Design and Test of a Magnetorheological Damper of a Multi-Layered Permanent Magnet

Abstract

To effectively suppress spindle vibrations in rotating machinery, magnetorheological (MR) dampers, as an ideal vibration control device, have attracted attention. To enhance the vibration damping effect, in the paper, a MR damper vibration with a multi-layered permanent magnet as the magnetic source is designed, and the self-made magnetorheological fluid is used as the damping medium. The mechanical properties of the MR damper were obtained through testing and calculation. On this base, both simulation and experimental methods are used to demonstrate the effectiveness of the multi-layered permanent-magnet MR damper. The simulation results show that the critical speed increases greatly for the first four modes. The experimental results show that the Y-direction displacement decreases greatly, especially at 1800 rpm and at 3400 rpm, after applying the MR damper. The vibration displacement at 1× frequency shows a 69.74% reduction at 2600 rpm and a 65.69% reduction at 3200 rpm in the Y-direction after applying the MR damper. The effectiveness of the multi-layered permanent magnet MR damper in rotor vibration suppression was confirmed.

1. Introduction

During operation of large rotating machinery, the mechanical unbalance caused by self-weight of machinery and magnetic pull unbalance appears, which severely interferes with normal unit operation [1,2,3], and dampers are required to suppress shaft vibration. Traditional passive dampers, such as hydraulic viscous dampers, solid viscous dampers, and friction dampers, have the problems of high energy consumption and time lag. In contrast, MR dampers have attracted attention due to their faster response speed and lower power consumption [4].

The MR damper, as a new type of intelligent vibration control device, contains magnetorheological fluid (MRF), which can undergo a rapid transformation from a liquid state to a semi-solid state under an applied magnetic field, and the process is continuous and reversible. Replacing traditional passive control components with MR dampers can effectively adjust system damping and stiffness, making them an ideal vibration control device [5]. The performance of MRFs plays a crucial role in the effectiveness of MR dampers. MRF is a suspension formed by dispersing micron-sized magnetic particles into a base liquid along with appropriate additives. The increase in temperature will make the viscosity of the MRF reduce, which is conducive to the stable operation of the rotor at low rotational speeds [6]. Sometimes the temperature will bring adverse effects and, at this time, it is necessary to increase the cooling system. It is of note that in the absence of a magnetic field, it behaves as a freely flowing Newtonian fluid. When a magnetic field is applied, the randomly distributed magnetic particles align into chain-like structures along the field direction, causing a sharp increase in the fluid’s viscosity [7] and exhibit Bingham plastic behavior similar to a semi-solid [8,9]. This magnetic field-controlled rheological property has the characteristics of low energy consumption [10], rapid response [11], and continuous apparent viscosity adjustment [12], which provides the foundation for the application of MR dampers in damping [13], braking [14], ferrofluid pump [15], polishing [16], and sealing [17].

MR dampers represents a primary application [18,19], which are widely applied in suspension systems and rotor systems. British professor Jeffcott proposed the Jeffcott rotor model, which subsequently garnered widespread attention from researchers [20]. It was demonstrated by Wang Jianxiao et al. [21] that the amplitude of the rotor system reduced greatly after applying an MR damper, in addition to adjusting the magnetic field intensity. It was shown by Zhao et al. [22] that after applying the MR dampers, the system’s vibration amplitude decreased from 0.0049–0.0055 m to 0.0014–0.0016 m, achieving a 70% reduction. It was reported by Zhang [23] that the amplitude variation of a rotor with a diameter of 1000 mm decreased from 6.5 × 10⁻³ m to 1.06 × 10⁻³ m after applying the MR damper. In general, these findings all indicate that the MR dampers effectively reduce vibration.

