Numerical Analysis of the Vehicle Damping Performance of a Magnetorheological Damper with an Additional Flow Energy Path

Abstract

Vehicles experience various frequency excitations from road surfaces. Recent research has focused on active dampers that adapt their damping forces according to these conditions. However, traditional magnetorheological (MR) dampers face a “block-up phenomenon” that limits their effectiveness. To address this, additional flow-type MR dampers have been proposed, although revised designs are required to accommodate changes in damping force characteristics. This study investigates the damping performance of MR dampers with an additional flow path to enhance the vehicle ride quality. An optimization model was developed based on fluid dynamics equations and analyzed using electromagnetic simulations in ANSYS Maxwell software. Vibration analysis was conducted using AMESim by applying a sinusoidal road surface model with various frequencies. Results show that the optimized diameter of the additional flow path obtained from the analysis was 1.1 mm, and it was shown that the total damping force variation at low piston velocities decreased by approximately 56% compared to conventional MR dampers. Additionally, vibration analysis of the MR damper with the optimized additional flow path diameter revealed that at 30 km/h, 37.9% acceleration control was achievable, at 60 km/h, 18.7%, and at 90 km/h, 7.73%. This demonstrated the resolution of the block-up phenomenon through the additional flow path and confirmed that the vehicle with the applied damper could control a wider range of vehicle upper displacement, velocity, and acceleration compared to conventional vehicles.

1. Introduction

1.1. Research Background and Conventional Study Status

Technological advances are constantly being made to improve the ride comfort of vehicles, and various suspension devices for vehicles are emerging. In particular, dampers, which are part of the vehicle suspension system, play a major role in controlling vehicle vibration. Dampers come in a wide range of shapes and functions, including active and passive types. Many studies have been conducted on active dampers, whose damping force changes depending on the road surface over which the vehicle passes, rather than existing passive dampers, which have limited damper performance.

However, fully active dampers are difficult to use based on their feasibility and high price. Therefore, semi-active dampers, which are a compromise between passive and active dampers, are favored in many devices that require vibration attenuation. A magnetorheological (MR) damper, which is a representative semi-active damper, uses an MR fluid, which is an intelligent material. MR fluids exhibit rheological properties whose physical properties change based on a magnetic field, which is a key factor in actively controlling the damping force of the damper. When a magnetic field is applied, the magnetic MR fluid particles form a chain shape along the direction of the magnetic field, and yield stress is generated because this chain structure must be broken for the fluid to flow. The magnitude of the yield stress is controlled by the magnitude of the applied magnetic field. The efficiency of the MR fluid is very high because the yield stress is generated only by the application of a magnetic field. Therefore, the reaction rate is fast, and very little energy is used. The characteristics of MR dampers that provide effective performance using high efficiency fluids have been confirmed in several studies.

Kwon et al. considered the performance of an MR damper based on the density, viscosity, and orifice dimensions of the MR fluid [1]. Park et al. proposed a viscous flow model for an MR fluid and analyzed the velocity distribution and damping performance of MR dampers. And the width dimensions of the annular orifice and pole piece and the action point and self-saturation of the hysteresis curve were considered according to the density change in an MR fluid, and an analytical design method for the magnetic circuit was proposed [2]. Liu et al. considered the increase in the damping force of an MR damper through shape optimization of the piston and pole piece [3].

Lee et al. examined the hysteresis of MR dampers using the cylindrical Bingham model, Bouc–Wen model, and polynomial model, and compared it with the experimental results [4,5]. Yoon et al. considered the riding and handling performance of large buses equipped with air springs and MR dampers [6]. Jin et al. considered the relationship between the damping force response speed of an MR damper and vehicle driving characteristics [7]. Kim et al. proposed a plan for comparing the performance characteristics of MR and passive dampers [8].

Despite these characteristics, MR dampers also have disadvantages. Generally, in structured MR dampers, the total damping force and piston speed are not perfectly linear. Even when the piston speed increases slightly, the damping force increases very rapidly. However, after a certain piston speed is reached, regardless of how much the piston speed increases, the damping force shows a slight increase. Because of the characteristics of these MR dampers, a large damping force can be generated even at very low piston speeds, effectively suppressing the vehicle vibration. However, if the direction of motion of the vehicle changes continuously within this low piston speed range, a sudden change in the damping force occurs. Therefore, when vertical vibration occurs at a low piston speed, the damping force rapidly changes from positive to negative, which significantly affects ride comfort. This is called the ‘block-up phenomenon’, and an MR damper with an additional flow path was suggested to solve this problem. Sohn et al. proposed an MR damper structure for passenger cars with an additional flow path inside the piston and examined the damping force characteristics based on changes in the number of additional flow paths [9]. The MR damper with additional flow path has an internal fluid flow pattern that is different from that of existing MR dampers. When a magnetic field is not applied, the MR fluid flows in both the additional orifice hole (additional flow path) and the basic flow path. However, when a magnetic field is applied, the MR fluid flows only through the additional orifice hole.

Figure 1 and Figure 2 show schematics of an MR damper with an additional flow path inside the piston that is not affected by magnetic fields. The piston separates the cylinder of the MR damper into the upper and lower chambers. The interior of the cylinder consists of an additional flow path inside the piston, a base gap, and a nitrogen accumulator. The nitrogen accumulator provides vibration absorption and stability to the damper through elastic force. The MR fluid in the inner chamber of the cylinder passes through an additional flow path and a basic gap when a magnetic field is not applied. The performance of MR dampers with an additional flow path in alleviating the shortcomings of existing MR dampers has been demonstrated, and the number of cases in which they are applied is increasing. Park et al. proposed an asymmetric MR damper with different damping forces in the tensile and compression directions by opening and closing additional flow paths [10]. Jeon et al. considered the sensitivity and purge control performance of an MR damper with an additional flow path [11].

