Jorge De-J Lozoya-Santos

Ruben Morales-Menendez

Ricardo Ramirez-Mendoza

PhD Juan C Tud Ón-Martínez

Olivier Sename
email: olivier.sename@gipsa-lab.grenoble-inp.fr

Luc Dugard
email: luc.dugard@gipsa-lab.grenoble-inp.fr

MAGNETO-RHEOLOGICAL DAMPER -AN EXPERIMENTAL STUDY

A Magneto-Rheological (MR) damper is evaluated under exhaustive experimental scenarios, generating a complete database. The obtained database includes classical tests and new proposals emphasizing the frequency contents. It also includes the impact of the electric current fluctuations. The variety of the performed experiments allows to study the MR damper force dynamics. A brief description of the damper behavior and a categorization of experiments based on driving conditions and target applications on vehicle dynamics is discussed. The identification of two MR damper models as well as their cross validation emphasize the importance of the persistence of experimental inputs and the combinations of rod displacement and electric current sequences for better modeling. New findings in Design of Experiments for model identification are presented.

1. Mass production: Mechanical simplicity given that it is

Introduction

A Magneto-Rheological (MR) damper is a hydraulic monotube damper whose oil has metallic particles and its damping coefficient varies according to the supplied electric current. This device performs actuation in a semi-active automotive control. Semi-active control strategies are used in automotive primary suspensions, [START_REF] Lam | Semi-Active Control of Automotive Suspension Systems with Magneto-Rheological Dampers[END_REF][START_REF] Song | An Adaptive Semiactive Control Algorithm for Magnetorheological Suspension Systems[END_REF][START_REF] Savaresi | Mixed Sky-Hook and ADD: Approaching the Filtering Limits of a Semi-Active Suspension[END_REF][START_REF] Poussot-Vassal | A New Semi-Active Suspension Control Strategy through LPV Technique[END_REF][START_REF] Choi | Vibration Control of Magnetorheological Damper System subjected to Parameter Variations[END_REF], aeronautics, [START_REF] Wei | Adaptive Control of Helicopter Ground Resonance with Magnetorheological Damper[END_REF], seismic mitigation in structures, [START_REF] Luo | Composite Semiactive Control of a Class of Seismically Excited Structures[END_REF][START_REF] Xu | Semi-Active Control of Structures incorporated with Magnetorheological Dampers using Neural Networks[END_REF][START_REF] Ikhouane | Adaptive Control of a Hysteretic Structural System[END_REF][START_REF] Cho | Implementation of Modal Seismically Excited Structures using Magnetorheological Dampers[END_REF], washing machines, [START_REF] Spelta | Control of Magnetorheological Dampers for Vibration Reduction in a Washing Machine[END_REF], human prosthetics, [START_REF] Herr | User-Adaptive Control of a Magnetorheological Prosthetic Knee[END_REF], among others. An MR damper has good industrial features, Lord1 , as: physically the same as a passive damper. 2. Efficiency: The device can change the damping coefficient in 40 ms, [START_REF] Koo | A Comprehensive Analysis of the Response Time of MR Dampers[END_REF], over high dynamic range of piston displacements (frequency bandwidth). It also offers a high dissipative force independent of velocity. 3. Energy: Low power requirements (12 volts @ 3 A) 4. Performance: Large force capacity (3 kN), robustness on heavy duty applications as well as consistent efficacy across extreme temperature variations, and inherent system stability (no active forces generated).

The research on MR damper modeling have been done using several approaches for more than 20 years. The main approaches are: physical meaning of the parameters, [START_REF] Spencer | Phenomenological Model of a MR Damper[END_REF], hysteresis based, [START_REF] Guo | Dynamical Modeling of Magneto-rheological Damper Behaviors[END_REF], blackbox models, [START_REF] Choi | A Hysteresis Model for Field-Dependent Damping Force of a Magnetorheological Damper[END_REF][START_REF] Savaresi | Identification of Semi-Physical and Black-Box Non-Linear Models: the Case of MR-Dampers for Vehicles Control[END_REF], damping and stiffness coefficients as main damping force contributors, [START_REF] Choi | Control Characteristics of a Continuously Variable ER Damper[END_REF], among others. The common drawback of the later models is that they are not adequate for controller synthesis.

Approaches in control strategies can be classified as: (a) linear controllers using linearized MR damper models, mixed sky-hook, [START_REF] Savaresi | Mixed Sky-Hook and ADD: Approaching the Filtering Limits of a Semi-Active Suspension[END_REF], H 2 + Linear-Gaussian-Quadratic (LGQ), [START_REF] Cho | Implementation of Modal Seismically Excited Structures using Magnetorheological Dampers[END_REF], H ∞ , [START_REF] Du | Semi-Active H 1 Control of Vehicle Suspension with Magneto-Rheological Dampers[END_REF][START_REF] Choi | Vibration Control of Magnetorheological Damper System subjected to Parameter Variations[END_REF], (b) nonlinear control techniques such as Lyapunov techniques, [START_REF] Luo | Composite Semiactive Control of a Class of Seismically Excited Structures[END_REF], sliding mode control, [START_REF] Lam | Semi-Active Control of Automotive Suspension Systems with Magneto-Rheological Dampers[END_REF], Linear Parameter Varying / H ∞ , [START_REF] Poussot-Vassal | A New Semi-Active Suspension Control Strategy through LPV Technique[END_REF], adaptive, [START_REF] Spelta | Control of Magnetorheological Dampers for Vibration Reduction in a Washing Machine[END_REF], and (c) intelligent control approaches as Artificial Neural Networks based, [START_REF] Wang | Neural Network Modeling and Controllers for Magneto-Rheological Fluid Dampers[END_REF][START_REF] Xu | Semi-Active Control of Structures incorporated with Magnetorheological Dampers using Neural Networks[END_REF][START_REF] De-J. Lozoya-Santos | MR-Damper based Control System[END_REF][START_REF] Wei | Adaptive Control of Helicopter Ground Resonance with Magnetorheological Damper[END_REF]) and heuristic rules, [START_REF] Herr | User-Adaptive Control of a Magnetorheological Prosthetic Knee[END_REF].

A Design of Experiments (DoE) that exhaustively explore the behavior of an automotive MR damper is proposed. Frequency of the piston displacement

f cq
The q th value of the frequency of the piston displacement within a set of Q frequency values The objectives are: to observe the MR damper under operating conditions, and to state the ideal DoE that identify the damper characteristics. These characteristics are: the transient response of the MR force due to an electric current change, the MR damping force relation with the dis-placement, velocity and acceleration of the damper rod, the hysteresis, and the effect of the manipulation shape on the damper case temperature and damping force.

The experimental setup consists of a particular Commercially available Off-The-Shelf ACDelco T M MR damper which is used for the suspension of a 2008 Cadillac T M vehicle. Even this, the methodology may be extended to any kind of shock absorbers. The identification of an MR damper model shows the added-value of the database.

This paper is structured as follows. The main concepts for MR damper modeling are presented in section 2. Section 3 reviews the experimental research work, and presents the experimental system as well as the DoE. The discussion of findings are presented in section 4. Section 5 presents a study case. Finally, section 6 concludes the research. Table 1 describes the nomenclature.

