Mechanical stress and deformation in the rotors of a high-speed PMSM and IM
DOI: 10.1007/s00502-021-00866-5
© The Author(s) 2021
Received: 15 December 2020
Accepted: 5 February 2021
Published: 2 March 2021
Abstract
High-speed electric machines are gaining importance in the field of traction drives and aviation due to their high power density. The evaluation of the mechanical stress in the rotor is one crucial part in the design process for this type of machines. The mechanical stress cannot be measured directly. Accordingly, a validation of the calculated mechanical stress is difficult and normally not performed. Instead of the mechanical stress, the deformation at the rotor surface can be measured using a spin test machine with distance sensors. The deformation can then be used to validate the calculation results.
In this paper, the mechanical load exerted on an IM rotor for a \(60\,\text{kW}/20000\,\frac{1}{\text{min}}\) high-speed electric machine and an PMSM rotor for a \(75~\text{kW}/25000\,\frac{1}{\text{min}}\) high-speed electric machine is analysed in detail. The mechanical stress and the deformation are calculated and analysed using a FEM simulation model. Then, a spin test is performed on the two rotors. First, the burst speed is determined by operating two rotor samples above their defined test speed. Then, the deformation is measured at the rotor surface for different operating speeds and the defined test speed. The measurement and the simulation results are compared and discussed.
It can be shown that the two designs do not exceed the maximum mechanical stress for the defined operating range. In the deformation measurement of the IM rotor, a plastic deformation up to \(\varepsilon _{\text{IM, pl}} = 8\) μm and elastic deformation up to \(\varepsilon _{\text{IM, el}}=22\) μm can be seen. In regards to plastics, PMSM rotor expands up to \(\varepsilon _{\text{PMSM, pl}}= 5\) μm. The maximum elastic deformation of the PMSM rotor is \(\varepsilon _{\text{PMSM, el}}=40\) μm. The comparison of the calculated and the measured elastic deformation shows good accordance for the two rotor types. Both models are capable of describing the deformation and the state of stress in the rotors. In burst tests, both rotors withstand rotational speeds far above the defined test speed.
Keywords
high-speed electric machines von Mises stress mechanical stress deformation overspeed test burst test permanent magnet machine induction machine machine designMechanischer Stress und Verformung in den Rotoren einer Hochdrehzahl-PMSM und einer Hochdrehzahl-IM
Zusammenfassung
Im Bereich der elektrischen Fahrzeug- und Flugzeugantriebe gewinnen Hochdrehzahlmaschinen aufgrund ihrer hohen Energiedichte an Bedeutung. Im Dimensionierungsprozess spielt die Bewertung der mechanischen Spannung im Rotor eine entscheidende Rolle. Die Validierung der Berechnungsergebnisse stellt eine Herausforderung dar, da die mechanische Spannung nicht direkt gemessen werden kann. Anstelle der mechanischen Spannung, kann die Verformung an der Rotoroberfläche bei einem Schleudertest mittels Abstandssensoren gemessen werden. Die Verformungen können zur Validierung des Berechnungsmodells verwendet werden und somit einen Rückschluss auf die mechanische Spannung im Rotor der Maschine geben.
In dieser Arbeit wird die mechanische Belastung in einer 60 kW/20000 \(\frac{1}{\mathrm{min}}\) Hochdrehzahl-Induktionsmaschine (IM) und einer 75 kW/25000 \(\frac{1}{\mathrm{min}}\) Hochdrehzahl-permanentmagneterregten Synchronmaschine (PMSM) untersucht. Die mechanische Spannung und die Verformung werden mittels eines FEM-Simulationsmodells berechnet und analysiert. Mit zwei Prototypen werden Schleudertests durchgeführt und die Berstdrehzahl des PMSM-Rotors und des IM-Rotors bestimmt. Danach wird an zwei weiteren Prototypen die Verformung an der Rotoroberfläche für unterschiedliche Betriebsdrehzahlen und die definierten Schleuderdrehzahlen gemessen, bevor auch für diese beiden Rotoren die Berstdrehzahl bestimmt wird. Die Berechnungs- und Messergebnisse werden anschließend miteinander verglichen und diskutiert.
