Design and optimization of permanent magnetic thrust bearing used in vessels

Abstract

To isolate the vibration transmitted from the propeller to the vessel’s hull, the main engine, the shaft together with the bearings are proposed to be installed on a large-scale isolation system. Permanent magnetic thrust bearing (PMTB) is applied to further reduce the vibration transmission. The Coulombian model is adopted to calculate the force and stiffness, which determine the application feasibility of the PMTB in vessels. Explicit computations are presented for stacked PMTB. The calculation result is compared with that obtained by finite element method (FEM) and experimentally tested. It is revealed that the Coulombian model is accurate enough and more economic than the FEM, which make it advantageous to the structure design and parameter optimization of PMTB. In the influence study by Coulombian model, it is found that the airgap width can exponentially change the maximum axial magnetic force and stiffness; the radial width is not a sensitive factor for both force and stiffness; the axial length should be 4/5 of the radial thickness to obtain the largest force and decent stiffness.

Keywords

Permanent magnetic thrust bearing Coulombian model finite element method parameter optimization magnetic force and stiffness

1. Introduction

In marine propulsion systems, the thrust bearing transforms the propeller’s axial movement directly to the hull, which exciting the hull to vibrate and radiate sound. This will seriously reduce the quietness of the vessels and also cause negative influence on the marine organism [1,2]. On another hand, traditional thrust bearing needs complicate lubrication system. To reduce the vibration and simplify the structure, the floating raft isolation system is adopted and the permanent magnetic thrust bearing (PMTB) is introduced to replace the traditional thrust bearing. As shown in Fig. 1, the shaft system and all bearings are installed on a large raft which is connected with the vessel hull by vibration isolators.

Fig. 1.

Schematics of the isolation system for marine propulsion system

Compared with traditional Michelle thrust bearings, the stiffness of PMTB decreases by one order of magnitude, from 10⁷ N/m to 10⁶ N/m. Lower stiffness means the shaft system will undergo larger displacement when transmitting the same thrust from the propeller. Furthermore, the stiffness of the thrust has great influence on the dynamic characteristics of the whole isolation system [3].

Permanent Magnetic Bearings (PMB), which are generally manufactured with axially or radially magnetized magnets, have the characteristics of no mechanical friction, low noise, low power-consumption and vibration reduction [4], so they are largely applied in the fields like space technology [5,6], mechanical processing [7] and so on. Yonnet [8] is the pioneer in the study upon PMB, who calculated the magnetostatic energy created by a system of two magnets and obtained the expressions for forces by differentiate the energy expression with respect to the displacement. Beleggia [9–11] derived the general expression for the magnetostatic energy of two magnetic elements with arbitrary shape. Vokoun [12] continued the work of Beleggia and obtained analytical expressions for calculating the attraction force between two cylindrical permanent. The research achievement is utilized to continue research work in fields like magnetic guns [13], nonlinear energy sink [14], ultra-broad-band nonlinear acoustic metamaterials [15] and energy harvesting [16]. Another popular way to determine the magnetic force is based on Coulombian approach. It uses fictitious magnetic charge to model the magnetic field intensity. Rakotoarison [17] computed the demagnetization field in each point inside the permanent magnet and the magnetic fields outside it. Ravaud [18–20] determined the magnetic force exerted between rings axially or radially magnetized using Coulombian model.

This paper firstly reviews the Coulombian approach for magnetostatic calculations and extends the results to obtain the magnetic force of large-scale stacked PMTB. Then, to verify the correctness of the calculating procedure, the numerical results from FEM and the tested results from a large-scale stacked PMTB are compared with that of Coulombian model and FEM method. Finally, the Coulombian model is used to analyze the influence of airgap width, radial thickness and layers on axial magnetic force and stiffness.

2. Force calculation of PMTB using Coulombian model

2.1. Coulombian model for single pair of magnetic ring

The derivative process is began with Maxwell’s equations $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\cdot \overset{\rightarrow }{E}={\rho}/{\varepsilon}_{0}\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\cdot \overset{\rightarrow }{B}=0.\end{eqnarray}$ (2)

Equation (1) is the Gauss’s law and Eq. (2) is the Gauss’s law for magnetism, both in differential form. The meaning of the symbols are listed in the appendix in alphabetical order.

