Theoretical research of mechanical behaviour of magneto-rheological fluid

A. Klevinskis*, V. Bučinskas**, D. Udris*** *Vilnius Gediminas Technical University, Basanavičiaus 28, 03224 Vilnius, Lithuania, E-mail: andrius.klevinskis@dok.vgtu.lt **Vilnius Gediminas Technical University, Basanavičiaus 28, 03224 Vilnius, Lithuania, E-mail: vytautas.bucinskas@vgtu.lt ***Vilnius Gediminas Technical University, Naugarduko 41, 03227 Vilnius, Lithuania, E-mail: dainius.udris@vgtu.lt


Introduction
Design of new features in machines and devices, reducing dimensions of equipment and wide use of computer control requires implementation of materials with special properties.In order to create smart computercontrolled drives with desired dynamic characteristics, smart materials became desired in many applications.One of successful computer controlled motion application is an implementation of magneto-rheological materials in such drives.Magneto-rheological fluids (MRF) are a class of materials whose rheological properties may be rapidly altered by applying a magnetic field [1].These materials usually are liquids with magnetically polarized ferrous particles suspended in a carrier liquid.The particle size in MRFs has on the order of micrometers and is 1-3 orders of magnitude larger than the same particles in ferromagnetic fluids [2].Main interest to MRF is caused due to their feature to change its viscosity from magnitude of applied external magnetic field as well as liquid shape change.This feature makes MRF attractive to various commercial applications, such as dampers [3,4] and brakes [4,5].This paper is intended to evaluate possibility to utilize MRF for use in computer controlled actuators.

Formulation of research using finite element method
Device designing with MRF requires estimation of behaviour of MRF in static and dynamic magnetic field and possibility to reach desired characteristics.Theoretical modelling of MRF is hardly possible due inability directly to solve non-linear differential equations.The way to do that is an application of finite element analysis.From the perspective of FEA methodology, there are necessary to perform such steps for FEA in case of MRF [6] (Fig. 1): 1. Building the geometry of object; 2. Building the finite element model; 3. Solving the finite element model; 4. Analysing the results.Mathematical modelling starts with geometry of object (Fig. 1).2D or 3D CAD programs are mostly used for geometrical modelling.CAD model should approximately, with some boundary simplification, correspond to modelled object.
CAD model is often sophisticated geometrically, therefore, in order to perform calculations using finite element method, it is idealized which helps to simplify its geometry and eliminate elements that do not affect calcula-tions and to exchange thin panes of model to surfaces, etc. Idealization of a model takes place in order to simplify the task and make the model able to mesh as well as to shorten calculation time.
Fig. 1 Algorithm of evaluation of liquid additional pressure from magnetic field influence, which is caused by current in the solenoid From magneto-rheological fluids documentation that provides Yield Stress -Magnetic Field Strength dependencies, we can calculate yield stress values in accordance to calculated magnetic field strength.Achieved values can be used for calculation of total pressure drop in a system.

The goal of the research
The main goal of an executed theoretical research is to determine pressure fluctuation change to magnetorheological fluids in the closed volume under influence of external magnetic field, thus causing fluid chamber to change their shape and deliver displacement to the system.In order to reach the goal, the following tasks have been set: to determine how external magnetic field influences general magneto-rheological fluid's pressure fluctuation; to determine parameters that define magneto-rheological fluid's property to change its state when exposed to a magnetic field.

