Performance Analysis of Magnetorheological Damper with Folded Resistance Gaps and Bending Magnetic Circuit

Abstract

The traditional magnetorheological (MR) damper subject to the limited space has shortcomings such as small damping force, narrow dynamic range and low adaptability. In this study, a new MR damper with folded resistance gaps and bending magnetic circuit was proposed for improving the damping performance. The length of the resistance gap was increased by configuring the multi-stage folded annular gap structure, and the magnetic circuit was established to activate the non-flux region. The mathematical model was established for the MR damper to analyze the damper force, magnetic circuit and dynamic performance. Subsequently, the finite element analysis (FEA) methodology was utilized to investigate the changes of magnetic flux densities in the folded resistance gaps. The test rig was setup to explore and verify the dynamic performance of the proposed MR damper under different excitation conditions. The results indicate the maximum damping force is approximately 4346 N at the current of 1.5 A, frequency of 0.25 Hz and amplitude of 7.5 mm. The damping force and dynamic range of the proposed MR damper are enhanced by 55.82% and 62.21% compared to that of the traditional MR damper at the applied current of 1.5 A, respectively, thus highlighting its high vibration control ability.

1. Introduction

Magnetorheological (MR) damper is widely employed for semi-active control in the vibration isolation systems, including the automobile [1,2,3,4], civil structures [5,6,7], aerospace [8,9,10] and prosthetic knees [11,12]. With the MR fluid as the working medium, the MR damper efficiently achieves the continuous damping control, fast response speed, and low energy consumption [13,14,15,16].

The structural configuration is an important property that affects the damping characteristics of the MR damper [17,18]. However, the traditional MR damper is unable to provide large damping forces under dimensional constraints due to the simple structure with low utilization rate of magnetic field and gap. Facing this challenge, strides have been made continuously on the structural design of the MR damper mainly focusing on the improving the magnetic supply system [19,20,21,22,23], selecting the working mode of the MR fluid [24,25,26] and optimizing the flow channel layout [27,28,29,30,31,32,33,34].

Improving the magnetic supply system includes the arrangement of multiple coils, the design of the embedded permanent magnet and the optimization of the magnetic circuit distribution. Bai et al. [19] developed an MR damper with five excitation coils equipped at the inner bypass valve of the concentric tube and led to six damping gaps. This design increased the dynamic range and reduced the field-off damping force over the operational velocity range of 0–5.2 m/s. Aydar et al. [21] designed an MR damper in which a permanent magnet was embedded. The simulation results show that the output damping force of the MR damper could be increased by 100% through enhancing the magnetic field. Cheng et al. [22] presented an MR damper featuring a meandering magnetic circuit with the damping force range increased by 1.8 times than that of the conventional MR damper. The experimental results show that this design could provide damping force as high as 3400 N with an excitation velocity of 0.0628 m/s. Idris [23] designed an MR damper with a serpentine flux external concentric bypass valve with the dynamic range increased by 1.5 times higher than that of the traditional structure.

The working mode of the MR damper contains the flow mode, the shear mode and the squeeze mode. Yazid et al. [24] suggested an MR damper combined with squeeze and shear modes, experimentally compared the damping performance in different conditions. The comparison results show that the mixed mode could obtain higher damping force than the single mode MR damper. Mughni et al. [25] proposed and tested a novel MR damper combining the flow and shear flow modes in series. The MR damper can obtain higher force by increasing the number of the effective areas. Ruan et al. [26] developed an MR damper using the squeeze flow mode. The experimental results show that the maximum damping force was approximately 6.5 kN.

Considering the flow channel types, the structure of the MR damper can be classified into the axial flow, radial flow and axial-radial hybrid flow. Bai et al. [29] modeled and tested the annular-radial-duct MR damper. Results show that this design could provide a large damping force of 3149 N with an excitation velocity of 0.19 m/s. In addition, a large damping force range (140–3149 N) was achieved under a corresponding excitation velocity range (0.025–0.19 m/s). Wang et al. [30] designed an MR damper with bifold valve as inner bypass, which had the axial and radial damping channels. The maximum damping force reaches 3600 N at the current of 1.5 A. However, the radial gap increases the piston radius to a certain extent, leading to the excessive radial size of the MR damper. Kim et al. [32] introduced a bifold flow mode MR damper with two axial flow gaps in the piston connecting through the feedback hole. The experimental damping force reaches 3150 N, and the equivalent damping force was more than twice that of the traditional MR damper. Hu et al. [34] proposed, manufactured and tested an MR damper with serial-type flow channels, which had multiple axial annular channels. The maximum positive damping force reaches 5486 N and the dynamic adjustable range reaches to 7.