Ferromagnetic materials have the property of $\begin{eqnarray}\displaystyle \overset{\rightarrow }{B}={\mu}_{0}\overset{\rightarrow }{H}+\overset{\rightarrow }{J}, & & \displaystyle\end{eqnarray}$ (3) where J is the magnetic polarization, which represents the vector sum of magnetic dipole moment in unit volume. By combining Eq. (2) and Eq. (3) $\begin{eqnarray}\displaystyle {\nabla}\cdot \overset{\rightarrow }{H}=-{\nabla}\cdot \overset{\rightarrow }{J}/{\varepsilon}_{0}. & & \displaystyle\end{eqnarray}$ (4)

From Eq. (2), it is known that there is no monopole magnetic charge which can independently create a magnetic field, but with regard to the electrical charge, we imagine fictitious magnetic charge σ distributing on the surface of and inside permanent magnets, which are indicated by σ_s and σ_v respectively. By comparing Eq. (4) with Eq. (1), the fictitious magnetic charge density is $\begin{eqnarray}\displaystyle {\sigma}=-{\nabla}\cdot \overset{\rightarrow }{J}. & & \displaystyle\end{eqnarray}$ (5)

If the magnetic rings of the PMTB are radially magnetized and the magnetic polarization J is uniform, then the expression of J in cylindrical coordinate system is $\begin{eqnarray}\displaystyle \overset{\rightarrow }{J}=(J,0,0), & & \displaystyle\end{eqnarray}$ (6) then the expression of σ_v is $\begin{eqnarray}\displaystyle {\sigma}_{v}=-{\nabla}\cdot \overset{\rightarrow }{J}=-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial (rJ_{r})}{\partial r}}-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial (J_{{\theta}})}{\partial {\theta}}}-{\displaystyle \frac{\partial (J_{z})}{\partial z}}=-{\displaystyle \frac{J}{r}}. & & \displaystyle\end{eqnarray}$ (7)

As discussed above, there is no monopole magnetic charge for an entire magnetic ring, so $\begin{eqnarray}\displaystyle \iint _{s}{\sigma}_{s}ds+\iiint _{v}{\sigma}_{v}dv=0. & & \displaystyle\end{eqnarray}$ (8) From Gauss’s theorem $\begin{eqnarray}\displaystyle \iiint _{{\Omega}}{\nabla}\cdot \overset{\rightarrow }{F}dV=\iint _{\partial {\Omega}}\overset{\rightarrow }{F}\cdot \overset{\rightarrow }{n}ds. & & \displaystyle\end{eqnarray}$ (9) Identified from Eq. (7), (8), (9), it is obtained that $\begin{eqnarray}\displaystyle {\sigma}_{s}=\overset{\rightarrow }{J}\cdot \overset{\rightarrow }{n}. & & \displaystyle\end{eqnarray}$ (10)

