im Journal of JET v°iume 10 (2°1?) p.p. 57-?0 Issue 4, December 2017 Type of article 1.01 Technology www.fe.um.si/en/jet.html ROTOR MECHANICAL STRESS ANALYSIS OF A DOUBLE-SIDED AXIAL FLUX PERMANENT MAGNET MACHINE MEHANSKA ANALIZA ROTORJEV DVOSTRANSKEGA SINHRONSKEGA STROJA Z AKSIALNIM MAGNETNIM PRETOKOM Franjo PranjičR, Peter Virtič1 Keywords: Axial flux permanent magnet machine (AFPMM), mechanical stress analysis (MSA), rotor thickness Abstract This paper presents the mechanical stress analysis (MSA) of a rotor disk in a double-sided axial flux permanent magnet machine (AFPMM). The analysis considers the rotor of a prototype AFPMM with a double external rotor and single internal stator. Rotor disks of the prototype AFPMM are constructed of two 11.6 mm-thick steel disks and represent around 50% of the total weight of the machine. The new rotor disk thickness was determined based on a rotor axial displacement due to the attractive force between the permanent magnets on opposite rotor disks. Povzetek Članek predstavlja mehansko analizo rotorjev dvostranskega sinhronskega stroja s trajnimi magneti in aksialnim magnetnim pretokom. Analiziran je rotor prototipa stroja, ki ima dvojni zunanji rotor ter notranji stator. Rotor analiziranega stroja je izdelan iz dveh 11,6 mm debelih R Corresponding author: Franjo Pranjič, Tel.: +386 3 777402, Mailing address: Koroška cesta 62a, E-mail address: franjo.pranjic@um.si 1 University of Maribor, Faculty of Energy Technology, Hočevarjev trg 1, 8270 Krško JET 57 Franjo Pranjič, Peter Virtič JET Vol. 10 (2017) Issue 4 jeklenih diskov, kar predstavlja približno 50% skupne teže stroja. Trajni magneti na nasproti ležečih rotorskih diskih povzročajo pritezne sile med rotorskimi diski, ki se posledično upognejo. Na podlagi upogiba rotorskih diskov pa je določena nova debelina le-teh. 1 INTRODUCTION Axial flux permanent magnet machines have been becoming increasingly popular lately due to their compactness, high degree of reliability, efficiency, simple construction and high-power density, [2-6]. This type of machine is also called "a disk-type machine" and has various topologies: • Single-sided (one stator and one rotor) • Double-sided (single stator-double rotor or single rotor-double stator) • Multistage (multiple rotors and stators). nc i QZ U < c DC 1 ad 1 □ C of I < £ £ ■K □ C t£ £ I < ar s ££ I < a: i u en e it u, K K ttL 4 C) Figure 1: Basic topologies of AFPMM: a) single-sided, b) double-sided, c) multistage All the above-mentioned topologies can be constructed with or without iron cores (coreless) and with surface-mounted or buried permanent magnets (PMs). Low power permanent magnet machines are usually constructed with coreless stators and steel rotors with surface mounted PMs, [1]. Each machine topology has its own strengths and weaknesses. Topologies without stator cores are used for low- and medium-power generators and have various advantages, including the absence of cogging torque, as well as their linear torque-current characteristics, high power density, and compact construction. Due to the absence of the core losses, these types of generators can operate with a higher efficiency compared to the conventional generators, [7]. Mechanical stress analysis (MSA) has been presented in several publications. In [9], the authors present the MSA for a high-speed AFPMM and analyse the stress level of the rotor disks due to the high-speed rotation, using the three-dimensional finite element method (3D FEM). Fei et al. present the simplified 2D and 3D FEM for analysis and design of rotor disks of high-speed AFPM generators in [10]. Rani et al. present the computational method of rotor stress analysis for conventional rotors using J-MAG software in [11]. In [14], the authors presented the structural analysis of low-speed axial-flux permanent-magnet machines. Virtic, [12], analysed the rotor disk thickness of the same prototype AFPMM concerning the magnetic flux density magnitudes. This article firstly presents the double-sided AFPMM with an internal coreless stator and two external rotors and