In traditional structural designs, obtaining larger damping forces requires increasing the length of piston, layout of radial flow channels, internal or external bypass arrangement, multiple exciting coil and embedding a permanent magnet. However, these methods increase the axial or radial dimensions of MR dampers, while causing challenges for obtaining much more superior damping characteristics under space constraints. In addition, the multiple exciting coil and embedding permanent magnet increase the complexity of the MR damper. Regarding to the compact configurations, some designs are widely used, such as the layout of meandering magnetic circuit and folded flow channels.

There is research using meandering magnetic circuit, but the method of serpentine flux path is almost always used in MR brakes and MR dampers with a bypass valve. For extending the effective gap length, the layout of folded flow channels needs further exploration. Combining the above two designs should be considered to achieve better performance, such as combining the axial folding flow channel and bending the magnetic circuit inside the piston. This is rarely considered in the above literature, wherein the research on MR dampers using the meandering flow and flux path method is not available.

In this work, a new MR damper configuration with folded resistance gaps and bending magnetic circuits was proposed, aiming to obtain a larger controllable damping force subject to the limited volume constraint. This configuration combines the characteristics of the folded resistance gaps and bending magnetic circuit, both of which were integrated in the damper piston instead of the bypass. The mathematical model of the MR damper was correspondingly established, and the electromagnetic properties were analyzed using the finite element method. Then, the damping force performance of the proposed MR damper was evaluated under different current values, and the simulated results were naturally compared with the traditional configuration. The test rig was established to validate the modeling results.

2. Design of the Proposed MR Damper

2.1. Structural Principle and Design

Figure 1 shows the structural principles of the traditional MR damper and the MR damper proposed in this work. The flow gap in which the magnetic circuit passes through vertically is the effective gap. As shown in Figure 1a, the magnetic circuit divides the flow gap of a traditional MR damper into two parts: two short effective gaps and a long non-flux gap. The controllable damping force rises accordingly with the rising length of effective gap. This design has both serpentine flow gaps and bending magnetic circuit, as shown in Figure 1b. The folded resistance gaps increase the length of the effective damping gap, while shortening the axial size of the piston. Compared to the MR damper with the radial damping gap, the radial dimension of the damper is also significantly decreased. In addition, the bending magnetic circuit activates the non-flux gap of the traditional extra, and increases the length of the second outer effective gap. This design improves the activation rate of the flow gap, and increases controllable damping force and dynamic range.

There are some essential differences between this configuration and others. The proposed MR damper improves in terms of both the distribution of the magnetic circuit and the flow path; however, the other dampers only have improvement in terms of one. A typical example is the MR damper with a serial-type flow path proposed in [32]. In addition, compared with the above damper, the proposed damper structure provides less non-flux gap and more coil turns.

Figure 2 illustrates the structure of the proposed MR damper with folded resistance gaps and bending magnetic circuits. This double-ended MR damper mainly composed of two piston rods, two end covers, cylinder, piston assembly and end shield. The piston assembly includes the inner magnetic sleeves, middle magnetic sleeves, outer magnetic sleeves, exciting coil, non-magnetic ring, magnetic semicircles, non-magnetic disks and magnetic core. The magnetic core and the magnetic sleeves are made of No. 10 steel. The non-magnetic ring, non-magnetic disks, end covers, cylinder and piston rods are made of non-magnetic materials of stainless steel. The inner gap is composed of a magnetic core and an inner magnetic sleeve. The middle gap is composed of an inner magnetic sleeve and a middle magnetic sleeve. The first outer gap is composed of a middle magnetic sleeve and an outer magnetic sleeve. The second outer gap is formed between the outer magnetic sleeve and the magnetic semicircle. The inner, middle and outer gaps are axial gaps. There are four convexities at the opposite end of the inner magnetic sleeves and the middle magnetic sleeves, which are used to form the radial gaps. These axial gaps and radial gaps are connected in series to form a folded resistance gap. In addition, the grooves and convexities of each part of the piston assembly are matched to ensure the uniform thickness of the flow gap. The two magnetic semicircles are external magnetic elements of the excitation coil, which is designed as two semicircular blocks to facilitate the winding of the excitation coil.