For a pair of magnetic rings, as shown in Fig. 2, the inner ring is attached to the rotor and the outer ring to the stator. The axial force is the sum of magnetic force between surface and surface, volume and surface, volume and volume, which can be written as $\begin{eqnarray}\displaystyle F_{z}=F_{z}(\text{s}-\text{s})+F_{z}(v-\text{s})+F_{z}(v-v), & & \displaystyle\end{eqnarray}$ (11) where $\begin{eqnarray}\displaystyle F_{z}(\text{s}-\text{s})=\mathop{\sum }_{i=1,2}\mathop{\sum }_{j=3,4}\int _{{\theta}_{1}=0}^{2{\pi}}\int _{z_{1}=0}^{h}\int _{{\theta}_{2}=0}^{2{\pi}}\int _{z_{2}=z_{a}}^{z_{b}}(-1)^{i+j}A(r_{i},r_{j})\times dz_{1}d{\theta}_{1}dz_{2}d{\theta}_{2} & & \displaystyle\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle F_{z}(v-\text{s}) & = & \displaystyle \mathop{\sum }_{j=3,4}\int _{r_{out}=r_{2}}^{r_{1}}\int _{{\theta}_{1}=0}^{2{\pi}}\int _{z_{1}=0}^{h}\int _{{\theta}_{2}=0}^{2{\pi}}\int _{z_{2}=z_{a}}^{z_{b}}(-1)^{j}\times B\left(r_{j},r_{out}\right)dr_{out}dz_{1}d{\theta}_{1}dz_{2}d{\theta}_{2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\mathop{\sum }_{i=1,2}\int _{r_{in}=r_{4}}^{r_{3}}\int _{{\theta}_{1}=0}^{2{\pi}}\int _{z_{1}=0}^{h}\int _{{\theta}_{2}=0}^{2{\pi}}\int _{z_{2}=z_{a}}^{z_{b}}(-1)^{i}\times B(r_{i},r_{in})dr_{in}dz_{1}d{\theta}_{1}dz_{2}d{\theta}_{2}\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle F_{z}(v-v)=\mathop{\sum }_{i=1,2}\int _{r_{out}=r_{2}}^{r_{1}}\int _{r_{in}=r_{4}}^{r_{3}}\int _{{\theta}_{1}=0}^{2{\pi}}\int _{z_{1}=0}^{h}\int _{{\theta}_{2}=0}^{2{\pi}}\int _{z_{2}=z_{a}}^{z_{b}}C(r_{i},r_{in})\times dr_{in}dz_{1}d{\theta}_{1}dz_{2}d{\theta}_{2}. & & \displaystyle\end{eqnarray}$ (14)

We first consider the situation when two fictitious magnetic charges located on the surfaces, as shown in Fig. 2. Compared with Coulomb’s law, the magnetic force is $\begin{eqnarray}\displaystyle A(m,n)={\displaystyle \frac{{\sigma}_{1}{\sigma}_{2}}{4{\pi}{\mu}_{0}d^{2}}}\cdot {\displaystyle \frac{z_{2}-z_{1}}{d}}\cdot mn, & & \displaystyle\end{eqnarray}$ (15) where $\begin{eqnarray}\displaystyle d=\sqrt{(m-n)^{2}+(z_{2}-\text{z}_{1})^{2}}, & & \displaystyle\end{eqnarray}$ (16) considering Eq. (7) and (10), it is obtained $\begin{eqnarray}\displaystyle B(m,n)={\displaystyle \frac{A(m,n)}{n}},\quad C(m,n)={\displaystyle \frac{A(m,n)}{mn}}, & & \displaystyle\end{eqnarray}$ (17) substitute Eq. (12)–(17) in to Eq. (11), we can obtain the entire axial force generated by a pair of magnetic rings. It is obvious that the expressions of Eq. (12), (13), (14) which include quadruple, quintuple or sextuple integrals, are very complicated and it is impossible to obtain the analytical results [19]. If we totally adopt numerical method, the computational cost will be too high, so the efficient and feasible way is to finish part of the multiple integrals analytically and utilize numerical integral to calculate the rest part.

Fig. 2.

Sectional view of a pair of magnetic ring. The outer ring is attached to the stator and the inner ring is attached to the rotor. The inner ring can move axially. Both rings are radially magnetized.

After calculation, we found that the results of Eq. (13) and (14) are much smaller than that of Eq. (12). The ratios of peak values are less than 1%, so to further reduce the computational cost, we only calculate the magnetic force between surface and surface and ignore the other two parts.

2.2. Force calculation for stacked PMTB

For the next step, we consider the stacked structure, as shown in Fig. 3. The magnetic rings of the stator (outer) are fixed on the bearing block and the magnetic rings of the rotor (inner) are connected with the shaft which can move axially. All the magnetic rings are radially and uniformly magnetized with the same orientation.

Fig. 3.

Schematic diagram of the permanent magnetic thrust bearing.