With the development of MR dampers, they can be categorized into coil-type [24] and permanent-magnet-type [25] MR dampers based on the magnetic field source. The damping force of coil-type MR dampers is regulated by varying the current in the electromagnetic coil. The force is affected by the magnetic field strength of the permanent-magnet-type MR dampers. For coil-type MR dampers, it was demonstrated by Wang et al. [26] that the amplitude of the rotor system can be reduced by increasing the current. However, when the current exceeded a certain range, the damper exhibited nonlinear vibrations. It was found by Ma et al. [27] that after applying the MR damper, the amplitude decreased effectively by 64.37% at the first critical speed. However, during the increase in current, the rotor system exhibited unstable and the nonlinear vibration behavior. Moreover, coil-type MR dampers will fail due to power interruption, which may result in severe accidents.

Therefore, permanent-magnet-type MR dampers attract widespread attention for their higher safety. It was reported by Liu et al. [28] that the damping force of an MR damper increased from 0.56 kN to 2.32 kN by embedding permanent magnets with different arrangement angles to alter the magnetic field direction and strength. It was reported by Kim et al. [29] that the magnetization zone was adjusted by modifying the MRF housing shape in a permanent-magnet MR damper, and the highest damping force of 28 N was obtained. It was shown by Khedkar et al. [30] that the damper force of the MR damper under the magnetic field of 7000 G is higher than that of 2000 G and 4000 G. These studies confirm that the rotor system vibrations can be effectively controlled by permanent-magnet MR dampers, and the factors such as magnetic pole orientation, housing geometry, and magnetic field strength all will significantly affect the damper force.

To obtain a permanent-magnet-type MR damper with simple design and strong vibration reduction effect, in this paper, a MR damper with a multi-layered permanent magnet is designed. The magnetic field strength increases by increasing the number of layered permanent magnets, and, thus, the damper force increases. The vibration reduction effect of the damper designed in this paper is verified through simulation and experiments.

2. Damping Strategies for Vibration Attenuation

Damping systems can be categorized into passive dampers, semi-active dampers, active dampers, and hybrid dampers based on different vibration control methods. However, these damping systems exhibit variations in power capacity and size constraints. To address this issue, the paper compared their key parameters such as motor power and dimensions, as summarized in the following

Table 1. The different damping systems and motor characteristic range.

Damper System	Damping Device	Applicable Motor Characteristic Range
Passive dampers	Hydraulic damper	Size range 50–500 mm Power range > 100 kW
	Rubber ring	Size range 1–500 mm Power range 1–10 kW
Semi-active damper	Current flow damper	Size range 5–150 mm Power range 1–100 kW
Active dampers	Piezoelectric actuators	Size range 5–200 mm Power range < 1 kW
Hybrid dampers	The piezoelectric actuator is combined with the rubber ring	The size can be adjusted by the piezoelectric actuator and rubber ring Power range > 100 kW

. These include hydraulic dampers, electrorheological dampers, piezoelectric actuators, and rubber rings

MR dampers can be divided into semi-active dampers and passive dampers according to the magnetic field control mode, and can also form hybrid dampers by combining with hydraulic dampers. They are suitable for a wide range of motor sizes and power ranges, making them a better vibration reduction solution. Therefore, this paper designs a permanent-magnet-type passive MR damper. Meanwhile, a rotor system with a power of 3.7 kW, a shaft diameter of 10 mm, and a shaft length of 820 mm is designed for vibration reduction experiments. This damper is suitable for low-power rotor systems, but the size and magnetic field strength of the designed MR damper can be adjusted according to actual motors, making it adaptable to rotor systems with larger power ranges and sizes, such as hydroturbines, hydrogenerators, and other mechanical equipment.

3. Design of MR Damper and Its Mechanical Properties Experiment

3.1. The Structure and Parameters of the MR Damper

The structural and physical diagram of the MR damper is shown in Figure 1. As illustrated, two moving damping plates and one stationary damping plate are contained in the MR damper The bearing is welded to the moving damping plates, while the stationary damping plate is fixed at the base. The moving damping plates are positioned at 5 mm above the base, with 1 mm gaps between the moving damping plates, stationary damping plate, and housing walls. These gaps are filled with MRF. During vibration of the rotating shaft, the MRF undergoes shear motion, and a damping force generates under a magnetic field. The structural parameters of the MR damper are listed in

Table 2. Structural parameters of MR damper.