MR Damper Modeling

The mechanical damping force of a damper must be described using the rod displacement x and piston velocity ẋ, [START_REF] Fukushima | Optimum Characteristics of Automotive Shock Absorbers under Various Driving Conditions and Road Surfaces[END_REF]. In an MR damper, the electric current I varies the oil viscosity, changing the generated mechanical damping force, [START_REF] Kordonsky | Magnetorheological Effect as a Base of New Devices and Technologies[END_REF]. The operating range of an MR damper for comfort and road holding conditions is shown in their force-velocity (F-V) characteristic under a fluctuating electric current. The F-V characteristic have yield points that consist of damping force and a given piston velocity. The yield points vary due to mechanical and electric current magnitudes changes and exhibits hysteresis.

MR damper modeling approaches can be classified as either static or dynamic. A static MR damper model estimates the generated force using the instantaneous electric current and displacement/velocity and not earlier values. In a dynamic MR damper modeling approach, each model variable can vary without direct exogenous influence. A typical classification of MR damper modeling has been based in all the known model approaches: phenomenological (parameters whose meaning is related to the mechanical parts, physical meaning), semi-phenomenological (parameters with physical meaning) and black-box (parameters without physical meaning). A new classification for modeling is proposed based on its original structure as: passive or I-driven.

A passive model is valid under the assumption that the damper is filled with an MR fluid, and its objective is to successfully model the hysteretic and nonlinear behavior of the device under constant electric current. Therefore, it does not include the electric current as an input variable. Representative passive models are: Bingham, [START_REF] Stanway | Non-Linear Modelling of an Electro-Rheological Vibration Damper[END_REF], Bouc-Wen Modified, [START_REF] Spencer | Phenomenological Model of a MR Damper[END_REF], Polynomial, [START_REF] Choi | A Hysteresis Model for Field-Dependent Damping Force of a Magnetorheological Damper[END_REF], Semi-Phenomenological, [START_REF] Guo | Dynamical Modeling of Magneto-rheological Damper Behaviors[END_REF], Phase-Transition, [START_REF] Wang | Modelling Hysteretic Behavior in MR Fluids and Dampers using Phase-Transition Theory[END_REF], and Three-parameters, [START_REF] Choi | Vibration Control of Magnetorheological Damper System subjected to Parameter Variations[END_REF]. These models study the force-velocity and transient response curves.

An I-driven model includes the electric current as input variable and its formulation is based on two assumptions: the damper includes an MR fluid as oil and the persistent fluctuation of electric current. Ideally, this model structure (1-1) of a damper with an MR fluid can be represented as equivalent to two dampers in parallel: a damper with constant shear stress (passive) and a damper with variable-shear stress (semi-active) due to the variation of the applied electric current. The sum of the two components yields the total MR damping force f MR .

f MR = f p (x, ẋ) + f I (x, ẋ, I)
(1-1)

f p = k p x + c p ẋ (1-2) f I = y MR (I) • g( ẋ, x) (1-3)
where g( ẋ, x) is a nonlinear function. It is not surprising that black-box approaches obtain the best results: Nonlinear Auto Regressive with eXogenous (NARX) input model, [START_REF] De-J. Lozoya-Santos | Magneto-Rheological Damper Modelling[END_REF], and NARX based on Neural Networks (NNARX) [START_REF] Sa Burton | Modeling the Response of ER Damper: Phenomenology and Emulation[END_REF]Wang andLiao, 2001, 2005;[START_REF] Savaresi | Identification of Semi-Physical and Black-Box Non-Linear Models: the Case of MR-Dampers for Vehicles Control[END_REF][START_REF] Boada | A New Non-Parametric Model Based on Neural Network for a MR Damper[END_REF][START_REF] De-J Lozoya-Santos | Design of Experiments for MR Damper Modelling[END_REF]. The state of the art model approaches are: (a) the modified Bouc Wen with polynomial dependence on electric current, [START_REF] Spencer | Phenomenological Model of a MR Damper[END_REF], and (b) the black-box approaches.

For designing an automotive suspension control system, an MR damper model can range from a complex one (high performance simulation) to a simple one (for controller synthesis). For controller synthesis, the model features are: (a) it must be I-driven, (b) comfort and road holding domains must be met, (c) experimental dataset must consider persistent inputs that represent the typical operating conditions.

A Modified Semi-Phenomenological Model

A modified version of the semi-phenomenological model [START_REF] Guo | Dynamical Modeling of Magneto-rheological Damper Behaviors[END_REF] where the force of the damper is described by: (a) the force due to the spring effect of the gas accumulator, (b) the damping force due to the oil viscosity, and (c) the MR force due to the electric current, (Lozoya-Santos et al., 2009a). This approach assumes a non significative effect of the electric current on damping coefficient for low velocities. In a tradeoff velocity value the magnetic links in the MR fluid are broken by the stress changing the damping coefficient. This coefficient remains the same until the links are joined, a phenomenon depending on the velocity, [START_REF] Jolly | Properties and Applications of Commercial Magnetorheological Fluids[END_REF]. When the magnetic links are rejoined, the magnetic hysteresis mixed with a increasing velocity in inverse direction generates hysteresis in the F-V characteristic. The observed consequence is a proportional effect of the electric current on the maximum f MR :

f MR = u c • c MR • β + c p ẋ + k p x (2-1) β = tanh (h ẋ ẋ + h x x) (2-2)
where u c is the controller output of a semi-active suspension, c MR is a proportional rate of damping force per ampere over the tradeoff velocity of piston, c p is a linear viscous damping coefficient, k p is a linear stiffness coefficient, h ẋ is a hysteresis coefficient due to the velocity of piston, h x , is a hysteresis

x f [N]
A specific damping coefficient is linked to a maximum velocity.

[m/s] . coefficient due to the displacement of piston, β describes the damping force behavior due to the friction and activation of the symmetrical blow-off valves.

The Maximum Deflection Velocity (VMax) model

The model is based on the variation of the postyield damping coefficient depending on the maximum deflection velocity, [START_REF] De-J. Lozoya-Santos | A LPV Quarter of Car with Semiactive Suspension Model including Dynamic Input Saturation[END_REF]. The maximum deflection velocity is a function of the maximum amplitude and the frequency of the deflection, [START_REF] Bastow | Car Suspension and Handling[END_REF]. Hence this measure can capture the dynamic behavior of the damping coefficient, such as,

ζ I = f I ẋ = y MR (I) • g( ẋ, x) ẋ ∝ I • c MR ||| ẋ||| ∞ i i-k (3-1) f I -→ ζ I • ẋ ∝ I • ẋ • c MR ||| ẋ||| ∞ i i-k (3-2)
where ζ I is the damping ratio depending on the electric current, f I is the damping force due to the change of oil viscosity caused by the electric current fluctuation, i is the instantaneous sample of the measured ẋ, k is an integer subscript meaning the receding horizon to compute the maximum absolute velocity represented by ||| ẋ||| ∞ i i-k , i. e. the infinite norm of the velocity of the piston, ẋ. Figure 1 shows how the damping coefficient changes with maximum velocity, where the dashed lines are the damping coefficient trajectories that depend on the interval of ẋ. The model is:

f MR = u c • c MR • ρ + c p ẋ + k p x (4-1) ρ = ẋ • 1 ||| ẋ||| ∞ i i-k + ε (4-2)
where ρ describes the damping force behavior due to the friction and the activation of the symmetrical blow-off valves.