In equilibrium (Fig. 3(a)), the corresponding outer and inner rings face to each other just right and provide no axial force. When the propeller thrust drives the rotor magnetic rings to move in the axial direction, as shown in Fig. 3(b), the induced magnetic attractive and repulsive forces make it produce axial thrust.

In the stacked structure, every two magnetic rings of inner and outer ones generate magnetic forces. If the PETB consists of T pairs of magnetic rings, then all the magnetic forces can be written in a matrix form as follows $\begin{eqnarray}\displaystyle \mathbf{F}=[F_{pq}]_{T\times T}, & & \displaystyle\end{eqnarray}$ (18) where $\begin{eqnarray}\displaystyle F_{pq}=(-1)^{p+q}\mathop{\sum }_{i=1,2}\mathop{\sum }_{j=3,4}\int _{{\theta}_{1}=0}^{2{\pi}}\int _{z_{1}=0}^{h}\int _{{\theta}_{2}=0}^{2{\pi}}\int _{z_{2}=z_{pq}}^{z_{pq}+b}(-1)^{i+j}\times A(r_{i},r_{j})dz_{1}d{\theta}_{1}dz_{2}d{\theta}_{2}, & & \displaystyle\end{eqnarray}$ (19) where $\begin{eqnarray}\displaystyle z_{pq}=z_{a}-h(p-1)+b(q-1). & & \displaystyle\end{eqnarray}$ (20) The total axial force exerted by the PMTB is the sum of F elements $\begin{eqnarray}\displaystyle F_{entire}=\mathop{\sum }_{p=1}^{T}\mathop{\sum }_{q=1}^{T}F_{pq}. & & \displaystyle\end{eqnarray}$ (21)

2.3. Stiffness calculation for stacked PMTB

The stiffness of PMTB is very important for the dynamic characteristics of marine propulsion systems. The value of stiffness can be obtained from computing the gradient of the force to displacement. When concentrating on the axial stiffness, it becomes calculating the derivative of axial force F_entire to axial displacement z_a $\begin{eqnarray}\displaystyle K_{z}=-\partial F_{entire}/\partial z_{a}. & & \displaystyle\end{eqnarray}$ (22)

3. Force calculation of PMTB using FEM

The FEM simulation is implemented by Comsol Multiphysics through the physic field of “Magnetic field, No current”. As shown in Fig. 4, the software takes advantage of the rotational symmetry of the PMTB and a 2D model is adopted to reduce the computational cost. The principle of the magnetic force calculation is to obtain the magnetic induction distribution through Maxwell stress tensor method [13,21]. Maxwell stress tensor T_ij is expressed as $\begin{eqnarray}\displaystyle T_{ij}={\displaystyle \frac{1}{{\mu}_{0}}}B_{i}B_{j}-{\displaystyle \frac{1}{2{\mu}_{0}}}B_{ij}^{2}{\delta}_{ij}. & & \displaystyle\end{eqnarray}$ (23) Where, δ_ij is the Kronecker delta function. Then the force is computed by $\begin{eqnarray}\displaystyle \mathbf{F}=\int _{\partial {\Omega}}2{\pi}r\overset{\rightarrow }{\mathbf{n}}TdS, & & \displaystyle\end{eqnarray}$ (24) where $\vec{\mathbf{n}}$ is the surface normal vector and the integration is conducted over the surface of region 𝛺. In Comsol, the stiffness calculation is also based on computing the gradient of magnetic force with respect to displacement vector. The physical field of “sensitivity” is applied to complete the stiffness calculation. It should be mentioned that we only calculate the axial magnetic stiffness, so the 2D model is sufficient. If calculating the magnetic stiffness in radial direction or coupled stiffness coefficients, a complete 3D model is required.

Fig. 4.

Magnetic force calculation of stacked PMTB using FEM in Comsol Multiphysics.

The force calculation results by two methods are illustrated in Fig. 5. It shows that both methods give almost identical force curves. Only at the peaks or valleys, the values of Coulombian model are a little smaller than that of FEM, but it is obvious that the FEM curve is not as smooth as the one generated by Coulombian model. In the FEM calculation, the 2D model is meshed into 6850 elements. If finer mesh is used, the FEM curve can be smoother, but more elements means more time-cost.

