Multiple-degree-of-freedom damping characteristics evaluation of dual halbach array eddy current damper for turbopump in cryogenic environment

Abstract

Liquid hydrogen turbopumps are used in large high-performance rockets. Stable high-speed rotation is required for rocket turbopumps. The damping mechanism of the pump must suppress vibration not only in the radial direction but also in the axial direction. However, conventional damping elements using oil or rubber cannot be used due to the cryogenic temperature of liquid hydrogen. Therefore, the application of eddy current dampers to liquid hydrogen turbopumps is focused on in this paper. Although various structures of eddy current dampers have been developed, the multi-degree-of-freedom damping characteristics of dual Halbach array type eddy current dampers for liquid hydrogen turbopumps have not yet been investigated. The variation of damping characteristics with temperature has also not yet been verified. In this paper, we propose a novel dual Halbach array type eddy current damper for liquid hydrogen turbopumps. The proposed damper can generate high damping force and can be operated maintenance-free at the cryogenic temperature. The analysis results show that the damping characteristics strongly depend on temperature and that the amplitude reduction effect is greater at low temperatures. It was also found that the proposed damper has a higher damping force density than conventional dampers.

Keywords

Eddy current damper magnetic damper dual Halbach array cryogenic environment turbopump

1. Introduction

In recent years, the increasing expectations for the space industry have focused attention on the reuse of rockets [1]. Large high-performance rockets utilize liquid hydrogen turbopumps [2], which are the heart of the rocket. Liquid hydrogen turbopumps require stable high-speed rotation in cryogenic environments [3]. The damping mechanism of the pump must suppress vibration not only in the radial direction but also in the axial direction. However, oil dampers and anti-vibration rubbers used as conventional damping elements cannot be utilized due to their characteristic deterioration at cryogenic temperatures. A squeeze film damper [4] using a squeeze effect is one damping method, but liquid hydrogen has very low viscosity, so this damper cannot be used. Conventional turbopumps for rockets adopted wire mesh dampers [5] that use friction damping, but they have a short life span due to wear.

Therefore, the application of eddy current dampers (ECDs) is considered in this paper. Various structures of an ECD have already been proposed; a perpendicular-motion-type ECD comprising a ring magnet and conducting disk [6], a tuned inerter ECD using a rack and gear mechanism [7], a dual-sided hybrid excitation ECD [8]. In contrast, we have developed an ECD for turbopumps [9]. The ECD has a conductor mover with two disks, and the disks are sandwiched between ring-shaped permanent magnets (PMs). The multi-degree-of-freedom (multi-DOF) damping characteristics are improved by auxiliary magnets [10]. However, as a damper for rockets, the damping characteristics per mass still remain an issue.

To solve this problem, this paper focuses on the dual Halbach array (DHA) [11]. The effectiveness of the Halbach array in ECDs in one direction (radial or axial direction) at room temperature has already been investigated [12]. In addition, a structure for high-speed compressors with ECD using DHA has already been proposed, and its characteristics at room temperature in axial direction have been shown [13]. However, a DHA-type ECD for turbopumps has not yet been proposed. The change in multi-DOF damping characteristics at cryogenic temperatures has also not yet been investigated. In addition, variation characteristics in eddy current loss distribution at cryogenic temperatures have not yet been verified.

In this paper, we propose a novel DHA structure ECD with high damping density at cryogenic temperatures. The multi-DOF damping performance and eddy current distribution at different temperatures are also clarified. First, the evaluated structure is shown. Radial and axial damping characteristics are evaluated by a dynamic analysis using a finite element method (FEM). Then, changes in the damping characteristics and eddy current loss distribution as the temperature changes are shown. From the analysis results, it is clarified that the design structure with DHA can improve the damping characteristics.

2. Proposal of dual halbach array eddy current damper for turbopump

The DHA ECD with the auxiliary PMs (ECD-APM) is shown in Fig. 1(a), as the proposed structure. Figure 1(b) shows the DHA ECD without the auxiliary PMs. The mover consists of a cylindrical conductor with two disks. It has a hollow structure to arrange the turbopump shaft in the center. The stator consists of only PMs. The mover is supported by a mechanical spring and has three-degree-of-freedom in translation. The stator of the conventional ECD [9] consists of a yoke and PMs, as shown in Fig. 1(c). To show the structural difference, one side of the cross-section is shown in Fig. 2. Because of the axisymmetric structure, the y–z cross section is the same as the x–z cross section.