When the MR damper works, the piston motion makes the MR fluid in the compression chamber squeeze into another chamber through the flow gap. When the current acts on the exciting coil, the MR effect of the MR fluid is produced, and the controllable damping force is established in the folded resistance gaps.

2.2. Magnetic Circuit of the Developed MR Damper

Figure 3 displays the magnetic circuit diagram of the proposed MR damper. The magnetic sleeves and non-magnetic ring are strategically placed on both sides of the outer gap, which forces the magnetic flux line to bend and pass through the outer gap four times. The magnetic flux line passes through the magnetic core, inner gap, inner magnetic sleeve, middle gap, middle magnetic sleeve, first outer gap and outer magnetic sleeve in sequence. The magnetic flux line moves axially in the outer magnetic sleeve, while it is bent by the non-magnetic ring. The magnetic flux line returns to the magnetic core through the second outer gap, the magnetic semicircle, and the symmetrical path at the other end of the piston assembly to form a closed magnetic circuit. Hence, the MR fluid with most of the length in the outer gap can be activated.

According to the Kirchhoff’s law, the magnetomotive force of the magnetic circuit gives:

$$N_{c}I={\displaystyle {\oint }_c\mathit{H}\mathit{d}\mathit{l}=}{\displaystyle \sum _{i=1}^n{H}_i{l}_i}$$

(1)

where N_c is the number of turns of exciting coil, I is the applied current, H_i and l_i represent the magnetic field intensity and the effective length of part i in magnetic circuit, respectively.

The total magnetic flux is expressed as:

$$\Phi ={\displaystyle \oint \mathit{B}\mathit{d}\mathit{S}=B_i{S}_i}$$

(2)

where B_i and S_i represent the magnetic flux density and cross-sectional area of part i in magnetic circuit, respectively.

The relationship between magnetic flux density B_i and magnetic field intensity H_i is given by:

$$B_i={\mu }_o{\mu }_i{H}_i$$

(3)

where μ_o is the air permeability and μ_i is the relative permeability of magnetic materials in each part.

The effective components magnetoresistance in the equivalent closed magnetic circuit can be expressed as:

$$R_i=\frac{l_i}{{\mu }_o{\mu }_i{S}_i}$$

(4)

According to the magnetic flux direction shown in Figure 3, the magnetic circuit can be equivalent to a series circuit, and the total reluctance is equal to the sum of the reluctance of each part. Since the reluctance is symmetrically arranged at both ends of the piston assembly, the total reluctance R_m can be expressed as:

$$R_m=R_1+2{\displaystyle \sum _2^9{R}_i}+R_{10}$$

(5)

Combining Equations (1)–(4):

$$N_{c}I={\displaystyle \sum _{i=1}^n\frac{B_i}{{\mu }_o{\mu }_i}{l}_i}={\displaystyle \sum _{i=1}^n\frac{l_i}{{\mu }_o{\mu }_i{S}_i}\Phi }={\displaystyle \sum _{i=1}^n{R}_i\Phi }$$

(6)

The magnetic flux density B_i of each part of the magnetic circuit is:

$$B_i=\frac{\Phi }{S_i}=\frac{N_{c}I}{S_i{\displaystyle \sum _{i=1}^n{R}_i}}$$

(7)

2.3. Design of Main Structural Dimensions

In this paper, the design of the MR damper with folded resistance gap and curved magnetic circuit is to change the magnetic field structure to make the magnetic flux line pass through the multi-stage meandering damping flow channel; consequently, the design of the flow channel is the design focus of this research. Since the shear yield stress that can be achieved by each effective damping gap in the multi-stage serpentine flow channel is closely related to the size of the damper, in order to make each effective damping gap reach the saturated shear stress under the same current and time, as far as possible, we make the magnetic flux densities generated in the flow channel approximately equal, and then it can be deduced that the cross-sectional area of each effective damping channel is also equal. After that, the structural parameters of the damper can be preliminarily determined according to the external dimensions of the traditional damper and fully considering the installation space of the damper and the strength of external parts such as the damper cylinder and piston rod. After that, considering the reliability of the assembly of each part, the main parameters of the proposed MR damper can be obtained.