Im vorgegeben Betriebsbereich liegen die berechneten mechanischen Spannungen unterhalb der vorgegebenen Materialwerte. Der IM-Rotor zeigt in der Messung eine plastische Verformung von bis zu \(\varepsilon_{\mathrm{IM,pl}} = 8\) μm und eine elastische Verformung von bis zu \(\varepsilon_{\mathrm{IM,pl}}= 22\) μ m auf. Der PMSM-Rotor weitet sich plastisch bis zu \(\varepsilon_{\mathrm{PMSM,pl}}= 5\) μm auf, die maximale elastische Verformung beträgt \(\varepsilon_{\mathrm{PMSM,pl}} = 40\) μ m. Der Vergleich zwischen der gemessenen und berechneten elastischen Verformung zeigt eine gute Übereinstimmung. Beide Modelle sind in der Lage, die Verformung und die mechanische Belastung im Betriebsbereich zu beschreiben. In den Schleuderversuchen erreichen die Rotoren Berstdrehzahlen, die weit über der definierten Schleuderdrehzahl liegen.
Schlüsselwörter
Hochdrehzahlmaschinen von Mises-Stress mechanischer Stress Verformung Schleudertest Bersttest Maschinendesign Hochdrehzahl-PMSM und Hochdrehzahl-IM1 Introduction
High speed electric machines are gaining importance not only for applications such as machine spindles, turbochargers and pumps, but also for traction drive applications for vehicles and aviation [1–4]. One main reason is the high power density of high speed electric machines. The development and research on SiC semiconductors, that are able to switch at higher frequencies, strengthens this trend [5, 6].
The design of high speed electric machines is limited by eddy current losses in the rotor, skin and proximity effect in the stator winding, rotor dynamics, and the mechanical stress in the rotor [7–11]. The mechanical stress limits the maximum circumferential speed and accordingly the outer diameter of the rotor, the bore volume and the power of the electric machine. Thus, the calculation of the mechanical stress is a crucial part in the design process.
In a large part of the published studies concerning the mechanical stress in high speed electric machines, the mechanical stress and the deformation are calculated using FEM simulation models. But, validation by measurements is missing [1, 12–15]. Karthaus performs a comparison for a Permanent Magnet Synchronous Machine (PMSM) with interior magnets, where he compares the calculated deformation at the rotor surface with a measurement, using laser-based distance sensors [16]. The mechanical stress can then be deduced from the measured deformations using a FEM simulation model. The mechanical stress cannot be measured directly.
Machines data
Machine data | ||
---|---|---|
Rotor length | \(l_{\text{2}}\) | 110 mm |
Air-gap | δ | 0,5 mm |
Height of bandage in PMSM | \(h_{\text{PMSM,ba,2}}\) | 4 mm |
Height of slot bridge in IM | \(h_{\text{IM,sb,2}}\) | 0,5 mm |
Outer diameter of shaft | \(D_{\text{o,shaft}}\) | 35 mm |
Inner diameter of shaft | \(D_{\text{i,shaft}}\) | 15 mm |
Outer diameter of rotor | \(D_{\text{o,2}}\) | 81 mm |
Outer diameter of stator | \(D_{\text{i,1}}\) | 160 mm |
Inner diameter of stator | \(D_{\text{o,1}}\) | 82 mm |
Number of stator slots | \(N_{\text{1}}\) | 24 |
Number of pole pairs | p | 2 |
Rated power PMSM | \(P_{\text{N,PMSM}}\) | 75 kW |
Rated power IM | \(P_{\text{N,IM}}\) | 60 kW |
Rated speed PMSM | \(n_{\text{N,PMSM}}\) | \(25000\,\frac{1}{\text{min}}\) |
Rated speed IM | \(n_{\text{N,IM}}\) | \(20000\,\frac{1}{\text{min}}\) |
Maximum speed PMSM | \(n_{\text{max,PMSM}}\) | \(50000\,\frac{1}{\text{min}}\) |
Maximum speed IM | \(n_{\text{max,IM}}\) | \(30000\,\frac{1}{\text{min}}\) |
2 Mechanical Stress in Electric Machines
In this section, the mechanical principles and the material behaviour are explained as basis for the simulation models in Sect. 3 and the discussion of the measured deformation during the spin test in Sect. 5.