Design of Experiments (DoE) for MR damper models

The DoE for the MR damper consists of a pair of training sequences: displacement and electric current. Table 2 summarizes various works from the point of view of experimentation. Important findings can be summarized as follows:

1. The DoE for identification of MR damper models have not precisely been focused on a specific application.

2. Commonly, the bandwidth of the damping manipulation signal has not been addressed. 3. The effect of inputs on the temperature and force of the MR damper has not usually been fully explored. 4. The DoE ordinarily exhibit a lack of high-frequency bandwidth in the piston displacements. 5. The persistent exploration of the force-velocity diagram mostly has not been an objective. 6. The number of experiments usually is high and only static information is retrieved, (9 in [START_REF] Sa Burton | Modeling the Response of ER Damper: Phenomenology and Emulation[END_REF], 392 in [START_REF] Nino | Minimizing the Frequency in a Black Box Model of a MR Damper[END_REF]). 7. The experiments commonly have long duration. 8. The good models for the F-V diagram mostly were identified with low displacement frequencies. 9. The number of experiments usually is less than 5 for persistent displacement signals.

The best practices could be:

1. It is possible a better identification of MR damper model parameters with the data obtained from random displacement sequences rather than harmonic displacements, [START_REF] Sa Burton | Modeling the Response of ER Damper: Phenomenology and Emulation[END_REF]. 2. For a displacement bandwidth below 2 Hz, the force is strongly dependent on the frequency. Over 4 Hz, the nonlinear force appears due to the inertial effect of the MR fluid, [START_REF] Li | Testing and Steady State Modeling of a Linear MR Damper under Sinusoidal Loading[END_REF]. 3. There is a nonlinear induced effect on the MR force caused by the velocity, [START_REF] Bastow | Car Suspension and Handling[END_REF]. 4. The amplitude of the displacement is not significant in the magnitude of the f MR , meanwhile the frequency of the displacement has a high nonlinear effect. The electric current has the strongest effect, [START_REF] Shivaram | Statistical Modeling of a MR Fluid Damper using the Design of Experiments Approach[END_REF].

Taking into account that the MR damper has a typical damper mechanical structure, some important observations on damper testing, [START_REF] Kowalski | The Effects of Different Input Excitations on the Dynamic Characterization of an Automotive Shock Absorber[END_REF], are extrapolated to the MR damper experimentation in order to define the DoE:

1. The temperature of the damper must be monitored throughout the experiment. 2. The stroke must be the same between experiments if the resultant identified models are going to be compared. 3. The sine-on-sine displacements identify frequency dependent parameters, hence the models tend not to be generalized. 4. The sine-on-sine displacements and the harmonic displacement show different parameters when the identification is applied to the same model. 5. The parameters derived from random and swept sinusoidal displacements converge to similar values and good models in restricted bandwidths. 6. The DoE with sequences of electric current in a bandwidth 0 -10 Hz and piston displacement in a bandwidth 0 -15 Hz need less experimentation time 7. A model without parameters skew can be found using one test.

Based on the literature review, twelve training sequences are presented (some are new proposals). The training sequences focus on the shape of the MR damper piston displacement and the electric current through the MR damper coil. The objectives of the DoE are: (a) define the interactions between the velocity, displacement, and electric current, (b) analyze the impact of the DoE in the identification process, (c) find the best training sequence for MR dampers modeling, and (d) design and identify models for controller synthesis, and (e) identify models for specific driving and target applications.

Displacement Sequences.

The common displacement features for the proposed sequences are (a) a bandwidth between 0.5 and 14.5 Hz, and (b) a maximal displacement of 7.6 mm for symmetric signals and 12.7 mm for random signals, Fig. 2.

1. Road Profile Smooth highway (RP). RPs have been utilized as a test signal for passive damper experiments, [START_REF] Kowalski | The Effects of Different Input Excitations on the Dynamic Characterization of an Automotive Shock Absorber[END_REF], and in the training of MR damper artificial neural network models, [START_REF] Savaresi | Identification of Semi-Physical and Black-Box Non-Linear Models: the Case of MR-Dampers for Vehicles Control[END_REF]. The objective is to test the MR damper under standard automotive conditions. The displacements were based on the specification of the spectral density of several road profiles, [START_REF] Wong | Theory of Ground Vehicles[END_REF]. These are computed using an algorithm based on the Fourier inverse transform, [START_REF] Da Silva | Dynamical Performance of Highway Bridge Decks with Irregular Pavement Surface[END_REF]. The RP smooth highway is the most common for commercial vehicles. 2. Sinusoidal signal with stepped Frequency and constant amplitude Signal (SFS). The SFS is an usual displacement sequence in the DoE for MR dampers, [START_REF] Sa Burton | Modeling the Response of ER Damper: Phenomenology and Emulation[END_REF], [START_REF] Spencer | Phenomenological Model of a MR Damper[END_REF], [START_REF] Li | Testing and Steady State Modeling of a Linear MR Damper under Sinusoidal Loading[END_REF], [START_REF] Kowalski | The Effects of Different Input Excitations on the Dynamic Characterization of an Automotive Shock Absorber[END_REF], [START_REF] Xia | An Inverse Model of MR Damper using Optimal Neural Network and System Identification[END_REF], [START_REF] Wang | Modelling Hysteretic Behavior in MR Fluids and Dampers using Phase-Transition Theory[END_REF], [START_REF] Guo | Dynamical Modeling of Magneto-rheological Damper Behaviors[END_REF], [START_REF] Nino | Application of Black Box Models for MR Damper Identification[END_REF][START_REF] Shivaram | Statistical Modeling of a MR Fluid Damper using the Design of Experiments Approach[END_REF][START_REF] Shivaram | Statistical Modeling of a MR Fluid Damper using the Design of Experiments Approach[END_REF]. The sinusoidal displacement has a constant frequency and a constant amplitude. Hence, the experimentation with a SFS sequence could derive in long duration experiments and a redundant dynamics exploration. The frequency of the signal is increased 0.5 Hz each 3 or 6 displacement periods. It is important to include the transient response due to abrupt frequency changes. This in order to capture a rich exploration of the specimen in short time. 3. Sinusoidal CHirp Signal (CHS). The CHS is a sinusoidal signal with a constant amplitude and linear frequency increments through the experiment. 4. Frequency Modulated -(FM). The frequency is varied uniformly in the bandwidth with a constant amplitude. This pattern is created using a voltage controlled oscillator. The supplied voltage is a stepped signal whose steps have finite duration, named Increased Clock Period Signal (ICPS), [START_REF] Söderström | System Identification[END_REF]. The step duration is 200 ms. This duration guarantees full cycles of displacement over 5 Hz. The signal frequency output is proportional to the supplied voltage.

Triangular wave with Positive and Negative Variable

Slopes Signal (TPNVS). This signal consists of a posi- tively sloped ramp followed by a negatively sloped one.

From origin (0 mm), the ramp with positive slope raises to 7.6 mm, then it continues with a negative slope until reaching -7.6 mm. It changes again to positive slope and returns to origin. The slope is increased every 3 or 6 periods, depending on the experiment. The slope increment stops when the TPNVS period is equal to 1/14.5 Hz. 6. Amplitude Modulated Signal (AM). The AM is a modification of the well know signal in communications engineering. The amplitude changes with a stepped shape each 3 cycles of the carrier frequency. Each sequence has 200 uniformly distributed amplitude levels. The ob-jective is to obtain the f MR on a fixed carrier frequency and random stepped amplitudes. This sequence is oriented to the identification of Linear Parameter-Varying MR damper models, where the scheduling variable is the oscillation frequency.