Table 1. Main structural dimensions of the proposed MR damper.

Parameter	Value (mm)	Parameter	Value (mm)
Inside radius of piston rod (ro)	7	Length of first outer gap (L1)	8
External radius of core (rc)	13	Length of second outer gap (L2)	7
Radius of piston (rp)	31	Length of non-magnetic ring (ta)	13
Radius of inner gap (ri)	19	Thickness of non-magnetic disk (tb)	3
External radius of piston rod (rd)	10	Thickness of sleeve (tc)	3
Thickness of gap (g)	1	Height of exciting coil (Hc)	11

lists the main structural dimensions of the proposed MR damper. The ratio of the effective damping gap length to the total gap length is defined as the MR gap activation rate. The MR gap activation rate of the proposed MR damper is 66.6% theoretically, which is 21.8% higher than that of the traditional MR damper with the same dimensions. The magnetic circuit achieves the dual effect of increasing the damping force and broadening the dynamic range.

3. Mathematic Modeling of the Proposed MR Damper

The MR damper adopts the flow mode as the working mode, and its damping force mainly comes from the viscous force and yield stress of the MR fluid flowing in the gap. Therefore, the total damping force can be expressed as:

$$F_d=(\Delta P_{\eta }+\Delta P_{\tau })A_p$$

(8)

where Δp_η is the viscosity pressure drop, Δp_τ is the field-dependent pressure drop due to the yield stress of the MR fluid flowing through the gap, A_p is the effective area of the piston, it can be expressed as:

$$A_p=\pi r_p{}^2-\pi \left[{\left(r_i+g\right)}^2-r_i{}^2\right]-\pi r_d{}^2$$

(9)

where r_p is the radius of piston, r_i is the radius of inner gap, r_d is the external radius of piston rod respectively, g is the thickness of gap.

The total pressure drop of the gap includes the field-dependent pressure drop and the viscosity pressure drop. The field-dependent pressure drop is caused by the MR effect of the MR fluid under the shear yield stress. Similarly, the viscosity pressure drop is caused by the MR fluid flow, which is changing basically with the viscosity and flow rate of the MR fluid [27].

Due to the radial gaps being very short, and the fact that the flow direction of the MR fluid in the gap is parallel to the magnetic flux line, only an insignificant damping force is generated, which can be ignored. Considering solely the axial gaps, three-stage gaps could be identified at the inner, middle and outer regions. The pressure drop of each gap can be expressed as:

$$\Delta p_{gap}=\Delta p_{\eta }+\Delta p_{\tau }=\frac{6\eta ql}{\pi g^{3}r}+\frac{c\tau l_e}{g}$$

(10)

where η is the viscosity of the MR fluid without magnetic field; q is the volumetric flow rate of the MR fluid through the flow gap; l is the length of the flow gap; r is the inner radius of the annular flow gap; c is the correction factor, with a value of 2–3; τ is the shear yield strength of the MR fluid in the flow gap, which is related to the magnetic flux density; le is the length of the effective flow gap.

Accordingly, the pressure drop of the inner annular gap can be expressed as:

$$\Delta p_i=\frac{6\eta q{L}_1}{\pi g^3{r}_i}+\frac{c{\tau }_i{L}_1}{g}$$

(11)

where τ_i is the shear yield strength of the MR fluid in the inner annular gap.

Similarly, the pressure drop of the middle annular gap can be expressed as:

$$\Delta p_m=\frac{6\eta q{L}_1}{\pi g^3{r}_m}+\frac{c{\tau }_m{L}_1}{g}$$

(12)

where r_m is the radius of the middle annular gap, τ_m is the shear yield strength of the MR fluid in the middle annular gap.

The viscosity pressure drop of the whole outer annular gap can be expressed as:

$$\Delta p_{o\eta }=\frac{6\eta q(2L_1+2L_2+t_a+2t_b)}{\pi g^3{r}_e}$$

(13)

where r_e is the radius of the outer annular gap.

