KEYWORDS: Magnetorheological (MR) fluid damper; Herschel-Bulkley model;
Magnetostatic analysis; Regression model; Artificial neural network;
Optimization.

The current study presents the optimal design of a magnetorheological (MR) damper
for vehicular application. The optimization includes the Herschel-Buckley (H-B) model

for defining the objective functions and a data-driven model for the magnetostatic anal-
ysis of the MR valve inside the damper.

For modelling the flow of the MR fluid inside the valve, a more sophisticated H-B
fluid model is used as it is capable of accounting for the shear-thinning behaviour in an
MR fluid at high shear rates a feature which the linear Bingham Plastic (BP) model
lacks. The flow modelling encompasses the effects of non-Newtonian and Newtonian
flow observed across the valve channel and the total pressure drop across the channel

is calculated accordingly. The developed formulation is then compared with computa-
tional fluid dynamics (CFD) model for different values of parameters of the H-B model

and satisfactory results one obtained. The parameters of the H-B model are estimated
using experimentally obtained characterization results of commercial MR fluid and the
resulting damping force is compared to that obtained with the BP model.
The pressure drop across the channel is dependent upon the parameters of the H-B
model, which in turn are functions of magnetic strength inside the MR valve. Besides
this, different objective functions in the optimization problem are also dependent upon
the magnetic strength. Therefore, conventionally, for calculating the magnetic strength
in various regions of the valve, a magnetostatic model is used. This conventional model
is linear, incorporating constant magnetic permeability, and is valid for only low-ranges
of current. In this work, a finite element (FE) model is made and it is seen that magnetic
strength holds a non-linear relationship with the applied current and gradually saturates

at higher current. It is also observed that the magnetic saturation limit changes for dif-
ferent geometric dimensions. For this, a polynomial model for magnetic permeability

of the MR fluid is proposed using FE results and is incorporated in the conventional
magnetostatic model. The proposed model is able to capture the non-linear behaviour;
however, it is valid only from low to mid-ranges of current and is unable to capture the
magnetic saturation limits. The FE analysis also revealed that the valve-core and outer housing regions of the
MR valve draws maximum magnetic flux density. Continuous exposure to high flux

may result in losing magnetic properties. Therefore, using this observation, the mag-
netic flux in these two regions are assumed constant and a volume-constrained geomet-
ric design is proposed in which two geometric parameters are proposed to define the

whole geometry. These geometric parameters are kept as two design variables (DVs)
for the geometric optimization problem.
Further, for calculating the magnetic strength in the active and valve-core regions,
regression models are fitted using the FE data generated from three different design of
experiments (DoE). For a particular DoE technique, improved regression models are
chosen after significance level test. The improved models resulting from different DoE
techniques are then compared and the most suitable models are chosen based on the

coefficients of determination. The fitted model is able to explain the non-linear behav-
iour; however, it is incapable of explaining the magnetic saturation effects observed at

high currents.

To model the non-linear behaviour and magnetic saturations at high currents for dif-
ferent geometric dimensions, an artificial neural network (ANN) model having feedfor-
ward architecture is proposed. The number of activation units (neurones) in the hidden

layer is decided through error analysis and the optimal weights are obtained using the

Levenberg-Marquardt algorithm using MATLAB® software. It is found that the pro-
posed ANN is capable of capturing the intriguing details shown by the magnetic

strength inside the MR valve.
Finally, for the optimization problem, all damping-related objective functions are
defined based on the H-B model formulation and the suitable data-driven models are
chosen for calculating the magnetic strength in the active and valve-core regions. Using
this combined formulation, four objective functions are defined, namely, maximising
damping force and dynamic range, and minimising inductive time constant and control
energy. These objective functions are combined with individual weights allotted to

them and formed into a single-objective minimization problem with three design vari-
ables (DVs), namely, coil width, active length and applied current. To keep the mag-
netic strength in the valve-core region less than the saturation limit, a constraint of 1.5

T is defined. Three sets of weight factors are decided according to vehicle applications.
The optimization problem is solved using a combined Genetic Algorithm (GA) and Sequential Quadratic Programming (SQP) approach. The reference values of objective
functions are chosen by generating 1000 uniformly distributed initial points of DVs and
employing SQP methods to all 1000 points. Among 1000 initial points, the candidate
value is chosen based on the minimum value of the objective function resulting from
the SQP method. Further, using suitable reference values, the GA method is employed
to obtain a global minimum. To guarantee a global minimum, the SQP methods is again
employed keeping optimal values of GA as the initial values and optimal values are

obtained accordingly for different sets of weights. The optimal results are then com-
pared and discussed for their optimality.