2.1 Force, stress and deformation – Hooke’s law
2.2 Von Mises equivalent stress
2.3 Stress-strain diagrams for ductile and for brittle materials
First, the material behaviour for ductile materials is explained (see Fig. 3 (a)): If a force is applied to the material, mechanical stress and deformation occur. The mechanical stress and the deformation are proportional until the mechanical stress reaches the yield strength \(R_{\text{p0,2}}\). The yield strength \(R_{\text{p0,2}}\) is defined as the mechanical stress at which a plastic deformation of \(\varepsilon =0,2\) % remains after releasing the mechanical load. Once the yield strength is reached, the deformation increases further, while the mechanical stress increases slower. After reaching the ultimate strength \(R_{\text{m}}\) at the plastic deformation \(A_{\text{g}}\), the stress decreases until the material breaks. As long as the mechanical stress is smaller than the yield strength \(R_{\text{p0,2}}\), the deformation in the material is almost elastic. When the force is increased further, the deformation becomes strongly plastic. The material changes its internal structure and does not return into its initial state [17, 22]. To describe the material behaviour in a material model, the stress-strain diagram is simplified. The stress-strain diagram is divided into an elastic area and a plastic area, even though small plastic deformation occurs in the elastic area and vice versa. By simplifying the material model, the behaviour can be linearised and the material is described by Hooke’s law with the elastic and the shear modulus from (2). In this work, the materials are just modelled for the elastic case since the rotors should be high-fatigue-resistant and no plastic deformation should occur.
Orthotropic materials, like fibre-based bandages, show different material properties in different spatial directions. In this case, elastic modulus, shear modulus and Poisson ratio have different values regarding the different spatial directions [23].
Brittle materials do not show a plastic deformation. If a force is applied to the material, the deformation and the mechanical stress are almost proportional to each other. Just before the material breaks at the ultimate strength \(R_{\text{m}}\), the deformation becomes slightly larger related to the mechanical stress. As shown in Fig. 3 (b), the deviations are small compared to the linearised material model. Thus, the material can be described by Hooke’s law with the elastic and the shear modulus according to (2) [17, 24].
3 Simulation model
In this section, the FEM simulation models of the PMSM and the IM rotor are introduced. They are used in order to calculate the deformation and the mechanical stress due to centrifugal forces and thermal expansion.
The mechanical simulation model consists of the examined structure with its individual components that are allocated to the corresponding material models. Different mechanical loads are assigned to the components such as centrifugal forces or press fits. Furthermore, the contact modelling between the components is an essential part in the mechanical simulation process. According to the contact definition, the exchange of forces between the different components is defined. The contact definition is also used to simulate the press fit. If two components interfere, a force is added to the model that is acting contrary to the interference [26]. In iterative calculation steps, an equilibrium between the assigned forces in the contact and the centrifugal forces is calculated. When the result converges under a certain level of error, the simulation is finished. For the calculation, the Augmented Lagrange Algorithm is used [27, 28]. First, the simulation model of the PMSM rotor is introduced, followed by the simulation model of the IM rotor.
3.1 PMSM rotor
3.1.1 Boundary and symmetry conditions
The structure of the rotor does not change along the axial direction of the rotor. Therefore, it is sufficient to model just one magnetic sheet layer of the rotor lamination with the corresponding axial part of the shaft, the magnets and the bandage. Due to symmetry conditions, it is sufficient to simulate only one pole of the rotor, using appropriate boundary conditions. The symmetry conditions in the axial direction are enabled as friction-free bearings at top and bottom of the model, marked as B1 in Fig. 4. The symmetry conditions in radial direction are enabled as friction-free bearing on the sides of the model (see Fig. 4 (a) B2). Inside the hollow shaft, a cylindrical bearing is assigned to enable the rotation.
3.1.2 Contact modelling
Contact definitions PMSM
Contact | Components | Friction coefficient | Press fit |
---|---|---|---|
C1: Frictional | RL and shaft | 0,15 | 35 μm |
C2: Bonded | RL and FM | – | – |
C3: Bonded | MA and FM | – | – |
C4: Frictional | BA and FM | 0,15 | 210 μm |
C5: Bonded | RL and glue | – | – |
C6: Bonded | glue and MA | – | – |
C7: Frictional | BA and MA | 0,15 | 210 μm |
BA \(\hat{=}\) bandage, FM \(\hat{=}\) fill material, RL \(\hat{=}\) rotor lamination, MA \(\hat{=}\) magnet |
3.1.3 Mesh
Before the model can be solved, it needs to be meshed. In axial direction, it is necessary to insert at least three mesh layers, to get sufficient computational accuracy, as described in [29]. In areas with high mechanical loads as well as in areas, where the gradient of the mechanical stress is high, e.g. in the areas of the press fits and contacts, the mesh needs to be refined. The thin layer of the glue requires a fine mesh. Its width of 0, \(1\,\text{mm}\) is divided into three mesh layers. Its mesh of the simulation model is shown in Fig. 4 (b).