The field-dependent pressure drop due to the yield stress of the outer annular gap can be expressed as:

$$\Delta p_{o\tau }=\frac{2c({\tau }_f{L}_1+{\tau }_s{L}_2)}{g}$$

(14)

where τ_f and τ_s are the shear yield strength of the MR fluid in the first effective outer gap and the second effective outer gap, respectively.

The total damping force of the proposed MR damper can be given by:

$$F_d=2[\frac{6\eta q}{\pi g^3}(\frac{L_1}{r_i}+\frac{L_1}{r_m}+\frac{L_1+L_2+t_a/2+t_b}{r_e})+\frac{c{L}_1({\tau }_i+{\tau }_m+{\tau }_f)}{g}+\frac{c{L}_2{\tau }_s}{g}]A_p$$

(15)

The dynamic range represents the damping performance of the MR damper, which can be approximately expressed by the ratio of the output damping force to the viscosity damping force. It can be written as:

$$D=\frac{F_d}{\Delta p_{\eta }{A}_p}$$

(16)

4. Modeling and Simulation Analysis

4.1. Properties of the MR Fluid

The MRF-J25T MR fluid was used in this work, which was produced by Chongqing Institute of Materials in China [4]. The relationship between magnetic flux density B and shear stress τ is nonlinear, as shown in Figure 4. By using the least square method, the τ-B curve of the MR fluid is fitted by polynomial, and the relationship between shear yield stress and magnetic flux density at the damping gap is obtained

$$\tau =a_3\times B^3+a_2\times B^2+a_1\times B$$

(17)

where a₁, a₂ and a₃ indicating the polynomial coefficients of the shear yield stress at the damping gap varying with the magnetic flux densities and a₁ = 11.25 kPa/T, a₂ = 58.92 kPa/T², a₃ = −24.52 kPa/T³.

4.2. Finite Element Analysis

The FEA of the proposed MR damper was carried out to obtain the magnetic flux density under different currents. The damping performance was further analyzed by combining the field-dependent yield stress of the MR fluid and the developed mathematical model. The flow gaps and the exciting coil of the proposed MR damper were integrated in the piston. Therefore, the electromagnetic performance of the piston determines the damping performance of the proposed MR damper. Since the proposed MR damper is an axisymmetric structure, half of the piston model was selected for FEA, as shown in Figure 5a. In addition, a FE model of the traditional MR damper with the same dimensions was established for comparison purposes, as shown in Figure 5b.

Figure 6 shows the magnetic flux of the designed MR damper and the traditional MR damper with the current of 1.5 A, respectively. In the proposed MR damper, there are inner effective gaps (P1), middle effective gaps (P2) and outer effective gaps (P3) and (P4), which increase the total length of the effective gaps, as shown in Figure 6a. Notably, when the bending magnetic circuit passed through the second outer gaps (P4), there are two sections to be activated effectively, which also contribute to the growth of the total length of the effective gaps. In contrast, there are only two short effective gaps in the traditional damper, as shown in Figure 6b.

Figure 7 shows the contours of magnetic flux density of the both MR damper at the current of 1.5 A. More than half of the flow path of the traditional MR damper is absent in the magnetic field. However, the inner, middle gaps and majority of outer gaps in Figure 7a are all exposed to the magnetic field. The magnetic flux density of the inner gaps is approximately 0.5 T. The maximum magnetic flux density of the proposed MR damper, which is at the magnetic core, is 1.4 T, while the larger value of 1.8 T is observed in the traditional one, which is close to the magnetic saturation of the magnetic material. The latter has lower magnetic resistance, leading to premature magnetic saturation that can be relatively avoided in the former, that is, the proposed MR damper could achieve a high utilization rate of the magnetic field.

Figure 8 shows the distribution of magnetic flux density value as a function of the annular gap paths of the two MR dampers. The gap path of traditional MR damper is divided into three sections. The magnetic flux density of the two ends of the gap path is nearly 0.5 T. There is no magnetic flux line passing through the middle section in the gap path, where the flux density is close to 0 T. Even though the magnetic flux density in the middle and outer gap of the proposed MR damper is lower to the traditional one, the length of the effective gap in the proposed one visibly extended, owing to the addition of extra inner, middle gaps and the activation of the second outer gaps.