1.2. Research Purpose

Unlike general MR dampers, the flow path of the MR fluid and the additional flow path are processed in the piston that makes up the magnetic circuit; thus, the pressure difference and magnetic field between the top and bottom of the piston generate a damping force change. In addition, in previous studies, tests were mainly conducted to apply single-bump excitation to the damper model to observe the transient response characteristics of MR dampers. However, vehicle dampers receive excitation from the road surface at various frequencies depending on the vehicle speed, which is not suitable as a test for designing MR dampers for vehicles. Therefore, in this study, an MR damper with an additional flow path was designed and combined with vehicle modeling to apply frequency excitation in various road surface environments and at various vehicle speeds, and the movement of the upper part of the vehicle was analyzed and compared with vehicle modeling with a general damper.

2. Research Method

2.1. Mathematical Modeling

2.1.1. Damping Force Model

The MR damper generates a magnetic field when a current is applied to the internal coil, and the MR fluid in the chamber generates yield stress. The generated yield stress causes a pressure drop between the upper and lower chambers, resulting in a damping force corresponding to the movement of the piston. The damping force of the proposed MR damper with an additional flow path can be expressed as follows:

$$F_{Damper}\equiv F_{viscosity}+F_{MR}+F_{accumulator}$$

(1)

The total damping force of the proposed MR damper with an additional flow path is defined as $F_{Damper}$, the viscous damping force due to fluid viscosity is defined as $F_{viscosity}$, the damping force due to the yield stress generated by the MR fluid is defined as $F_{MR}$, and the elastic force of the MR damper accumulator is defined as $F_{accumulator}$. $F_{accumulator}$ can be interpreted as an experimental measurement equation, which is formulated as follows:

$$F_{accumulator}=k_e{x}_{piston}$$

(2)

In Equation (2), $k_e$ is the accumulator coil spring constant, and $x_{piston}$ is the piston displacement. $k_e$ can be indicated as in Equation (3).

$$k_e={\displaystyle \frac{A_r^2}{C_g}}={\displaystyle \frac{P_0\kappa }{V_0}}{A}_{rod}^2$$

(3)

In Equation (3), $A_{rod}$ is the area of the silver piston rod, $P_0$ is the initial pressure, $V_0$ is the volume of the accumulator, $C_g$ is the compliance of the accumulator gas, and κ is the specific heat ratio. However, the elastic force of the MR damper accumulator is not included in $F_{Damper}$ because $F_{MR}$ has an effect similar to that of the elastic element of the nitrogen accumulator. Therefore, $F_{Damper}$, the total damping power of the MR damper with an additional flow path used in this study, is assumed as follows.

$$F_{Damper}\cong F_{viscosity}+F_{MR}$$

(4)

2.1.2. Viscous Damping Force Model

To mathematically model $F_{viscosity}$, which corresponds to the first term in Equation (4), the following assumptions based on hydrodynamic theory are applied to use Bernoulli’s equation.

• The MR fluid is assumed to be an incompressible fluid due to its high density and minimal volume change even under operating input;
• The flow within the MR damper is considered steady-state, assuming laminar flow in the low-velocity, low-Reynolds-number region. Therefore, it is treated as steady flow;
• The flow in the additional flow path and upper and lower chambers is friction-free;
• The flow in the additional flow path and upper and lower chambers is the flow along the streamline;
• To simplify the energy conservation analysis, the control volume is divided into sections A and B, as shown in Figure 3;
• It is assumed that the MR fluid conserves energy within each section, so the pressure drop between the inlet and outlet of each MR fluid section is the same;
• To reflect that the kinetic energy of the piston is fully transferred to the fluid, it is assumed that the piston velocity $V_{piston}$ and the MR fluid velocity $V_{inlet}$ at the inlet are the same.

Based on the above assumptions, the values in the lower and upper chambers of the MR fluid are expressed using the Bernoulli equation.

$${\displaystyle \frac{P_1}{\rho g}}+{\displaystyle \frac{V_1^2}{2g}}+z_1+h={\displaystyle \frac{P_2}{\rho g}}+{\displaystyle \frac{V_2^2}{2g}}+z_2$$

(5)

In Equation (5), $P_1$ and $P$ are the pressures in the lower and upper chambers, respectively, $V_1$ and $V_2$ are the average flow rates in the lower and upper chambers, respectively, $z_1$ and $z_2$ are the gross pump heads in each chamber, and h is the total head. Ρ and g are the density and gravitational acceleration of the MR fluid, respectively, and in incompressible flow, the following holds true:

$$V_1\approx V_2$$

(6)

Because Equation (6) is established, Equation (5) can be reorganized as follows:

$$∆P=P_2-P_1=\rho gh$$

(7)

Additionally, the following theory is applied to $F_{viscosity}$ to formulate it in the same manner as $F_{MR}$. In the proposed MR damper, the additional flow path can be interpreted as a fully developed laminar flow in a circular pipe covered by hydrodynamics. In the proposed additional flow-path-type MR damper, the flow in the non-magnetic orifice is shown in Figure 4. This can be interpreted as fully developed laminar flow within a circular pipe in fluid dynamics, which provides structural characteristics that may help alleviate the block-up phenomenon.