Electric Current Sequences.

The electric current span is between 0 to 2.5 A for the proposed sequences. Six sequences for this input are described in Figure 3. tion over MR dampers allows for the observation of the f MR operation zones. A stepped increment of the electric current value covering the full span is proposed. The stepped increment will depend on the experiment. The duration of the step is defined by the duration of the displacement pattern. 2. Positive variable slope Ramp (RC). The RC begins from zero and finishes in the maximum electric current, then it starts over, [START_REF] Chang | Neural Network Emulation of Inverse Dynamics for a MR Damper[END_REF]. The coil excitation with several slopes allows for the exploration of the effect of the ratio A/secs over the MR force.

Ramp with Positive and Negative Variable Slopes (RP-NVS).

The RPNVS begins from zero raising to the maximum electric current with a specified positive slope. The signal continues with the same slope but with a negative sign until reaching 0 A. The MR fluid has a hysteretic behavior on the magnetization of the metallic particles. The combination of this magnetization with the accumulator effect over the MR damper could be emphasized using this sequence. 4. Increased Clock Period Signal (ICPS). This signal is the same used in the FM signal generation. In this case, the step duration must be at least equal to the settling time of the f MR , 4 • τ MR ms. τ MR is the time constant of the first order transfer function f MR (s)/I(s), [START_REF] Koo | A Comprehensive Analysis of the Response Time of MR Dampers[END_REF].

The damping change reaches is steady state at 4 • τ MR ms. The objective is to extract the steady and transient behavior of the f MR due to the persistence magnetic excitation of the MR fluid. 5. Amplitude Pseudo Random Binary Signal (APRBS). This is a stepped signal where the amplitude and the step duration change randomly, [START_REF] Söderström | System Identification[END_REF]. This electric current sequence has been used in order to fit an Artificial Neural Network model of an MR damper, [START_REF] Savaresi | Identification of Semi-Physical and Black-Box Non-Linear Models: the Case of MR-Dampers for Vehicles Control[END_REF]. When the MR damper is subjected to an APRBS electric current, the f MR could not get to a steady state due to the magnetic links of the metallic particles not being fully aligned. This is true only if the minimum APRBS step duration is not, at least, equal to the settling time of the f MR .

6. Pseudorandom Binary Signal (PRBS). This test is usually exploited for process identification in domains such as MR dampers. The common units are voltage but in this work they are proposed as amperes because these are the more realistic manipulation units in a semi-active control strategy. Several DoE were proposed, Table 3. Each proposed experiment has the following scope: a. To perform an analysis of the effect of the input variables on the f MR by employing the surface response method and to define their level of significance for the estimation of the output. The analysis includes the linear, quadratic, and crossed interaction of the inputs, (Experiment # 1). b. To analyze f MR response when submitted to step changes of the electric current and define the bandwidth of this transfer function, (Experiment # 2). c. To observe the MR damper behavior under constant electric currents in the frequency bandwidth of displacement for automotive applications, (Experiments # 3 and # 21-29). d. To explore the f MR gain with regard to a constant ratio change in the applied electric current when a harmonic displacement is applied, (Experiment #4). Emphasis is made on the magnetic hysteresis effect in the MR fluid, (Experiment # 5). e. The exhaustive exploration of the f MR in low displacement velocities, in order, to determine if this dynamic information identifies generalized models for several types of excitation (Experiment #6 with ICPS and Experiment # 7 with PRBS). f. To compare the f MR obtained from a repetitive electric current sequence in order to validate the independence of the damping variation from the frequency of the displacement for several types of excitation (Experiment # 8 with ICPS and Experiment # 9 with PRBS).

AM SC

Cycles ∆( fc) =4, see Table 5 16

∆(I)
PT IC(I) = 1, see Table 5 1

TPNVS SC d(x) dt = 32
fc i A t 1 fc i SFS RPNVS Cycles ∆( fc) =6, f c = {0.5, 1, . . . , 14.5}, x pp = 16mm 47 | ± d(I) dt | = 2.5 fc i A t 1 fc i CHS ICPS f c ∈ {0.5, 6}, x pp = 16mm 30 τ I = 0.07s 1 28 CHS PRBS f c ∈ {0.5, 6}, x pp = 16mm 30 τ I ∈ {0.05, 0.5}s 1 28 SFS ICPS Cycles ∆( fc) = 2 fc i
, f c = {0.5, 1, . . . , 14.5}, x pp = 16mm 58 τ I = 0.100s

1 fc i SFS PRBS Cycles ∆( fc) = 2 fc i
, f c = {0.5, 1, . . . , 14.5}, x pp = 16mm 58 τ I ∈ {0.05, 0.5}s h. To validate the model identification quality given a realistic displacement and several types of electric current (Experiments # 12, # 13 and #14), when this quality is compared with harmonic-displacement model identifi-cation (for example, Experiments # 8-11). i. To observe the effects on the temperature of the f MR when the device is submitted to realistic displacements and several excitation changes over long periods of time, (Experiments # 15 and # 16). j. To define the influence of the piston position as a factor in the delivered f MR due to the effect of the internal accumulator, when submitted to several types of displacements and excitations, (Experiments # 17-20). k. To emphasize the effect of the amplitude displacement of rod and the electric current randomness when the frequency of the piston displacement is constant (Experiment # 21) and compared with rich frequency displacements and constant amplitude (Experiments # 1-11). l. To have a database with the classical experimentation for MR dampers (Experiments # 21-29) and to do a comparison with those obtained with a persistent electric current.

1 fc i FM ICPS τ(vco) = 0.10s, f c = {0.5, 14.5}, x pp = 16mm 30 τ I = 0.10s 1 FM PRBS τ(vco) = 0.10s, f c = {0.

Experimental System

The experimental system consists of three sections: acquisition system, actuation system, and control system. The acquisition system captures three signals: (a) the displacement, (b) the generated force, and (c) the electric current through the MR damper coil. The operation of the systems is done via a Supervisory Control System (SCS), which is based on National Instruments T M equipment and LabView T M . The actuation system is an MTS T M equipment that includes: an actuator of 25 kN with a 15 Hz bandwidth, a controller hardware unit Flextest GT, a software Station Manager T M , and a MultiPurpose TestWare T M , see Fig. 4. The load capacity is 25 kN at 20,690 kPa. The stroke is 150 mm. The span of the piston deflection is ±12.5 mm. An electric current driver adjusts the command from 0 to 2.5 A.

Experimental results

Various important results were found. These results show the potential application of this research. The factors are: (a) the displacement variation, (b) the effect of the electric current and the frequency of the displacement on hysteresis, (c) the linearity of the generated force with the electric current, and (d) the temperature of the damper. Said phenomena were observed in Experiment # 21 (persistent inputs), Experiment # 22 (classical DoE), and Experiment # 26 (persistent frequency of the piston displacement with constant current).

The velocity of damper rod is computed with central differentiation algorithm. Both, rod displacement and velocity are filtered with a Butterworth filter with cut-off frequency of 16 Hz. Figure 5a shows the phase plane obtained after the proposed procedure for Experiment # 21. For displacements in Experiments # 22-29, the central differentiation is enough for the computation.