Fig. 5.

Diagram of the magnetic force versus axial displacement calculated by FEM and Coulombia model.

The stiffness calculation results comparation is shown in Fig. 6. It is apparent that the unevenness of FEM force curve leads to the stiffness curve much more unsmooth. This also causes the FEM failing to predict the magnet stiffness at the valleys, namely when the axial displacement equals to an integral multiple of the axial length of magnetic ring h (b = h). But apart from the parts around the valleys, the two curves have similar values for most axial displacements.

Fig. 6.

Diagram of the magnetic stiffness versus axial displacement calculated by FEM and Coulombia model.

Moreover, even though using a 2D model, the computing process of FEM still costs around 2 hours, while it costs only 6 minutes by Coulombian model by the same computer. We can conclude that the Coulombian model yields physically reasonable force calculation results in much smaller time-cost.

4. Experimental results

NdFeB is adopted as the permanent magnetic material to manufacture an experimental prototype of the PMTB. Fig. 7 illustrates the experimental setup for the axial magnetic force measurement. The outer magnetic rings are fixed to the stator (bearing block) of the PMTB and the inner rings are attached to the rotor (shaft) of the PMTB. One end of the PMTB rotor is attached with the motor through a flexible coupling. The other end is connected to an axial loading, which is a hydraulic pump. At the beginning of the experiment, the motor drove the shaft to rotate and the hydraulic pressure was set to zero. During the measurement, we recorded the exerted force every 1 mm of the rotor axial displacement. The measurement was conducted three times and the relationship between the applied load and the axial displacement was obtained by averaging the data.

Fig. 7.

Schematic diagram of the experimental apparatus for the measurement of PMTB axial magnetic force.

The tested axial magnetic force compared with the negative magnetic forces calculated by Coulombian model and FEM is shown in Fig. 8, in which only the comparison above the x axis is illustrated because of the symmetry. It is shown that the values obtained by both the Coulombian model and the FEM are reasonably close to the corresponding measurements. Deviations are expected and acceptable because the calculations by Coulombian approach and FEM are based on the uniform magnetization approximation. It proves that both the Coulombian model and FEM simulation have sufficient accuracy for the axial force prediction of the PMTB.

Fig. 8.

Diagram of the magnetic force versus axial displacement calculated by FEM, Coulombia model and tested results.

Central difference method is applied to deal with the tested data and obtain the magnetic stiffness. The results comparison is shown in Fig. 9. It is revealed that for most values, the result of FEM is smaller than that of tested and the values computed by Comlombian is closer to the ones by tested, so the Coulombian model has better prediction upon magnetic stiffness than the FEM with less time-cost.

Fig. 9.

Diagram of the magnetic stiffness versus axial displacement calculated by FEM, Coulombia model and tested results.

After comparing the results of FEM, Coulombian model and tested, we conclude that both calculation methods can well determine the magnetic force and the Coulombian model can get smoother curve than FEM. Upon stiffness determination, Coulombian model is preciser than FEM. It also reveals that in Coulombian model only considering the reaction between faces is accurate enough in the PMTB design.

5. Parameter optimization of PMTB

In this section, the Coulombian model is used to study the influence of PMTB structural parameters on the magnetic force and stiffness, including the airgap width, the radial thickness difference and the axial length of magnetic ring.

5.1. Influence of airgap width

When calculating the magnetic force and stiffness for different airgap widths w_a between the outer and inner rings, the influence of other structural parameters should be minimized. Equation (12) reveals that the surface area of the magnetic rings is a determinative factor in Coulombian model, so to reduce its influence we keep the middle position of the airgap r_d constant, in which way the outer and inner rings variation of surface area can offset each other.

