek LIPI Gedung 20 Lantai 2 Bandung, 40135, Indonesia c Directorate of Technology for Manufacturing Industry, Agency for Assessment and Application of Technology, Gedung Teknologi 2 Puspitek Serpong, Tangerang, Indonesia Received 23 October 2012; Received in revised form 01 December 2012; Accepted 03 December 2012 Published online 18 December 2012 Abstract Analytical and numerical studies of the deflection in the structure of 10 kW low speed permanent magnet generator (PMG) have been discussed in this paper. This study is intended to prevent failure of the structure when the prototype is made. Numerical analysis was performed with the finite-element method (FEM). Flux density, weight and temperature of the components are the required input parameters. Deflection observed were the movements of the two main rotor components, namely the rim and shaft, where the maximum deflection allowed at the air gap between rotor and stator should be between 10% to 20% of the air gap clearance or 0.1000 mm to 0.2000 mm. Base on the analysis, total deflection of the analytic calculation was 0.0553 mm, and numerical simulation was 0.0314 mm. Both values were in the acceptable level because it was still below the maximum allowed deflection. These results indicate that the structure of a permanent magnet generator (rim and shaft) can be used safely. Key words: permanent magnet generator, finite element, air gap, deflection. Abstrak Studi secara analitis dan numerik mengenai defleksi pada struktur generator magnet permanen (GMP) kecepatan rendah kapasitas 10 kW telah dibahas dalam makalah ini. Studi ini dimaksudkan untuk mencegah kegagalan struktur saat prototipe sudah dibuat. Analisis numerik dilakukan dengan metode elemen hingga (MEH). Kerapatan fluks, berat dan suhu komponen merupakan parameter-parameter masukan. Defleksi yang diamati adalah gerakan dua komponen utama rotor yaitu rim dan poros, di sini defleksi maksimum yang diizinkan pada celah udara antara rotor dan stator harus berkisar antara 10% sampai 20% dari clearance celah udara atau 0,1000 mm sampai 0,2000 mm. Berdasarkan hasil analisis, defleksi total hasil perhitungan analitis adalah 0,0553 mm sedangkan simulasi numerik adalah 0,0314 mm. Kedua nilai tersebut memenuhi persyaratan karena masih di bawah defleksi maksimum yang diizinkan. Hasil tersebut menunjukkan bahwa struktur generator magnet permanen (rim dan poros) dapat digunakan secara aman. Kata kunci: generator magnet permanen, elemen hingga, celah udara, defleksi. I. INTRODUCTION In general, the constructions of low-speed high torque permanent magnet generators (PMG) tend to have large dimension, heavy and expensive. The major costs are caused by materials, installation and transportation. The construction of the radial flux PMG is dominated by the weight of inactive components that is equal to 2/3 of the total weight, while the rest is that of the active components (iron, copper, and permanent magnet) [1]. They serve as a support to keep the clearance at the air gap and to hold * Corresponding Author. Tel: +62-81394297528 E-mail: alam_hilman@yahoo.com http://dx.doi.org/10.14203/j.mev.2012.v3.87-94 H.S. Alam et al. / Mechatronics, Electrical Power, and Vehicular Technology 03 (2012) 87-94 88 the active components to stay in place when subjected to normal force, shear force and thermal effects. Reducing the weight of generator is an issue of interest to designers and manufacturers. This is because the inactive structure of direct drive generator is directly connected