3.1.4 Material data and modelling
Component | Magnet | Rotor lamination | Glue | Fill material | Bandage | Shaft |
---|---|---|---|---|---|---|
Material | Sm2Co17 | M250-35A | Loctite 326 | Araldite | STS40 24k | 42CrMo4 |
α in 1/K | 1⋅10−5 | 1,28⋅10−5 | 8⋅10−5 | – | 6,31⋅10−7 | 1,1⋅10−5 |
E in GPa | 150 | 185 | 300 | 4,5 | 12,49 (rz)/ 145,82 (ϕ) | 210 |
ρ in kg/m3 | 8400 | 7600 | 1600 | 1600 | 1550 | 7720 |
ν | 0,27 | 0,28 | 0,3 | 0,125 | 0,257(rϕ)/ 0,019(ϕ z)/ 0,352(rz) | 0,3 |
\(R_{\text{p}}\) in MPa | – | 455 | – | – | – | 500 |
\(R_{\text{m}}\) in MPa | 35 (+), 800 (−) | 575 | 34 | – | 2400 (ϕ) | 825 |
+ marks a tension stress, – marks a compressive stress; r, ϕ and z are spatial directions in the cylindrical coordinate system. |
The mechanical stress of the ductile material should be below the yield strength \(R_{\text{p}}\). To ensure this, a safety factor \(\gamma = 1,05\) is defined, which allows a maximum mechanical stress \(\sigma _{\text{max,RL}}= 433,3\,\text{MPa}\) for the rotor lamination and a maximum mechanical stress \(\sigma _{\text{max,shaft}}= 476,2\,\text{MPa}\) for the shaft.
The magnets are expected to be loaded with compressive stress. For Sm2Co17, a safety factor \(\gamma = 2 \) is defined. Accordingly, the maximum mechanical compressive stress for the magnets is set to \(\sigma _{\text{max,MA}}= 400\,\text{MPa}\).
The fibres of the bandage are brittle as well, yet the bandage, as a composite material, is considered as a ductile material. At \(0,5\cdot R_{\text{m}}\), the first fibres are expected to break. This weakens the component, but does not destroy it completely. Therefore, the maximum stress for the bandage is defined to \(\sigma _{\text{max,BA}}= 1200\,\text{MPa}\). Nevertheless, fibre composites have an additional safety factor regarding the maximum relative elongation, also known as breaking elongation. The breaking elongation for the bandage material is set to \(\Delta \varepsilon _{\text{max}} = 0,5\) %.
The fill material in the PMSM is not dimensioned to support any mechanical load. It is just used to create a smooth surface with the magnets to mount the bandage. Therefore, it is not further considered in the examination of the mechanical stability of the rotor.
3.1.5 Mechanical load profile and temperature
Mechanical load profile for the PMSM and the IM simulation model
Calculation step | Rotational speed n in 1/min | |
---|---|---|
IM | PMSM | |
1 | 0 (press fit) | 0 (press fit) |
2 | 5000 | 5000 |
3 | 10000 | 10000 |
4 | 15000 | 20000 |
5 | 20000 | 30000 |
6 | 30000 | 40000 |
7 | 31000 | 45000 |
8 | 32000 | 50000 |
9 | 33000 | 51000 |
10 | – | 52000 |
11 | – | 53000 |
12 | – | 54000 |
13 | – | 55000 |
Rotor temperature: \(T = 20\,^{\circ }\text{C}\) |
3.2 IM rotor
3.2.1 Boundary and symmetry conditions
Due to symmetry conditions, it is sufficient to simulate only a part of the rotor using appropriate boundary conditions. Along the circumference, the structure is repeated with the number of rotor slots, in this case \(N_{\text{2}} = 30\). Therefore, only a \(1/30\) of the rotor needs to be simulated. Along the axial direction, another symmetry condition can be found. The short-circuit ring and the retaining ring on both sides are identical and accordingly just half the length of the rotor needs to be simulated. To consider these symmetry conditions, the surfaces on which the model is mirrored need to be friction-free bearings (see Fig. 5 (a), B1 and B2). Inside the hollow shaft, a cylindrical bearing is assigned to enable the rotation of the model.
3.2.2 Contact modelling
Contact definitions IM
Contact | Components | Friction coefficient | Press fit |
---|---|---|---|
C1: Bonded | CB and SCR | – | – |
C2: Frictional | CB and RL | 0,15 | – |
C3: Frictional | CB and RR | 0,15 | – |
C4: Friction free | SCR and RL | 0 | – |
C5: Friction free | SCR and RR | 0 | – |
C6: Frictional | RR and shaft | 0,15 | 16 μm |
C7: Frictional | RL and shaft | 0,15 | 16 μm |
C8: Friction free | SCR and shaft | 0 | – |
CB \(\hat{=}\) copper bar, SCR \(\hat{=}\) short-circuit ring, RR \(\hat{=}\) retaining ring, RL \(\hat{=}\) rotor lamination |
3.2.3 Mesh
To calculate the FEM simulation model, the geometry needs to be divided into a mesh. The used mesh is depicted in Fig. 5 (b). The mesh consists of oktaeder and tetraeder elements. The mesh is refined in the area of the slot bridges and the teeth of the rotor, since the biggest gradient in the mechanical stress is expected in these areas.
3.2.4 Material data and modelling
Component | Copper bar | Short-circuit ring | Rotor lamination | Retaining ring | Shaft |
---|---|---|---|---|---|
Material | E-Cu57 | CuCr1Zr | M250-35A | 34CrNiMo6 | 42CrMo4 |
α in 1/K | 1,68⋅10−5 | 1,8⋅10−5 | 1,28⋅10−5 | 1,21⋅10−5 | 1,1⋅10−5 |
E in GPa | 110 | 190 | 185 (rϕ)/ 90 (z) | 210 | 210 |
ρ in kg/m3 | 8930 | 8910 | 7600 | 7730 | 7720 |
ν | 0,34 | 0,34 | 0,28 | 0,3 | 0,3 |
\(R_{\text{p}}\) in MPa | 180 | 310 | 455 | 800 | 500 |
\(R_{\text{m}}\) in MPa | 300 | 550 | 575 | 900 | 825 |
+ marks a tension stress, – marks a compressive stress; r, ϕ and z are spatial directions in cylindrical coordinate system. |
3.2.5 Mechanical load profile and temperature
The load profile is set similar to the earlier introduced load profile of the PMSM and can be seen in Table 4. The rotational speed is increased in steps of \(\Delta n = 5000\,\frac{1}{\text{min}}\) until the maximum operational speed of \(n_{\text{max}} = 30000\,\frac{1}{\text{min}}\) is reached. The rotational speed is then further increased in steps of \(\Delta n = 1000\, \frac{1}{\text{min}}\) until the test speed of \(n_{\text{test}} = 1,1 \cdot n_{\text{max}}=33000\,\frac{1}{\text{min}}\) is reached. The temperature is set to the ambient temperature of \(T = 20\,^{\circ }\text{C}\) to be able to compare the results to the measurement results of the spin test.
4 Measurement Setup
The expansion of the rotor is determined using a spin test, that measures the deformation of the rotor surface for different rotational speeds. The measurement is similar to the setup used in [16].
4.1 Test assembly
4.2 Sensor information and positioning
The used distance sensors operate on a capacitive measurement principle. The accuracy of the measurements depends strongly on the measurement assembly and the used materials of the test specimen. The accuracy of the sensors is specified better than \(\Delta \varepsilon = 1~\upmu \text{m}\). To examine the measurement accuracy during the measurement, the deformation is measured three times for each speed. The measurement accuracy can be determined by comparing the three deformation values. The measurement accuracy in the used test assembly is smaller than \(\Delta \varepsilon = 0,5~\upmu \text{m}\). To reduce the error due to measurement accuracy further, the deformation is averaged over the three measurements for each speed step. In the following the measured deformation at one speed is referred as the mean values of the three measurements.
4.3 Rotor specimen
4.4 Load profile
Different tests are carried out to analyse the mechanical strength and the deformation of the rotor specimen. First, the burst tests of the IM and the PMSM rotor are made to validate the mechanical strength of the rotors.
5 Results and Discussion
In this section, the results of the simulation model and the spin test are presented. In Sect. 5.1, von Mises stress is used to analyse the mechanical stress in the rotor for different rotational speeds. The calculated deformation at the rotor surface is then compared to the measurement results of the spin test in Sect. 5.2. The results are analysed and discussed. In Sect. 5.3, the results of the burst test are presented for the two rotor types.
5.1 Von Mises stress
First, the state of stress in the rotor designs is analysed using the von Mises criterion. In order to analyse the influence of the press fit, the state of standstill is shown. Furthermore, the state of stress due to centrifugal forces is depicted for three different rotational speeds.
