Analysis

Written by: Ricky Gill

Introduction

Here a detailed electromagnetic and structural analyses are conducted to ensure the AS-RW operates reliably. Electromagnetic simulations and calculations determine the torque and momentum for the reaction wheel, while structural analysis verify the relaiblity of the device and critical speeds. 

Electromagnetic Analysis

Torque & Back-EMF Analysis

FEA simulations can be used to design brushless motors with specific specifications such as torque and speed. Here the 3D CAD model is simplified into a 2D model to reduce the dimensionality of the problem (reduces computation time for running a simulation).  This is done by taking a cross-section of the motor at the average diameter radius of the motor (essentially midway between the coil windings). That surface is then unraveled to form the simplified 2D model of the motor. 

Electromagnetic material properties were set consistent with designed part. The rotor is set as 316 stainless steel, the stator (CCA) is defined as FR-4, the magnets are defined as neodymium (grade N52), and coil windings are set as copper. Steady state simulations are performed where the rotor speed is set using a linear speed converted from a rotational speed of 5000 RPM and the part geometry. The coil windings are excited using equations defined from field-oriented control (FOC). FOC uses sinusoidal current waveforms to commutate the motor and generate torque. Equations for FOC can be defined as follows (see this reference for more details: https://microchiptech.github.io/mcaf-doc/7.0.2/algorithms/foc/fundamentals.html?

Results are shown below. We can see the average torque and torque ripple (defined as the peak to peak torque divided by the average torque) for different RMS phase current values. 

The supply voltage is 12 V. From this we can calculate power using the equations for a star connected system. At 0.2A the required power would be approximately 4.15 W. 

Since sinusoidal currents are being fed into the windings, we can calculate the torque constant using the following equation.

Structural Analysis

Material Properties Sources

Material properties used for each material are outlined below:

• 316L Stainless Steel

  • Density: 8000 kg/m3

  • Young's Modulus: 193 GPa

  • Poisson's Ratio: 0.29

  • Yield Stress: 380 MPa

  • Ultimate Stress: 585 MPa

• AISI 1010 Steel

  • Density: 7870 kg/m3

  • Young's Modulus: 205 GPa

  • Poisson's Ratio: 0.29

  • Yield Stress: 305 MPa

  • Ultimate Stress: 365 MPa

• Neodymium

  • Density: 7010 kg/m3

  • Young's Modulus: 41.4 GPa

  • Poisson's Ratio: 0.28

  • Yield Stress: 165 MPa

  • Ultimate Stress: 170 MPa

• 6061-T6 Aluminum

  • Density: 2700 kg/m3

  • Young's Modulus: 68.9 GPa

  • Poisson's Ratio: 0.33

  • Yield Stress: 276 MPa

  • Ultimate Stress: 310 MPa

• FR-4

  • Default Ansys material library values used (orthotropic elasticity model, refer to Ansys documentation)

• ZR02

  • Density: 5070 kg/m3

  • Young's Modulus: 190 GPa

  • Poisson's Ratio: 0.25

  • Yield Stress: 410 MPa

  • Ultimate Stress: 410 MPa

• TECAPEEK - CF30 black

  • Density: 1410 kg/m3

  • Young's Modulus: 6.34 GPa

  • Poisson's Ratio: 0.3

  • Yield Stress: 120 MPa

  • Ultimate Stress: 120 MPa

Critical Speed

The AS-RW is engineered for a maximum operating speed of 10,000 RPM. However, following an extensive Phase D test campaign, its speed may be restricted to lower values if necessary. Theoretical and numerical Rotordynamics calculations are done to estimate the critical speed of the reaction wheel. Understanding the rotor's critical speed is essential to ensure that the reaction wheel operates at safe speeds, preventing resonance-related issues.

Analytical Calculations

Analytical calculations are shown below outlining how the reaction wheel's critical speed is estimated.

From the calculations above, the critical speed of the reaction wheel is around 6000 rad/s (approximately 60000 RPM). In reality, sub-harmonics will be present in the physical system which will cause enhanced vibrations at lower frequencies. These sub-harmonics will be identified during Phase D testing. 