to the prime mover and its weight can reach 80% of the total [1]. It is needed to counteract the magnetic attractive force between the stationary and moving parts and it is influenced by the type and nature of the material used. Tensile stress is generated by normal/Maxwell force that could reach ten times the shear stress. Distance or clearance between the rotor and the stator must be maintained to avoid damage to the PMG [2,3]. Research to find potential solutions in terms of geometry, materials and sources of excitation becomes an important issue to increase market competition, reducing prices and weights of components so that the efficiency and reliability of PMG can be improved [3]. Design of the generator in this study is focused on the rotor shaft that serves as a support structure of active components. One important step in the design is stress analysis on the structure, which will find the number of iterations to meet allowable rotor deflection before manufacturing of prototypes. Stress analysis based on an analytical approach then is validated by numerical methods or also called finite element method (FEM). The method has been applied to the analysis of electromagnetic in some earlier studies [1-10]. Several advantages over other numerical methods are: it gives detailed and exact analysis and computation [5][9]; it provides an efficient solution [6]; it is able to analyze various types of electrical machines, including permanent magnet parametric geometry, post processing and visualization of results [10]. II. METHODOLOGY Clearance on generator is generally calculated as 1/1000 of the air gap diameter [1]. For 10 kW of capacity, the PMG with air gap radius, rg = 168.5 mm will need 0.1685 mm of clearance. The allowed deflection of the rotor ranges from 10- 20% [1]. Figure 1 shows a cross-section of radial flux PMG. Air gap between the stator and rotor is designed to be 1 mm, greater than it should be (0.1685 mm) due to manufacturing consideration, then the maximum deflection of the air gap clearance to be ranged between 0.100 mm to 0.200 mm. The parameters, which affect the deflection of the structure consist of flux density 𝐡𝐡�, mass of the component M and the temperature difference Ξ”T. They are used as inputs to calculate the normal stress, weight of the components and the thermal expansion of the material. Design is analytically calculated and the results are validated using FEM. The allowable maximum deflection is used as a consideration in the design iteration. Input data is obtained from the calculation using the FEM. Figure 2 shows the method used in designing the structure of PMG to target 10-20% total deflection of the air gap clearance. A. The Existence of Forces on PMG The distance between the rotor and stator in the PMG, is one of the most critical factors in the design considerations. Besides affected by weight of the part itself, deflection is also influenced by the air gap flux density. If it increases, the normal Figure 1. Cross section of the radial flux PMG. Figure 2. Design methodology of the PMG. H.S. Alam et al. / Mechatronics, Electrical Power, and Vehicular Technology 03 (2012) 87-94 89 stress and deflection in the rotor will rise higher. In addition subjected to normal stresses, PMG is subjected to a shear stress as well. Shear stress is one of the important factors in the design as it relates to the torque to be generated [1]. Normal stresses on PMG occur in the normal direction or lead directly to the air gap, see Figure 