The hydrodynamic theoretical equation used for a fully developed laminar flow is as follows:

$$∆P_{orifice}={\displaystyle \frac{L_{orifice}}{D_{orifice}}}{\displaystyle \frac{{\rho V}_{orifice}^2}{2}}{f}_{orifice}$$

(8)

In Equation (9), $∆P_{orifice}$ is the pressure drop in the additional passage, $L_{orifice}$ and $D_{orifice}$ are the length and hydraulic diameter of the additional passage, respectively, $V_{orifice}$ is the average flow velocity of the MR fluid while flowing in the additional passage, and $f_{orifice}$ is the friction in the additional passage and is a coefficient. The friction coefficient in laminar flow in a fully developed circular pipe can be expressed in terms of the Reynolds number as follows:

$$f_{orifice}={\displaystyle \frac{64}{Re}}={\displaystyle \frac{64\mu }{\rho V_{orifice}{D}_{orficie}}}$$

(9)

Using this approach, the flow in the primary gap is shown in Figure 5, which can be considered as fully developed laminar flow between two parallel plates. Using this approach, the flow in the underlying gap can be regarded as a laminar flow between two fully developed plates. The basic hydrodynamic theoretical equations used for this flow are as follows:

$$∆P_{gap}={\displaystyle \frac{L_{gap}}{D_{gap}}}{\displaystyle \frac{{\rho V}_{gap}^2}{2}}{f}_{gap}$$

(10)

In this case, similar to the additional flow path, the friction coefficient is expressed in terms of the Reynolds number as follows:

$$f_{gap}={\displaystyle \frac{96}{Re}}={\displaystyle \frac{96\mu }{\rho V_{gap}{D}_{gap}}}$$

(11)

In addition, according to this assumption and Figure 3, the flow rate in and out of the damper is the same and is defined as follows:

$$V_{piston}\left(A_{piston}-A_{rod}\right)=V_{gap}{A}_{gap}=V_{orifice}{A}_{orifice}$$

(12)

Therefore, $F_{viscosity\_orrifice}$ and $F_{viscosity\_gap}$ can be expressed as follows.

$$F_{viscosity\_gap}={\displaystyle \frac{48\mu L_{gap}}{{\left(2D_{gap}\right)}^2}}{\displaystyle \frac{{\left(A_{piston}-A_{rod}\right)}^2}{A_{gap}}}{V}_{piston}$$

(13)

$$F_{viscosit{y}_{orifice}}={\displaystyle \frac{32\mu L_{orifice}}{{\left(D_{orifice}\right)}^2}}{\displaystyle \frac{{\left(A_{piston}-A_{rod}\right)}^2}{A_{orifice}}}{V}_{piston}$$

(14)

2.1.3. Magnetic Damping Force Models

The formula for calculating the $F_{MR}$ corresponding to the last term in Equation (4) is expressed as follows:

$$F_{MR}=(A_{piston}-A_{rod})∆P_{MR}$$

(15)

In Equation (16), $A_{piston}$ is the cross-sectional area of the silver piston, and $∆P_{MR}$ is the magnitude of the internal pressure drop due to the characteristics of the MR fluid. $∆P_{MR}$ can be expressed as follows:

$$∆P_{MR}={\displaystyle \frac{L_{gap}}{D_{gap}}}C{\tau }_y$$

(16)

In Equation (17), $L_{gap}$ is the length of the basic gap, $D_{gap}$ is the width of the basic gap, C is the flow rate coefficient of the MR fluid, and ${\tau }_y$ is the yield stress of the MR fluid generated by the application of current which can be calculated using Equation (18):

$${\tau }_y=144+138708B+158790B^2-176510B^3+52962B^4$$

(17)

In Equation (18), the yield stress of the MR fluid ${\tau }_y$ is determined by the magnetic flux density B generated in the magnetic circuit of the MR damper. The magnetic flux density B is generated by the magnetic field that occurs when current flows in the solenoid coil according to Ampere’s law. The relationship is shown in Equation (19):

$$\oint B\cdot dl=\mu 0\int J\cdot da$$

(18)

To calculate the magnetic flux density B according to Equation (19), the magnetic circuit of the MR damper is expressed mathematically, and the permeance method is used to calculate the magnetic flux density. Permeability is the opposite of magnetoresistance R, indicates the degree to which the magnetic flux flows well in a magnetic material, and can be expressed by Equation (20):

$$\int {\displaystyle \frac{{\mu }_{0}uda}{l}}={\displaystyle \frac{1}{R}}$$

(19)

The magnetic circuit of a general MR damper is composed of a solenoid coil, core, flux ring, and an MR fluid that fills the flow path. Each permeance is obtained from the shape parameters of the magnetic circuit components using Equation (20) and is calculated from the magnetic circuit. Because electric circuits are similar, we can calculate the magnetic flux of the desired part of the circuit. The similarities between the magnetic circuits and electric circuits are listed in

Table 1. Similarity of electric and magnetic circuit.

Magnetics (Unit)	Symbol	Electronics (Unit)	Symbol
MMF (At)	F	EMF (V)	V
Reluctance (A/Weber)	R	Resistance (ohm)	R
Flux (Weber)	Ø	Current (A)	I
Flux density (Tesla)	B	Current density (A/${\mathrm{m}}^2$)	J
Field Intensity	H	Field intensity (V/m)	E
Permeability (H/m)	μ	Resistivity (Ohm∙m)	ρ
Permeance (Weber/A)	P

.

Based on the above calculations for the proposed MR damper with an additional flow path, an orifice hole is formed inside the coil such that the fluid flow is not affected when the MR fluid passes through the orifice. When a magnetic field is applied to these dampers, the fluid friction caused by the yield stress causes the MR fluid to flow through the orifice hole. Thus, when the piston speed is high, the viscous damping force generated by the orifice is greater than the damping force caused by the basic gap and MR fluid. This can be observed in Figure 6 and Figure 7.

For the proposed MR damper with an additional flow path, the total damping force is shown in Figure 7. When the piston speed is high in this damper shape, the viscous damping force caused by the flow through the orifice is greater than the damping force of the basic MR damper with yield stress. The damper operates based on the smaller of the two damping forces, and this small value determines its overall damping force. This allows the damper system to balance the damping force to provide more stable damping performance (25). Accordingly, the total damping force of the proposed MR damper with an additional flow path, $F_{damper}$, is defined as follows:

$$F_{damper}=\min (F_{viscosit{y}_{orifice}}, F_{viscosity\_gap}+F_{MR})$$

(20)

This confirms that Equation (4) can be expressed as Equation (20) for the MR damper with an additional flow path proposed in this study, and it was used to examine the total damping force based on the piston speed and applied current.