In the displacement sequence, the amplitude variation is a stronger factor than the frequency is. The vertical uniformity of the covering is caused by the amplitude variation and the applied electric current. The maximum achieved velocity or horizontal covering is a consequence of the carrier frequency, Fig. 5b. The maximum MR damping force is quasi linear with the applied electric current. Experiment # 21 and # 26 show that when the electric current is increased, the maximum force is also increased in a quasi linear way. Fig. 6a shows how the larger the electric current, the greater the maximum F MR is. The variations between maximum and minimum forces are consequence of the displacement. Fig. 6b shows the proportional increment on the maximum force when the electric current is increased. The effect of the applied electric current and the frequency of the piston displacement on the hysteresis. Experiment # 22 shows that the electric current is inversely proportional to the hysteresis for frequencies below that of 7 Hz (Fig. 7a-7b), and proportional for frequencies above 7 Hz (see difference from Fig. 7b-7c). The electric current increases the yield point, which is the MR force value for a specific velocity when the transition from pre-yield to postyield behavior happens. By contrast, the frequency of the piston displacement is proportional to the hysteresis, and the hysteresis increases with the frequency when no electric current is present, Fig. 7a.

Body temperature of the MR damper. A procedure is performed over the database in order to determine the effect of the DoE on the body temperature. The procedure is as follows: (a) obtain the increase rate of the body temperature for each experiment and their replicas; (b) perform a descendent sort on the DoE dataset by the increase rate of damper cylinder temperature and create a table; and (c) identify the displacement and electric current sequences that increase the body temperature. damper body for Experiments # 10 and # 21 during all replicas. The rate of damper body temperature is also shown. Experiment # 21 shows that a carrier frequency over 6 Hz raises the rate of change of damper body temperature. On the other hand, for frequencies below 6 Hz, the rate is considerable lower. The results shows that the temperature is strongly dependent on the frequency of piston displacement , and in a minor way on the displacement amplitude. This is shown in Experiments # 17-19 that although they have a bandwidth of 0-15 Hz, their displacement magnitudes are small.

A second procedure studies the distribution of the measured damping force when the damper body is cold (first replica) and hot (last replica). This procedure uses the hypothesis Levene's test, [START_REF] Brown | Robust Tests for the Equality of Variances[END_REF], which is an inferential statistic used to assess the equality of variances in different samples. One advantage of Levene's test is that it does not require normality of the underlying data or when these data are prone to outliers. The hypothesis test is done to the Experiments # 6 (it has the highest damper body temperature increase rate), # 3 (it has a middle damper body temperature increase rate), and # 12 (it has the lowest damper body temperature increase rate). In the three experiments, the null hypothesis is not rejected, hence although certainly the viscosity and temperature of the oil are well correlated, the temperature effect is negligible because the narrow range of operation. Given this result, any replica of the proposed DoEs can be used for MR damper modeling. As a summary, the experimental database can expose the phenomena from the behavior of the MR damper in a wide manner and from several functional scenarios.

The experimentation of MR dampers aimed at obtaining models for simulation of driving conditions is not well addressed in literature. In fact, it is not a recurrent practice for typical damper. In vehicle dynamics simulation and evaluation, the basic damper as a linear device has only a damping coefficient. who is selected using an analysis of the whole vehicle. These damping coefficients can be: (a) low for comfort, (b) medium for straight driving on rough roads, and (c) high for best handling, [START_REF] Dixon | The Shock Absorber Handbook[END_REF]. When using a passive damper, there is a comfort/handling compromise which defines the baseline damper configuration. When using an MR damper, the vehicle suspension is called semiactive and the mentioned compromise takes place in a control law in order to obtain the optimum damping according to the situation. In semiactive suspensions, the MR damper can be a Continuously Controlled Damper (CCD) (continuous variation of the manipulation), and an On-Off Controlled Damper (On-Off CD) (manipulation varies between two fixed values). Figure 10, shows a categorization of expected strokes and velocities in a piston damper for the specific driving conditions of a commercial vehicle, [START_REF] Fukushima | Optimum Characteristics of Automotive Shock Absorbers under Various Driving Conditions and Road Surfaces[END_REF]. This categorization can be extended to the MR damper using the stroke and velocity spans of the damper rod and their frequency content in order to propose subsets of experiments focused on the driving condition. According to the vehicle dynamics terminology SAE J670 (Jan 2008), besides the driving conditions specified in Fig. 10, the target applications body control, wheel hop (best handling), and wheel tramp mitigation are of main interest when using semiactive suspensions. In addition, there is a lack of MR damper experimentation in order to obtain proper experimental data to be used in these target applications. Operating range where damping force is not required Fig. 10. Damping operating range for several driving conditions, [START_REF] Dixon | The Shock Absorber Handbook[END_REF].

The body control refers to control of the bounce, pitch, and roll of a vehicle (called primary ride comfort). Wheel hop is the vertical oscillatory motion of a wheel between the road surface and the sprung mass. Wheel tramp is the form of wheel hop in which the wheels on an axle hop in opposite phase and it is present in solid axles. The resonant frequencies for most passenger cars range from 1-1.2 Hz for bounce motion, from 1.2-1.5 Hz for pitch motion, from 1.5-2 Hz for roll motion, and from 10-14 Hz for wheel hop [START_REF] Barak | Magic Numbers in Design of Suspension for Passenger Cars[END_REF]. In the case of wheel tramp, a more typical rear suspension in pickups and rear-wheeled vehicles, the resonant frequency can be between 14-17 Hz, [START_REF] Kramer | Optimized Damping to Control Rear End Breakaway in Light Trucks[END_REF]. Using such categorization in Figure 10 and the optimum damper characteristics presented in [START_REF] Fukushima | Optimum Characteristics of Automotive Shock Absorbers under Various Driving Conditions and Road Surfaces[END_REF] for a base-line damper, the characteristics of the tested MR damper are suggested. Based on Figure 11, an extension of the results [START_REF] Fukushima | Optimum Characteristics of Automotive Shock Absorbers under Various Driving Conditions and Road Surfaces[END_REF] to a MR damper must be considered because the experimental velocities and strokes. Figure 11 shows the optimum MR damping characteristics for the categories shown in Figure 10 using a experimental dataset. As the short and long strokes require considerable differences in forces, the increase of electric current can achieve these differences. Once the resonance frequencies are known, as well as the stroke-velocity relations in several driving conditions, the next step is to know the domain of stroke, velocity, and frequency of the piston displacement for each experiment. The determination of the stroke and velocity domains requires the force-velocity and force-stroke (displacement) plots while the frequency content is obtained from the power spectral density computation. Table 6 shows the results of the categorization for driving conditions and target applications in vehicle dynamics analysis.

Case study

A set of experiments has been selected in order to perform identification and cross validation procedures. The modified Semi-Phenomenological (SP) and maximum velocity MR damper models, eqns (2-1) and (4-1) were identified using the least squares estimation. The identification procedure allows to obtain the model parameters for each selected experiment. The cross validation test consists on the evaluation of an identified model with a different experimental data. The Error-to-Signal Ratio (ESR) (5) [START_REF] Savaresi | Identification of Semi-Physical and Black-Box Non-Linear Models: the Case of MR-Dampers for Vehicles Control[END_REF], and the Relative Root-Mean-Square Error (RRMSE) (6) [START_REF] Witters | Black-box Model Identification for a Continuously Variable Electro-Hydraulic Semi-Active Damper[END_REF] are two classical index performances.