3. Normal stresses on the stator and rotor move in and out radially and are larger than the shear stress. When the flux density, 𝐡𝐡� in the air gap rises (during operation: 𝐡𝐡� >0.8T), the normal stress, which will be produced is around 10 times the shear stress. The normal stress π‘žπ‘ž is a function of the square of the air gap flux density [1]: π‘žπ‘ž = 𝐡𝐡 οΏ½2 2πœ‡πœ‡π‘‘π‘‘ [Pa] (1) where: 𝐡𝐡�: air gap flux density [T or N/Am] and Β΅o: permeability of free space (1.26 Γ— 10-6 N/A2). Shear stress 𝜎𝜎� [Pa] is the most important variable in the design and is proportional to the generated torque as represented by the equation [1]: 𝑇𝑇 = 2πœ‹πœ‹πœŽπœŽοΏ½π‘…π‘…2𝑙𝑙[N.m] (2) where: R = radius of PMG [m] and l = axial length of the PMG [m]. Shear stress 𝜎𝜎� [Pa] acting on the PMG is perpendicular or cut the air gap, see Figure 4. When the wave of flux density 𝐡𝐡� [T or N/Am] and the electric load 𝐾𝐾� is sinusoidal at an angle Ξ΄ [A/m], the shear stress is [1]: 𝜎𝜎� = 1 2 𝐡𝐡�𝐾𝐾�[Pa] (3) B. Thermal Expansion Besides influenced by the normal force and weight of the components, the air gap clearance is affected by dimensional changes due to heating. The heat arising from the losses occurred in the PMG causes the temperature rise and the expansion of the components. The difference in temperature rise between the stator and the rotor will cause changes in the air gap clearance, see Figure 5. Dimensional changes due to thermal expansion are calculated as [1]: βˆ†πΏπΏ = πΏπΏπ‘‘π‘‘π›Όπ›Όβˆ†π‘‡π‘‡ (4) where βˆ†π‘™π‘™ : dimensional changes [m], 𝛼𝛼 : coefficient of thermal expansion of the material [Β°C-1], 𝑙𝑙𝑑𝑑: initial length [m], and βˆ†π‘‡π‘‡: temperature rise [Β°C]. C. Designing Rim Rim on the rotor serves as a retaining structure and the permanent magnet holder as demonstrated in Figure 6. In this study, it uses steel with Young's modulus of 200 GPa. Deflection on the rim, π‘ˆπ‘ˆπ΄π΄ is calculated using [1, 10]: Figure 3. Cross-section of the PMG with normal voltage along the air gap. Figure 5. Expansion due to temperature rise in the stator βˆ†Ts and rotor βˆ†Tr. Figure 4. Cross section of the PMG with shear stress along the air gap. Figure 6. Cross section area of the rim. H.S. Alam et al. / Mechatronics, Electrical Power, and Vehicular Technology 03 (2012) 87-94 90 π‘ˆπ‘ˆπ΄π΄ = π‘žπ‘žπ‘…π‘…2 πΈπΈβ„Žπ‘¦π‘¦π‘Ÿπ‘Ÿ ⎩ βŽͺ ⎨ βŽͺ ⎧ 1 + 𝑅𝑅3οΏ½π‘˜π‘˜1 (𝑠𝑠𝑙𝑙𝑛𝑛𝑠𝑠 βˆ’π‘ π‘ π‘π‘π‘‘π‘‘π‘ π‘ π‘ π‘  ) 4𝑠𝑠𝑙𝑙𝑛𝑛 2 𝑠𝑠 βˆ’ π‘˜π‘˜2 2𝑠𝑠𝑙𝑙𝑛𝑛𝑠𝑠 +π‘˜π‘˜2 2 2𝑠𝑠 οΏ½ 𝐼𝐼�� 𝑠𝑠 𝑠𝑠𝑙𝑙𝑛𝑛 2 𝑠𝑠 + 1 𝑑𝑑𝑝𝑝𝑛𝑛𝑠𝑠 οΏ½οΏ½ 𝑅𝑅 4𝐴𝐴 +𝑅𝑅 3 4𝐼𝐼 οΏ½βˆ’π‘…π‘… 3 2𝐼𝐼𝑠𝑠 οΏ½ 1 οΏ½π‘˜π‘˜π‘…π‘…οΏ½ 2 +1 οΏ½+𝑅𝑅1βˆ’π‘…π‘…π‘‘π‘‘ 𝑝𝑝 οΏ½ ⎭ βŽͺ ⎬ βŽͺ ⎫ [mm] (5) where, R : radius of the neutral axis at the rim (143.5 mm), Ro: radius of the shaft (50 mm), Ri: radius of the rim surface (135 mm), ΞΈ: angle between the two rim (30Β°), A: cross-section area of the arm retaining rim (43,975 mm2), a: cross- section area of the rim, 20,781 mm2, I: moment of inertia rim (258,575,015 mm4), k: radius of the rim gyration (π‘˜π‘˜ = �𝐼𝐼/𝐴𝐴 = 76.68 π‘šπ‘šπ‘šπ‘š), k1 and k2: the correction factor of stress concentration due to momen and shear stress, and hyr: thickness of the rim (10 mm). Stress concentration at the retaining structure of the rim will result in a correction factor due to the moments and shear forces, see Figure 7. By assuming the geometry of the structure is in the form of ellipse, the correction factor can be calculated with the equation [10]: π‘˜π‘˜1 = 𝐢𝐢1 + 𝐢𝐢2 οΏ½ 2𝑝𝑝 𝐷𝐷 οΏ½ + 𝐢𝐢3 οΏ½ 2𝑝𝑝 𝐷𝐷 οΏ½ 2 (6) for 0.4 ≀ 2𝑝𝑝/𝐷𝐷 ≀ 1.0, then : 𝐢𝐢1 = 3.465 βˆ’ 3.739 Γ— οΏ½ 𝑝𝑝 𝑐𝑐 + 2.274 Γ— 𝑝𝑝 𝑐𝑐 (7) 𝐢𝐢2 = βˆ’3.841 + 5.582 Γ— οΏ½ 𝑝𝑝 𝑐𝑐 βˆ’ 1.741 Γ— 𝑝𝑝 𝑐𝑐 (8) 𝐢𝐢3 = 2.376 βˆ’ 1.843 Γ— οΏ½ 𝑝𝑝 𝑐𝑐 βˆ’ 0.534 Γ— 𝑝𝑝 𝑐𝑐 (9) The correction factor due to the influence of shear force is represented by [10]: π‘˜π‘˜2 = 𝐢𝐢1 + 𝐢𝐢2 οΏ½ 2𝑝𝑝 𝐷𝐷 οΏ½ + 𝐢𝐢3 οΏ½ 2𝑝𝑝 𝐷𝐷 οΏ½ 2 + 𝐢𝐢4 οΏ½ 2𝑝𝑝 𝐷𝐷 οΏ½ 3 (10) for 0.5 ≀ 𝑝𝑝/𝑐𝑐 ≀ 10.0, then: 𝐢𝐢1 = 1.000 + 2.000 Γ— 𝑝𝑝 𝑐𝑐 (11) 𝐢𝐢2 = βˆ’0.351 βˆ’ 0.021 Γ— οΏ½ 𝑝𝑝 𝑐𝑐 βˆ’ 2.483 Γ— 𝑝𝑝 𝑐𝑐 (12) 𝐢𝐢3 = 3.621 βˆ’ 5.183 Γ— οΏ½ 𝑝𝑝 𝑐𝑐 + 4.494 Γ— 𝑝𝑝 𝑐𝑐 (13) 𝐢𝐢4 = βˆ’2.270 + 5.204 Γ— οΏ½ 𝑝𝑝 𝑐𝑐 βˆ’ 4.011 Γ— 𝑝𝑝 𝑐𝑐 (14) D. Shaft Design Shaft diameter is designed based on analytical calculations with the data inputs are weight of the components. Figure 8 shows the design of the rotor shaft construction of the PMG. The mass that must be supported by the shaft is obtained from the data of geometry and density of the permanent magnet material and rim. Based on Figure 8, diagram of a simple free- body can be modeled to calculate the load and deflection in the shaft (see Figure 9). Force F1 is generated from the input torque to rotate the generator while the force F2 is generated from normal weight of the rim and the magnet. These two forces are detained by the reaction forces on the bearing RA and RB. To find out-loading on the shaft, the following static equilibrium equations are used [12]: �Σ𝐹𝐹 = 0 Σ𝑀𝑀 = 0 οΏ½ (15) The maximum shear stress on the shaft 𝜏𝜏 [MPa] is [12]: 𝜏𝜏 = 0.5πœŽπœŽπ‘¦π‘¦π‘π‘ 𝐹𝐹𝑠𝑠 = 16 πœ‹πœ‹π‘‘π‘‘3 οΏ½((πΆπΆπ‘šπ‘šπ‘€π‘€)2 + (𝐢𝐢𝑑𝑑𝑇𝑇)2 ) (16) where: πœŽπœŽπ‘¦π‘¦π‘π‘ : yield strength of the material (250 MPa for steel ST 45), 𝐹𝐹𝑠𝑠 : safety factor, M: the maximum moment on the shaft [N.m], T: the maximum torque on the shaft from the design power [N.m], Cm: the fatigue life factor and shock loads (1.5 for initial sudden load), Ct: Figure 7. Correction factor due to: (a) moment and (b) shear force. Figure 8. Rotor shaft construction. H.S. Alam et al. / Mechatronics, Electrical Power, and Vehicular Technology 03 (2012) 87-94 91 factor influenced by torque/twist (1.0 for small vibration) [11]. Based on the relationship between strain and torque curves for elastic material, shaft deflection is calculated by the equation [12]: 𝑑𝑑2𝑣𝑣 𝑑𝑑π‘₯π‘₯ 2 = 𝑀𝑀 𝐸𝐸𝐼𝐼 (17) where 𝑣𝑣: the elastic deflection curve [m], M: the maximum moment on the shaft [MPa], E: modulus of elasticity of the shaft material [MPa] and I: moment of inertia of the shaft [m4]. By integrating the equation and inserting the boundary conditions on the support or bearing then the maximum deflection on the shaft can be known. E. Finite Element Method In general, analysis procedure using FEM consists of three steps: preprocessing, field solution, and post processing. Preprocessing comprises meshing and defining the material sand problems. In the meshing process, the continuum or area is divided into a set of finite number of elements. Defining the materials includes determination of the types of material in the sub-region, while defining the problems is the determination of the boundary conditions required as input in the calculation. Field processing is the solution of partial differential equation based on minimizing the energy function, which is a mathematical function that relates to the potential energy stored in the system. Post processing is the extraction of the solution into a quantitative value in the form of graphs that contains all the parameters, and critical value [4]. III. RESULT AND DISCUSSION A. Rim Deflection Input data to get the rim deflection is the air gap flux density, which is calculated using the finite element solver FEMM4.2 [17]. The maximum flux density 𝐡𝐡� to one pole is 1.05 T, see Figure 10. With reference to Eq.