A method for modeling Electro-Rheological (ER) dampers is proposed. It consists in two sequential steps: Characterization and Customization. Both steps are. This study presents nondimensional analysis of an Eyring constitutive model to describe the field-dependent behavior of an electrorheological. This paper presents the design, analysis, testing and modeling of an electrorheological (ER) fluid damper developed for vibration and seismic.

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A method for modeling an Electrorheological ER damper is proposed. The modeling method comprehends two simple steps: 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.

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: A qualitative comparison based on density plots highlights the advantages of this proposal. 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: 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 [ 23 ]. However, most of them are highly dependent on internal physical properties of the damper usually confidential informationdemand 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: 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: Section 4 describes the proposed method. Section 5 presents the modeling procedure.

Electroryeological 6 shows the results and evaluates the performance of the customized model. Finally, Section 7 concludes the paper. There are many mathematical models to reproduce the characteristic behavior of the ER damper. The major efforts have been aimed eoectrorheological parametric models. 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.

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 electrorheologifal field is applied.

### An electrorheological fluid vibration damper – IOPscience

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 dampdr 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.

This model represents the da,per behavior of the ER damper in postyield zone and its increment due to the frequency, but the assessment of the electrorheollgical is done with constant conditions of frequency displacement and electric field. Another model is based on electroeheological 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.

## ERF damper

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 eamper 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 daper 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.

Most of the electdorheological are dependent on internal physical properties of the damper, ER fluid, and its design; this makes the implementation of these models very restricted i.

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. A commercial ER damper was used, Figure 1 a.

The experimental setup, Figure 2 aconsists of three modules: A series of displacement sequences and actuation signals damped used to capture the static and dynamic relations between velocity, displacement, actuation signal, and the damper force [ 14 ]. These sequences ensure the ER damper will be tested in the automotive domain. The sequences used for the displacement of the piston are road profile RPFigure 2 b[ 15 ], and decreasing-amplitude stepped frequency sinusoidal DSFS signal, Figure 2 c[ 16 ].

The RP sequence is used to test the ER damper under standard automotive conditions and represents the motion in a vehicle suspension when the car is driven through a specific surface.

The RP smooth highway is the most common road eldctrorheological commercial vehicles. Electroreological the other hand, the DSFS signal is used to analyze the transient response and the hysteresis loops when changes in magnitude and frequency are present. In addition, DSFS sequence has a similar frequency spectrum experienced by automotive suspension systems.

For the PWM duty cycle, the Stepped inCrements SC signal, Figure 3 ais used to study the effect of the actuation signal under different displacements sequences. Increased clock period signal ICPSFigure 3 b eamper, and pseudorandom binary signal PRBSFigure 3 csequences are used to analyze the damper epectrorheological response under actuation signal variations.

The ICPS is a signal with random amplitude variations, whereas the PRBS is a signal whose amplitude switches between two constant values with a random frequency. The DoE consists of a combination of displacement and actuation sequences i. For the actuation signal the use of a PRBS signal shows how the damper behaves electrorheologicak operated at its limit conditions; for the case of ICPS the full range of force was shown.

The proposed method does not need a priori knowledge i. In the DoE step, the experiments are defined on the automotive range of operation. In the characterization step, the dynamical characteristics of the damper behavior are analyzed based on the characteristic force-displacement FD and force-velocity FV diagrams.

Then the general model is customized. The ER damper force can be represented by two components: The SA force component depends on the actuation signal as where is the added force due to the manipulation signal to dakper damper force if a voltage is applied and is the measured damper force in an experiment with zero or minimum manipulation.

The passive Eledtrorheological and FD experimental diagrams, Figures 5 a and 5 bare analyzed and the following characteristics can be identified: Afterwards the SA diagram, Figure 5 cis analyzed using 1. The SA phenomena include preyield and postyield regions and hysteresis. At the yield point eelctrorheological damper fluid behavior changes from a pseudoplastic to a quasisolid [ 17 ]. In the FV electrorheologicla the yield point is a Cartesian point where the damping force becomes independent of the velocity.

The yield point defines where the SA damper operates: Also, at the postyield zone, an average force gain FM is obtained, based on the average value in which the yield of the force occurs at each manipulation value.

The characterization procedure was applied to the damper, using experimentTable 2. Figure 6 shows the significant effects that are present in this ER damper operating in passive mode.

From the FV diagram, Figure 6 ait can be seen that this ER damper is asymmetrical; the maximum force in extension positive velocity is greater than the force generated in compression negative velocity.

The force dmaper a velocity dependent component, which is also different in extension and compression. This damper presents significant hysteresis in all its operational range, being more notorious at low speed and in positive velocity. This ER damper is subjected to the stick-slip phenomenon, especially in positive velocity; according to [ 5 ] this phenomenon appears in the ER damper as a force overshot when the flow changes its direction in the annular duct.

In electroeheological FD diagram, Figure 6 ban abnormal stick-slip appears as a peak, as well as effect of the frequency in the electrorheoloyical stiffness. The behavior of the SA component of the force is presented in Figure 7. The relation between the SA force and the PWM duty cycle becomes evident; this relationship is asymmetrical, Figures 7 a and 7 b. In the postyield region the force is almost independent of the piston velocity, but in the preyield zone the force is velocity dependent.

Also at low electrorheologicl the hysteresis loop in SA force is not significant, but as the velocity and the PWM electrorheologicak cycle rise, the hysteresis rises electrorhelogical, Figure 7 a. In Figure dxmper b the stiffness of the damper is affected when the frequency is incremented; also it is notorious how the stick-slip phenomenon became greater electrorheologicl the manipulation increases.

The average FM diagram, Figure 7 cshows that the average force gain for this particular ER damper has a linear behavior. Equations 23aand 3b are a general SA model which includes almost all the phenomena observed in SA dampers. Consider where with Equation 3a describes the passive force. The passive force component is an initial compensation force generated by the accumulator, is the viscous damping coefficient which describes the linear dsmper damping of the Newtonian fluids, is the stiffness coefficient which is the characteristic of linear elastomers, is the virtual damper mass, is the damping force due to friction, andmodel the hysteresis [ 18 — 20 ].

Equation 3b represents the SA forcewhere is the manipulation applied to the damper, is the force gain due to manipulation, anddescribe the behavior of the damper in the preyield zone. If the SA damper has an asymmetric behavior the model needs to have different coefficients for positive and negative velocities. In these equations, the use of the tanh function is replaced with the so-called squash function: