2.1. Magnetic Bearings
To maintain a high-speed rotating rotor in a non-contact state, the appropriate load for magnetic bearings must be calculated. The wind load for the magnetic bearing design was considered as follows.
The wind load analysis considers both rotational and non-rotational situations of the rotor sail. The non-rotational scenario represents the ship being anchored. Due to a storm forecast, ships are prohibited from leaving the harbor. Therefore, the structural stability is assessed under a wind speed of 25 m/s, corresponding to the typhoon warning, acting on the rotor frame. The load applied to the non-rotational rotor sail due to wind load, denoted as ‘
’ is calculated using Bernoulli’s principle in Equation (1):
where
is the density of air, assumed to be 1.225 at 15 °C;
A is the projected area of the rotor; and
is the wind velocity acting on the rotor. In the case of rotation, a wind speed of 16 m/s was assumed to calculate the stability of the rotor sail. This wind speed corresponds to the high surf advisory that imposes restrictions on departure. The rotational speed was determined by applying a speed ratio (SR) of 4 to maximize the efficiency of the rotor sail. A lift coefficient of 9.383 and a drag coefficient of 2.863 were employed [
16] to calculate the loads exerted by lift and drag during rotor rotation, according to Equations (2) and (3).
where
and
represent the lift and drag coefficients of the rotor, respectively.
The rotational torque is analyzed by dividing it into constant and acceleration sections. In the case of constant rotation, the stress applied to the rotor frame is calculated as shown in Equations (4) and (5).
Here,
represents the Poisson’s ratio of GFRP, assumed to be 0.25, and
denotes the density of GFRP, assumed to be 2500 kg/m
3. The rotational angular velocity,
ω, is calculated considering a high surf advisory of 16 m/s and a speed ratio of 4. When the wind speed reaches 16 m/s, the apparent wind speed (
) acting on the rotor, while sailing at a velocity of 6 knots, can reach up to 19 m/s. Additionally, when employing a rotor diameter of 1.5 m, the rotational speed exceeds 720 rpm. To ensure stability, an angular velocity of 83.78 rad/s, corresponding to 800 rpm, was employed.
and
represent the inner and outer diameters of the rotor, respectively. Since there is no force in the
z direction, the von Mises stresses (
) for the radial (
r), tangential (
t), and axial (
z) directions are determined using Equation (6):
In case of acceleration, the torque force (
T) applied on the rotor frame is calculated as shown in Equation (7):
where
I is the moment of inertia of the rotor and
is the angular acceleration. The mass of the rotor was calculated by substituting the density of Glass-Fiber-Reinforced Plastic (
ρ), while the moment of inertia is defined using Equation (8):
where
M represents the mass of the rotor, which is calculated by multiplying the volume of the rotor by its density (
ρ). To ensure appropriate power and stability of the motor, the angular acceleration was set to 8.378 rad/s
2, assuming a maximum speed attainment time of 10 s.
When considering the wind load and rotational torque acting on the rotor sail, it is determined that the applied RMB for the prototype should be able to withstand a maximum load of 6000 N. Accordingly, the RMB applied to the prototype is designed with a straight-fill structure, considering manufacturability, and the coil arrangement and key structures are depicted in
Figure 2.
The AMB must withstand the vertical load from the rotor sail’s own weight and the vertical acceleration of the ship’s movement. Inadequate load capacity may lead to collisions and damage between the support base and rotor sail. Simultaneously, a trade-off between power consumption for AMB operation and the drift generated via the rotor sail should be achieved to maximize efficiency. In this study, considering the prototype specifications, an AMB capable of handling a maximum load of 2100 N (=212 kgf) is designed. The AMB is configured in a form that combines electromagnets (EM) and permanent magnets (PM) to minimize power consumption. The key specifications are illustrated in
Figure 3.
The designed RMB and AMB each exhibit electromagnetic forces in the horizontal (x-direction) and vertical (z-direction) distances, as shown in
Figure 4. The RMB has a magnetic flux density of 1.5 T at a maximum current of 10 A. Considering electromagnetic forces based on current and gap, it is determined that a current of 4–6 A is appropriate. For the AMB, the analysis of magnetic field control at a maximum current of 10 A reveals similar force characteristics in the up and down coils. To support the load of the rotating rotor, a current of 5 A is deemed suitable.
Figure 4 illustrates the electromagnetic forces based on the distances for the designed RMB and AMB.
2.2. Lightweight Structure Rotor Design
Several conditions need to be considered for the theoretical analysis of the prototype structure design. The wind interacting with the rotor sail is assumed to be an incompressible and ideal flow fluid to estimate and calculate the Magnus effect. Since the rotor sail will be applied to a magnetic bearing, frictional forces between the rotor sail and the bearing are neglected. The SR, defined as the ratio of the rotor’s rotational speed to the wind speed acting on the rotor, is set to 4, which has been identified as the optimal point of efficiency compared to rotational speed [
11,
15,
16]. It is assumed that the rotor sail should be controlled to maintain the speed ratio as the wind speed changes.
The schematic design of the rotor sail is illustrated in
Figure 5. The rotor sail, with dimensions of 1.5 × 9 m
2, incorporates RMB and AMB, with the rotor frame being rotated using a permanent magnet (PM) motor.
Commercial rotor frames typically consist of sandwich-structured composite materials made of GFRP/Carbon-Fiber-Reinforced Plastic (CFRP) and polyurethane (PU) foam [
18]. GFRP/CFRP is used for the outer wall, while polyurethane foam serves as the core material. Considering the external forces experienced during ocean navigation, the commercial frames are estimated to have a composite thickness of 8 mm. When making a judgment about structural strength, we put in an S.F. of 3.0 to see if this value exceeds the structural strength.
In this paper, we assumed and examined the rotor design, considering the load capacity of magnetic bearings. Additionally, we considered the design of a lightweight rotor structure to minimize power consumption for rotor sail propulsion while withstanding wind load and rotational torque. With a GFRP rotor of 2 mm thickness, a size of approximately 1.5 m by 9 m results in a weight of about 212 kg. This weight seems suitable considering the load capacities of 6000 N for RMB and 2100 N for AMB. However, further verification is needed to assess whether it can withstand wind load and rotational torque at a level of approximately one-fourth compared to the typical thickness of commercial rotors, which is 8 mm.
Given the thinness of the frame, a single material is used instead of a sandwich-structured composite. GFRP, which has lower strength compared to CFRP, is selected as the material for a thorough stability analysis.
The structural stability of the rotor sail is analyzed using the ANSYS 2022 R1 static structural module in both non-rotational and rotational situations. Since the lower RMB is positioned above the AMB, the position of the lower RMB is fixed. Subsequently, the stress and deformation of the rotor frame are analyzed based on the position (z) of the upper RMB. The mesh for the FEM analysis is created in a hexahedral shape for the rotor frame, while the endplate is generated in a tetrahedral shape based on the rotor frame, as shown in
Figure 6. The total number of elements used is 98,471. When a wind load compresses the rotor frame, the RMB provides internal support for the rotor frame. Since the RMB prevents the deformation of the rotor frame, the corresponding parts are set as fixed supports.
Figure 7a illustrates the distribution of stress and deformation resulting from wind loads in a non-rotational situation, simulated through FEM analysis. The position of the upper RMB significantly affects stress and deformation. Notably, the point (
z/H) corresponding to 20% exhibits the minimum stress, while the minimum deformation occurs at the 40% location (
Figure 7c). Even in rotation, the distribution of stress and deformation tends to exhibit similar patterns (
Figure 7b).
Typically, it is expected that minimum stress and subsequent deformation occur when the upper RMB is centered. However, our observations reveal that the minimum stress is concentrated within the 20–40% range. This phenomenon arises due to deformation resulting from external forces acting on an object. When the deformation is constrained by a fixed support, stress becomes concentrated instead of being uniformly distributed. Consequently, while the deformation of the rotor frame limited by the fixed support decreases, the stress intensifies rather than diminishes.
Regarding deformation, it was confirmed that the least deformation occurs near the point where the difference in cross-sectional area between the upper and lower rotor frames of the upper RMB is minimized, given that the lower RMB is fixed at a distance of 600 mm from the bottom. Furthermore, as the rotor frame situated above the upper RMB experiences significant deformation, it becomes influenced by the weight of the endplate. Consequently, the position of the maximum stress shifts from the bottom to the top.
In the case of non-rotation, the maximum stress occurs when the rotor frame at the top of the upper RMB reaches 70%, resulting in a stress value of 7.945 MPa. During rotation, the maximum stress value is 31.653 MPa.
The structural stability of the rotor sail was analyzed using the ANSYS static structural module under acceleration conditions. The FEM analysis employed a mesh configuration, as illustrated in
Figure 8. The mesh consists of hexahedral elements surrounding the rotor, totaling 36,264 elements. When torque is applied during rotor rotation, it induces tension on the rotor, rendering the support provided by the RMB ineffective. Therefore, the mesh was generated without subdividing the parts.
Figure 9 presents the results of the FEM analysis, illustrating the stress and deformation caused by torque applied to the rotor frame under maximum acceleration conditions (a = 8.378 rad/s
2). Since the upper section is constrained by the endplate, while the lower part is free, the highest stress is observed in the region where deformation is restricted, as previously mentioned. In summary, torque during maximum acceleration induces a stress of 0.15144 MPa. The torque applied during the rotational motion at a constant maximum speed is calculated as the maximum stress of 9.86 MPa using Equations (4)–(6).