Numerical Simulations

Using ANSYS Mechanical, a modal analysis was performed to calculate the rotor's critical speed. Cylindrical constraint boundary conditions were applied to the rotor shaft-bearing interface with radial and axial displacements fixed and tangential displacements free. Results from the numerical simulations are shown in the table below and in the Campbell diagram. The critical speed found using FEA is 6321 rad/s (approximate 6% difference between analytical results). This gives more confidence that the first natural frequency of the reaction wheel is six times larger than the maximum operating speed.  As mentioned previously, sub-harmonics and the actual vibration characteristics of the reaction wheel will be determined in Phase D. 

Campbell Diagram

Campbell Diagram results

Mode 2 Visualization

Centrifugal Loading Conditions

Disks spinning at high rotational speeds are subjected centrifugal forces. These forces induce stresses within the rotor and can cause material failure and excessive deformation which can lead to failure while in operation. A series of calculations and simulations are performed to ensure that the reaction wheel is designed with a safety factor of 2 against permanent deformation. 

Analytical Calculations

Radial and tangential stresses in the rotor are calculated from first principles below:

Analytical equations indicate that the stress values in the rotor/flywheel at 1000 rad/s are well below the yield limits off materials used in the flywheel assembly. Among them, Neodymium has the lowest yield stress of 165 MPa, which remains considerably higher than the maximum calculated stresses above. 

Numerical Simulations

Using Ansys Mechanical, a quasi-static simulation was conducted to estimate the stress induced in the reaction wheel operating at a speed of 1000 rad/s. A fixed cylindrical constrain is applied at the two shaft-bearing interfaces. Maximum Von-Mises stresses and deflections are shown in the images below. Stress values are lower than the analytical calculations above due to the rotor core thickness being accounted for in addition to the flywheel thickness. The average Von-Mises stress is 1.76 MPa. The highest stress values appear at the magnet-rotor core interface, but they primarily result from stress singularities. Maxmium deflection values are sub-micrometer levels. 

Modal Analysis

A simplified modal analysis model is created of the AS-RW assembly which consists of the flywheel, upper housing structure, and CCA. The bottom housing of the reaction wheel is not included to get a worse case estimate of the vibrational modes and deflections inherent to the structural design.  Fixed constraints are applied to the mounting hole interfaces on the CCA and upper housing structure. Cylindrical constraints are applied to the two flywheel shaft-bearing interfaces with only deflection in the tangential direction free. 

Mode numbers and corresponding frequencies are listed/shown below:

• Mode 1: 0 (Due to the flywheel being able to rotate)

• Mode 2: 1483.2 Hz

• Mode 3: 1485.5 Hz 

• Mode 4: 2726.8 Hz

• Mode 5: 3290.0 Hz

• Mode 6: 3383.1 Hz

• Mode 7: 4482.4 Hz 

• Mode 8: 4564.2 Hz 

• Mode 9: 4590.2 Hz

• Mode 10: 4762.6 Hz

• Mode 11: 4765.2 Hz 

• Mode 12: 5156.2 Hz 

• Mode 13: 5571.1 Hz

• Mode 14: 5574.4 Hz 

• Mode 15: 6351.9 Hz

• Mode 16: 6402.1 Hz 

• Mode 17: 6427.7 Hz 

• Mode 18: 6454.3 Hz

• Mode 19: 6519.7 Hz

• Mode 20: 6548.1 Hz

Mode numbers and natural frequencies (left). 

Random Vibration Analysis

Random vibration analysis is performed here to determine if the clearance between components in the AS-RW assembly is sufficient when subjected to environmental conditions during launch. The modal analysis from the previous section can be used with the random vibration MPE  (SpaceX Falcon 9 Rocket) as shown in the figure below to obtain probabilistic estimates of the deflection of each component.

 SpaceX random vibration MPE

Results from the random vibration analysis are shown in the figures below. Worst case deflection values (3-sigma 99.73% probability) in the Y-direction are just above a micron (0.0011 mm). This means that there is adequate clearance between the CCA and the stator/flywheel (nominally 2 mm). Note the CCA components will need to be less than 1.5 mm at a maximum directly under the flywheel or stator. This will allow for enough clearance with margin based on this analysis. There is also adequate clearance between the flywheel and bottom surface of the housing structure (nominally 1.5 mm). Stress values induced during random vibration are significantly lower than yield stresses used in the AS-RW structure (<1 MPa).  