(1), the normal stress q is 438.67 kN/m2. Then based on Eq.(6) to (9), the constants C1, C2 and C3 are 2.771, 0.582 and 1.355 respectively. The correction factor due to the moment k1 is 2.666. From Eq.(10) to (14), the constants C1, C2 and C3 each is 5.095, -5.465 and 5.406. The correction factor due to the shear stress, k2 is 3.184. By substituting all constants and correction factors above into equation (5), 0.00452 mm of the total deflection at the rim UA is obtained. The result of analytical calculation of the rim deflection is further validated using FEM. Figure 11 shows the result of post processing. The maximum deflection at the rim is equal to 0.00791 mm, found at the end of the rim marked in red color. Deflection on other areas of the rim is clearly visible in accordance with subdomains or elements. B. Shaft Deflection Based on the geometry and material density data, the total mass that must be supported by the shaft is 29.833 kg. Multiplying the gravitational acceleration 9.8 m/s2, thereby the load on the shaft due to the weight of the component F2 is 292.36 N. In designing the shaft, normal stresses arising from the flux density in the air gap could be ignored because the direction is opposite to each other thus eliminating one another. From design data, the maximum power transmitted P = 10 kW and speed n = 600 rpm, as a result the maximum torque of the shaft is 159.26 N.m. Assuming that 100 mm of a pulley diameter is mounted on the rotor then the force generated F1 is 3,184.8 N. According to moment equilibrium law (Eq.(15)), the maximum moment acting on the shaft lies in the pedestal of the bearing A, with M = 462.75 N.m. By using Eq.(16), the maximum stress in the shaft is 62.5 MPa. Referring to the calculation results, the diameter of the shaft, which is safe from the aspect of loading with the minimum safety factor of 2 is 40 mm. However, for the ease of manufacturing, the shaft diameter is adapted to the diameter of the Figure 9. Free-body diagram on the shaft. Figure 10. Flux density distributions at air gap region for one pole. H.S. Alam et al. / Mechatronics, Electrical Power, and Vehicular Technology 03 (2012) 87-94 92 common bearing in the market and 55 mm of the closest diameter is determined. By entering the boundary conditions of the bearing deflection equals to 0 then according to Eq.(17) shaft deflection is 0.0518 mm, which is found at the end part or in the area of F1. Shaft deflection resulted from FEM simulation reaches its maximum value at 0.0246 mm or about 50% lower than the analytical calculation, see Fig.12. While the deflection at the middle or on the rim support is close to zero. Thus, 55 mm of shaft diameter is safe to use as rotor retaining structure. C. Deflection due to thermal expansion of the material Maximum permissible temperature of the permanent magnet (100Β°C) is a reference to calculate the thermal expansion of material between the stator and rotor. If the thermal expansion coefficient of steel is 2.13Γ—10-6/Β°C, the ambient temperature is 25Β°C, rotor radius Rr is 0.1680 m, and stator radius Rs is 0.1685 m, then referring to Eq.(4), the changes of rotor and stator diameters (Ξ”Rr and Ξ”Rs ) are 0.166 mm and 0.167 Figure 11. Deflection on the rim resulted from numerical simulation. Figure 12. Shaft deflection resulted from numerical simulation. H.S. Alam et al. / Mechatronics, Electrical Power, and Vehicular Technology 03 (2012) 87-94 93 mm respectively. Thereby the change in diameter due to thermal expansion will cause the increase of air gap opening of 0.001 mm. D. Total Deflection on the Structure of PMG Total deflection on the structure of PMG is the sum of the rim and shaft deflection and deflection due to thermal expansion of the material. Table 1 shows the maximum deflection for each component and the total deflection on all structures. There is a little difference in the deflection between the analytical and simulation results; however, both are allowable because the values are still below the maximum allowable deflection (10% to 20% of the air gap clearance). IV. CONCLUSION β€’ Analytical and numerical analysis of the deflection of generator structure have been discussed in this paper. β€’ Both analyses are each producing 0.0553 mm and 0.0314 mm of the total d e f l e c t i o n or about 43% di f f er e n t . Nevertheless both are included in the safe deflection categories because the values are still below 10% to 20% of the air gap clearance. β€’ Based on the results, it can be concluded that 55 mm of the selected shaft rotor diameter can be safely used. ACKNOWLEDGEMENT The authors grateful to Directorate of Technology for Manufacturing Industry, Agency for Assessment and Application of Technology, for allowing numerical simulation using CATIA. Moreover thanks to all the team members of electric machines in The Research Center for Electrical Power and Mechatronics, Indonesian Institute of Sciences forany assistance that has been given. REFERENCES [1] A. S. McDonald, M. A. Mueller and H. Polinder, β€œComparison of Generator Topologies for Direct-Drive Wind Turbines Including Structural Mass,” Proceedings of the International Conference on Electrical Machines (ICEM06), Chania, Crete, 2-5 September 2006. [2] A. S. McDonald, M. A. Mueller and H. Polinder, β€œStructural Mass in Direct-Drive Permanent Magnet Electrical Generators,” Renewable Power Generation, Vol. 2, pp. 3- 15, 2008. [3] M. A. Mueller and A. S. McDonald, β€œA Lightweight Low Speed Permanent Magnet Electrical Generator for Direct-Drive Wind Turbines,” Wind Energy, Vol. 12, pp. 768- 780, 2009. [4] L. Ovacik, β€œOptimal Design of Nonlinier Magnetic Systems Using Finite Elements,” Journal of Engineering and Natural Sciences, pp. 1-39, 2004. [5] Y. Dou, Y. Gu, J. 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Mirzaei, β€œDesign, Prototyping and Analysis of Low-Cost Disc Permanent Magnet Generator with Rectangular Flat-Shaped Magnets,” Iranian Journal of Science & Technology, Transaction B, Engineering, Vol. 32, No. B3, pp 191-203, 2008. [9] E. Schlemmer, β€œFinite Element Analysis of Electrical Machines Used in Two-Frequency Indirect Temperature Rise Tests,” International Conference on Renewable Energies and Power Quality, 2009. [10] A. Reinap, D. Hagstedt, F. MΓ‘rquez, Y. Loayza, and M. AlakΓΌla, β€œDevelopment of A Radial Flux Machine Design Environment,” IEEE International Conference on Electrical Machines, 2008. Table 1. Total maximum deflection on the structure of PMG. Component Max. Deflection (mm) Analytic Numeric Rim 0.0045 0.0078 Shaft 0.0518 0.0246 Termal Expansion - 0.0010 (Analytic) Total Deflection 0.0553 0.0314 H.S. Alam et al. / Mechatronics, Electrical Power, and Vehicular Technology 03 (2012) 87-94 94 [11] R.J. Roark and W.C. 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