2.1.4. Quarter-Car Model

The equation of motion of the two-degree-of-freedom (2-DOF) suspension system shown in Figure 8 is expressed as a linear spring constant $k_s$ and a linear damping rate $c_s$. The tire was modeled using the spring constant $k_t$. The linear equation of motion is expressed as follows:

$$-c_s({\dot{z}}_s-{\dot{z}}_u)-k_s(z_s-z_u)=m_s{\ddot{z}}_s$$

(21)

$$c_s({\dot{z}}_s-{\dot{z}}_u)+k_s(z_s-z_u)-k_t(z_u-z_r)=m_u{\ddot{z}}_u$$

(22)

$${\ddot{z}}_s+{\displaystyle \frac{c_s}{m_s}}({\dot{z}}_s-{\dot{z}}_u)+{\displaystyle \frac{k_s}{m_s}}(z_s-z_u)=0$$

(23)

$${\ddot{z}}_u-{\displaystyle \frac{c_s}{m_u}}({\dot{z}}_s-{\dot{z}}_u)-{\displaystyle \frac{k_s}{m_u}}(z_s-z_u)+{\displaystyle \frac{k_t}{m_u}}(z_u-z_r)=0$$

(24)

The Laplace transform of Equations (23) and (24) are as follows:

$$s^2{Z}_s(s)+s{\displaystyle \frac{c_s}{m_s}}(Z_s(s)-Z_u(s))+{\displaystyle \frac{k_s}{m_s}}(Z_s(s)-Z_u(s))=0$$

(25)

$$s^2{Z}_u(s)-s{\displaystyle \frac{c_s}{m_u}}(Z_s(s)-Z_u(s))-{\displaystyle \frac{k_s}{m_u}}(Z_s(s)-Z_u(s)+{\displaystyle \frac{k_t}{m_u}}(Z_u(s)-Z_r(s))=0$$

(26)

This can be written in matrix form as in Equation (27).

$$\left[\begin{array}{cc}{s}^2+s{\displaystyle \frac{c_s}{m_u}}+{\displaystyle \frac{k_s}{m_u}}+{\displaystyle \frac{k_t}{m_u}}& -s{\displaystyle \frac{c_s}{m_u}}-{\displaystyle \frac{k_s}{m_u}}\\ -s{\displaystyle \frac{c_s}{m_s}}-{\displaystyle \frac{k_s}{m_s}}& s^2+s{\displaystyle \frac{c_s}{m_s}}+{\displaystyle \frac{k_s}{m_s}}\end{array}\right]\left[\begin{array}{c}{Z}_u(s)\\ Z_s(s)\end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{k_t}{m_u}}\\ 0\end{array}\right]Z_r(s)$$

(27)

The transfer function can be obtained using Cramer’s formula.

$${\displaystyle \frac{Z_s\left(s\right)}{Z_r\left(s\right)}}={\displaystyle \frac{\frac{k_t{c}_s}{m_u{m}_s}\left(s+\frac{k_s}{c_s}\right)}{\begin{array}{c}{s}^4+\left(\frac{c_s}{m_u}+\frac{c_s}{m_s}\right)s^3+\left(\frac{k_s}{m_u}+\frac{k_s}{m_s}+\frac{k_t}{m_u}\right)s^2\\ +\left(\frac{k_t{c}_s}{m_u{m}_s}\right)s+\frac{k_t{k}_s}{m_u{m}_s}\end{array}}}$$

(28)

2.2. Numerical Analysis Modeling

2.2.1. Additional Flow Path MR Damper Design

To analyze the damping performance of the MR damper with an additional flow path, a numerical analysis model was designed based on the mathematical model described in Section 2.1, which can evaluate the change in damping performance by simulating the pressure drop and magnetic field intensity change of the MR damper. Prior to designing the damper, a popular and widely used MR fluid was chosen. MRF-132 DG, the MR fluid used in this study, is recognized for its usefulness in many fields. In addition, because vehicle dampers must be designed within the operating range required by the vehicle, the quarter-car model of the vehicle on which the damper will be installed was modeled using AMESim 4.3.0.

Based on this, the length of the piston rod that can be used in the magnetic circuit was determined, and an electromagnetic analysis was performed using ANSYS Maxwell software 2024 R3. The main variables of the electromagnetic analysis were the number of coil turns, current range, and the diameter and length of the main flow path, which were repeated until each operating point was located in the linear section of the B-H graph. Because the main flow path has a significant influence on both electromagnetic and flow analyses, and its design range is closely related to saturation of the magnetic circuit, the optimal value was selected through electromagnetic analysis prior to flow analysis. The above process is a modified design guide provided by the L Company and is shown in Figure 9; the manufacturer of the MR fluid and its effectiveness have already been proven.

Conversely, because the additional flow path was almost unaffected by the magnetic circuit, the optimal value was selected based on a theoretical fluid dynamics equation. Subsequently, the anti-rod stress of the MR fluid was derived using the optimal value selected through electromagnetic analysis. Through the above process, the damping force model of the optimally designed additional passage-type MR damper was transferred to AMESim, and the performance of the MR damper was analyzed by applying road surface excitation modeled with bumps and random frequencies. The above processes and the result designed according to the processes are expressed in Figure 10 and Figure 11.

2.2.2. Vibration Analysis Method Using the Quarter-Car Model

For vehicle modeling to be used in the performance analysis of an MR damper with an additional flow path, a linear suspension quarter car is modeled using AMESim based on Company M’s medium-sized vehicle data. This is illustrated in Figure 12. Because it is a quarter-car model, 317.5 kg (one-quarter of the total spring upper mass) was set as the sprung mass because the upper spring mass was distributed and transmitted to the four tires. The suspension spring, unsung mass, and tire spring coefficients used in this study are listed in

Table 2. Specification of test vehicle.

Parameter	Value	Units
Whole sprung mass (M)	1270	kg
Unsprung mass (mf, mr)	35.3	kg
Spring stiffness (kf, fr)	28,500	N/m
Damping coefficient (c)	1850	Ns/m
Damper response time (t)	7	ms
Tire radius (r)	0.335	m
Tire stiffness (ktf, ktr)	268,000	N/m
CG to front wheel	1.02	m
CG to rear wheel	1.9	m
Vehicle tread	1.7	m

.

To compare a vehicle with a general damper and a vehicle with an MR damper, the damping coefficient of the general damper was set to 1850 Ns/m, and the damping coefficient of the MR damper was set to the results derived from theoretical fluid dynamics equations.

2.2.3. Magnetic Circuit Model

A magnetic circuit that generates the damping force $F_{MR}$ generated by the yield stress of the fluid consists of a solenoid coil, core, and flux ring. The MR fluid inside the flow path acts as a magnetic path of the magnetic field generated by the solenoid coil. However, when magnetic saturation occurs in the magnetic path, the magnetic flux does not increase, and the remaining current is converted into Joule heat, which reduces the performance of the MR fluid. Therefore, an action point must be selected in the linear interval of the B-H diagram where self-saturation does not occur.

For these calculations, dampers containing MR fluids were modeled using ANSYS Maxwell 2D. In the electromagnetic analysis setting, the shape mode was cylindrical based on the Z-axis, the analysis mode was magneto-static, and the damper was located at the center of a vacuum sphere with a diameter of 1 m. The electromagnetic force generated by the solenoid core was modeled as the product of the number of coil windings and the current. The modeled MR damper was designed as a two-road type to eliminate deviations based on the accumulator and operating direction. The cylinder and piston rod used a non-magnetic material, AL6061, to prevent magnetic flux from flowing out of the magnetic circuit and to secure mechanical rigidity. In contrast, AISI 1008, a ferromagnetic material, was used as the core and flux ring corresponding to the magnetic circuit, and copper was used as the solenoid coil. The empty space inside the flow path and damper was input by extracting several action points from the B-H graph of the fluid provided by the manufacturer of MRF-132DG. The designed magnetic circuit and numerical values are shown in

Table 3. Design parameters for the proposed MR damper.

Parameters	Values
$D_{coil}$	0.42 mm
$A_{piston}$	388.09 mm2
$A_{rod}$	36 mm2
$D_{gap}$	1.1 mm2
$L_{gap}$	26.1 mm
$L_{cylinder}$	80 mm
$D_{orifice}$	1.1 mm
$A_{core}$	84.8 mm2
$L_{orifice}$	26.1 mm
No. turns per coil	162 turns
current	0.2~1.0 A

and Figure 13, and the physical properties and B-H diagrams of the materials constituting the magnetic circuit are shown in

Table 4. Magnetic circuit materials.

Core and Flux Ring	AISI 1008
MR fluid	LOAD Cooperation MRF-132DG
Cylinders and piston rod	AL6061

and

Table 5. Properties of the MR fluids (MRF-132 DG) used in this study.

Properties	Values
$\rho$	$2950 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
$C$	2.5
$\mu$	0.03

and Figure 14 and Figure 15, respectively.

2.2.4. Road Surface Model

An excitation function using a sinusoidal road surface was used for the performance evaluation and vibration analysis of MR dampers. Sinusoidal excitation was designed with an emphasis on evaluating the damper response according to the frequency band. The overall performance of the MR damper and an existing commercial damper were analyzed and compared at various angles.

Table 6. Excitation parameters.

Parameter	Velocity [km/h]	Frequency [Hz]
Sine wave excitation frequency	30	0.833
	60	1.667
	90	2.5

summarizes the parameters used in the analysis.

The sinusoidal road surface was modeled to move along a sinusoidal wave with an amplitude of 10 mm and a period of 10 m, as shown in Figure 16. The vertical displacement from the road surface $z_{sin}$ according to speed can be expressed as follows:

$$z_{sin}=A_{sin}\sin \left({\omega }_{sin}t\right)$$

(29)

$${\omega }_{sin}=2\pi f_{sin}$$

(30)

$$f_{sin}={\displaystyle \frac{V_{sin}}{D_{sin}}}$$

(31)

When the speed is 30 km/h, the sine wave $z_r$ is defined as follows:

$$z_r=0.01\sin \left(5.24t\right)$$

(32)

In Equations (29)–(32), $A_{sin}$ is the amplitude of the sinusoidal road surface, $V_{sin}$ is the vehicle speed, and $D_{sin}$ is the length of one cycle of the sinusoidal road surface. The road surface difference at each speed is shown in Figure 17.

3. Results and Discussion

3.1. Electromagnetic Analysis Results

The magnetic flux density graphs in the main flow path, additional flow path, and core calculated using Maxwell’s electromagnetic analysis in ANSYS Electronics Desktop are shown in Figure 18, Figure 19, and Figure 20, respectively. For the magnetic flux density graph of the main flow path and the additional flow path, the value at the center of the flow path was extracted based on the damper cross-sectional view, and for the core, the value was extracted at the center of the piston considering the effective area and length of the piston, which is shown in Figure 18.

Through the analysis, the following results were obtained. The maximum magnetic flux density of the main flow path was 0.1 T when the current was 0.2 A, but as the current increased to 1.0 A, the magnetic flux density was 0.3 T, and it was found to be evenly distributed over the distances of 9~20 mm and 26~37 mm, which is shown in Figure 19. Since this area is the area between the piston and the flux ring, it was found that the damping force due to magnetic force would be stably generated.

