# Analysis of a Shaftless Semi-Hard Magnetic Material Flywheel on Radial Hysteresis Self-Bearing Drives

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## Abstract

**:**

## 1. Introduction

## 2. System Layout

## 3. Mechanical Design, Energy and Rotordynamic Analysis

#### 3.1. Mechanical Design and Energy Analysis

^{®}Mechanical APDL 15.0 (Pennsylvania, Canonsburg, USA). Due to the axisymmetry of both geometry and loads, a two-dimensional model is sufficient. The axisymmetric formulation of the PLANE82 element is used to reduce the size of the problem. The rotor, spun at 18,000 rpm, is constrained only in the axial direction, since other translations are constrained by the axial symmetric formulation of the problem. Resulting stresses from the plane stress and the FE approaches are evaluated along the radius and compared in Figure 4. The difference between the two solutions is lower than the $0.5\%$. This comparison demonstrates that the analytical solution well describes the stress field under the centrifugal loading.

#### 3.2. Rotordynamics

^{®}Mechanical APDL 15.0. Even if the rotor is axisymmetric, the three-dimensional model is needed to evaluate the three-dimensional mode shapes. ANSYS Solid95 elements (twenty-node brick elements) are used because they embed both gyroscopic and Coriolis effects. The latter fulfills an important role in the dynamics of thin-walled rotating cylinders.

## 4. Radial Active Magnetic Suspension

#### Modeling

## 5. Results

#### 5.1. Response to the External Disturbance at Standstill

#### 5.2. Response to the Unbalance at Maximum Speed

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbol | Description | Unit |

${A}_{p}$ | Pole area | m^{2} |

${A}_{s}$ | Cross-sectional area of the stator | m^{2} |

$\beta $ | Ratio between the inner and outer radius | - |

${\mathbf{B}}_{\mathbf{T}}$ | Transformation matrix for the AMBs forces | - |

B | Magnetic flux density | T |

$\chi ({I}_{p}-{I}_{t})$ | Couple unbalance | kgm^{2} |

${E}_{Max}$ | Storage capability | J |

${\mathbf{f}}_{\mathbf{AMB}}$ | Vector of the AMBs forces | - |

${\mathbf{f}}_{\mathbf{ext}}$ | Vector of the external forces | - |

${\mathbf{f}}_{\mathbf{unb}}$ | Vector of the unbalance force | - |

${f}_{e}$ | Electromagnetic force | N |

${f}_{e,\phantom{\rule{0.166667em}{0ex}}ij}^{+}$ | Force provided by the electromagnet supplied with the current ${i}_{0}+{i}_{c,\phantom{\rule{0.166667em}{0ex}}ij}$ | N |

${f}_{e,\phantom{\rule{0.166667em}{0ex}}ij}^{-}$ | Force provided by the electromagnet supplied with the current ${i}_{0}-{i}_{c,\phantom{\rule{0.166667em}{0ex}}ij}$ | N |

${f}_{ij}$ | Force of the AMB i along the j-axis | N |

$\mathbf{G}$ | Gyroscopic matrix | - |

g | Magnetic gap length | m |

${g}_{0}$ | Nominal magnetic gap | m |

${\overline{H}}_{r},\overline{B}$ | Magnetic operating point of the rotor material | A/m |

h | Flywheel height | m |

${H}_{g}$ | Magnetic field strength in the magnetic gap | A/m |

${H}_{r}$ | Magnetic field strength in the rotor | A/m |

${H}_{s}$ | Magnetic field strength in the stator | A/m |

i | Current | A |

${i}_{0}$ | Bias current | A |

${I}_{p}$ | Polar moment of inertia | kgm${}^{2}$ |

${I}_{t}$ | Transversal moment of inertia | kgm${}^{2}$ |

${i}_{c,\phantom{\rule{0.166667em}{0ex}}ij}$ | Control current at the j-axis of the AMB i | A |

