Finite Element Analysis of Linear Magneto-Rheological

Damper

Raju Ahamed, M. M. Rashid and Hazlina Bint Yusuf

Department of Mechatronics Engineering, International Islamic University Malaysia, 53100, Kuala Lumpur, Malaysia

Email: raju.ahamed@live.iium.edu.my

Abstract

In this paper a linear MR damper model is designed using COMSOL finite element simulation

software. In conventional finite element MR damper modeling, selecting the MR fluid property

is a big challenge, which has been resolved entirely in this COMSOL MR damper modeling. The

overall design procedure of the damper model is discussed briefly in this work. This dynamic

simulation clearly illustrates the magnetic flux generation around the piston coil. By this

COMSOL simulation the induced magnetic flux density, magnetic field intensity, applied current

density, temperature gradient etc. are analyzed.

Keywords: COMSOL Multiphysics, MR damper, MR fluid, Vehicle suspension

1. Introduction

The usage of MR damper is vast for both low- and high-speed application [1, 2]. They are mainly

used in automobiles [3], civil construction such as buildings, bridges and frame structure [4, 5].

MR dampers consist a piston rod, a piston head, hydraulic and pneumatic reservoirs. These

hydraulic and pneumatic reservoirs are separated by a floating piston or diaphragm [6, 7]. The

piston head is attached by piston rod and it contains magnetic circuit (coil on bobbin concentric

to a tubular flux return). When the piston rod assembly moves inside the MR damper, the fluid

flows through a circular gap in the piston head. When current supplied to the coil in the piston

head, a magnetic field is induced in the gap and raises the yield stress of the MR fluid in the

circulation gap. Increasing yield stress raises the pressure drop down in piston head and changes

the velocity profile of the MR fluid in the gap. In this way, MR dampers produce controllable

field-dependent yield force, in addition to passive velocity dependent viscous damping force.

The MR damper has been analyzed in finite element method by many researches. Yasrebi et al

presented an analytical and experimental modeling of a Magneto-rheological damper and used

ANSYS software for finite element analysis [8]. In their research only the magnetic field effect

on MR damper fluid that flows in the annular gap between cylinder and piston head is modeled.

Zekeriya Parlak et al proposed a MR damper design optimization method and used ANSYS

finite element analysis and CFD tools to obtain optimal value of design parameters [9]. In this

research the MR damper finite element analysis and CFD analysis have done separately. Khan et

al analyzed different 2-D Axisymmetric MR models by finite element method using ANSYS

software. The design optimization has done only for piston design variation for analyzing the

vibration controlling force. According to the previous finite element analysis, all researchers

have used ANSYS software where the MR fluid property selection is the big challenge. For

solving this problem, in this study an MR damper model is designed and COMSOL Multiphysics

software is used for finite element analysis. Finally, the analytical study is compared with

experimental results for the proposed model’s validation.

2. COMSOL Simulation Physics for the Proposed MR Damper

Finite element analysis of the proposed MR damper is performed with COMSOL Multiphysics 5,

which has a powerful, interactive environment for modeling and solving all kinds of scientific

and engineering problems based on partial differential equations (PDEs). The Finite Element

Analysis (FEA) of the proposed MR damper model is analyzed by utilizing the Magnetic field

and Laminar Fluid Flow physics modules. At the beginning of a time-dependent and stationary

solver, the relations are well-defined by Maxwell’s equations, including the equation of

continuity for constant electric charge density as defined in Eq. (1), and Ampire’s law is defined

by Eqs. (1a) and (1b).

∇ . J=0 ; (1)

( jωσ−ω2 ϵ 0ϵ r ) A+∇× (μ0−1 μr

−1 B )−σV × B=Je ; (1a)

B=∇× A . (1b)

here, H is the magnetic field intensity and J the current density. Constitutive relations are

established between magnetic field intensity and magnetic flux density (B) in Eq. (1b), and

between current density and electric field intensity (E) in Eq (2).

J=σE+Je (2)

In the proposed MR damper simulation, the magnetic field is produced by an electromagnet.