On the other hand, the magnetic flux density of the additional flow path was up to 0.005 T when the current was 0.2 A and up to 0.05 T when the current was 1.0 A, which was only 6 to 1/20th of the main flow path. In addition, it was confirmed that the maximum magnetic flux density was limited to some areas where the magnetic flux density was concentrated due to rapid changes in shape, such as the corners of the piston, such as 10 mm, 23 mm, and 35 mm, and this is shown in Figure 20. This is a result that conforms to the design intention of having the additional flow path generate a larger damping force due to the flow than the damping force due to the magnetic field in order to reduce the initial damping force.

In addition, as the current increases, the piston’s magnetic flux density increases, but as the current increases, the increase gradually decreases due to saturation of the magnetic circuit. The magnetic saturation phenomenon mainly occurred in the middle of the piston and the center of the coil, at a distance of 10 to 16 mm, and had a magnetic flux density of up to 1.75 T when 1.0 A was applied. However, when the current increases from 0.8 A to 1.0 A, the magnetic flux density increase is 0.1 T, which is half of the increase of 0.2 T when the current increases from 0.6 A to 0.8 A. At this time, the saturated magnetic flux increases at a distance of 16 to 29 mm of the additional flow path. It could be seen by referring to Figure 21 that the leakage occurred in an area of 29 mm. In order to intuitively check the effect of magnetic flux saturation, the magnetic flux density images of the piston, plus ring, and MR fluid are shown in Figure 22.

The following results were obtained through analysis. It was confirmed that when the main flow path had a magnetic flux density of about 0.3 T to 1.0 A, the additional flow path had a small value of about 0.05 T. This is because the additional flow path is designed to generate a larger damping force due to the flow than the damping force due to the magnetic field to reduce the initial damping force.

In addition, as the current increased, the magnetic flux density of the flow path increased; however, as the current value increased, the increase gradually decreased owing to the saturation of the magnetic circuit. Magnetic saturation mainly occurred at the corners where the cross-sectional area of the circuit changed rapidly, and the highest magnetic flux density was observed at the corners at both ends of the core, as shown in Figure 22. Additionally, the magnetic flux density of the flux ring calculated using the permeance method was compared with that calculated using ANSYS Maxwell, as listed in

Table 7. Comparison of ANSYS Maxwell and the permeance model.

Ampere [A]	Magnetic Flux Density [T]		Error [%]
	ANSYS Maxwell Model	Permeance Model
0.6	1.408	1.532	8.08%
0.8	1.598	1.642	2.67%
1.0	1.713	1.702	−0.646%

. It was confirmed that the magnetic flux density of the two models had an error of up to 8.08%.

Based on the above results, the relationship between the current applied to the MR damper and the yield stress of the MR fluid is shown in Figure 23. It was found that F_MR had a damping force of at least 3.37 kPa at 0.2 A and up to 16.38 kPa at 1.0 A. As the current increased, the damping force also increased. However, because the increase in the magnetic flux density gradually decreased, the slope of the damping force F_MR in the form of a high-order polynomial for the magnetic flux density also gradually decreased.

3.2. Flow Analysis Results

The $V_{piston}$ of the proposed MR damper was set to the range of −0.4 m/s ≤ $V_{piston}$ ≤ 0.4 m/s to calculate the damping force in the main flow path, which is shown in

Table 8. Damping force according to the MR.

Current [A]	Damping Force [N]
$0$	0
$0.2$	66.83
$0.6$	258.98
$1$	374.06

and Figure 24 and Figure 25.

Figure 24 and Figure 25 demonstrate the effect of the non-magnetic orifice on the damping force of the MR damper. In the case of a conventional MR damper without flow through a non-magnetic orifice, where only viscous damping in the gap and the MR fluid’s damping effect due to the applied current are present, a significant variation in the total damping force is observed within the very low piston velocity range of −0.1 m/s ≤ $V_{piston}$ ≤ 0.1 m/s, spanning approximately −479.39 N ≤ $F_{damper}$ ≤ 479.39 N. Conversely, as seen in Figure 25, an MR damper with an auxiliary flow path containing a non-magnetic orifice exhibits a much smoother variation in total damping force within the same low piston velocity range, maintaining a range of approximately −270.22 N ≤ $F_{damper}$ ≤ 270.22 N. This indicates that the damping force variation is reduced by about 56% compared to the conventional MR damper. Therefore, in cases where vertical vibrations occur at low piston velocities, the “block-up phenomenon”, characterized by abrupt shifts in damping force from positive to negative damping that degrade ride comfort, can be mitigated by incorporating a non-magnetic orifice. From Figure 24 and Figure 25, it is possible to confirm the effect of the presence or absence of an orifice that is not affected by the magnetic field on the increase or decrease in the damping force of the damper. When only the viscous damping force in the gap and the damping force of the MR fluid according to the applied current are shown in Figure 24, the damping force of the damper changes rapidly at very low piston speeds. However, as shown in Figure 25, which shows the total damping force of the damper proposed in this study, the damping force changed slightly at low piston speeds with or without the additional flow path.

In this study, to observe the damping force characteristics according to the shape of the additional flow path, the shape of the additional flow path was modified in the proposed MR damper, and the total damping force characteristics of the MR damper were confirmed accordingly. If the length of the additional flow path, $L_{orifice}$, changed, then there was an overall change in the shape of the proposed MR damper. To prevent this, we checked the change in the damping force characteristics through changes in the diameter $D_{orifice}$ of the additional flow path. The $D_{orifice}$ of the proposed MR damper with an additional flow path was 1.1 mm. Observe the change when $D_{orifice}$ increases or decreases, and calculate the total damping when $D_{orifice}$ = 1.2 mm and $D_{orifice}$ = 1.0 mm, which are values in an appropriate range that are not significantly affected by the magnetic field. This is illustrated in Figure 26.