Fig. 1.

Evaluated ECDs.

Fig. 2.

One side of the cross-section.

The generating principle of the damping force is shown in Fig. 3. Figure 3(a)–(c) illustrates the generating principle of the eddy current in x–z cross section. The support configuration of the mover in the x–y section is shown in Fig. 3(d). The mover is supported by springs in the x-, y-, and z-directions, enabling the generation of damping force in three-degree-of-freedom. When a mover vibrates, a change in the magnetic flux through the mover generates an electromagnetic force in the opposite direction of motion, which is used for damping. The DHA concentrates the magnetic flux on the disk part of the conductor, which increases the amount of the magnetic flux variation. Therefore, the advantage of the DHA is that it does not require a back yoke, allowing it to be smaller and lighter with the same damping performance compared with the conventional structure.

Fig. 3.

Generating principle of the magnetic damping force.

3. Damping characteristics analysis

First, the analysis condition and design specification are described. Next, the analysis results of the damping characteristics are shown. The property comparison at some temperatures is investigated, and the structural comparison regarding the damping coefficient is shown.

3.1. Analysis conditions

This section describes the analysis condition. The damping characteristics are calculated by the two-dimensional (2D) FEM [14,15]. The outer diameter of the disk is 115 mm, the inner diameter of the moving conductor is 74 mm, and its length is 40 mm. The PM thickness is 3.5 mm. The elements and nodes of the 2D mesh model (see Fig. 4) are 23,086 and 11,617, respectively. The PM is a samarium cobalt magnet (Sm2Co17) that retains its magnetic force even at cryogenic temperature, compared with NdFeB magnets [16]. The BH curve used in the analysis is shown in Fig. 5. The detail of BH curve is described in the Appendix.

Fig. 4.

2D mesh model.

Fig. 5.

BH curve of the samarium-cobalt magnet.

The resistivities of aluminum at −250 °C, −100 °C, and 20 °C are 1.0 ×10⁻¹⁰, 1.4 ×10⁻⁸, and 2.8 × 10⁻⁸ Ω⋅m, respectively. An initial spring property is 7895.7 N/m. To verify the damping characteristics, the damping response is calculated with an initial velocity of 5 mm/s. The time step is 0.5 ms. The weight of the mover is set to 2 kg.

The axial motion analysis was performed as an axisymmetric model. For axial motion analysis, 2D analysis is possible under axisymmetric conditions. However, for radial motion analysis, 2D analysis cannot be performed under axisymmetric conditions. Therefore, a flat plate model is used for the radial 2D analysis. The model has the length of the cylindrical model when it is cut open. The analysis results are non-dimensionalized.

3.2. Analysis results and discussion

First, the comparison of the response with and without the DHA ECD-APM is shown in Fig. 6. The temperature was considered as −250 °C. It can be confirmed that the damper suppresses vibration in the radial and thrust directions. The maximum amplitude was reduced by 1/16 and 1/19 in the radial and axial directions, respectively. The results show that the ECD acts as a damping element.

Fig. 6.

Comparison of the response with and without the DHA ECD-APM.

Next, the damping characteristics of the DHA ECD-APM at −250 °C and −100 °C and 20 °C are shown in Fig. 7. Figures 6 and 7 are nondimensionalized within each figure, and the values that divide the raw data are different. From Fig. 7, it can be seen that the maximum amplitude decreases as the temperature decreases. This characteristic is the same in both radial and axial directions. At −100 °C, the vibration state was overdamped and underdamped at −250 °C.

Fig. 7.

Comparison of the response at some temperatures.

Finally, the comparison of the DHA ECD-APM, DHA ECD, and conventional ECD is shown in Fig. 8. The temperature was considered as −250 °C. The damping coefficient in the radial and axial direction are shown in Fig. 9. Damping coefficients were calculated by calculating log damping ratio and stiffness using time history data. The model with the best damping performance was the DHA ECD-APM. The convergence was fast while suppressing the amplitude, comparing the conventional ECD. This is due to the large magnetic flux density change in the conductor of the DHA ECD-APM, as shown in Fig. 10. On the other hand, convergence was slightly improved by the auxiliary PM, comparing the DHA ECD with and without the auxiliary PM. This is because the magnetic flux generated by the auxiliary PM interferes with the magnetic field of the DHA. To improve the damping force density per mass, one strategy is to omit the auxiliary PM, which has a mass of 0.093 kg.