5.1.1 PMSM rotor
At standstill, the influence of the press fits can be seen. The press fit between shaft and rotor lamination leads to increased mechanical stress at the inner radius of the rotor lamination and the shaft. The stress increases up to \(\sigma _{\text{vM},\text{RL}, n_{0}} = 332\,\text{MPa}\) and \(\sigma _{\text{vM},\text{shaft}, n_{0}} = 370\,\text{MPa}\). Shrinking the bandage on the fill material and the magnets results in a large expansion of the bandage. Accordingly, the von Mises stress in the bandage is increased up to \(\sigma _{\text{vM},\text{BA}, n_{0}} = 679\,\text{MPa}\). The stress in the fill material and at the outer diameter of the rotor lamination is relatively small. The magnets show a mechanical stress of \(\sigma _{\text{vM},\text{MA}, n_{0}} \approx 40\,\text{MPa}\) due to the press fit. At the magnet edges, the mechanical stress is even higher. When the rotor is accelerated, the mechanical stress in the rotor yoke increases while the stress in the shaft decreases. At \(n = 50000\,\frac{1}{\text{min}}\), the maximum von Mises stress in the rotor lamination is \(\sigma _{\text{vM},\text{RL}, n_{0}} = 365,5\,\text{MPa}\) and is within the permissible mechanical stress defined in Sect. 3. The stress in the bandage increases up to \(\sigma _{\text{vM},\text{BA}, n_{0}} = 780\,\text{MPa}\). The magnets are pressed on the bandage and a small mechanical stress of \(\sigma _{\text{vM},\text{MA},n_{0}} \approx 50~\text{MPa}\) arises at the outer contour of the magnets. Since the mechanical stress at the magnets is dominated by a compressive stress component, the magnets do not break (see Table 3). At test speed (see Fig. 10 (d)), the von Mises stress in the bandage reaches \(\sigma _{\text{vM},\text{BA}, n_{\text{spin}}} = 842\,\text{MPa}\). The von Mises stress in the glue exceeds at some positions the ultimate strength. The mechanical load in the magnets increases up to \(\sigma _{\text{vM},\text{MA}, n_{\text{spin}}} \approx 80\,\text{MPa}\). In the analysed speed range, the mechanical stress in the components is still within the defined limits of the materials, as described in Sect. 3. For the bandage, the maximum deformation in radial direction needs to be examined. The elongation in tangential direction is \(\Delta \varepsilon = 0,242\,\text{mm}\) and corresponds to an expansion of the bandage of \(\Delta \varepsilon = 0,57\) %. This expansion is smaller than the permissible expansion defined in Sect. 3.
5.1.2 IM rotor
The influence of the press fit can be seen in Fig. 11 a). Due to the press fit, stress arises in particular in the yoke of the rotor lamination, the retaining ring and the short-circuit ring. The maximum mechanical stress in the rotor lamination is \(\sigma _{\text{vM},\text{RL}, n_{0}} = 370\,\text{MPa}\). The mechanical stress in the shaft increases slightly. In the copper bar, no change in the mechanical load can be seen.
When the rotor is accelerated, mechanical stress arises in the slot bridges and in the teeth of the rotor lamination and the retaining ring, as marked in Fig. 11 (d). The width of the slot bridge at the top of the copper bar is thinner than the tooth width, and accordingly, the mechanical stress in the slot bridge is significantly higher. The slot bridge is the crucial point in the rotor design. The mechanical stress in the short-circuit ring increases as well at the outer diameter. Though, the copper bar shows just a small mechanical load. The mechanical stress in the rotor lamination at maximum operational speed is \(\sigma _{\text{vM}, \text{RL}, n_{\text{max}}} = 420\,\text{MPa}\). In the retaining ring, the stress is higher and reaches \(\sigma _{\text{vM}, \text{RR}, n_{\text{max}}} = 440\,\text{MPa}\). The stress values are smaller than the maximum permissible mechanical stress defined by the safety factor and the yield strength of the used material in Sect. 3. At test speed, the mechanical stress in the rotor lamination increases up to \(\sigma _{\text{vM}, \text{RL}, n_{\text{spin}}} = 480\,\text{MPa}\) and in the retaining ring up to \(\sigma _{\text{vM}, \text{RR}, n_{\text{spin}}} = \)\(500\,\text{MPa}\). These stress values exceed the yield strength and already lead to a plastic deformation. The increase of the mechanical stress can be seen in Fig. 11 (b), (c) and (d).