It was confirmed that when the diameter of the orifice is large, the damping force changes rapidly at low piston speeds, but it can produce a fast-damping effect, and when the diameter is small, the opposite effect can be produced, as shown in the Figure 26. Based on the results, the total damping force is compared when $D_{orifice}$ = 1.0 mm, $D_{orifice}$ = 1.2 mm, and $D_{orifice}$ = 1.1 mm, as shown in Figure 27. The comparison is based on an applied current of 1.0 A, which is the most severe change in the damping force and is a basic characteristic of the MR damper.

It is difficult to infer the most appropriate diameter of the additional flow path based on the results shown in Figure 26, because the advantages and disadvantages of each case are clear. Therefore, the most appropriate diameter of the additional flow path is selected through the average attenuation coefficient in the previously set range of −0.4 m/s ≤ $V_{piston}$ ≤ 0.4 m/s.

Table 9. Average damping coefficient.

${\mathit{D}}_{\mathit{o}\mathit{r}\mathit{i}\mathit{f}\mathit{i}\mathit{c}\mathit{e}}$ [mm]	Average Damping Coefficient [N·s/m]
1.0	1945
1.1	1949
1.2	1939

confirms that the average attenuation coefficient of $D_{orifice}$ = 1.1 mm had the highest value of 1949 N∙s/m. Therefore, in this model, the final attenuation coefficient (

Table 10. Final damping coefficient.

Current [A]	Damping Coefficient [N·s/m]
0.0	1013
0.2	1180
0.6	1676
1.0	1949

) based on the current is calculated as follows.

3.3. Vibration Analysis Results

The vibration analysis results obtained using AMESim are presented in

Table 11. Coordinates of peak points in the 30 km/h scenario.

Current [A]		0	0.6	1.0	Vehicle
Kinematic	Variables	Displacement
Peak Point	(x, y)	(0.414, 0.015)	(0.399, 0.014)	(0.394, 0.0136)	(0.395, 0.0138)
		Velocity
		(1.15, 0.0844)	(1.20, 0.0769)	(1.21, 0.0754)	(1.21, 0.0759)
		Acceleration
		(0.871, 0.622)	(0.884, 0.485)	(0.892, 0.451)	(0.889, 0.462)

,

Table 12. Coordinates of peak points in the 60 km/h scenario.

Current [A]		0	0.6	1.0	Vehicle
Kinematic	Variables	Displacement
Peak Point	(x, y)	(0.287, 0.0124)	(0.269, 0.0119)	(0.262, 0.0117)	(0.265, 0.0118)
		Velocity
		(0.766, 0.219)	(0.74, 0.184)	(0.7305, 0.175)	(0.7335, 0.178)
		Acceleration
		(0.621, 1.97)	(0.5935, 1.73)	(0.584, 1.66)	(0.5875, 1.69)

and

Table 13. Coordinates of peak points in the 90 km/h scenario.

Current [A]		0	0.6	1.0	Vehicle
Kinematic	Variables	Displacement
Peak Point	(x, y)	(0.676, 0.00608)	(0.6445, 0.00672)	(0.6335, 0.00708)	(0.638, 0.00695)
		Velocity
		(0.5635, 0.118)	(0.536, 0.123)	(0.527, 0.127)	(0.535, 0.125)
		Acceleration
		(0.4535, 1.79)	(0.43, 1.88)	(0.422, 1.94)	(0.425, 1.92)

and Figure 28, Figure 29, Figure 30, Figure 31, Figure 32, Figure 33, Figure 34, Figure 35 and Figure 36.

For a displacement of 30 km/h, when the added current was 0 A, 0.6 A, and 1 A, the maximum values were 0.015 m, 0.014 m, and 0.0136 m, and the times to reach them were 0.414 s, 0.399 s, and 0.394 s, respectively. For the test vehicle, the maximum values and reaching times were 0.0138 m and 0.395 s, respectively.

Concerning speed, when the added current was 0 A, 0.6 A, and 1 A, the maximum values were 0.0844 m/s, 0.0769 m/s, and 0.0754 m/s, and the times to reach them were 1.15 s, 1.20 s, and 1.21 s, respectively. For the test vehicle, the maximum values and reaching times were 0.0759 m/s and 1.21 s, respectively.

For acceleration, when the added current was 0 A, 0.6 A, and 1 A, the maximum values were 0.622 m/s², 0.485 m/s², and 0.451 m/s², and the time to reach them was 0.871 s, 0.884 s, and 0.892 s, respectively. For the test vehicle, the maximum values and reaching times were 0.462 m/s² and 0.889 s, respectively.

For a displacement of 60 km/h, when the added current was 0 A, 0.6 A, and 1 A, the maximum values were 0.0124 m, 0.0119 m, and 0.0117 m, and the time to reach them was 0.287 s, 0.269 s, and 0.262 s, respectively. For the test vehicle, the maximum values and reaching times were 0.0118 m and 0.265 s, respectively.

When the added current was 0 A, 0.6 A, and 1 A, the maximum values of speed were 0.219 m/s, 0.184 m/s, and 0.175 m/s, and the time to reach them was 0.766 s, 0.74 s, and 0.7305 s, respectively. For the test vehicle, the maximum values and reaching times were 0.178 m/s and 0.7335 s, respectively.

Concerning acceleration, when the added current was 0 A, 0.6 A, and 1 A, the maximum values were 1.97 m/s², 1.73 m/s², and 1.66 m/s², and the time to reach them was 0.621 s, 0.5935 s, and 0.584 s, respectively. For the test vehicle, the maximum values and reaching times were 1.69 m/s² and 0.5875 s, respectively. It was confirmed that when the vehicle speed was 30 km/h and 60 km/h, the higher the added current, the smaller the maximum value and the shorter the time to reach it.