Here at the COMSOL simulation model, the wound coil is considered as electromagnet and the

magnetic field provided by this wound coil is necessary to energize the MR fluid. By varying the

current through the electrical coil the magnetic flux density can be varied and the MR fluid can

be energized accordingly. In this COMSOL MR damper model, the DC current is applied in the

form of current density i.e. current in the coil area. In COMSOL the current density is calculated

by Eq. (3).

Je=N I coil

Aecoil

(3)

In Eq. (3), N is the number of turns, I coil is the current, A is the coil area and Je is the current

density. Static magnetic analysis is done by 2D axisymmetric model as shown in Fig. (2). Eq. (4)

is used to find out the damping force (F v).

−ρ ω2u2=∇ . s+Fv ; (4)

s=s0+C : (ϵ−ϵ 0−ϵ inel ) ; (4a)

ϵ=12

(∇u2+(∇u2 )T ); (4b)

−ρ ω2u+ jωα dM ρu=∇ . ( s+ jω βdK s )+F v . (4c)

However, the piecewise mathematical description for solving the multi-dimensional problem of

the fluid Eq. (5) is used.

ρ (u .∇ ) u=∇ .[−pl+μ(∇u+ (∇u )T )−23

μ (∇ .u )l ]+F ; (5)

∇ . ( ρ . u )=0; (5a)

σ i j=( σ y

γ+η0) ϵ i j σ>σ y ;

(5b)

ϵ y=0σ>σ y ; (5c)

γ=√ 12

˙ϵ i j . ˙ϵ i y .(5d)

In Eq. (5b), σ i j Represents the deviatoric stress tensor, σ y is the yield stress of the fluid, and η0

is its base viscosity. In the Eq. (5d) the term ϵ i j is the rate of strain tensor, and γ is the strain rate

magnitude.

3. COMSOL Finite Element Modeling of the Proposed Linear MR Damper

The proposed MR Damper model for Finite Element Analysis (FEA) is shown in Fig. (1a). The

MR damper is analyzed as a 2D axisymmetric model, as shown in Fig. (1c). For a given current,

the magnetic flux density at the damper piston, MR Fluid and the damper housing is determined.

The damper inner piston, MR fluid gap, outer piston and the damper housing are the components

that complete the magnetic circuit around the coil. The magnetic flux density can be varied by

changing the current through the coil, so the flux passing through the MR fluid will also be

changed and the MR fluid will be energized accordingly. A plastic liner gap exists among the

inner piston, MR fluid gap and the electrical coil, which is a thin rectangular region.

(a)

(b) (c)

Figure 1: (a) MR Damper model (b) Cross sectional view of MR Damper and (c) 2-D axisymmetric MR Damper model.

3.1 COMSOL MR Damper Model Construction and Meshing

In COMSOL, the MR damper model is constructed by selecting model wizard from the new

window and the 2D axisymmetric is selected from model wizard window. Magnetic Field (mf)

and Fluid Flow physics are selected from the select physics tree and from the select study tree,

stationary and time dependent study is selected as selected physics Interfaces and the required

parameters are inserted in the parameter table. For the MR damper 2D model simulation, suitable

a Pistonb Air gapc Damper walld Coile MR fluid

a

d

b

c

e

materials are selected from the Add materials window and at that time various boundary

conditions are selected.

After building the model and assigning Physics and material attributes, the subsequent stage is

the meshing of the model, which is an important part of the COMSOL simulation. The meshed

models are shown in Fig. (2a).

3.2 Characterization of MR Damper’s Finite Element Numerical Model

In the COMSOL software there are different kinds of solver such as Stationary, Time dependent,

Frequency domain etc. used for finite element analysis. Different solvers have different kinds of

advantages. The time dependent study step is selected and solves the problem in the time

domain. It automatically takes the initial values for the vector potential of the stationary solution.

In contour plot the 2D Magnetic Flux Lines option is used to generate the 2D magnetic flux lines

around the electrical coil of the damper’s axisymmetric model as shown in Fig. (4) below.