Figure 9 shows the magnetic flux density of each effective gap path for the designed damper with the various current. For all effective gaps, the magnetic flux density of each gap increases with the increase in the applied current. The maximum flux density in the inner gap, middle gap, first outer gap and second outer gap are about 0.49 T, 0.41 T, 0.3 T and 0.3 T with the current of 1.5 A, respectively.

Figure 10 shows the average magnetic flux density obtained by integrating the magnetic flux density of the gap. In the first place, the average magnetic flux density of the folded resistance gaps increases from the outside to the inside because of the decrease in the magnetic flux area under a specific current. That is why the average magnetic flux density of the first outer gap and the second outer gap is approximately the same, as they share identical radii. In addition, as the applied current increases, the average magnetic flux density of each gap increases. Under the applied current of 0.25 A to 1.5 A, the value of the inner gap is about from 0.09 T to 0.46 T, and the middle gap from 0.08 T to 0.38 T, and the outer gaps from 0.06 T to 0.29 T, respectively.

The damping forces for both MR dampers are evaluated by combining the mathematical models and FEA results, as shown Figure 11. Since the total flow gap length of the proposed MR damper is longer than that of the traditional one, the damping force of the former is slightly greater than that of the latter at the current of 0. Meanwhile, though the damping force of the latter is slightly higher than that of the former in the small current range (from 0.25 A to 0.75 A), the output damping force of the proposed one is much greater than the traditional one when the current is larger than 0.75 A. When the current is 1.5 A, the output damping force of the proposed MR damper reaches 4585 N, which is about 1.64 times that of the traditional. Generally speaking, the damping force of the proposed MR damper is approximately linear to the current within the range of 0–1.5 A, while the damping force curve of traditional damper is nonlinear. These indicate that the output damping range of the proposed MR damper is much larger than that of the traditional one.

5. Experimental Analysis and Discussions

5.1. Prototype and Test System Setup

The prototype of the MR damper with folded resistance gaps and bending magnetic circuit was manufactured. The prototype and its components are shown in Figure 12, and the test rig is shown in Figure 13.

The MR damper is installed on the damper test rig equipped with the force sensor and displacement sensor. The DC power supply is used to provide different excitation currents to the MR damper. The electrohydraulic servo controller controlled the vibration excitation platform driving the damper to make the corresponding vibration. The computer that can transmit data with the electrohydraulic servo controller is used to set test parameters, and monitor and store test data.

5.2. Damping Performance Analysis

Figure 14a shows the damping force of different amplitudes with the frequency of 0.25 Hz and the current of 1 A; the maximum damping force increases with amplitudes. Figure 14b shows the damping force of different frequencies with the amplitude of 7.5 mm and the current of 1 A; the maximum damping force increases with frequencies. The piston vibration velocity of piston is proportional to the volumetric flow rate. For the positive correlation applies equally to the relation between the volumetric flow rate and viscous damping force, which is explained in Equation (10). Considering the viscous damping force accounts for a small proportion of the total damping force under the low speed excitation, the damping force increases slightly with the amplitude and frequency, as is shown in Figure 14.

Figure 15 shows the damping force at three frequency levels of 0.5 Hz, 0.75 Hz and 1 Hz with the current of 1 A and the amplitude of 10 mm. Figure 15a verifies the statement in Figure 13 that the variation in damping force and frequency is consistent. When the frequency is 1 Hz, the maximum damping force of the damper is 4739 N. Figure 15b illustrates the relationship between damping force and velocity. The variation curve of maximum damping forces at different frequencies but at the same velocity almost overlaps, namely if the speed and the current are fixed, the output damping forces are roughly the same. The results indicate that the MR damper has a stable damping force and controllability under different exciting conditions.

Moreover, it can be seen from Figure 14 and Figure 15 that the curves of the damping force and displacement have some distortion and are missing in the upper left and lower right parts. This is due to a certain volume of air being mixed into the MR fluid during fluid filling, which leads to a certain empty stroke of the MR damper due to the failure of the timely volume compensation during the working process.

Figure 16 shows the tested output damping force values at a frequency of 0.25 Hz and an amplitude of 7.5 mm for five current levels, namely 0 A, 0.5 A, 0.75 A, 1 A, and 1.5 A. The output damping force increases with the increasing current. When the input current is set to be 0 A and 1.5 A, the corresponding output damping force are 225 N and 4346 N, respectively. The damping force of the proposed MR damper is 1.56 times that of the traditional at the current of 1.5 A. This indicates that the MR damper is able to utilize the magnetic field generated by the excitation coil more effectively, and can provide higher controllable damping force.

As shown in Figure 17, the simulation and experimental results under different currents were compared to verify the feasibility of the established damping force model. The variation curve of the simulated damping force with current is approximately linear while the linear relationship is not obvious in the experiments. The difference between the model and the experiment was caused by the errors of manufacturing and experiment, and the established model cannot represent the small change in pressure drop. The maximum error is 366 N when the current is 1 A. However, the errors are within the allowable range, and the growth trend of the damping force predicted by the established model is almost the same as measured in the experiments. The damping force of the experimental model and the established model increases with the current. The predicted model established by the Bingham model is simple and easy to implement, so it is suitable for designing the MR damper proposed in this study.

Figure 18 compares the damping force and the dynamic range of the proposed MR damper to that of the traditional MR damper with a frequency of 0.25 Hz and amplitudes of 7.5 mm. It can be seen from the figure that the traditional MR damper can have better damping performance when the applied current is small, because the magnetic flux density of the traditional damper at the damping gap is larger in the state of small current. With the gradual increase in the applied current, the traditional MR damper gradually reaches the magnetic saturation state, and the proposed MR damper continues to increase gradually with an upward trend. Figure 18a shows that the field-off damping force of the former is approximately equal to that of the latter at the current of 0 A. However, the design can provide a significantly larger damping force and dynamic range than the traditional one with both input current of 1.0 A and 1.5 A. The damping force of the proposed MR damper is enhanced by 55.82% compared with the traditional one at the applied current of 1.5 A. In addition, the dynamic range of the former can reach 19.3, enhanced by 62.21% compared with the latter, leading to better damping performance.

To evaluate the damping performance of dampers subject to limited space, it is more feasible and significant to compare the output force per unit volume of dampers than to compare its dimensions directly.

Table 2. Comparison of the damping force per unit volume of the proposed MR damper with other MR dampers.

MR Damper	Radius of Piston (mm)	Maximum Positive Damping Force (N)	Damping Force Per Unit Volume (kN/m3)	Dynamic Range
Traditional MR damper	31	2789	2.4 × 104	11.8
MR damper in Reference [32]	35.7	5486	2.92 × 104	7
Proposed MR damper	31	4346	2.93 × 104	19.3

compares the damping force per unit volume of the proposed MR damper with that of the traditional MR damper and the structures MR damper reported in Reference [32] under sinusoidal displacement excitations with current of 1.5 A. The damping force per unit volume is the ratio between the output damping force and the piston volume of the MR damper. The table shows that the proposed damper has a smaller radius and has a larger damping force per unit volume (2.93 × 10⁴ kN/m³) and dynamic range (19.3).

6. Conclusions

A new MR damper configuration for better damping performance by generating a larger damping force and dynamic range was proposed. The internal structure of folded resistance gaps was designed by placing concentric sleeves based on the constraint of the axial and radial size. The bending magnetic circuit could be guided by laying the magnetic and non-magnetic elements. These two features were combined to improve the magnetic field utilization and the MR gap activation rate. A mathematic model of the proposed MR damper was derived; the magnetic properties were obtained by FEA and compared with the traditional one. Then, the damping force performance of the proposed MR damper was obtained by the mathematic model with magnetic properties. The experimental and simulation results validated the feasibility of the design methodology, indicating a preferable way to design MR dampers under the volume constraint.

A longer effective damping length was designed, leading to an MR activation rate of the gap of 66.6%, which is 21.8% higher than that of the traditional MR damper with the same dimensions. Under the excitation with different velocities and different input currents, the damping performance of the proposed MR damper was tested and analyzed. The experimental results show that the damping force of the MR damper increases with the increasing input current. At the current of 1.5 A, the maximum damping force of the MR damper reaches 4346 N, which enhanced the traditional configuration by 55.82%. Compared to the traditional and referenced configurations, it has a larger damping force per unit volume (2.93 × 10⁴ kN/m³) and the dynamic range, up to 19.3, extended by 62.21% than the traditional one.