Structure	Parameters
Dynamic damping plate	Length: 37.5 mm; width: 22 mm; thickness: 2 mm
Static damping plate	Length: 27 mm; width: 28 mm; thickness: 2 mm
Permanent magnet	Length: 24 mm; width: 33 mm; thickness: 5 mm
Septum	Length: 24 mm; width: 33 mm; thickness: 4 mm
Bearing	Outer diameter: 8.5 mm; inner diameter: 5 mm

.

The performance parameters of the MRF are listed in

Table 3. Performance parameters of MRF.

Physical Quantity	Parameters
Particle size	5 μm
Mass fraction of particles	60 wt.%
Density of coated CIP	5.036 g/cm3
Zero-field viscosity (25 °C)	19.47 Pa·s
Field-induced shear yield stress (175 kA/m, 25 °C, 1000 s−1)	18.95 kPa

. The main steps for preparing magnetorheological fluid (MRF) are as follows. Firstly, the carbonyl iron particles (CIP) are activated with dilute hydrochloric acid, and the concentration of dilute hydrochloric acid is 0.5 mol/L. Secondly, the activated particles are coated with a silane coupling agent (Dodecyltrimethoxysilane), and the mass ratio of silane coupling agent to particles is 7:24. Finally, the coated particles are dispersed in polyalphaolefin synthetic oil to prepare the MRF, and the mass fraction of particles is 60 wt.%. The obtained MRF has good dispersion stability. For more detailed procedures, please refer to References [11,12], our published papers.

Figure 2 shows the schematic diagram of the vibration reduction system of the MR damper. As illustrated, the rotor system consists of a flexible shaft, balancing disc, electric motor, MR damper, and sensors. The parameters of each component in the rotor system are listed in

Table 4. The parameters of each component of the rotor system.

System Device	Parameters
Pivot	Diameter: 10 mm; length: 820 mm
Counterweight plate 1	Diameter: 74 mm; thickness: 25 mm; mass: 500 g
Counterweight plate 2	Diameter: 74 mm; thickness: 15 mm; mass; 800 g
Motor	Rotational speed: 0~10,000 r/min
Displacement sensor	Diameter: 5 mm; sensitivity: −8 V/mm; range: 0~10,000 Hz
Tachometer sensor	Test range: 1~20,000 r/min

. The electric motor drives the main shaft rotation, motor speed signal is received by the rotational speed sensor, and the vibration signal is acquired by the displacement sensors. The signals are collected and recorded through the DHDAS dynamic signal acquisition system.

3.2. The Test and Calculation of Mechanical Properties of the MR Damper

Figure 3 shows the test bench of mechanical properties. On this test bench, the mechanical properties of the MR damper are tested by varying external excitations. The vibration displacement is set within the range of −10 to 10 mm, with vibration velocities ranging from 0.05 to 0.60 m/s. A computer system is employed to acquire data including damping force, displacement, and velocity.

Figure 4 presents the mechanical properties of the MR damper. As shown in Figure 4a, as the vibration displacement x varies from −10 mm to 10 mm, the damping force of the MR damper first increases and then decreases, and it reaches the maximum value when the x is 6 mm. Figure 4b demonstrates that the damping force gradually increases with increasing shear velocity, and approaches a maximum value of about 70 N when the shear velocity reaches 0.5 m/s. Based on these results, the damping coefficient k and stiffness coefficient c of the damper can be calculated using Equations (1) and (2) [31].

$$k={\displaystyle \frac{F_d}{x}}$$

(1)

where k is the stiffness coefficient, F_d is the damping force, and x is the displacement of the damping plate.

$$c={\displaystyle \frac{F_d}{\upsilon }}$$

(2)

where c represents the damping coefficient, F_d denotes the damping force, and υ indicates the velocity of the damping. Based on the formula calculations, the stiffness coefficient k₁ is calculated as 70 N/mm, and damping coefficient c₁ is calculated as 0.12 N·s/mm, which are indispensable for the later subsequent damper simulation.

