Mathematical Problems in Engineering

Volume 2015, Article ID 905731, 15 pages

http://dx.doi.org/10.1155/2015/905731

## Method for Modeling Electrorheological Dampers Using Its Dynamic Characteristics

ITESM Campus Monterrey, Avenue Eugenio Garza Sada 2501, Col. Tecnológico, 64849 Monterrey, NL, Mexico

Received 6 June 2014; Accepted 30 August 2014

Academic Editor: Xingsheng Gu

Copyright © 2015 Carlos A. Vivas-Lopez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A method for modeling an *Electrorheological* (ER) damper is proposed. The modeling method comprehends two simple steps: characterization and model customization. These steps are based on the experimental data of the damper behavior. Experiments were designed to explore the nonlinear behavior of the damper at different frequencies and actuation signals (i.e., automotive domain). The resulting model has low computational complexity. The method was experimentally validated with a commercial damper. The *error-to-signal Ratio* (ESR) performance index was used to evaluate the model accuracy. The results were quantitatively compared with two well-known ER damper models: the *Choi* parametric model and the *Eyring-plastic* model. The new proposed model has a 44% better ESR index than the *Choi* parametric model and 28% for the *Eyring-plastic* model. A qualitative comparison based on density plots highlights the advantages of this proposal.

#### 1. Introduction

In an automotive suspension system the shock absorber has the purpose of dissipating the energy of the motion of the vehicle caused by the road disturbances. This energy dissipation allows the suspension to achieve two important objectives: decrease the vertical acceleration and maintain the tires in contact with the ground. Passive suspension systems are tailored to achieve a tradeoff of these objectives [1]. Semiactive (SA) suspension systems use a particular type of shock absorber which is capable of online modifying the amount of energy that can dissipate. This change on the damper needs to be controlled, to achieve the desired objectives.

The electrorheological (ER) damper is a hydraulic device, which is filled with a mixture of low viscosity oil and particles that are sensitive to an electric field. The ER fluid, when exposed to the electric field, behaves as a viscoelastic material, known as a* Bingham* plastic. This means that ideally it behaves as a solid at low stress efforts, but it flows as a viscous fluid when this force reaches its yield stress. Furthermore, the yield stress is field dependent; it increases as the electric field does.

An accurate mathematical model to predict the nonlinear dynamic behavior of the ER damper is needed in order to get a better control of the SA suspension system. There are several contributions in this topic [2, 3]. However, most of them are highly dependent on internal physical properties of the damper (usually confidential information), demand too much computational effort, or fail to capture the nonlinear behavior of the ER damper.

A new method to model an ER damper is proposed. The method comprehends two main steps: (1) a characterization procedure where the dynamical response of the damper is analyzed and (2) a model customization where a general model is tailored. This method requires experimental data of the ER damper. The resulting model is light enough to be implemented in an embedded system. The method is validated with intensive experimental data and compared to others published.

This paper is organized as follows: in Section 2 a bibliographic review of ER damper modeling is presented. In Section 3 the experimental system and the* Design of Experiments* (DoE) are shown. Section 4 describes the proposed method. Section 5 presents the modeling procedure. Section 6 shows the results and evaluates the performance of the customized model. Finally, Section 7 concludes the paper.

#### 2. ER Damper Models: State of the Art

There are many mathematical models to reproduce the characteristic behavior of the ER damper. The major efforts have been aimed at parametric models. [4] proposes a model based on the viscoelastic characteristics of the ER damper; this model contemplates a linear passive damping force that depends on the piston velocity and a lineal SA effect that only depends on the electric field applied to the ER fluid, this is a simple model that does not contemplate hysteresis and the damping force is completely linear.

A model based on the pressure drop in the ER channel is presented by [5]; this pressure drop considers an effect that depends on the damper velocity and others that just depend on the electric field; the coefficients for this model are based on physical dimensions of the damper and physical properties of the ER fluid. [6] proposes a model using a bond graph to model de governing equation of motion of the damper, but the physical dimensions and properties of the damper and ER fluid are needed. Later [7] shows two different types of ER damper configurations. The first one is the most common, the cylindrical type, in which the ER fluid flows through an annular channel where the electric field is applied. The second one is the orifice type; this type has a mechanism located inside the piston of the damper, which regulates the flow of the ER fluid through its chambers; two models were proposed, one for each type of damper, but the ones of physical parameters are needed.

Following the same line in terms of parametric models, [8] describes a hydromechanical based model. This model divides the damper in different zones where the pressure drop is calculated and the damper force depends on those pressure drops. This model captures the damper behavior in the preyield zone, in terms of the hysteresis. The authors do not evaluate the effects of the frequency in the model and the transient behavior of the force during changes in intensity of the electric field, which is important for control purposes. [9] presents a parametric dynamic model in which the pressure drops in the annular duct are calculated with respect to time. This model represents the hysteretic behavior of the ER damper in postyield zone and its increment due to the frequency, but the assessment of the model is done with constant conditions of frequency displacement and electric field. Another model is based on a lumped parameter method, in which the sections of the ER damper (upper chamber, lower chamber, annular duct, and connecting pipe) are divided into lumps and modeled with differential equations. This model predicts the nonlinear behavior of the ER damper in the preyield and postyield zones but depends on physical properties of the damper and it is sensitive to the initial conditions, [10].

Regarding the nonparametric models, [11] presents a polynomial equation with only three constants that can be fitted by least square estimation (LSE) methods; the force of the damper only depends on the velocity of the piston. The advantage of this model is the few number of constants but it does not seem to be very accurate; also it needs a set of constants for every field manipulation interval. Another approach is the Eyring model [12] which uses an Arcsinh function with shape parameters that depends on the electric field intensity and the frequency. This model can represent the behavior in both the preyield and the postyield zone but needs the identification of every parameter in each combination of frequency and field intensity; the accuracy of the model depends on how small are the considered intervals of the variables, but when changing the between this levels the model does not consider a transient response of the force. [13] introduces a neurofuzzy training algorithm to model the force of the ER damper, using the values of the acceleration and velocity of the damper; this model captures the nonlinear behavior of the ER damper with high level of accuracy, but the evaluation was done under very limited conditions.

Most of the models are dependent on internal physical properties of the damper, ER fluid, and its design; this makes the implementation of these models very restricted (i.e., confidential information). Our proposal considers general model that is customized based only on experimental data of the ER damper. The experimental system and Design of Experiments are shown in the next section.

#### 3. Design of Experiments

A commercial ER damper was used, Figure 1(a). The damper has a stroke of ±150 mm and a force range of N. The damper is actuated by a 2 module which is controlled by a 25 kHz pulse-width modulated (PWM) signal. The PWM duty cycle range was 10–80%. Since the ER damper needs to be operated with a voltage signal of 0–5 kV the 2 module proportionally transforms the duty cycle of PWM signal to voltage. Figure 1(b) shows the characteristic force-velocity (FV) diagrams at different PWM duty cycles.