its characteristics, with a focus on the selected dimensions of rotor disks. The 58 JET Rotor mechanical stress analysis of a double-sided axial flux permanent magnet machine prototype AFPMM was analytically analysed in [7] and optimized in [8] by using evolutionary optimization with a genetic algorithm and an analytical evaluation of objective functions. Since the thickness of the rotor disks was not included in the optimization (due to the assumed infinite permeability), this article presents the mechanical stress analysis (MSA) of the rotor disks used in the prototype machine and, based on the results, a new rotor disk thickness is determined. The mechanical stress analysis in this article is accomplished by: 1. analytically calculating the pressure caused by the PMs on opposite disks and the attractive force between them, 2. simulating the stress distribution and deflection of the disks with Solidworks software based on the calculated magnetic pressure and force, The primary reason for the rotor optimization lies in the fact that the weight of the two rotor disks represents about 50% of the total weight of the machine, [1]. 2 AFPMM PROTOTYPE The AFPMM considered in this article is a double-sided AFPMM with two external rotors and one internal coreless stator. Figure 2 shows the geometric parameters, and Table 1 shows the optimized data of the analysed prototype AFPMM. d s Figure 2, [8].- Geometric parameters of the AFPMM The PMs used in the prototype AFPMM are neodymium magnets (NdFeB). Figure 3 shows the PMs; their characteristics are presented in Table 2, where: • Br is the remanent magnetic flux density, • Hcb is the coercive magnetic field intensity of the magnetic flux density, • Hcj is the coercive magnetic field intensity of the polarization, • (BH)max is the maximum energy product, and • Tmax is the maximum working temperature of PMs JET 59 Franjo Pranjič, Peter Virtič JET Vol. 10 (2017) Issue 4 gfl Figure 3, [17]:NdFnB pnemnnnnj magnets usnd in jhn prototype AFPMM Table 1: GEOMETRY AND PARAMETERS OF ANALYSED AFPMM Symbol Quantity Value/Unit R Rotor disk radius 150 mm dFe Rotor disk thickness 11,6 mm dM Permanent magnet thickness 5 mm Tm Magnetic pole pitch 25° aL O 1- q RiPM Inner radius of PM 80 mm Cd RoPM Outer radius of PM 150 mm Br Remanent magnetic flux density 1,22 T Tp Pole pitch 36 ° P Number of pole pairs 5 i Rated phase current 12,3 A A Electrical current density 5 A/mm2 a Rated power at 1500 min1 4,4 kW N Number of turns per coil 50 Number of coils 12 (2x6) cc dc Coil width 20 mm O 1— < ds Stator thickness 15 mm uo Tc Coil pitch 30° m Number of phases 3 dag Air-gap thickness 1mm Sw Copper wire cross section 2,46 mm2 60 JET Rotor mechanical stress analysis of a double-sided axial flux permanent magnet machine Table 2: PROPERTIES OF PERMANENT MAGNETS USED IN PROTOTYPE AFPMM Type of Br HcB HcJ BHmax Tmax PM (kJ/m3) (T) (kA/m) (kA/m) (°C) min max min max 38SH 1,22 1,25 907 1592 287 310 150 2.1 Stator design The internal stator is constructed from non-magnetic polypropylene square plate with dimensions of 400x400x15mm. Each side of the plate has a carved space for six coils, four thermocouples, and slots for the conductors (Figure 4a). After the conductors are inserted in the slots, a varnish is applied, and the stator is ready for mounting (Figure 4b). Figure 4, [17]: a) Stator support structure, b) Stator ready for mounting 2.1 Rotor design Rotor disks are constructed from structural steel (St52), which has adequate magnetic properties and a suitable price. From the safety point of view, the thickness selected for the disks was 12 mm. After balancing, the final thickness was 11.6 mm. Figure 5, [17]/ a) Unbalanced rotor disks, b) Balanced rotor disk with au accessory for gluiug the PMs ou the rotor disk JET 61 Franjo Pranjic, Peter Virtic JET Vol. 10 (2017) Issue 4 Figure 6, [7]; n) Stntoe nnd double eojoe with the shnft, b) Rotoe disk with PMs 3 METHODS AND RESULTS The mechanical stress analysis was performed numerically and analytically, using the Solidworks simulation tool. Solidworks software simulates the magnetic pressure of the PMs on the rotor disk and determines the stress distribution and deflection using the Finite Element method (FFEPlus, i.e. Fourier Finite Element Plus algorithm). In finite element analysis, a problem is represented by a set of algebraic equations that must be solved simultaneously. FFEPlus is an iterative method that solves the equations using approximate techniques; a solution is assumed for each iteration, and the associated errors are evaluated. The iterations continue until the errors become acceptable, [13]. Since the attractive forces of PMs are high, the deflection of the disk must not be too high due to the safety reasons, such as preventing the PMs from crashing into stator surface, preventing distortion of the air gap and consequently the characteristics of the prototype AFPMM. 3.1 Parameter selection and calculation Maxwell stress is the link between electromagnetic and structural designs. It is represented by the magnetic attraction force acting between the rotor disks. Classical analysis of magnetic equivalent circuits can be used to determine the airgap flux density and hence the Maxwell stress is given by [14]: = Bl_ (3.1) q where Bd is the airgap flux density and jUo - permeability of free space (4n*10-7 Vs/Am) [14]. The magnetic flux density in the air gap is determined by equation (3.2) [1]: Bd =_B__<3-2> " 1 + ( + 0,5d s ))«- d M k = 1 ■ (3.3) sat (( + 0,5dFe) 62 JET Rotor mechanical stress analysis of a double-sided axial flux permanent magnet machine M 1 A B Mo AH (3.4) where Bd is the magnetic flux density in the air gap, Br is the remenent magnetic flux density of the PM, dag is the air gap thickness, ds is the stator thickness, dFe is the rotor disk thickness, dM is the PM thickness, ksat is the saturation factor for iron, ¡j.r is the permeability of the steel, ¡j.rec is the relative recoil permeability, which is determined with the data of the magnets in Table 2. The attractive force between PMs on opposite disks can be calculated as magnetic pressure multiplied by the active surface area of all PMs SPM as shown in [1]: B 2 F = (S PM ) Spm = «, f (( " Dn ) œ = ■ aPu2V 360 (3.5) (3.6) (3.7) Where ai is the coefficient that is calculated with the angle of PMs multiplied by the number of PMs per rotor disk (poles) and divided by 360 degrees. Using the previously-described equations, data needed for the simulation was determined as shown in Table 3. Table 3: GEOMETRY AND PARAMETERS OF ANALYSED AFPMM Symbol Quantity Value/Unit q Magnetic pressure caused by the PMs 74496 Pa Spm Active area of all PMs 0,0351 m2 F Attractive force between rotor disks 2615 N dm Permanent magnet thickness 5 mm Bd Peak value of magnetic flux density in the air gap 0,4327 T dag Air-gap thickness 1mm ds Stator thickness 15 mm jrec Relative recoil permeability 1,0704 ksat Saturation factor 1,02 3.2 Simulation First, the simulation of the stress analysis and deflection was performed for the 11.6 mm rotor disk thickness. The simulation itself included the entire rotor for the accuracy of the results since in many articles the analysis includes only a segment of the rotor. JET 63 Franjo Pranjic, Peter Virtic JET Vol. 10 (2017) Issue 4 The force between PMs on opposite rotor disks was applied on each magnet on the simulated rotor disk. Figures 7a and 7b show the Von Mises stress distribution on the rotor disk and the displacement for 11.6 mm rotor thickness, respectively. It is clear that the rotor thickness can be reduced from the mechanical point of view since the maximum deflection is only 0.0053 mm. After a few simulations, it was determined that the 7 mm rotor thickness would be sufficient to withhold the forces between the adjacent PMs on opposite disks in such a way that the deflection remains acceptable. Figures 8a and 8b show the Von Mises stress distribution on the rotor disk and the displacement of 7 mm rotor disk thickness, respectively. It can be seen from Figure 8a that the simulated deflection is 0.2171mm. In [12], the author analysed the rotor disk thickness for this prototype AFPMM concerning the magnetic characteristics of the machine and determined that the characteristics are acceptable at 7 mm rotor thickness since there is practically no difference between magnetic flux density magnitudes calculated at 7 mm and 11.6 mm of rotor disk thickness. The simulation in Solidworks shows the same result for the mechanical point of view. Figure 7:11,6 mm rotor thickness: n) deflection, b) Von Mises stress distribution Figure 8: 7 mm rotor thickness: n) deflection, b) Von Mises stress distribution 64 JET Rotor mechanical stress analysis of a double-sided axial flux permanent magnet machine 3.3 Analytical verification Equations for bending circular plates are derived in [15] and [16]. Timoshenko, [15], derived the differential equations for symmetrical bending of circular plates from observing the symmetrically distributed load acting on a circular plate. In [16], the authors presented equations for various types of loads on a circular plate. Figure 10 shows the case that is suitable for a rotor disk of AFPMM with surface mounted PMs. ya JET 65 Franjo Pranjič, Peter Virtič JET Vol. 10 (2017) Issue 4 Equations for deflection calculations are: A = Mrb DC2 + QbDC3 + qDL11 Mrb = -qa 2 i G v 2arbr (a2 - r2)" 47 Qb = (( - ro2) C2=- ■ ( w\ 1 + 2ln I (3.8) (3.9) (3.10) (3.11) ((, C, =■ 4 a r C8 =-8 2 ,2 1 + v + (1 - v)I 2 ( C 9 =■ 1 + v . ( a r | 1 - v -ln I —!- I +- 2 V br I 4 L11 = — 11 64 L17 = -17 4 1 + 41-^1 - 5 I Hl I - 4 liiL 1 + (1 + v )ln I r0 (3.12) (3.13) (3.14) (3.15) (3.16) D = Et3 12 (1 - v2) (3.17) 66 JET Rotor mechanical stress analysis of a double-sided axial flux permanent magnet machine Table 4 presents the variables used in the set of equations (3.8)-(3.17) and their values. Values of abovementioned variables are presented in Table 4 for a 7 mm rotor disk thickness. Table 4: Variables used for deflection calculation Symbol Quantity Value/Unit ya M rb Or br ro Qb Deflection of rotor disks Bending moment Outer radius of the disk Inner radius of the disk 0.20129 mm 469 Nm 150 mm 15 mm Radial location of unit line loading or start of a 80 mm distributed load Constant termed the "flexural stiffness" or "flexural 6513 Pa mm3 rigidity", Unit shear force (force per unit of circumferential 39979 Pa mm2/mm length) q Magnetic pressure C2, C3 C9 Plate constants dependent upon the ratio a=b 74496 Pa C2 0.2425 ¿11, L17 Loading constants dependent upon the ratio a=r0 v Poisson's ratio E Elastic module of the material used for rotor disks t Thickness of the circular plate (rotor disk). C3 C9 ¿11 ¿17 0,28 210 GPa 7 mm 0.0005 0.0818 0.015994568091594 0.112680595325043 D Using the equations described above, 0.20129 mm deflection was calculated for the 7 mm thick rotor disk. Magnetic pressure q was reduced by a factor that takes into account the area of magnets (multiplied by a coefficient ai) since it is not constant over the area as marked on Figure 9. Compared to the results obtained via the simulation, we can see that there is only 2.66% difference which is acceptable. JET 67 Franjo Pranjič, Peter Virtič JET Vol. 10 (2017) Issue 4 3 CONCLUSION Using the Solidworks software and a set of analytical equations a new rotor disks thickness was determined for the analysed prototype AFPMM. MSA showed that, from a mechanical point of view, the existing rotor disks thickness can be reduced to 7 mm and maintain sufficient stiffness, so the air gap does not change significantly. By changing the thickness of the rotor disks, the weight of disks is reduced by approximately 40%. References [1] Gieras JF, Wang RJ, Kamper MJ: Axinl Flux Permanent Magnet Beushlnss Machines, Springer Verlag, 2008 [2] W. Fei, P. C. K. Luk, and K. Jinupun: Design and analysis of high-spnnd coenlnss nxinl flux pnemnjnjj magnet generator with circular magnets and coils, Electr. Power Appl. IET, vol. 4, no. 9, pp. 739-747, 2010 [3] Xue, Y., Han, L., Li, H., Xie, L.: Optimal design and comparison of different PM synchronous generator systems for wind turbines, Int. Conf. Electrical Machines and Systems, pp. 2448-2453, 2008 [4] Pinilla, M., Martinez, S.: Selection of main design variables for low-speed permanent magnet machines devoted to renewable energy conversion, IEEE Trans. Energy Convers., 26, (3), pp. 940-945, 2011 [5] M. Mirsalim, R. Yazdanpanah, and P. Hekmati: Design and analysis of double-sided slotless axial-flux permanent magnet machines with conventional and new stator core, IET Electr. Power Appl., vol. 9, no. 3, pp. 193-202, 2015 [6] H. Hatami, M. Bagher, B. Sharifian, and M. Sabahi: A New Design Method for Low-Speed Torus Type Afpm Machine for Hev Applications, IJRET, Volume: 02 Issue: 12, pp. 396406, 2013 [7] P. Virtic, P. Pisek, T. Marcic, M. Hadziselimovic and B. Stumberger: Analytical Analysis of Magnetic Field and Back Electromotive Force Calculation of an Axial-Flux Permanent Magnet Synchronous Generator with Core^s Stator, IEEE Transactions on Magnetics, vol. 44, no. 11, pp. 4333-4336, 2008 [8] P. Virtic, M. Vrazic, and G. Papa: Design of an axial flux permanent Magnet synchronous machine using analytical method and evolutionary optimization, IEEE Trans. Energy Convers., vol. 31, no. 1, pp. 150-158, 2016 [9] S. Kumar, T. A. Lipo, and B. Kwon: A 32,000 rev/min axial flux permanent magnet machine for energy storage with mechanical stress analysis, IEEE Trans. Magn., vol. 52, no. 7, pp. 1-1, 2016 [10] W. Fei, P. C. K. Luk, and T. S. El-Hasan: Rotor integrity design for a high-speed modular air-cored axial-flux permanent-magnet generator, IEEE Trans. Ind. Electron., vol. 58, no. 9, pp. 3848-3858, 2011 68 JET Rotor mechanical stress analysis of a double-sided axial flux permanent magnet machine [11] J. A. Rani, E. Sulaiman, M. F. Omar, M. Z. Ahmad, and F. Khan: Somputational Method of Rotor Stress Analysis for Vorious Flux Switching Machine Using J-MAG, IEEE Stude nt Conference on Research and Development (SCOReD), 721-726, 2015 [12] P. Virtic: Analysis of rotor dish thickness in corelest stator axial flux prrsnrnrnt mnenet synchrouous machine, PRPEGR/\D ELERTROTECHNICZNY, v/ol. ISSN 003n, d7, 12, p1, 1215, 2022 [10] 1015 SalidworEs E-lelp ICoeuonentation [1 4] M.A. Mlueller, A.E. McDonald and D.E. IMacpherson: Structural analysis oo low-speed ax-al-flux pbrmbnbnt-mubnbt machines, IEE Procaedings-Eleetric Power Appl., v/ol. 602, n7. p, 27. 2417-6426, 2000 [10] E. Timonhenko: borory of Plates dnd Shrlb, Second edition, 1181, McGraw-Hill oook Company, ICBC 0-01-414719-1 [17] W. C. Young and R. G. Bunynas: Roarh's Formulao for Stress dnd Straino vol. 1, no. 1 th Editio2. 0002 [11] P. PPirtic: Nacrtovanjr in analiza sin0ronshi0 strojrv s tras'nimi sutu^'i in ahsialnim mubnbtnim prrtohomo Doctoral thes is, University of Mar^o^ 0707 Nomenclature (Symbol meaning) rotor disk ra dius rotor disk th ickness permanent magnet th ickness magnetic pitch inner radius of PM outer radius of PM remanent magaetic flux density pole pitch number of pole pairs rated phase cucrent electricalnucredt density rated power at 1000 min-1 number of tu rns per coil coil wiPth (Symbols) R dFe dM Tm RiPM Ropm Br Tp P I A P N dc JET 69 Franjo Pranjič, Peter Virtič JET Vol. 10 (2017) Issue 4 "s Tc m dag Sw Sd dd ksat ^rec Spm ai q F Ya Mrb ar far ro D Qb CfrCi C9 ¿11, L17 V E t stator thickness coil pitch number of phases air-gap thickness Copper wire cross section airgap flux density permeability of free space fictitious air gap thickness saturation factor for iron permeability of the steel relative recoil permeability active surface area of all PMs coefficient that is calculated with angle of PMs multiplied by the number of PMs per rotor disk (poles) and divided by 360 degrees magnetic pressure attractive force between adjacent magnets deflection of rotor disks bending moment outer radius of the disk inner radius of the disk radial location of unit line loading or start of a distributed load stiffness factor of the material unit shear force (force per unit of circumferential length) plate constants dependent upon the ratio a=b loading constants dependent upon the ratio a=r0 Poisson's ratio elastic module of the material used, thickness of the circular plate (rotor disk) 70 JET