## 1 Introduction

The research on stabilizing devices both at home and abroad mainly concentrated in fin stabilizers [1,2,3], anti-rolling tank stabilizers [4,5,6], joint stabilizers, new technology stabilizers, and so on. Joint stabilizers and new technology stabilizers are hotspots of ship stabilizing technology research at the present stage. Kallstrom has demonstrated excellent performance in course control and stabilizing [7]. The stabilizing-anti-overturning integrated balance technique first adopted by INTERING company has been widely used in previous studies [8,9]. Chen and Lai have studied dual-tank stabilizers [10], and Zhao et al. have studied the theory of integrated balance system of large ship [11]. Park et al. suspended stabilization weight via the application of magnetic suspension technology, through the swaying parameters to control the stabilization weight, so as to change the center of gravity of structure and achieve stabilization [12]. Hirakawa et al. have designed a vertical moving stabilization weight on the boat model, by applying the Coriolis effect to achieve stabilization [13]. This study first puts forward another magnetic suspension stabilization technology, describes its working principle, establishes the mathematical model of the magnetic suspension stabilizer to carry out simulation, and then designs the magnetic suspension stabilizer and implements experiments.

## 2 Constitution and working principle of a magnetic suspension stabilizer

The constitution of a magnetic suspension stabilizer is shown in Figure 1. It consists of the iron core, the electromagnetic coil (stator), the stabilization mass block (rotor), and its control system, of which the stator is fixed on the ship and the rotor (stabilization mass block) is suspended between upper and lower stators in a stable state. Assuming that the ship is suffering from anticlockwise disturbance, anticlockwise rolling, the stabilization mass block will maintain its space position unchanged due to the inertia effect, so the position of the left and right stabilization mass blocks in the stator has changed relatively. The sensors detect the changing positions and send the changing positions to the controller by reference signals; therefore, the current of the stator changed, and then the stabilization mass blocks generate clockwise torque on the stator (hull deck) according to the theory of magnetic levitation. According to the theory of vibration, when the natural frequency of stabilization mass blocks is equal to or approximately the frequency of the disturbance, most of the vibration energy will be concentrated in the stabilization mass blocks, and the hull deck will not vibrate. The ship can be stabilized through the above two functions and *vice versa*.

## 3 Mathematical model and control

### 3.1 Mathematical model of the magnetic suspension stator and the stabilization mass block

In Figure 2, upper and lower images are magnetic suspension stators, where *M* is the quality of the stators, and the stabilization mass block is in the middle, where *m* is the quality of the block. When the mass block is stably suspended between stators, according to the theory of magnetic suspension stiffness coefficient [14], the displacement stiffness coefficient is calculated as follows:

*μ*

_{0}is the absolute permeability of vacuum,

*N*is the turns of the electromagnet coil,

*A*is the cross-sectional area of the iron core,

*x*

_{0}is the gap between the stabilization mass block and the damping mass stator. To overcome anti-rolling mass, the upper bias current should be higher than the lower bias current in order to overcome the weight of the stabilization mass block, namely

*k*. Assuming that the stator is suffering from upward disturbing force

*x*

_{2}(

*t*), the displacement is ignored due to less influence of damping on this system, and the equations of motion are obtained as follows:

The velocities of the magnetic suspension stator and the stabilization mass block can be expressed as below, and their simulation diagrams are shown in Figure 4.

The accelerations of the magnetic suspension stator and the stabilization mass block can be expressed as below, and their simulation diagrams are shown in Figure 5.

### 3.2 Mathematical model of the stabilizer and the ship

When the stabilizer is applied on the ship, its simplified mathematical model is shown in Figure 6, where *g* is the acceleration of gravity, *V* is the volume of displacement, *s* is the cross-sectional area at the ship draft, damping influence is ignored, and the equations of motion are obtained as follows:

The displacements of the stabilizer and the ship can be expressed as below, and their simulation diagrams are shown in Figure 7.

The velocities and accelerations of the stabilizer and the ship can be expressed as (4) and (5), and the acceleration simulation diagrams are shown in Figure 8.

*k*

_{1}= 8 × 10

^{5}N/m,

*k*

_{2}= 2 × 10

^{5}N/m,

*M*= 1,600 kg, m = 400 kg,

*F*= 100 N in Figures 7 and 8.)