Research methodology
The pressure drop developed in a device based on pressure driven flow mode is commonly assumed to result from the sum of a viscous component  P and a field de- pendent induced yield stress component  P , and can by approximated by [7]: where L is the length, g is the gap between fluid walls, w is the width of the flow channel between the fixed poles, Q is the volumetric flow rate, η is the viscosity with no applied field, y  is the yield stress developed in response to an applied field H.The parameter c is a function of the flow velocity profile and has a value ranging from 2 (for less than ~1) to a maximum value of 3 (for Behavior of controllable fluids is often represented as a Bingham plastic having variable yield strength [7]: where total  is the total yield stress,   H  is the strength caused by the applied magnetic field H, p  is the magnetic fieldindependent plastic viscosity defined as the slope of the shear stress versus shear strain rate relationship,  is the shear rate.
Below the yield stress, material behaves viscoelastically [7]: where G is the complex material modulus.
From the formula (1) it is evident, that in order to determine how external magnetic field influences magneto-rheological fluid's pressure loss, it is needed to determine yield stress y  .In majority of magneto-rheological fluid's paperwork there is graphs presenting yield stress vs. magnetic field strength, therefore, once magnetorheological fluid's magnetic field strength is calculated, yield stress figures are available.
To determine magneto-rheological fluid's and Magnetic field Strength (H) a Femm 4.2 program was used.The program enables solving low frequency electromagnetic problems on two-dimensional planar and axisymmetric domains [9].Femm 4.2 program has three modules: 1. Femm.exe program is designed for creation of geometry, defining of physical properties of materials and boundary conditions of model; 2. Triangle.exeprogram splits the analysed mathematical model into triangle elements; 3. Fkern.exe is a solver that solves differential equations in order to obtain a solution.
In magnetostatic problems time is invariant, therefore [9]: where H is field intensity, J is divergence, B is flux density.
Connection between B and H in each material is expressed as follows [9]: where  is permeability.
For nonlinear materials, the permeability is funcof B [9]: Femm goes about finding a field that satisfies (4) and ( 6) via magnetic vector potential approach.Flux density is written in terms of the vector potential, A, as [9]: where A is vector potential.This definition of B always satisfies (5).The, (4) can be rewritten as: For linear isotropic materials (assuming the Coulomb gauge, 0 A  ), (9) reduces to: Femm retains the form (9), so that magnetostatic problems with a nonlinear B -H relationship can be solved.
To execute the research with Femm.exe program there was a two-dimensional solenoid model created and above the solenoid there were MRF fluids situated in an aluminium box.Fig. 2 Principal scheme of solenoid: h -displacement under the influence of magnetic field Internal height of the box is 1 mm.Magnetorheological fluid is isolated from external environment by 1 mm aluminium walls.In the research solenoid, MRF and electromagnetic forces were used that were arranged symmetrically and vertically to the axis, therefore the research is axisymmetric meaning that there was a half of the model used for the research.Accurate model dimensions are provided in a Fig. 2. To strengthen magnetic field of solenoid there was an iron core used which had 2000 copper coils of 0.63 mm in diameter winded around iron core.
The analysed model with Triangle.exeprogram was split into 11852 elements.In total 6133 nodes were generated.Elements number of finite elements mesh was enlarged in those areas, where measures of the model were smaller (Fig. 3).The research was executed under changing strength of solenoid's flow from 0 to 2 A. After each experiment an average magnetic field strength H and magnetic flux density B, crossing notional horizontal fluid's symmetry axis was calculated.
After comparing magnetic lines arrangement in Figs. 4 and 5 that are formed under influence of electric streams flowing through solenoid's strands, we can see that magneto-rheological fluids distorts magnetic field's lines of force (Fig. 6).  1 and 2 as well as Figs.7 and 8.The Fig. 8 shows, that in range of current strength from 0 to 2 A, magnetic induction changes almost linearly.
From researched magneto-rheological fluids documentation that provides Yield Stress -Magnetic Field Strength dependencies, we can take the figures of yield stress in accordance to the figures of magnetic field strength.The results are presented in Table 3 and Fig. 9.Additional pressure generated by magnetic field strength causes change of magneto-rheological fluid surface, therefore the system can be treated as mechanical.

S u m m a r y
This paper is intended to research of mechanical behaviour of magneto-rheological fluid in magnetic field.This liquid is with highly non-linear properties in magnetic field and their mechanical behaviour cannot be obtained directly.Additional pressure generated by magnetic field strength causes change of magneto-rheological fluid surface, therefore the system can be treated as mechanical.There is presented algorithm of evaluation of researched liquid additional pressure from magnetic field influence, which is caused by current in the solenoid.There is algorithm and method proposed how finite element method can be used to find additional pressure.The results of this solution give understanding of additional pressure characteristics of magneto-rheological fluid and possibility to use it for magnetically driven mechanical drive.Finally, conclusions on results are made.

Fig. 3
Fig. 3 Calculation of the magnetic field strength: a -meshed solenoid; bdensity plot of solenoid In the research there were four Lord Corporation magneto-reological fluids of different characteristics used that are displayed in documentation [10-13].The research was executed under changing strength of solenoid's flow from 0 to 2 A. After each experiment an average magnetic field strength H and magnetic flux density B, crossing notional horizontal fluid's symmetry axis was calculated.After comparing magnetic lines arrangement in Figs.4 and 5that are formed under influence of electric streams flowing through solenoid's strands, we can see that magneto-rheological fluids distorts magnetic field's lines of force (Fig.6).

Fig. 4 Fig. 6
Fig. 4 Simulated magnetic flux lines cross the air

Table 1
Calculated magnetic field strength values of different magneto-rheological fluids