3.3. Simulation and Experiment of Vibration Reduction System of MR Damper

3.3.1. Simulation of Vibration Reduction Effect of MR Damper

ANSYS software (2021 version) was used to simulate the vibration reduction effects of MR dampers in the rotor system. The dimensional parameters of the 3D model of the rotor system are listed in

Table 5. The dimensional parameters of the 3D model of the rotor system.

Character Radical	Parameters
Spindle diameter	10 mm
Spindle length	820 mm
Counterweight disk diameter	74 mm
Counterweight disk length	5 mm

. The simulation of three major components, such as the main shaft, the magnetorheological (MR) damper, and the supports, are included in the entire simulation of the rotor system. The material of the three components is selected as structural steel, and the density is 7850 kg/m³. For the simulation of the damper, the stiffness coefficient k₁ calculated as 70 N/mm and damping coefficient c₁ calculated as 0.12 N·s/mm need to be substituted. For the simulation of the fixed supports (as shown in Figure 2), the bearing constraint command was used, and to simplify analysis, the stiffness coefficient k₂ and the damping coefficient c₂ also needs to be substituted; the value of k₂ is 65 N/mm and the value of c₂ is 1 × 10⁻⁴ N·s/mm, which are selected according to the reference [32].

3.3.2. Experiment Procedures of Vibration Reduction Effect of MR Damper

The vibration reduction effect experiment of the MR damper was conducted with and without the MR damper using the rotor system, as shown in Figure 2. Before the experiment, the test system and sensors were first calibrated. After the calibration, the rotational speed of the shaft varied from 1000 rpm to 5000 rpm, at 400 rpm intervals. The sampling frequency was set at 1000 Hz with a sampling duration of 60 s. The shaft rotational speed, and shaft displacements in both X-direction and Y-direction, were recorded by the DHDAS.

4. Simulation and Experiment of Vibration Reduction Effect of Rotor System

4.1. The Analysis of Simulation Results

Figure 5 and Figure 6 show the first six intrinsic frequencies and mode shapes of the rotor system without applying the MR damper and with applying the MR damper, respectively. The critical speeds and intrinsic frequencies of the rotor system with and without applying the MR damper are presented in

Table 6. Critical speed and intrinsic frequency of rotor system with or without MR damper.

Variable	Modal Order	Rotate Direction	Critical Speed (rpm)	Intrinsic Frequency (Hz)
Undamped	1	BW	2452.5	40.969
	2	FW	2463.7	40.970
	3	BW	3437.7	57.404
	4	FW	3450.8	57.405
	5	BW	-	146.700
	6	FW	-	146.700
Damping	1	BW	3104.7	51.768
	2	FW	3107.5	51.768
	3	BW	3637.0	60.724
	4	FW	3649.9	60.727
	5	BW	-	143.760
	6	FW	-	143.760

. As shown in Figure 5 and Figure 6 and Table 6, after applying the MR damper, the intrinsic frequencies of the first four modes are higher than those without the MR damper, indicating the increase in the system’s stiffness and damping force; however, for the fifth and sixth modes are close to that without the MR damper, attributed to the MR damper introducing energy dissipation to the rotor system, thereby attenuating its vibrations [33]. Moreover, resonance occurs when the system’s rotational frequency approaches or coincides with its intrinsic frequency; the rotational speed of the shaft at this intrinsic frequency is the critical speed [34]. From Table 6, it is revealed that for the first four modes, these critical speeds all increase greatly, indicating the effectiveness of the MR damper in vibration reduction.