Fig. 10 and Fig. 11 illustrate the axial magnetic force and stiffness versus displacement when w_a varies from 1.5 mm to 19.5 mm, which means the ratio between w_a and radial thickness of magnetic rings t_out (t_in) varies from 5% to 65%. It is obvious that the increment of w_a can reduce the axial force and stiffness greatly. The maximum forces and stiffnesses are extracted and ploted in Fig. 12 versus w_a. Through curve fitting, an function of $\begin{eqnarray}\displaystyle f(x)=2.058\text{e}^{5}\times \text{e}^{-0.09565x} & & \displaystyle\end{eqnarray}$ (25) is obtained whose curve (the blue line in Fig. 12) can perfectly cover all maximum force points. This gives the conclusion that the axial magnetic force decreases exponentially with the increment of airgap width. We can see that the stiffness has the similar tendency with the force.

Fig. 10.

Diagram of the magnetic force versus axial displacement exerted between 9-layer stacked magnetic rings for different air gaps: r_mid = 0.29825 m, t_out = t_in = 0.03 m, σ_s = 1.25 T, h = b = 0.03 m.

Fig. 11.

Diagram of the magnetic stiffness versus axial displacement exerted between 9-layer stacked magnetic rings for different air gaps: r_mid = 0.29825 m, t_out = t_in = 0.03 m, σ_s = 1.25 T, h = b = 0.03 m.

In Fig. 12, the axial position where the maximum forces occur are labeled near the corresponding point which reveals that increasing airgap width can slightly alter the maximum force towards larger displacement. It is also found that increasing the number of layers T can minimize the position alternation. Generally, a PMTB is comprised of more than ten layers of magnetic rings, so we believe that this influence can be neglected in the PMTB design.

Fig. 12.

Diagram of the maximum magnetic force and stiffness versus airgap width exerted between 9-layer stacked magnetic rings: r_mid = 0.29825 m, t_out = t_in = 0.03 m, σ_s = 1.25 T, h = b = 0.03 m. The axial displacement where the maximum forces occur are labeled near the corresponding point.

5.2. Influence of radial thickness

During the calculation, the total volume of all magnetic rings is kept constant, namely t_in + t_out = 0.06 m and w_a = 1.5 mm. In the study of this section, the value of radial thickness difference r_d = t_in − t_out changes from −30 mm to 30 mm. Fig. 13 and Fig. 14 illustrate the axial magnetic force and stiffness versus displacement for r_d varies from −30 mm to 0 mm and from 0 mm to 30 mm, respectively. Actually, for inner or outer magnetic ring, r_d = 30 mm or −30 mm means the radial thickness has increased or decreased 50%, which is a huge variation, but the magnetic force and stiffness don’t have dramatic change, so the radial thickness change is not a sensitive factor for magnetic force and stiffness.

Fig. 13.

Diagram of the magnetic force versus axial displacement exerted between 9-layer stacked magnetic rings for different r_d (the unit is mm): r₁ = 0.329 m, r₄ = 0.2675 m, w_a = 0.0015 m, σ_s = 1.25 T, h = b = 0.03 m.

Fig. 14.

Diagram of the magnetic stiffness versus axial displacement exerted between 9-layer stacked magnetic rings for different r_d (the unit is mm): r₁= 0.329 m, r_{4 =}0.2675 m, w_a = 0.0015 m, σ_s = 1.25 T, h = b = 0.03 m.

Fig. 15.

Diagram of the maximum magnetic force and stiffness versus the thickness difference r_d: r₁ = 0.329 m, r₄ = 0.2675 m, w_a = 0.0015 m, σ_s = 1.25 T, h = b = 0.03 m.

From Fig. 15, we find that the largest magnetic force does not appear at r_d = 0, namely t_in = t_out, but occurs where the inner ring thickness a bit larger than the outer one. The tendency of the stiffness is unexpected. The maximum stiffness versus r_d appears an inverse tendency compared with the one of force.