5.2 Surface deformation
5.2.1 PMSM rotor
The deformation at position 1 of the PMSM rotor during the spin test is shown in Fig. 12. The deformation increases with increasing speed. For rotational speed \(n< 40000\,\frac{1}{\text{min}}\), the deformation is elastic and the rotor returns to its initial state. At \(n \geq 40000\,\frac{1}{\text{min}}\), the deformation becomes plastic. A plastic deformation of \(\varepsilon _{\text{BA,pl}}= 3...5\) μm remains at the outer diameter of the rotor. The maximum deformation and the minimum deformation around the circumference are depicted. The maximum deformation appears above the magnets and the minimum deformation above the fill material. The minimum deformation increases up to \(n = 55000\,\frac{1}{\text{min}}\) and reaches a value of \(\varepsilon _{\text{BA,min}}= 12\) μm. The maximum deformation increases stronger and reaches \(\varepsilon _{\text{BA,max}}= 40\) μm. The measured plastic deformation cannot be calculated in the simulation model, since the material model is ideal and no residual plastic deformation is considered. The reason for this measured plastic deformation is an initial movement among the components during the spin test and the non-ideal material behaviour in the elastic area (see Sect. 2.2).
Elastic deformation in radial direction of the PMSM rotor is depicted in Fig. 13 as function of the circumference for four different rotational speeds. First, the deformation at the test speed is compared between the measurement and the simulation model. The measured deformation shows a different behaviour for the four poles of the rotor. The maximum deformation varies between \(\varepsilon _{\text{BA,max,el}}= 40\) μm and \(\varepsilon _{\text{BA,min}}= 25\) μm and the shape of the deformation differs. The magnets are glued to the rotor lamination. The glue is used to fix the magnets before adding the fill material. The glue is not able to secure the magnets against centrifugal forces. Therefore, the bandage is added. As described in Sect. 5.1, the von Mises stress in the glue exceeds the ultimate strength at some positions and it can be assumed that there are local spots where the glue breaks. The difference in the deformation might occur due to this inaccuracy. Another reason might be an anisotropic material behaviour in the rotor lamination or in the bandage. It can be seen, that the poles opposing each other show a similar behaviour. The deformation in the simulation model is the same for the four poles, since the deformation is extracted using symmetry conditions without considering material anisotropy. For test speed and the maximum operational speed, the maximum calculated deformation is close to the maximum measured deformation and the shape is reproduced well. For the speeds \(n = 40000\,\frac{1}{\text{min}}\) and \(n = 30000\,\frac{1}{\text{min}}\), the calculated and the measured deformation show some deviations in amplitude. The deformation is calculated higher as it actually is. Though, the deformation is mapped well. The deviation between the measurement results and the simulation model might occur due to the modelling of the glue between the magnet and the rotor lamination. In [16], the influence of the glue is analysed for a PMSM with buried magnets and a strong impact on the deformation can be noticed. If the model used here for the surface-mounted PMSM is simulated without the glue and a frictional or bonded contact between the rotor lamination and the magnet is used, the deformations are significantly smaller. Accordingly, the expansion behaviour for the model without glue is not capable to reproduce the actual behaviour of the rotor. The magnets are also kept at the rotor lamination due to magnetic forces between the magnet and the rotor lamination. These forces are not considered in the simulation model and should be analysed more in details.
The deviation can be seen as well in Fig. 14, where the maximum and the minimum elastic deformations are depicted over the speed. The deviation between the simulated deformation and the measured deformation between \(n = 10000\,\frac{1}{\text{min}}\) and \(n = 45000\,\frac{1}{\text{min}}\) can be seen here as well. Though it must be said, that the deviation shown in Fig. 14 appears to be large, since the maximum deformation of the pole at \(\phi = 30\,^{\circ }\) is depicted, which shows a higher deformation than the other poles.
To sum up, the measurement and the simulation model show good accordance. In the measurements themselves, we can observe a different deformation for the four poles. Thus, the comparison with the simulation model is difficult. Further, the simulation model shows a slightly higher deformation than the measurements. The deformation depends strongly on the modelling of the glue between magnet and rotor lamination. The simulation model is suitable to evaluate the maximum elastic mechanical stress and the deformation of the PMSM rotor.
5.2.2 IM rotor
In Fig. 16 (b), the deformation at position 2 at the short-circuit ring is depicted. The influence of the copper bars is less distinct in the measurement results than at position 1. The short-circuit ring and the copper bars expand together, and the short-circuit ring does not secure the copper bars as strongly as the rotor lamination. In the simulation model, this effect is not considered. The influence of the copper bars on the short-circuit ring is as strong as in position 1. But the magnitude of the deformation fits to the measurement results and describes the expansion of the IM rotor well.
