The mathematical model of a magnetorheological shock-absorber taking into account of the dependence of rheological properties from the magnetic field is developed. Performance characteristics of the MR shock-absorber (dependences of force on value of control electric signal taking into account of shock-absorber geometry, rod displacement, rheological properties of MR fluid) are calculated. The analysis of influence of control electric signal value on MR shock-absorber performance characteristics depending on amplitudes and frequencies of piston movement is carried out

E. V. Korobko

V. A. Bilyk

V. A. Kuzmin

English

Vibroengineering PROCEDIA

  ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533 19 
Simulation of Performance Characteristics of a 
Magnetorheological Shock-Absorber at the Dependence of 
Rheological Properties from the Magnetic Field 
V A Bilyk, E V Korobko and V A Kuzmin 
SIMULATION OF PERFORMANCE CHARACTERISTICS OF A MAGNETORHEOLOGICAL SHOCK-ABSORBER AT THE DEPENDENCE OF RHEOLOGICAL PROPERTIES FROM THE MAGNETIC FIELD. V A BILYK, E V KOROBKO AND V A KUZMIN 
V A Bilyk1, E V Korobko and V A Kuzmin 
Luikov A.V. Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus,  
15 P Brovki st., Minsk, 220072, Belarus 
 
E-mail: snowsoft@tut.by 
Abstract. The mathematical model of a magnetorheological shock-absorber taking into account of the 
dependence of rheological properties from the magnetic field is developed. Performance characteristics of the 
MR shock-absorber (dependences of force on value of control electric signal taking into account of shock-
absorber geometry, rod displacement, rheological properties of MR fluid) are calculated. The analysis of 
influence of control electric signal value on MR shock-absorber performance characteristics depending on 
amplitudes and frequencies of piston movement is carried out. 
1. Introduction 
Now active development of new compositions of magnetorheological (MR) fluids, theoretical and 
experimental research of their rheological properties and search of their practical application in the 
technician are conducted [1-3]. The most demanded application area of MR fluid is the machinery 
construction [3, 4]. In the scientific and technical literature various variants of active and semi-active 
cushion systems are considered [1, 3, 4]. In such systems one of the main elements is the controlled 
shock-absorber with a MR fluid, named MR shock-absorber. Last years necessity of development of 
MR shock-absorber mathematical models and numerical estimation of its performance characteristics 
increases by reason of occurrence of large quantity of MR fluid compositions with various rheological 
properties [1, 2, 4]. By development of controlled cushion systems of the concrete vehicle is required 
the designing of new geometry of a MR shock-absorber and its elements [4, 5]. 
The purpose of this work is simulation and analysis of performance characteristics of the designed 
MR shock-absorber taking into account rheological properties of the MR fluid, influence of the 
magnetic field and such as modes of dynamic rod load under the harmonic law. 
2. Problem statement 
In this work, performance characteristics of the magnetorheological shock-absorber, which is used for 
a controlled vibroprotective system, are modelled and numerically evaluated. Figure 1 shows the 
scheme of the magnetorheological shock-absorber and its basic elements. Hydraulic resistance is 
created in the annular channels 5 (Figure 1), thus the area 3 of the channels 5 defines a regulation zone 
of MR fluid viscosity at influence by the magnetic field (Figure 2). At electric current, giving on the 
solenoid (the inductance coil) 6, creates the magnetic field with the flux 7 passing through the core 
which represents the piston 4 rigidly connected with the rod 8 and located in the shock-absorber 
cylinder 1. 
Resistance force of the telescopic MR shock-absorber, depending on time t, is defined from the 
equation system [5]: 
 ( ) ( ) ( ) ( ), , ,mra fr gas fF t B F t B F t F t B= + +   (1) 
where 
                                                     
1  To whom any correspondence should be addressed. 
SIMULATION OF PERFORMANCE CHARACTERISTICS OF A MAGNETORHEOLOGICAL SHOCK-ABSORBER AT THE DEPENDENCE OF RHEOLOGICAL PROPERTIES FROM THE 
MAGNETIC FIELD. V A BILYK, E V KOROBKO AND V A KUZMIN 
20 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533  
 ( ) ( ) ( )( )0 1, · ),(fr pF t B F с P t B sgn v t= + ∆   (2) 
 ( ) ( )( )( )0 0 0/ – – ·
m
gas r r r
F t P V V S l z t S 
 
=   (3) 
 ( ) ( ) ( ), –  ,f p rF t B S S P t B= ∆    (4) 
 
 
Figure 1. The scheme of the MR shock-absorber:  
1 – the cylinder; 2 – a MR fluid without the 
magnetic field; 3 – a MR fluid in the magnetic 
field; 4 – the piston; 5 – the annular channel;  
6 – the solenoid; 7 – the magnetic field lines;  
8 – the rod; 9 – the pneumatic camera with a gas. 
 Figure 2. The geometry of the annular 
channel of the piston of the controlled shock-
absorber with a MR fluid in the cylindrical 
coordinate system. 
 
Ffr, Fgas, Ff – the forces of dry friction, gas friction in a pneumatic camera and hydraulic resistance 
of MR fluid in an annular channel. The inertia force of a MR shock-absorber piston is neglected [4, 5]; 
Sp and Sr – the area of cross-section section of a piston and a rod accordingly; F0 and с1 – the 
parameters, which define dry force from an experiment; m – the index of power; t – the time; ΔP=P1–
P2 – the pressure drop; vp – the piston velocity; z, r – the coordinates; Rp, Rr – the piston and rod radii; 
R1, R2 – the internal and external radii of a channel; lr – the initial piston position; B – the magnetic 
flux density. 
Let's define the pressure drop in an orifice channel of the piston. A flow of an incompressible 
viscoplastic MR fluid in the annular channel at the cylindrical coordinate system is described by the 
equation system, which contains movement equation, rheological equation and continuity equation: 
 ( )
( )
( )
( )
( ) ( ) ( )
2
1
1
01
, , , , 2 ,
n B R
R
Bu p u u
r B r z B u r t r dr Q t V t
t z r r r u r r
τ
ρ η η µ pi
−
∂ ∂ ∂ ∂ ∂   
= − + = + = =   ∂ ∂ ∂ ∂ ∂ ∂ ∂   
∫ ɺ  (5) 
where η – the apparent viscosity; u – the velocity; p – the pressure; ρ – the density of MR fluid; Q – 
the volume flow. Volume flow of MR fluid in the channel is defined [4] from the sinusoidal law of 
piston displacement x(t)=Am·sin(2pift) with defined values of frequency f and amplitude Am: 
 ( ) ( ) ( ) ( )2 2 2p r p r2 cos(2 )mQ t S S x t R R fA ftpi pi= − = −ɺ   (6) 
We use net method [6] for calculation of equation system (5).  
The MR fluid is considered motionless, and no-slip conditions are set on walls of an annular 
channel: 
 ( ) ( ) ( )1 2, 0 0  and    , 0,   , 0u r t u r R t u r R t= = = = = =   (7) 
MR fluid has been developed in A.V. Luikov Heat and mass transfer institute of the National 
academy of sciences of Belarus [2]. Measurements of rheological properties of the MR fluid have been 
SIMULATION OF PERFORMANCE CHARACTERISTICS OF A MAGNETORHEOLOGICAL SHOCK-ABSORBER AT THE DEPENDENCE OF RHEOLOGICAL PROPERTIES FROM THE 
MAGNETIC FIELD. V A BILYK, E V KOROBKO AND V A KUZMIN 
 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533 21 
executed in a range of shear rates 0.01–1000s-1 on Rheometer “Physica MCR 301” of manufacturer 
“Anton Paar” with a measuring cell “MRD70”.  
3. Results and discussion 
Rheological curves are constructed for the experiments at various values of magnetic flux density 
(Figure 3). All of them can be described by viscoplastic model Herschel-Bulkley (Figure 3 and 4): 
 ( ) ( ) ( ) ( )0, ·
n B
B B Bτ γ τ µ γ= +ɺ ɺ   (8) 
where parameters of dynamic yield stress τ0, plastic viscosity μ and index of power n depend on 
magnetic flux density B [in scale of mT] accordingly: 
 ( ) 20 140.0 19.273 0.1023B B Bτ = + + , [Pa]  (9) 
 ( ) 2200.1 8.5611· 0.1146·B B Bµ = + + , [Pa·s]  (10) 
 ( ) ( )0.3 1 0.00216·n B B= +   (11) 
 
 
 
Figure 3. The dependence of shear stress τ of the MR 
fluid on shear rate γɺ  at different values of magnetic 
flux density B:  
1 – B=0mT; 2 – 50; 3 – 100; 4 – 200; 5 – 300;  
6 – 500. Points – experiment; lines – the model (8). 
 Figure 4. The dependence of shear stress τ 
of the MR fluid on shear rate γɺ  and 
different values of magnetic flux density B 
by the model (8). 
 
Overall relative variation coefficient of Herschel-Bulkley model is equal 9.8%, that precisely 
enough describes viscoplastic behaviour of the MR fluid at different magnetic flux density B in the 
range 0–500mT (Figure 3). As a result, this model (8)–(11) allows to define the general calculated 
dependence of shear stress on shear rate and magnetic flux density (Figure 4). Having defined 
rheological state equation in a form of Herschel-Bulkley model and dependences its parameters (yield 
stress, plastic viscosity, index of power) on magnetic flux density, the problem of MR fluid flow in the 
annular channel of the controlled shock-absorber taking into account construction geometry, external 
influence of a magnetic field and conditions of dynamic loading of rod can be solved. 
For calculation of performance characteristics of the MR shock-absorber we use following data: 
Rr=0.008m; Rp=0.02m; R1=0.016m; R2=0.017m; L=0.02m; L1=L2=0.01m; P0=10MPa; V0=0.00009m3; 
ρ=2600kg/m3; F0=60Н; с1=3·10-6N/Pa; lr=0.08m. 
Results of numerical modelling of MR shock-absorber force are resulted in Figure 5 at rod motion 
under the harmonic law with amplitude of 20mm, frequency of 1 and 3Hz. For the purpose of 
SIMULATION OF PERFORMANCE CHARACTERISTICS OF A MAGNETORHEOLOGICAL SHOCK-ABSORBER AT THE DEPENDENCE OF RHEOLOGICAL PROPERTIES FROM THE 
MAGNETIC FIELD. V A BILYK, E V KOROBKO AND V A KUZMIN 
22 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533  
simplification it is admissible that the force component Fgas is equal to zero.  
 
 
a) Am=20mm, f=1 Hz; 
 
 
b) Am=20mm, f=3Hz. 
Figure 5. The dependence of MR shock-absorber force from rod displacement z at different values of 
magnetic flux density B: 1 – B=0 mT; 2 – 50; 3 – 100; 4 – 200; 5 – 300; 6 – 500. 
 
The force Fmra increases with growth of magnetic flux density: Fmra=263N (Figure 5-a) and  
Fmra=395N (Figure 5-b) at the absence of a magnetic field, Fmra=8781N (Figure 5-a) and  
Fmra=10198N (Figure 5-b) at magnetic flux density B=500mT. Thus, relative increase of force is equal 
33.4 (Figure 5-a) and 25.8 times (Figure 5-b).  
4. Conclusions 
Thus, the mathematical model of a MR shock-absorber taking into account dependence rheological 
properties of the MR fluid on shear rate and magnetic flux density is developed. Performance 
characteristics of the MR shock-absorber (dependence of force on rod displacement taking into 
account shock-absorber geometry, rheological properties of the MR fluid, magnetic flux density) are 
calculated. The analysis of performance characteristics is made for different loading conditions. 
Dependences of MR shock-absorber resistance force on magnetic flux density are defined. These 
dependences can be used by development of control algorithm of shock-absorber performance 
characteristics for electronic control units. 
Acknowledgments 
This research was performed under Belarusian project “Mechanika 1.03.03” of government program 
of scientific research “Mechanics, technical diagnostics, metallurgy” (2011–2013 years). 
References 
[1] Electrorheological Fluids and Magnetorheological Suspensions: proceedings of the 12th Int. 
Conf. 2011 Philadelphia USA ed R Tao 
[2] Korobko E V, Novikova Z A and Zhurauski M A 2011 Conf. book and Book of abstracts of 7th 
Annual European Rheology Conf. (Suzdal Russia) 114 
[3] Wang X and Gordaninejad F 2001 Proc. of SPIE Conf. on Smart Materials and Struct. 4331 
82–91 
[4] Dixon J C 2007 The Shock Absorber Handbook: Second Edition John Wiley & Sons 
[5] Derbaremdiker A D 1985 Vehicle shock-absorbers (Moscow: Mashinostroenie) (in Russian) 
[6] Paskonov V M, Polezhaev V I and Chudov L A 1984 Numerical simulation of heat and mass 
transfer processes (Moscow: Nauka) (in Russian) 


Simulation of Performance Characteristics of a Magnetorheological Shock-Absorber at the Dependence of Rheological Properties from the Magnetic Field

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Simulation of Performance Characteristics of a Magnetorheological Shock-Absorber at the Dependence of Rheological Properties from the Magnetic Field

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