5.3. Influence of axial length of single magnetic ring

In the study of this part, the total volume of all magnetic rings is still kept constant. At the same time, we kept h = b, t_in = t_out = 0.03 m and the total axial length of magnetic rings L = 0.24 m. In the calculation, the rings are divided into different layers to calculate the axial magnetic force and stiffness. ss, the ratio between axial length and radial width of a single magnetic layer, is used to describe the change of axial length of a single layer. Fig. 16 and Fig. 17 demonstrate the force and stiffness versus axial displacement exerted between magnetic rings of different ss. From Fig. 12 it is concluded that the maximum magnetic force appears at almost half of h (b), so the variation of the axial length can dramatically change the position where the maximum occurs, which will further determine the range of axial displacement of the rotor. From Fig. 17, it is revealed that increasing T, decreasing h and b can improve the stiffness. For a PMTB, the available axial displacement is strictly limited, so we can meet the demand through increasing layers and decreasing axial length of magnetic rings.

Fig. 16.

Diagram of the magnetic force versus axial displacement exerted between magnetic rings of different layers: r₁ = 0.329 m, r₂ = 0.299 m, r₃ = 0.2975 m, r₄ = 0.2675 m, σ_s = 1.25 T, t_in = t_out = 0.03 m, L = 0.24 m T = [24,12,8,6,4,3]. ss are labeled near the corresponding curves.

Fig. 17.

Diagram of the magnetic stiffness versus axial displacement exerted between magnetic rings of different layers: r₁ = 0.329 m, r₂ = 0.299 m, r₃ = 0.2975 m, r₄ = 0.2675 m, σ_s = 1.25 T, t_in = t_out = 0.03 m, L = 0.24 m, T = [24,12,8,6,4,3]. ss are labeled near the corresponding curves.

Fig. 18.

Diagram of the maximum magnetic force and stiffness versus the ratios between axial length and radial width of single magnetic layer ss: r₁ = 0.329 m, r₂ = 0.299 m, r₃ = 0.2975 m, r₄ = 0.2675 m, σ_s = 1.25 T, t_in = t_out = 0.03 m, L = 0.24 m, T = [24,12,8,6,4,3].

Maximum magnetic force and stiffness versus ss are shown in Fig. 18. It is found that the maximum magnetic force occurs at the point where the h (b) equals around 4∕5×t_out(t_in). Besides, the maximum stiffness monotonically decreases with increasing ss. When ss is less than 1, the stiffness will change dramatically, so in this range, ss is a sensitive factor for magnetic stiffness.

5.4. Obtaining the optimal structural parameters

From the above analysis, it is concluded that the airgap width w_a can reduce the magnetic force and stiffness exponentially, so smaller w_a is favorable for PMTB design. The specific value of w_a should be determined based on the dimension of magnetic rings. For example, the w_a should be among 1 mm to 3 mm for rings with 30 mm square section.

The radial thickness (t_out, t_in) is not a sensitive factor for both force and stiffness, so PMTB can be designed as t_out = t_in.

The layers or the axial length is a sensitive factor for both the force and stiffness. Especially, it determines the axial moving range of rotor when bearing thrust from the propeller. To maximize the axial magnetic force and meanwhile get large enough stiffness, the ratio between h (b) and t_out (t_in) should be around 4/5.

6. Discussions

This paper provides a feasible method, the Coulombian approach, for the design of PMTB. To verify the accuracy of the Coulombian model, the calculated results are first compared with that from FEM. It is revealed that both methods present almost identical force curves. However, the unevenness of FEM force curve not only leads to the stiffness curve much more unsmooth, but also fails to predict the magnet stiffness at the values around curve valleys. A large-scale stacked PMTB was manufactured and the tested results from the experiment further verify the correctness of Coulombian model. So it can be concluded that the Coulombian model has precise enough prediction of magnetic force and stiffness in much smaller time-cost than FEM, which makes it advantageous to the structure design and parameter optimization of PMTB.

In the influence study by Coulombian model, it is found that the airgap width and axial length of magnetic rings are pretty sensitive parameters for the force and stiffness. The former one has an exponential relationship with the force and stiffness and the latter one can change the range of available axial displacement for a PMTB. However, the impact of radial thickness is not so obvious and can choose the same thickness of outer and inner rings. All these conclusions can be very important in the design of PMTB.

Footnotes

Acknowledgements

This work was financially supported by Natural Science Fund 51705529 and National Science Key Lab Fund 61422040601162204004.