Fig. 8.

Comparison of the damping characteristics between the evaluated ECDs.

Fig. 9.

Comparison of the damping coefficient.

Fig. 10.

Magnetic flux density distribution of the ECDs.

The conventional ECD requires a yoke, but the DHA ECD requires no yoke, resulting in a weight reduction of 1.24 kg, as shown in Fig. 11. In Fig. 11, the total weight of the PM and yoke is shown. The total mass of the stator in the conventional ECD is 1.83 kg, but the stator weight of the DHA ECD is 0.5 kg by eliminating the yoke. Therefore, the DHA ECD achieved a weight reduction of about 73%. In the actual design, the weight will increase slightly because the PMs of the DHA need to be attached to non-magnetic support parts to increase their strength. However, it is clarified that the high damping force density per mass can be achieved by the DHA in a cryogenic environment.

Fig. 11.

Comparison of the stator weight.

4. Eddy current loss distribution

A comparison of eddy current loss distribution at −250 °C and 20 °C is shown in Fig. 12. From the results, it can be seen that the eddy current losses are generated in the flux concentration area. At −250 °C, eddy current losses are concentrated on the conductor surface. This is due to the skin effect. The skin thickness is expressed by the following equation: $\begin{eqnarray}d=\sqrt{\frac{2}{{\pi}f{\mu}{\sigma}}}\end{eqnarray}$ (1) where f is frequency, 𝜇 is permeability, and 𝜎 is conductivity. At −250 °C, the skin thickness is considered to have become thinner because of the increase in conductivity. The same trend was observed in both radial and axial directions.

Fig. 12.

Eddy current loss distribution.

5. Conclusion

In this paper, the ECD with a dual Halbach array was proposed as a damping element for liquid hydrogen turbopumps. The damping characteristics were investigated by the FEM analysis, and it was found that the damping force was higher than that of the conventional ECD. It was also found that the vibration characteristics change when the temperature changes. In addition, it is shown that the eddy current loss in the eddy current damper is concentrated on the conductor surface at cryogenic temperatures. The damping performance will be verified through experiments using the prototype.

Footnotes

Acknowledgements

This work was supported by FOUNDATION OF PUBLIC INTEREST OF TATEMATSU.

Appendix

This section describes the interpolation of BH curves at −250 °C and −100 °C. Arnold Magnetic Technology provides BH curves of the samarium-cobalt magnet (Recoma 33E) at various temperatures (see Fig. 13(a)), and the data (−40 °C to 350 °C) is shown in material library of JMAG (commercial FEM software, JSOL Corp.) [17]. The BH curves at −250 °C and −100 °C were obtained using the values of residual magnetization, knee point, and endpoint (see Fig. 13(b)) at each temperature in the BH curve. In this paper, the minimum value of B in the provided data was defined as the endpoint. Specifically, it was divided into two parts: residual magnetization and knee point, and knee point to the endpoints, and linear interpolation was performed for each temperature. Figure 5 is the interpolated BH curve. The values of residual magnetization at temperature T were calculated as follows: (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H_{r}(T)=0\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{r}(T)=1.16+0.000406(20-T)\end{eqnarray}$ where H_r and B_r are magnetic field and magnetic flux density of residual magnetization. The values of knee point at temperature T are calculated as follows: (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H_{k}(T)=-2044805.7-5070.076(20-T)\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{k}(T)=0.00000124032\times H_{k}(T)+0.781886\end{eqnarray}$ where H_k and B_k are magnetic field and magnetic flux density of knee point. The values of endpoint at temperature T are calculated as follows: (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H_{e}(T)=-2100827.7-5208.982(20-T)\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{e}(T)=0.00000130647\times H_{e}(T)-0.0645117\end{eqnarray}$ where H_e and B_e are magnetic field and magnetic flux density of endpoint. −250 °C was chosen because it is the liquid hydrogen temperature, 20 °C is room temperature, and −100 °C was chosen as a value in between.

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