For a displacement of 90 km/h, when the added current was 0 A, 0.6 A, and 1 A, the maximum values were 0.00608 m, 0.00672 m, and 0.00708 m, and the time to reach them was 0.676 s, 0.6445 s, and 0.6335 s, respectively. For the test vehicle, the maximum values and reaching times were 0.00695 m and 0.638 s, respectively.

When the added current was 0 A, 0.6 A, and 1 A, the maximum values were 0.118 m/s, 0.123 m/s, and 0.127 m/s, and the time to reach them was 0.5635 s, 0.536 s, and 0.527 s, respectively. For the test vehicle, the maximum value and reaching time were 0.125 m/s and 0.535 s, respectively.

Concerning acceleration, when the added current was 0 A, 0.6 A, and 1 A, the maximum values were 1.79 m/s², 1.88 m/s², and 1.94 m/s², and the time to reach them was 0.4535 s, 0.43 s, and 0.422 s, respectively. For the test vehicle, the maximum value and reaching time were 1.92 m/s² and 0.425 s, respectively. When the vehicle speed was 90 km/h, it was confirmed that the higher the added current, the larger the extreme value and the longer the time to reach it.

Based on the vibration analysis, the following conclusions were reached. In the case of sinusoidal excitation, at speeds of 30 km/h and 60 km/h, relatively small maximum values of displacement, speed, and acceleration and a small response time were observed at 1.0 A, which has a damping force similar to that of the suspension of a general vehicle. As the applied current decreases, the maximum values and response times increased, as shown in Table 11 and Table 12.

However, at a speed of 90 km/h, large displacement, speed, and acceleration maximum values and a slow response time were seen at 1.0 A, which has a damping force similar to that of a general vehicle suspension. The maximum value and response time decrease as the applied current decreases.

To verify the AMESim program analysis results, the Laplace transform of sinusoidal function Equation (29) is as follows:

$$Z_r(s)={\displaystyle \frac{0.01\times 5.24}{s^2+{5.24}^2}}$$

(33)

Substituting the vehicle parameters from Table 2 into Equation (33), the transfer function is as follows:

$${\displaystyle \frac{Z_s\left(s\right)}{Z_r\left(s\right)}}={\displaystyle \frac{4698.83\times (s+15.41)}{\begin{array}{c}{s}^4+58.24s^3+1740.59s^2\\ +44583.3s+2545997.18\end{array}}}$$

(34)

To find the function $Z_s\left(t\right)$, rearrange Equation (34) for $Z_s\left(s\right)$.

$$Z_s(s)={\displaystyle \frac{4698.83\times \left(s+15.41\right)}{\begin{array}{c}{s}^4+58.24s^3+1740.59s^2\\ +44583.3s+2545997.18\end{array}}}\times {\displaystyle \frac{0.01\times 5.24}{s^2+{5.24}^2}}$$

(35)

A comparison between $Z_s\left(t\right)$ calculated by the inverse Laplace transform using Python and the displacement graph obtained from AMESim for periods of 6 to 8 is shown in Figure 37. The error rates of the maximum amplitude and period were 5.23% and 0.01%, respectively, as shown in

Table 14. Comparison of AMESim and theoretical model.

	Max Amplitude [m]	Period [s]
AMESim	0.0143	1.20000
Theorical Model	0.013589	1.20012
Error [%]	5.23	0.01

. This indicates that the AMESim model used in the numerical analysis is reasonable.

4. Conclusions

This study optimally designed an MR damper with an additional flow path for vehicle and road surface model excitation with sinusoidal frequency excitation. Using ANSYS and AMESim software, the movement of the upper part of a vehicle equipped with an MR damper with an additional flow path was observed to examine its damping performance. The following conclusions were obtained in this study:

(1)

The MR damper with a non-magnetic orifice significantly reduces abrupt variations in damping force at very low piston velocities, decreasing force variation by approximately 56% compared to a conventional MR damper. This design modification effectively resolves the “block-up phenomenon,” resulting in smoother damping transitions and improved ride comfort during low-speed vertical vibrations;

(2)

The MR damper exhibits excellent performance in controlling the acceleration of the upper part of the vehicle by adjusting the applied current when the vehicle is driven at low and medium speeds. At 30 km/h, it could be adjusted by up to 37.9% from 0.0451 m/s² to 0.0622 m/s², and at 60 km/h it could be adjusted by up to 18.7% from 1.66 m/s² to 1.97 m/s²;

(3)

The MR damper can significantly reduce the displacement, speed, and acceleration of the upper part of the vehicle owing to road excitation when the vehicle is driven at a high speed. The displacement decreased by up to 14.1% from 0.00708 m to 0.00608 m, the velocity decreased by up to 7.09% from 0.127 m/s to 0.118 m/s, and the acceleration decreased by up to 7.73% from 1.94 m/s² to 1.79 m/s².

The MR damper has resolved the block-up phenomenon through the application of the additional flow path. Additionally, it was confirmed that the proposed MR damper has excellent performance in controlling the range of displacement, speed, and acceleration of the upper part of the vehicle at various vehicle speeds and sinusoidal road surface excitations. Future research will focus on validating the design method proposed in this study through the fabrication and testing of actual dampers, thereby enabling a more thorough evaluation of their performance in real-world applications. Additionally, further studies will be conducted on bump characteristics to analyze the response behavior under various driving conditions. Building upon these findings, the application of MR dampers to vehicle suspension systems is expected to enhance system performance and increase stability, leading to synergistic effects. These future research endeavors and their potential applications are anticipated to make a significant contribution to accelerating the practical implementation of MR damper technology.