Response Spectrum Analysis (Shock)

Shock loading conditions can also be modeled and used in conjunction with the modal analysis presented above in order to predict maximum deflection and stresses.  Shock boundary conditions are provided by the SpaceX Falcon 9 documentation and is shown below. These are severely overestimated since the interfaces between launch vehicle shock sources and the launch container plus the interface between the container and the CubeSat will help reduce the severity of the launch vehicle shock environment [NASA GEVS].  Despite that, a worst case scenario is considered where there is no attenuation and all of the energy is transmitted to the reaction wheel itself.

Shock loading conditions at the payload separation plane (Falcon 9)

Deformation in the Y-axis and stress magnitudes are shown in the images below for when the applied shock conditions are applied in the Y-axis. Maximum deflection is approximately 0.058 mm, located on the upper part of the reaction wheel housing structure. The Y-axis deflection of the flywheel itself is around 0.026 mm. Considering the worst case where the flywheel and housing structure are deforming towards each other,  there is still no appreciable concern of those two parts coming into contact. The stress values are relatively low throughout the structure except at the shaft portion of the flywheel. There is a stress singularity at the split line where the cylindrical constraint is applied, and it therefore can be ignored. However there still non-negligible stress at the filleted region. The magntiude of stress in this region is around 90 MPa. With the yeild stress of 316L steel being 380 MPa, the factor of safety is approximately 4.2.

Deflection and stress results for are shown below when the shock loading conditions are applied in the X-axis. Here the maximum stress and deflection values are relatively larger. The deflection of 0.2 mm in the lateral direction is acceptable based on tolerances between the flywheel and the stator/coils and the housing structure. The maximum stress again is due to stress singularities at the applied boundary condition. However a maximum stress of around 295 MPa is at the filleted region of the shafts at both the top and bottom interfaces of the flywheel. This results in a safety factor of  1.29. Again, the actual shock transmitted to the reaction wheel will be considerable lower than the values applied here since there will be attenuation provided by the dispenser and the ArcticSat Bus itself.

Shock (X-axis) stress results with label.

G-Force (Coupled Load) Analysis

Coupled load analysis is performed according to the recommended values as per Section 4.1.2.1 of the SpaceX 2025 Rideshare Payload User's Guide. Values listed in the aforementioned document as shown below.

Coupled loading conditions quasi-static load factors.

Deflection and stress results were computed for two cases. In the first case only the axial force was applied. In the second case, the axial and lateral loading conditions were applied. Note here that axial and lateral directions are with respect to how the reaction wheel is mounted on the ArcticSat structure. Worst case results are shown in the figures below. Maximum stress and deflection values are negligible in the housing structure. Maximum values occur in the flywheel and are also largely negligible.  

Radiation Tolerances

An analysis in SPENVIS was performed with a 575 km altitude with an inclination of 90 degrees for a total of 5 years. Standard trapped proton and electron fluxes were used. Standard galactic cosmic ray flux settings were also implemented. Results from the simulation are shown below. The expected radiation dosage at a nominal wall thickness of 2mm is less than 3 krad. A nominal wall thickness of 2 mm is implemented on the AS-RW housing structure to mitigate against radiation related electronics failures. This nominal wall thickness is expected to be sufficient for the AS-RW for the orbital parameters and total mission life mentioned above.  

The torque constant is also equal to the back-EMF constant from theory. Therefore the maximum theoretical speed can be obtained by dividing the supply voltage by the back-EMF constant. The targeted momentum storage is 0.004 Nms. Considering the moment of inertia of the flywheel (1.09E-5 kgm2), the theoretical back-EMF will not limit the reaction wheel momentum below 0.004 Nms. Doing so, the designed motor is expected to meet the momentum storage and torque requirements as outlined by the ADCS subsystem. Phase D testing will be required to experimentally validate the analysis above.