SSE = N ∑ i ( fMR i -f MR i ) 2 , MSE = SSE/N ESR = MSE σ f MR • 100% (5)
RRMSE = √ SSE ∑ N i ( f MR i ) 2 • 100% (6)
The cross validation intends to show the added-value of the DoE categorization for both models. For every model, the recommended DoE for a target applications with a continuous, on-off and constant damping are evaluated. Table 7 shows the results for DoEs with an electric current sequence in order to generate continuous and on-off controlled damping. These results are replicated when the index performance RRMSE is used, Table 9.

Another interesting result is the generalization of an MR damper model parameters when the identification is done with a specific DoE. Such DoE identifies parameters that allows high performances in f MR estimation in a cross val-idation procedure. Fig. 13(b). This suggests a necessity of a persistent electric current in MR damper experimentation for a better f MR estimation, Fig. 13(c). This qualitative evaluation suggests the categorization of DoE can be a valid tool in vehicle model evaluation.

Conclusions

An experimental database with several experiments for MR dampers has been presented. An experiment categoriza- tion was presented based on specific automotive applications.

In order to show the potential of the generated experimental database, three experiments are analyzed showing the effects of the variation of the displacement amplitude and electric current when the frequency of piston displacement is con- stant, the exploration of all the modes of operation of the damper when the electric current and the amplitude displacement remain constant, and the excitation of the hysteresis due to a random frequency under sinusoidal displacement and a several levels of electric current. These specific experiments show the information of the MR damper force ( f MR ) with the objective of a better understanding of the dynamic of this device and modeling it.

The identification of two MR damper models using nineteen experiments validates the results. A cross validation test shows the categorization of the experiments as a tool for modeling. Also, it shows one of the proposed DoE gives better cross validation results in both validated models. The main contributions of this research are:

1. A Design of Experiments (DoE) for an MR damper modeling considering the target application of the model. The domain of these applications considers the control of semi-active suspensions where comfort and handling are goals under several drive conditions. 2. New guidelines of DoE for MR damper modelling that consider displacement and electric current sequences are validated. 3. The methodology can be tailored to another application or type of damper.

Further work

This study does not accomplish the correlation of the phenomenological relations and parameters design of dampers with the experimental results. This correlation is the subject of a further study. Tests in an industrial laboratory are mandatory.

Achnowledgements

ε

  Hysteresis coefficient due to piston displacementh ẋ Hysteresis coefficient due to piston velocity i i th sample of experimental signal k Receding horizon for computing ||| ẋ||| ∞ k MR Stiffness coefficient depending on the electric current The level change of the electric current PT IC(•) Pattern Training Input where (•) can be I or x β, ρ Nonlinear function used to estimate variable force dissipation ζ I Damping ratio depending on the applied current τ I Duration in seconds of the applied current value τ MR Time constant, 1 st order transfer function f MR (s)/I(s) Constant for bounding the maximum damping coefficient ||| ẋ||| ∞ The infinite norm of the absolute deflection velocity ẋ

Fig. 1 .

 1 Fig. 1. The damping coefficient versus maximum deflection velocity.

Fig. 2 .

 2 Fig. 2. Training displacement sequences for model identification.

Fig. 3 .

 3 Fig. 3. Training electric current sequences for model identification.

∆

  ∆( fc) =3, f c = {0.5, 1, . . . , 14.5}, x pp = 16mm 24 ) =3, f c = {0.5, 1, . . . , 14.5}, x pp = 16mm 237 ∆(I) PT IC(I) = 8, I = {0, 0.4, . . . , ) =3, f c = {0.5, 1, . . . , 14.5}, x pp = 16mm 24 d(I) dt = 2.5

Fig. 4 .Fig. 5 .

 45 Fig. 4. Experimental system. The lines with arrows show the signals interaction between the experimental setup components. DoE: design of experiment; MR: magnetorheological; HCT: High Country Tek Inc; LVDT: Linear Variable Differential Transformer; ANNARX: Artificial Neural Networks and Auto Regressive with eXogenous input; I-PWM: Electric Current via Pulse Width Modulation.

Fig. 6 .

 6 Fig. 6. Maximum f MR versus electric current (a) and velocity (b).

Fig. 7 .

 7 Fig. 7. Influence of the electric current and the frequency of the piston displacement on the hysteresis for Experiment # 22. The chosen frequencies are f c = {1.5, 7, 14} Hz corresponding to the a, b and c plots. The applied currents are I = {0, 1.2, 2.5} A in each plot.

Fig. 9 .

 9 Fig. 9. Temperatures on the MR damper case and environmental.

  Figure-8 turn Rapid steering Lane change Bump passing Braking hard Rapid acceleration Smooth road Rough road Compression Piston velocity Tension Figure-8 turn Rapid steering Lane change Bump passing Braking hard Rapid acceleration Smooth road Rough road

  Fig. 11. Optimum MR damper characteristics, modified from (Dixon, 2007).

  Fig. 12 shows the ESRs boxplot of both models. The validation DoE were those DoE used in tables 7 and 8. The DoEs in the horizontal axis were sorted in ascendent way according to the average ESR and only the firsts nine DoE are shown in the horizontal rank. The f MR estimation with model parameters identified with the DoEs 4, 5, 21 4 shows a consistent low ESR average as well as the classic DoE 3 for the identified models. Several advantages of the new DoE proposals can be enumerated over the classical DoE: (a) the experimentation time is shorter in a 90% (DoEs 4, 5, 21 4 ), (b) the rate of temperature increase of the damper body are equal than classic DoE allowing a longer damper time life (DoE 4, 5), (c) they are easy implementable (DoEs 4, 5, 21 4 ). Fig 13 shows a qualitative comparison when model parameters identified with Experiments # 3 and # 21 estimate the f MR of Experiment #12. Subfigure (d) shows the piston velocity versus time. The subfigures array allows: to show the f MR response when the electric current changes (subfigures (a) and (b)), the relation of transient piston velocity with the F-V plot (subfigures (c) and (d)), and finally, the estimation precision when the modified SP model is identified with the suggested experiment 21 2 and the estimation error when it is identified with the classical DoE, Experiment # 3, (subfigures (a) and (b)). The maximum velocities for the damper rod, Fig. 13(d), allows to evaluate in a qualitative manner the MR damping estimation error in transient response, Fig. 13(a), and Force-Velocity characteristic, Figure 13(c). Later figures show that the model identification using Experiment # 21 2 allows a more accurate damper modeling. The model identification using Experiment # 3 computes a wrong MR damping force in peak velocities when the model estimates the damping

Fig. 12 .

 12 Fig. 12. Statistical performance of the ESR for cross validation of model parameters when they are identified with the new DoE and classical Experiment # 3. (a) shows the results for modified SP model and (b) shows VMax model. The horizontal axis shows the DoE. The vertical axis shows the ESR when the model estimates the f MR .

Fig. 13 .

 13 Fig. 13. The f MR estimation using the modified SP model versus data from Experiment # 12 (smooth road). The modified SP model estimates the f MR using two sets of parameters from Experiments # 21 2 and # 12. Subfigure (a) shows the f MR transient response, subfigure (b) shows the applied electric current in the MR damper versus time, subfigure (c) presents the F-V plot of the estimated forces versus the measured one, and subfigure (c) shows the piston velocity versus time describing by dotted lines its relation with the maximum damping force.

Table 1 .

 1 Nomenclature

	Var	Description
	c MR	Proportional rate of damping in post-yield region
	c p	Viscous damping coefficient, without electric current
	||e||	Error norm
	f c

Table 2 .

 2 Comparison of training inputs. First column is the model name, second column specifies inputs. Columns F-V and F-D are for Force-Velocity and Force-Displacement diagrams of predicted force versus experimental force. Cost function is the index fitting performance; d [mm] refers to amplitude in displacement; Hz is the frequency bandwidth and eXogenous input being current or voltage. Each one could have two values, first is the maximum tested value and second is the number of tested values. Other case is if there is only one maximum value reached for the respective signal. Finally, two values separated by a dash means the span of continuous variations.

	Model	Training	F-V	F-D	Cost function	d[mm]	Hz	eXogenous
	Bingham and Bouc Wen (1997)	Sinusoidal displacement with increasing	yes	yes	e	25	4,4	1A,4
	3 parameters (1998; 2008)	frequency and constant current	yes	yes	e	25	2,3	2A,6
	Phase transition (2006)		yes	no	Sum of	25	4,5	1A,4
					Square Errors
					(SSE)
	Statistical (2007)		no	no	Root Mean	3,1	8,2	60 and 120A-t
					Square (RMS)
	Statistical (2007)		no	yes	RMS	3	8,3	40, 80
								and 120A-t
	Semi-physical (2006)		yes	yes	SSE	20,3	4,5	2A, 5
	NARX (2006)		yes	yes	ESR	40,3	14.5,28	4A,14
	ANNARX (1996)	Band-limited white	no	yes	RMS	10	1.5,1	3 kV mm ,2
		noise as displacement	no	yes	RMS	5	4,1	3 kV mm ,2
		and constant current	no	yes	RMS	5	4,1	3 kV mm ,2
	ANNARX (2005)		no	no	SSE	25	3,2	12V,3
	ANNARX (2005)		no	no	ESR	25	15	1.2A,3
	Phase transition (2006)	Sinusoidal displacement and constant	yes	no	SSE	25	2.5	0.5A
		current
	Blind Hammerstein (2007)	ICPS (1989) and constant current	no	no	e	0-15	0-5	1.25V
	ANNARX (2002)	Band-limited white	no	no	SSE	22	1-5	0-3V
		noise as displacement	no	no	SSE	0-1	3-6	0-2.25V
	ANNARX (2005)	and current	no	no	SSE	0-25	0-3	0-6V
	ANNARX (2005)		yes	yes	SSE	0-20	1-5	0-2.25V
	ANNARX (2003)	Sinusoidal displacement and ramp current	no	no	e	15	2.5	0-2.25V
	ANNARX (2005)	Band white noise limited displacement	no	no	SSE	25	3,2	12V
		and stepped current
	ANNARX (2005)	Band white noise limited displacement	no	no	ESR	0-25	0-15	0-1.2A
		and APRBS (1989)

Table 3 .

 3 Design of Experiments. First column enumerates the DoE. The second and third columns define the sequences for displacement and electric current. The fourth and fifth columns specify the characteristics of each sequence. Time specifies the experiment duration. Table1describes the nomenclature.

	Displacement [mm]	Electric

Table 4 .

 4 Experiment 21 i . Carrier frequencies and the duration.

	Subscript i	f c	secs	Subscript i	f c	secs
	1	1.40	143.36	6	8.38	24.22
	2	2.80	71.87	7	9.78	20.70
	3	4.19	48.05	8	11.18 17.97
	4	5.59	35.94	9	12.57 16.01
	5	6.99	28.09	10	13.97 14.45

Table 5 .

 5 The DoE for Experiment #1 is a central composite design. The factors are: the electric current through MR damper coil, the frequency of the piston displacement and the amplitude of displacement.

	Input	Current(=)A	f c (=)Hz	x peak (=)mm
	levels
	combination
	1	0.5100	3.8985	1.9500
	2	0.5100	3.8985	7.4400
	3	0.5100	12.2040	1.9500
	4	0.5100	12.2040	7.4400
	5	2.0000	3.8985	1.9500
	6	2.0000	3.8985	7.4400
	7	2.0000	12.2040	1.9500
	8	2.0000	12.2040	7.4400
	9	0.0000	8.0513	4.7000
	10	2.5000	8.0513	4.7000
	11	1.2550	1.0662	4.7000
	12	1.2550	15.0362	4.7000
	13	1.2550	8.0513	0.6200
	14	1.2550	8.0513	12.5141
	15	1.2550	8.0513	4.7000
	16	1.2550	8.0513	4.7000

Table 6 .

 6 Categorization of experiments based on the target application. First column shows the application and driving conditions. Columns two and three define the recommend experiments for model identification to be utilized in close loop simulations of semiactive suspensions using CCD or On-Off CD manipulation. For open loop simulations, fourth column specifies the experiments using constant manipulations and fifth column shows the preferred electric current for each driving condition in order to assure comfort / handling. The numbers in second, third and fourth columns means the number of experiment. The r-p notation means from r-experiment and p-finishing.

	Application	Close loop simulations	Open loop simulations
	Simulation / Modeling objective	CCD	On-Off CD Constant damping	I [A]
	Body Control (BC)	4,12, 21 2	5, 14	27-29	Variable
	Wheel Hop (WH)	10,21 6 -21 9	11	26	≥ 1.25
	Wheel Tramp (WT)	10, 21 9 , 21 10	11	26	≥ 1.25
	Braking hard / Rapid acceleration (BH / RA)	4,5, 21 2	2	27-29	≥ 1.25
	Figure-8 Turn (F8T)	8, 21 2	9	22, 27-29	≥ 1.25
	Rapid Steering (RS)	8, 21 2	9	22, 27-29	≥ 1.25
	Bump Passing (BP)	10, 21 4	11	26	0
	Smooth Road (SR)	12, 19, 20, 21 2	14	18, 27-29	0
	Rough Road (RR)	10, 21 6	11	26	≥ 1.25
	Lane CHange (LCH)	12, 21 2	14	27-29	≥ 1.25
	Normal driving conditions	12, 15, 16	14	27-29	Variable

  Table 8 presents the results for DoEs with a constant electric current in order to generate a constant damping (soft or hard). Target application as row defines target application of the validation DoE. Target application as column defines the target application linked to the used DoE in the model identification. The column DoE specifies the used DoE in the model identification. For instance, the DoEs 12 and 21 2 are recommended for the target applications Body Control, Smooth Road, and Land Change (BC / SR / LCH) when the expected electric current sequence generates a Controlled Continuous Damping. The validity of this recommendation is shown for the cross validation test of both models when they are identified with DoE 12 and 21 2 in Table 7, and they simulate the MR damping force generated several other experiments, including the Classical DoE (CDoE) 3. The recommended DoE has the lowest ESR while the others present higher ESR.

Table 7 .

 7 Cross validation for the modified SP and VMax MR damper model parameters for continuous and On-Off damping. The results show the percentage of error for the ESR index performance. The rows define the DoE in the identification. The columns show the DoE for validation. A row with double line indicates when the DoE change the type of damping. The target applications are described by its acronyms according to Table6.

					Modified semi-phenomenological model
	Target application	WT		WH / WT		WH / RR	BP	BC / SR / LCH BH / RA	F8T / RS	CDoE
	Application					DoE with continuous controlled damping
		DoE	21 10	10	21 7	21 9	21 6	21 4	12	21 2	4		3
	WT	21 10	2.7	26.6	10.7	4.1	13.6	32.1 132.7	97.9	91		89.4
	WH / WT	10	9.8	6.4	4.6	7.1	3.5	3.7	16.1	9.9	9.9		10.6
	WH / WT		7.2	7.8	4.0	5.2	3.6	6.8	32.4	22.6	21.1	21.8
	WH / WT		3.3	17.1	6.7	3.4	8.1	19.6	86.9	61.9	59		58.6
	BC / SR / LCH	12	60.3	45.5	24.1	41.9	17.2	6.6	6.8	5.1	25.5	27.1
	BC / SR / LCH		178.2 157.3	73.9 123.7	51.5	19.4	10.6	6.9	83.1	96.1
	BP		15.7	8.7	6.3	11.2	4.7	2.9	10.4	6.7	8.1		8.3
	WH / RR		8.7	6.7	4.1	6.3	3.3	4.5	21.2	14.1	13.4	14.3
	BH / RA		13.5	7.6	6.5	10.0	5.2	4.4	11.2	8.0	6.8		7.1
	CDoE		15.8	8.3	7.9	11.9	6.4	5.5	12.0	9.4	7.3		6.7
	Application					DoE with an on-off controlled damping
		DoE			11			14	2		9	3
	BH / RA				10.6			17.1	8.6		13.5	7.3
	F8T / RS				6.8			17.3	8.9		8.8	12.4
	WH / WT / BP	11			6.8			17.4	12.9		9.0	12.3
	BC / SR	14			28.8			7.7	13.1		40.4	13.8
	CDoE				10.2			19.7	12.6		12.0	6.7
						Maximum velocity model
	Target application	WT		WH / WT		WH / RR	BP	BC / SR / LCH BH / RA	F8T / RS	CDoE
	Application					DoE with continuous controlled damping
		DoE	21 10	10	21 7	21 9	21 6	21 4	12	21 2	4		3
	WT	21 10	11.7	26.3	16.0	14.0	13.8	23.4	85.7	84.8	70.1	64.6
	WH / WT	10	17.2	15.8	13.6	16.3	11.3	18.4	52.4	79.7	48.5	39.2
	WH / WT		15.6	16.3	12.3	15.0	9.2	11.1	35.2	45.4	32.4	28.0
	WH / WT		12.4	21.2	14.0	13.5	11.5	17.7	63.9	65.3	53.1	48.4
	BC / SR / LCH	12	84.8	69.4	36.1	58.8	25.6	11.1	9.3	8.3	39.6	43.9
	BC / SR / LCH		140.8 121.3	59.4	97.6	41.8	16.6	10.4	7.2	66.5	75.6
	BP		27.7	21.5	15.1	21.7	11.0	7.3	13.9	17.1	18.2	17.3
	WH / RR		16.4	16.3	12.4	15.4	9.1	9.8	30.1	38.7	28.3	24.6
	BH / RA		24.2	20.7	16.8	20.9	13.9	10.8	14.3	12.2	13.5	14.9
	CDoE		23.2	20.0	15.8	20.2	12.6	9.2	13.7	13.2	13.9	14.5
	Application					DoE with an on-off controlled damping
		DoE			11			14	2		9	3
	BH / RA				32.3			12.4	13.6		37.8	16.2
	F8T / RS				18.7			25.1	22.3		12.7	19.0
	WH / WT / BP	11			15.7			64.7	48.1		18.6	33.1
	BC / SR	14			251.2			27.4	54.6		377.6	96.8
	CDoE				23.7			12.5	15.1		22.1	14.5

Table 8 .

 8 Cross validation for the modified SP and VMax MR damper model parameters for constant damping. The results show the percentage of error for the ESR index performance. The rows define the used DoE in the identification. The columns show the used DoE for validation. A row with double line indicates when the DoEs change the type of damping. The target applications are described by its acronyms according to Table6. Emphasis is made in the Experiment # 3 has different and constant electric currents, and Experiment # 21 2 uses an ICPS sequence as electric current,

	Modified semi-phenomenological model
	Target application	BP	BC / SR, LCH	CDoE
	Application		DoEs with soft damping
		DoE	26 1	27 1	3
	BP	26 1	12.6	44.5	52.7
	SR	27 1	93.7	23.5	41.0
	CDoE	3	27.3	75.4	6.7
	Target		WH / WT	BH / RA	CDoE
	application		RR	F8T / RS
	Application		DoEs with hard damping
		DoE	26 5	27 4	3
	WH / WT / RR	26 5	0.7	3.2	8.6
	F8T / LCH / RS	27 4	3.7	1.5	9.0
	CDoE	3	1.5	2.9	6.7
		Maximum velocity model
	Target application	BP	BC / SR, LCH	CDoE
	Application		DoEs with soft damping
		DoE	26 1	27 1	3
	BP	26 1	12.6	44.5	64.1
	SR	27 1	94.4	23.5	44.2
	CDoE	3	48.4	33.7	14.5
	Target		WH / WT	BH / RA	CDoE
	application		RR	F8T / RS
	Application		DoEs with hard damping
		DoE	26 5	27 4	3
	WH / WT / RR	26 5	11.6	16.7	30.5
	F8T / LCH / RS	27 4	95.0	4.5	75.3
	CDoE	3	11.8	7.5	14.5
	from Experiment # 12.

Table 9 .

 9 Cross validation for the modified SP and VMax MR damper model parameters for constant damping using the RRMSE as index performance. The rows define the used DoE in the identification. The columns show the used DoE for validation. The target applications are described by its acronyms according to Table6.

			Modified SP model
	Target		BH	BC / SR	WH / WT	BH	CDoE
	application		RA	LCH	RR	RA
		DoE	4	21 2	26 5	2	3
	BH / RA	4	26.1	28.3	11.1	29.8	26.7
	BC / SR	21 2	51.7	22.3	57.1	44.6	53.0
	LCH
	WH / WT	26 5	28.8	26.3	8.3	32.9	29.2
	RR
	BH / RA	2	27.0	29.7	13.8	29.3	26.9
	CDoE	3	27.0	30.6	12.1	30.0	25.8
				VMax model
	Target		BH	BC / SR	WH / WT	BH	CDoE
	application		RA	LCH	RR	RA
		DoE	4	21 2	26 5	2	3
	BH / RA	4	36.7	34.8	34.3	37.7	38.5
	BC / SR	21 2	91.8	27.8	111.7	73.8	98.2
	LCH
	WH / WT	26 5	61.6	79.1	34.1	68.4	55.1
	RR
	BH / RA	2	37.8	34.0	35.0	36.8	40.1
	CDoE	3	37.3	36.2	34.3	38.9	38.0

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Authors thank the PCP 2007-2011 (Programa de Cooperación del Posgrado, CONACYT) that partially supports the Tecnológico de Monterrey (Mexican side) and the Institut National Polytechnique de Grenoble (French side) during this research as an international collaborative project.