(a)

(b) (c)

Figure 2: (a) Mesh plots of the MR Damper, (b) and (c) Magnet flux density norm (surface plot)

of axisymmetric MR damper model

Fig. (2b) is a representation of the damper’s magnetic flux density value is in tesla for an

excitation current of 0.5 A, where the flux density varies from 0 to 1.25 Tesla in the damper

body. Observing closely, the production of higher flux density around the piston coil area is

found and expressed by the color variation. For a clearer inspection of the magnetic flux density

distribution, the vector plot results are observed by the phi component which is illustrated in Fig.

(3). Figs (4) and (5) show the relation between the applied current with magnetic field density.

(a) (b)

(c)

Figure 3: Magnetic Flax Density (a) line plot, (b) contour plot and (c) both

contour and Surface plot

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.5

1

1.5

2

2.5

3

3.5

Applied Current (A)

Mag

netic

Flu

x De

nsity

(T)

Figure 4: Magnetic Flux density vs. Applied current

0.1 A1%

0.2 A2%

0.3A2% 0.4 A

3% 0.5 A4%

0.6A5%

0.7A6%

0.8 A7%

0.9 A7%1.0 A

8%

1.1 A9%1.2 A

10%

1.3 A11%

1.4 A12%

1.5 A13%

Figure 5: Magnetic flux density and current density relation

From Fig. (4) it can be said that, with the increasing of applied current the magnetic flux density

increases. When applied current is 0.1 A than the magnetic flux density is 0.22 T. Finally, with

the increasing of applied current the magnetic flux density rises to 3.5 T. With the growing of

applied current the magnetic flux density increases and reaches to 13%, as shown in Fig. (5).

With the increasing magnetic flux density, the main properties of the MR fluid change rapidly

and within millisecond become semisolid and resulting an increment in damping force, which is

shown in Fig. (9). Fig. (6) presents the surface Magnetic fields for z and r component.

(a) (b) Figure 6: Surface Magnetic Field (a) z component (b) r component

From Fig. (6) it is more clear that, magnetic field created around the piston head coil in the

presence of applied current as shown by red and blue color. The induced magnetic field inside

the MR fluid is shown by a blue color and inside the piston coil is displayed by red color. Fig. (7)

Shows the surface electric energy density and electric displacement field norm.

(a) (b)

Figure 7: (a) Surface electric energy density and (b) Surface electric displacement

The electric current is supplied inside the piston coil and this current energy density is spread

gradually through the overall coil turns which is shown in Fig. (7a), whereas the electric

displacement is presented in Fig. (7b). The color disparity shows the electric energy density and

electric displacement variation. Fig. (8) presents the effect of MR fluid under magnetic field.

(a) (b)

(c) (d)

Figure 8: Effect of MR fluid (a) arrow line plot, (b) surface plot, (c) Surface velocity and (d)

surface dynamic viscosity.

According to Fig. (8), these fluxes vary with respect to the applied current to the coil. As the

current increases, the magnetic flux line value increases where the maximum magnetic flux line

is located around the piston coil. With the implication of excitation current to the MR damper

magnetic field of the piston head change the property of the MR fluid which is shown by the red

color in Fig. (8).

CONCLUSION

Design simplicity and versatile applicability make the MR damper an interesting research topic

for researchers. So researchers are trying to come out the design concepts and characterizing the

nonlinear behavior of the MR damper accurately. Finite element method is one of the recent

technique for designing and characterizing MR damper. Most of the researchers are using

ANSYS finite element analysis software for MR damper design, where the accurate selection of

the MR fluid’s property is impossible. As a solution to this problem in this paper the MR is

developed by COMSOL Multiphysics 5 finite element simulation software by choosing the MR

fluid parameters accurately. Here a linear MR damper has been designed in solidWorks and

characterized in COMSOL, which has a suitable applicability in the vehicle suspension system.

For characterizing perfectly, the MR dampers magnetic flux density, forces have been observed

for various current.

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