Appendix

References

Qiang

Lei

and Wei

, Transmission characteristics of propeller excitation for naval marine propulsion shafting, Chinese Journal of Ship Research10(6) (2015), 81–86, 94.

Pan

Farag

Lin

and Juniper

, Propeller Induced Structural Vibration through the Thrust Beraing, Adelaide, Australia, 2002.

Jiayou

, Research on Characteristics of Fluid Film Stiffness of Water-Lubricated Bearings and the Influence on Shafting Vibration, Ph.D. Dissertation, Harbin Engineering University, 2013.

Lulin

Yan

Shan-Shi

and Jing

, Research on magnetism engineering analytical calculation method for bi-barrel-shaped radial permanent magnetic bearings, Proceedings of the CSEE27(6) (2007).

Ohji

Mukhopadhyay

S.C.

Iwahara

and Yamada

, Performance of repulsive type magnetic bearing system under non-uniform magnetization of permanent magnet, IEEE Transactions on Magnetics36(5) (2000), 3696–3698.

Paden

Groom

and Antaki

, Design formulas for permanent magnet bearing, Journal of Mechanical Design125 (2003), 734–738.

Ohji

Ichiyama

Amei

Sakui

and Yamada

, A New conveyor system based on a passive magnetic levitation unit having repulsive-type magnetic bearings, Journal of Magnetism and Magnetic Materials272 (2004), 1731–1733.

Yonnet

, Permanent magnetic bearings and couplings, IEEE Transactions on Magnetics17(1) (1981), 1169–1173.

Beleggia

and De Graef

, General magnetostatic shape–shape interactions, Journal of Magnetism And Magnetic Materials285 (2005), L1–L10.

10.

Beleggia

Vokoun

and De Graef

, Demagnetization factors for cylindrical shells and related shapes, Journal of Magnetism and Magnetic Materials321 (2009), 1306–1315.

11.

Beleggia

Tandon

Zhu

and De Graef

, On the magnetostatic interactions between nanoparticles of arbitrary shape, Journal of Magnetism and Magnetic Materials278 (2004), 270–284.

12.

Vokoun

Beleggia

Heller

and Sittner

, Magnetostatic interactions and forces between cylindrical permanent magnets, Journal of Magnetism and Magnetic Materials321 (2009), 3758–3763.

13.

Vokoun

Beleggia

and Heller

, Magnetic guns with cylindrical permanent magnets, Journal of Magnetism and Magnetic Materials324 (2012), 1715–1719.

14.

Fang

Wen

Yin

and Yu

, Highly efficient continuous bistable nonlinear energy sink Highly efficient continuous bistable nonlinear energy sink, Nolinear Dynamics87(4) (2016), 1–19.

15.

Fang

Wen

Bonello

Yin

and Yu

, Ultra-low and ultra-broad-band nonlinear acoustic metamaterials, Nature Communications8(1) (2017), 1–11.

16.

Avvari

P.V.

Tang

Yang

and Soh

C.K.

, Enhancement of piezoelectric energy harvesting with multi-stable nonlinear vibrations, Processings of SPIE8688 (2013), 86881H–86882H.

17.

Rakotoarison

H.L.

Yonnet

J.P.

and Delinchant

, Using Coulombian approach for modeling scalar potential and magnetic field of a permanent magnet with radial polarization, IEEE Transactions on Magnetics43(4) (2007), 1261–1264.

18.

Ravaud

Lemarquand

and Lemarquand

, Force and stiness of passive magnetic bearings using permanent magnets. Part 1: Axial magnetization, IEEE Transactions on Magnetics45(7) (2009), 2996.

19.

Ravaud

Lemarquand

and Lemarquand

, Force and stiffness of passive magnetic bearings using permanent magnets. Part 2: radial magnetization, IEEE Transactions on Magnetics45(9) (2009), 3334–3342.

20.

Ravaud

and Lemarquand

, Halbach structures for permanent magnets bearings, Progress in Electromagnetics Research M14 (2010), 263–277.

21.

Furlani

E.P.

, Permanent Magnet and Electromechanical Devices, Acamedic Press, 2001.