Overall, the simulation model shows a good accordance with the measurements and is suitable for evaluating the maximum elastic mechanical stress of the IM rotor.
5.3 Burst tests
In this section, the results of the burst test are presented for the PMSM rotors and then for the IM rotors.
5.3.1 PMSM rotor
Results of the burst test of PMSM rotor
Rotor | Max. rotational speed \(n_{\text{burst}}\) | Termination criterion | Comments |
---|---|---|---|
PMSM Rotor 1 | \(62982\,\frac{1}{\text{min}}\) | Max. speed of gearbox | No visible deformation |
PMSM Rotor 2 | \(62990\,\frac{1}{\text{min}}\) | Max. speed of gearbox | No visible deformation |
5.3.2 IM rotor
Results of the burst test of IM rotor
Rotor | Max. rotational speed \(n_{\text{burst}}\) | Termination criterion | Comments |
---|---|---|---|
IM Rotor 1 | \(52402\,\frac{1}{\text{min}}\) | Burst of rotor | Destruction of the lamination sheet |
IM Rotor 2 | \(51874\,\frac{1}{\text{min}}\) | Shaft vibrations | Strongly deformed rotor, shape of the bars visible |
6 Conclusion
In this work, the mechanical load on a PMSM rotor and on a IM rotor of a high speed electric machine are examined. The mechanical stress and the deformation are examined, using two FEM simulation models. The results are compared to the measurements of a spin test. During the spin test, the deformation at the surface of the rotors is measured. Additionally, four burst tests of the examined rotors are performed.
The mechanical stress in the rotors is analysed using the von Mises stress and compared to the yield strength of the used materials. In both rotors, the maximum permissible stress in the used materials is not exceeded for \(n< n_{\text{max}}\). For the test speed, the mechanical stress in the rotor lamination of the IM rotor is close to the yield strength.
During the spin test, plastic deformation at the rotor surface can be observed for both rotor types. The deformation occurs possibly due to initial movement during the first run-up and due to a non-linear material behaviour in the elastic area.
The deformation at the rotor surface is depicted around the circumference for different rotational speeds. In the measurements, a behaviour varying around the circumference can be observed for the two rotor types. The PMSM rotor shows different deformations for the four poles. The maximum deformation at the test speed varies from \(\varepsilon _{\text{BA,max,el}}= 25\) μm to \(\varepsilon _{\text{BA,max,el}}= 40\) μm. This deviation is possibly due to the different glue connection of the magnets and the rotor lamination. At the rotor lamination of the IM rotor, the deformation changes as well. Some copper bars are connected better to the rotor lamination than others. The deformation at the test speed varies from \(\varepsilon _{\text{RL,max,el}}= 12\) μm to \(\varepsilon _{\text{BA,max,el}}= 20\) μm. The measured deformation is compared to the calculated deformation of the FEM models.
The calculation model of the PMSM shows some deviations compared to the measurements. For the speed range of \(10000\,\frac{1}{\text{min}}< n<45000\,\frac{1}{\text{min}}\), the deformation of the calculation model is slightly higher than the deformation in the measurement. At maximum operating speed, and at the test speed, the deformation fits the measurement results well. Overall, the shape of the deformation can be reproduced in the calculation model. It can be concluded, that the simulation model can be used to evaluate the mechanical stress and the deformation of the PMSM rotor.
The calculated deformation of the IM rotor reproduces the measurement in the spin test. The deformations calculated at the rotor lamination and at the short-circuit ring match the maximum deformation of the measurement. The shape of the rotor bars pressing to the rotor lamination can be recognized in the simulation model and in the measurements. The simulation model is capable of describing the mechanical stress and the deformation in the IM rotor.
The measurements and the simulation are carried out at the ambient temperature of \(T = 20\,^{\circ }\text{C}\) due to the predefined criteria of the spin test. Though, the temperature has a strong impact on the deformation and accordingly to the mechanical stress in the rotor. This influence needs to be determined in more details in future works.
Finally, the results of the burst test are presented. The two rotor specimen of the IM rotor reach rotational speeds that are 70 % higher than the maximum operating speed. The two PMSM rotors withstand the centrifugal forces up to more than 25 % of the maximum operating speed. A burst could not be reached due to the limited maximum speed of the spin test machine.
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