4.2. The Analysis of Experimental Results

Axis loci appear at different critical speeds of the rotor system, as shown in Figure 7. As shown in Figure 7, no significant difference in the peak-to-peak value was observed along the X-direction before and after applying the MR damper. All axis loci exhibit non-circular patterns, indicating the asymmetric stiffness in the elastic supports. However, the axis loci exhibit approximate circular shapes after applying the MR damper. Moreover, as shown in Figure 7a–c, a significant decrease trend of the peak-to-peak value was observed along the Y-direction before and after applying the MR damper, indicating that the vibration reduction effect of this magnetorheological (MR) damper is only effective in the Y-direction. As shown in Figure 7d, the Y-direction displacement has not changed significantly after applying the MR damper, attributed to the decrease in yield stress of MRF during vibration. It can be seen from the results that rotor system vibrations effectively reduce after applying the MR damper at critical speeds [35].

Figure 8 shows the peak-to-peak values of the rotor system at different speeds with and without applying the MR damper. As shown in Figure 8a, no significant difference in the peak-to-peak value was observed along the X-direction before and after applying the MR damper. In Figure 8b, as the rotational speed increases, it shows that the vibration peak-to-peak value has an obvious trend of significant decrease. The Y-direction displacement decreases from 40.13 μm to 28.08 μm after applying the MR damper at 1800 rpm. At 3400 rpm, the Y-direction displacement decreases from 47.5 μm to 26.4 μm, a 44.42% reduction. These results demonstrate that MR dampers can effectively suppress rotor system vibrations.

Figure 9 shows the waterfall plots of rotor vibration displacement at different speeds. As seen in Figure 9, during the rotor vibration at various rotational speeds, multiple vibration frequencies generate. Only the first two vibration frequencies are typically considered, and the amplitudes at other vibration frequencies are very small and it is usually not given key attention. The 1× frequency of rotor vibration is related to rotor unbalance, while the 2× frequency is associated with initial shaft bending [36]. By comparing Figure 9a,c, it can be seen that the vibration frequency characteristics in the X-direction show no significant changes after applying the damper. By comparing Figure 9b,d, it can be seen that vibration displacement at 1× frequency decreases from 9.98 μm to 3.02 μm after applying the MR damper, representing a 69.74% reduction, in the Y-direction, at 2600 rpm. The vibration displacement at 1× frequency decreases from more than 11.89 μm to 4.08 μm after applying the MR damper, representing a 65.69% reduction, in the Y-direction, at 3200 rpm. Overall, the amplitude at 1× frequency becomes smoother than without the MR damper. In addition, the amplitude at 2× frequency also reduced significantly, especially in the speed range of 850–1800 r/min. These findings indicate that MR damper effectively reduces vibrations caused by rotor unbalance and initial shaft bending, especially in the Y-direction.

5. Conclusions

In the paper, a MR damper with multi-layered permanent magnets was designed. The mechanical properties of the MR damper were obtained through testing and calculation. The stiffness coefficient k₁ is calculated as 70 N/mm, the damping coefficient c₁ is calculated as 0.12 N·s/mm. The simulation results show that the critical speeds all increase greatly for the first four modes. The experimental results show that the Y-direction displacement decreases from 40.13 μm to 28.08 μm at 1800 rpm and decreases from 47.5 μm to 26.4 μm at 3400 rpm after applying the MR damper. The vibration displacement at 1× frequency shows a 69.74% reduction at 2600 rpm and a 65.69% reduction at 3200 rpm in the Y-direction after applying the MR damper. Both simulation and experimental results demonstrate the effectiveness of the multi-layered permanent-magnet MR damper. Moreover, the layered permanent magnet structure demonstrates simplicity, safety, and reliability. The design of this layered permanent-magnet MR damper provides important theoretical foundation and practical value for spindle vibration control strategies.