### 3.3 Adaptive control based on frequency modulation

The displacement stiffness coefficient of the stabilization mass block is as follows:

Considering the aforementioned formulas and the simulation of two systems, it can be known when the coil number of turns, pole area size, and the value of the bias current or air gap size of the stabilization mass block will be changed, the displacement stiffness coefficient of the stabilization mass block will be changed, and the natural frequency of the stabilization mass block will be changed too [15,16,17,18,19]. When a device is assembled, the coil number of turns and the size of the magnetic pole cannot be changed easily. In this study, the displacement stiffness coefficient is changed by changing the value of the bias current and the size of the air gap. By changing the thickness of the gasket between the shell and the cover, the air gap size of the device is adjustable. The thicknesses of gaskets are 2, 3, and 4 mm, respectively, in the experiment device.

Through the reference adaptive control, the adaptive controller is driven by the adaptive control error which is a different value between the object output and the reference value. By adjusting the parameters of the controller, the errors of the system are reduced, eventually the actual output of the controlled object is adjusted to the desired output. The diagram of the control system is shown in Figure 9, where

From Figures 3–5, 7, and 8, it can be known that three frequency bands deserve our attention. In Figures 3(c), 4(c), 5(c), 7(b), and 8(b), the vibration of the stabilization mass block (stabilizer) is more intense in this frequency band, but the base (hull) hardly vibrates. It is known that this frequency band is near the natural frequency of the stabilization mass block (stabilizer). That is, when the natural frequency of the stabilization mass block (stabilizer) is equal to or approximately the frequency of the disturbance, the external interference force that is the vibration energy is basically concentrated on the stabilization mass block.

In Figures 3(b), 4(b), 5(b), 7(a), and 8(a), the vibration of the stabilization mass block (stabilizer) and the base (hull) is more intense in this frequency band. It is obtained that the frequency band is near the natural frequency of the base (hull). When the natural frequency of the base (hull) is equal to or approximately the external disturbance frequency, the stabilization mass block (stabilizer) and the base (hull) synchronously vibrate.

It can be seen from Figures 3(d), 4(d), 5(d), 7(c), and 8(c) that the stabilization mass block (stabilize) and the base (hull) are vibrated very violently and that the frequency band is nearly equal to the natural frequency of the whole system. The system resonates when the natural frequency of the whole system is equal to or approximately the external disturbance frequency.

## 4 Experiments

On the basis of theoretical calculation, the authors have designed a set of magnetic suspension stabilizers and implemented component processing and assembling. The component diagram and assembly diagram are shown in Figure 10. Stable suspension experiment is first carried out when the device is assembled. On the basis of the stable suspension, acceleration values of the stator and the stabilization mass block are, respectively, measured by loading 0–500 Hz sine voltage excitation signal on electromagnetic coil. The block diagram of the experiment is shown in Figure 11, and its specific parameters are shown in Table 1; field test and acceleration oscillograms are shown in Figures 12–15.

Parameter list of experiment

Parameter | Value |
---|---|

N | 138 |

Wire diameter | 1 mm^{2} |

Upper bias current | 1.5 A |

Lower bias current | 1.5 A |

Stator | 1259 g |

Rump mass | 225 g |

Gap | 0.5 mm |

Excitation amplitude | 500 mV |

Excitation frequency | 0–500 Hz |

Accelerometer | ADXL335B |

It can be seen from Figure 13 that the stabilization mass block and the base hardly vibrate. Figure 14 shows that the stabilization mass block vibrated very violently, and its acceleration value is approximately 0.94 g. Figure 15 shows that both the stabilization mass block and the base vibrated hardly, and their acceleration values are approximately 0.48 and 0.33 g.

## 5 Conclusions

- (1)According to the simulation and test data, when the natural frequency of the stabilization mass block is equal to or approximately the disturbance frequency of the outside, the base (hull) almost do not vibrate, and the external interference force that is the vibration energy is basically concentrated on the stabilization mass block.
- (2)When the disturbing force frequency changed, through the adaptive control algorithm, the natural frequency of the stabilization mass block can be adjusted to be similar to the outside disturbance frequency. Then the system will have a very good stabilizing effect.
- (3)When the magnetic suspension stabilization technology is applied on the ship, as long as the natural frequency of the stabilization mass block is equal to the wave frequency, it will have a good stabilizing effect regardless of the ship at the zero speed, low speed, and high speed.

This project was supported by the National Natural Science Foundation of China (Grant No. 51675261) and the Natural Science Foundation of Jiangsu Province (BK20161102).

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