${i}_{max}$ | Maximum continuous current | A |

K | Flywheel shape factor | - |

${l}_{r}$ | Length of the magnetic flux path in the rotor | m |

${l}_{s}$ | Length of the magnetic flux path in the stator | m |

${\mu}_{0}$ | Magnetic permeability of the free-space | H/m |

${\mu}_{r}$ | Magnetic permeability of the linearized rotor material | - |

${\mu}_{s}$ | Magnetic permeability of the stator material | - |

$\mathbf{M}$ | Mass matrix | - |

m_{ϵ} | Static unbalance | kgm |

m | Mass | kg |

$\nu $ | Poisson ratio | - |

N | Turns per electromagnet | - |

$\mathsf{\Omega}$ | Spin speed | rad/s |

$\omega $ | Natural frequency | rad/s |

${\mathsf{\Omega}}_{Burst}$ | Burst spin speed | rad/s |

${\mathsf{\Omega}}_{Max}$ | Maximum spin speed | rad/s |

$\mathsf{\Phi}$ | Magnetic flux | Wb |

$\mathbf{q}$ | Vector of the generalized coordinates | - |

$\rho $ | Mass density | kg/m${}^{3}$ |

${r}_{i}$ | Flywheel inner radius | m |

${r}_{o}$ | Flywheel outer radius | m |

r | Flywheel radius | m |

${\sigma}_{h}$ | Hoop stress | Pa |

${\sigma}_{eq}$ | Von Mises equivalent stress | Pa |

${\sigma}_{lim}$ | Maximum equivalent stress in normal operation | Pa |

$S{F}_{\mathsf{\Omega}}$ | Safety factor on the spin speed | - |

${\sigma}_{r}$ | Radial stress | Pa |

${S}_{UT}$ | Tensile strength | Pa |

t | Simulation time | s |

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**Figure 1.**3D cross-section view of the proposed FESS layout: ① Rotor; ② Hysteresis self-bearing drive; ③ Thrust AMB; ④ Flange; ⑤ Backup bearing; ⑥ Inner stator.

**Figure 2.**Schematic of the self-bearing hysteresis drive. (

**a**) View in the $xy$-plane (the transversal plane of the flywheel) with the three-phase (A, B, C) winding scheme of the motor-generator function. (

**b**) Partial cross-section in the $xz$-plane (the longitudinal plane of the flywheel) with the winding scheme responsible for the suspension flux generation only. (

**c**) Isometric view.

**Figure 3.**Polar and transversal moment of inertia ratio of the flywheel versus inner and outer radius ratio (

**a**). Flywheel height versus inner and outer radius ratio (

**b**).

**Figure 4.**Von Mises equivalent stress vs radius of the flywheel: analytical plane stress (solid); numerical finite-element (dash-dotted).

**Figure 5.**Campbell diagram of the flywheel: conical rigid body mode (solid); forward (dashed with squares) and backward (dashed) pure radial modes; forward (dash-dotted with circles) and backward (dash-dotted) radial shearing modes; synchronous whirling (dotted). The forward and backward pure radial modes (dashed with squares and dashed, respectively) are superimposed.

**Figure 6.**Pure radial (

**a**) and radial shearing (

**b**) mode shapes in the top, lateral and isometric view.

**Figure 9.**Magnetic characteristic of the FeCrCo 48/5 alloy: experimental (solid) and numerical (dash-dotted) loop.

**Figure 11.**Response of the system with hysteretic (solid) and linearized (dash-dotted) rotor material to external force steps: displacement of the flywheel at the location of the radial AMBs for the 100-N (

**a**) and the 300-N (

**b**) step amplitude; force provided by each AMB for the 100-N (

**c**) and the 300-N (

**d**) step amplitude.

**Figure 12.**Response of the system with hysteretic (solid) and linearized (dash-dotted) rotor material to external force steps: currents ${i}_{0}+{i}_{c}$ (

**a**) and ${i}_{0}-{i}_{c}$ (

**c**) for the 100-N step amplitude; currents ${i}_{0}+{i}_{c}$ (

**b**) and ${i}_{0}-{i}_{c}$ (

**d**) for the 300-N step amplitude.

**Figure 13.**(

**a**) Force provided by the most loaded electromagnet at steady state vs external force step applied to the rotor center of mass. Solid: total force (hysteretic rotor material case). Dash-dotted: residual force (hysteretic). Dashed: current-related force (hysteretic). Solid with circles: total force (linearized rotor material case). (

**b**) Current supplied to the most loaded electromagnet at steady state vs external force step applied to the rotor center of mass. Solid: current (hysteretic rotor material case). Dash-dotted: maximum current overshoot (hysteretic). Solid with circles: current (linearized rotor material case).

**Figure 14.**Responses to the unbalance: maximum displacement at the location of the AMB A (

**a**) and the AMB B (

**b**); force provided by the AMB A (

**c**) and the AMB B (

**d**); current supplied to the electromagnets of the AMB A (

**e**) and the AMB B (

**f**).

Symbol | Quantity | Value |
---|---|---|

$\nu $ | Poisson ratio | 0.3 |

${S}_{UT}$ | Tensile strength | 1150 MPa |

$\rho $ | Mass density | 7600 kg/m${}^{3}$ |

Symbol | Quantity | Value |
---|---|---|

${r}_{i}$ | Inner radius | 135 mm |

${r}_{o}$ | Outer radius | 150 mm |

h | Height | 600 mm |

Symbol | Quantity | Value |
---|---|---|

${A}_{p}$ | Pole area | 2010 mm${}^{2}$ |

${g}_{0}$ | Nominal magnetic gap | 0.8 mm |

${l}_{r}$ | Length of the magnetic flux path in the rotor | 36 mm |

N | Turns per electromagnet | 174 |

${i}_{0}$ | Bias current | 2.5 A |

${i}_{max}$ | Maximum continuous current | 5 A |

Symbol | Quantity | Value |
---|---|---|

m | Mass | 61.2 kg |

${I}_{t}$ | Transversal moment of inertia | 2.46 kgm${}^{2}$ |

${I}_{p}$ | Polar moment of inertia | 1.25 kgm${}^{2}$ |

$\chi ({I}_{p}-{I}_{t})$ | Couple unbalance | 4.50 × 10^{−5} kgm^{2} |

m_{ϵ} | Static unbalance | 2.05 × 10^{−4} kgm |

**Table 5.**Parameters of the PD controller in the form of $C\left(s\right)=P+\frac{D}{1+{\tau}_{d}s}$.

Symbol | Quantity | Value |
---|---|---|

P | Proportional gain | 12,039 $\mathrm{A}/\mathrm{m}$ |

D | Derivative gain | $111\phantom{\rule{0.166667em}{0ex}}\mathrm{A}\mathrm{s}/\mathrm{m}$ |

${\tau}_{d}$ | Time constant | $3.3\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\mathrm{s}$ |

Symbol | Quantity | Value |
---|---|---|

$\alpha $ | Local field factor | $0.15$ |

c | Domain rotation loss | $0.20$ |

a | Langevin parameter | $9.94\times {10}^{4}\phantom{\rule{0.166667em}{0ex}}\mathrm{A}/\mathrm{m}$ |

k | Pinning | $5.16\times {10}^{4}\phantom{\rule{0.166667em}{0ex}}\mathrm{A}/\mathrm{m}$ |

${M}_{S}$ | Saturation magnetization | $2.02\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}\mathrm{A}/\mathrm{m}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Circosta, S.; Bonfitto, A.; Lusty, C.; Keogh, P.; Amati, N.; Tonoli, A.
Analysis of a Shaftless Semi-Hard Magnetic Material Flywheel on Radial Hysteresis Self-Bearing Drives. *Actuators* **2018**, *7*, 87.
https://doi.org/10.3390/act7040087

**AMA Style**

Circosta S, Bonfitto A, Lusty C, Keogh P, Amati N, Tonoli A.
Analysis of a Shaftless Semi-Hard Magnetic Material Flywheel on Radial Hysteresis Self-Bearing Drives. *Actuators*. 2018; 7(4):87.
https://doi.org/10.3390/act7040087

**Chicago/Turabian Style**

Circosta, Salvatore, Angelo Bonfitto, Christopher Lusty, Patrick Keogh, Nicola Amati, and Andrea Tonoli.
2018. "Analysis of a Shaftless Semi-Hard Magnetic Material Flywheel on Radial Hysteresis Self-Bearing Drives" *Actuators* 7, no. 4: 87.
https://doi.org/10.3390/act7040087