Magnetorheological Fluid Dampers: A Close Look at Efficient Parametric Models

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Introduction
One of the ongoing challenges in civil engineering is the protection of facilities from destructive forces caused by wind and earthquakes.Te traditional seismic design assumes that an earthquake acts on a building through a solid base.To ensure partial dissipation of the induced energy, plastic deformation occurs in certain building components and structural damage occurs to some extent.Tis drawback can be avoided using structural control strategies.Te concept of control applications for improving the seismic performance of structures has been considered for several years.Passive supplemental damping strategies are well known and widely accepted by engineers for mitigating the efects of dynamic loading on structures [1][2][3][4].However, preliminary studies indicate that appropriately implemented semiactive systems perform signifcantly better than passive devices and have the potential to achieve most of the performance outcomes of fully active systems.Tis enables an efective reduction in responses under a wide range of dynamic loading conditions.Te reasonable efciency, high reliability, and minimal maintenance requirements of semiactive control algorithms render this form of structural control, the most successful method in terms of theoretical and practical considerations.Additionally, signifcant advances have been made in structural control devices.Among these, dampers, particularly magnetorheological (MR) dampers, appear to have signifcant potential to advance the acceptance of structural control as a viable means of mitigating dynamic hazards and have accounted for a large portion of studies in this area.
Jacob Rainbow [5] developed an MR fuid smart material in the 1940s.An MR damper comprises this solution, which can reversibly transform from a free-fowing linear viscous fuid to a semisolid with a controllable yield point in the presence of a magnetic feld [6].Tis feature provides simple, quiet, and responsive interfaces between the electronic controls and mechanical systems.Typically, MR fuids fow freely with a consistency similar to that of a motor oil [7].However, in the presence of an applied magnetic feld, iron particles (carbonyl iron) acquire a dipole moment aligned with the external magnetic feld, causing them to form linear chains parallel to the feld.Tis phenomenon solidifes suspended iron particles and restricts the motion of the liquid.Consequently, the yield strength develops within the fuid.Tis change is refected in the change in the damping force when MR fuids are used.Te magnitude of the change depends on the strength of the applied magnetic feld and can occur within a few milliseconds.
Unlike their electrical counterparts' electrorheological (ER) fuids, MR fuids are not extremely sensitive to moisture or other contaminants that may arise during their manufacture and use [8,9].Because the mechanism of magnetic polarization is not afected by temperature, the performance of devices based on MR is relatively insensitive to temperature over a wide range of temperatures (including automotive use) [7].Magnetorheological fuids can be used in three ways, all of which can be applied to MR devices depending on the intended use of the damper.Tese modes of operation are referred to as pinch, valve (pressurecontrolled fow), and shear modes.Te last mode of operation of the MR steamer, the valve mode, is the most commonly used among the three modes [10].An MR device operates in the valve mode when the MR fuid is used to impede its fow from one reservoir to another.Most devices that use controllable fuids can be classifed as either fxedpole devices (which often operate in pressure-controlled fow mode) or relatively movable-pole devices (which operate in direct shear mode) [10].
Te commercialization of MR technology began in 1995, with the use of rotational brakes in aerobic equipment.Since then, the application of MR material technology to actual systems has steadily increased [11].Magnetorheological fuids operating in the valve mode with fxed magnetic poles are suitable for hydraulic controls, servo valves, shock absorbers, and dampers (including models referred to as tubular/linear MR dampers).Te direct shear mode with a moving pole is suitable for clutches and brakes, clamping/ locking devices, dampers (including models referred to as the shear mode or rotating MR dampers), breakaway devices, and structural composites [9].
In recent years, several commercially available products have been developed or are close to commercialization [12,13]: (i) MR dampers for real-time active control systems in heavy duty trucks (ii) Linear and rotary brakes for low-cost, accurate, positional, and velocity control of pneumatic actuator systems (iii) Rotary brakes to provide tactile force-feedback in steer-by wire systems (iv) Linear dampers for real-time gait control in advanced prosthetic devices (v) Adjustable real-time controlled shock absorbers for automobiles (vi) MR sponge dampers for washing machines (vii) Magnetorheological fuid polishing tools (viii) Large MR fuid dampers to control wind-induced vibrations in cable-stayed bridges (ix) Very large MR fuid dampers for seismic damage mitigation in civil engineering structures Magnetorheological dampers provide an attractive solution for energy absorption in structures and considered fail-safe devices and are inexpensive [6], have few moving parts, and are reliable.Tese characteristics render MR dampers promising voltage-or current-controlled actuators that can be used in various engineering applications [7,14].In civil engineering applications, the expected damping forces are considerably high.For MR dampers, a memorydependent multivalued relationship between force, deformation, and hysteresis is observed.Many mathematical models have been developed to efciently describe this behavior and use them for time history and random vibration analysis.High-precision models for MR dampers can be developed using two families of models: semiphysical [15,16] and black-box models [17,18].Semiphysical models use a simplifed model of the physical device and thereafter use some type of measurement to identify its free parameters.Te black-box model, on the other hand, is a strategy for studying a complex object or device without any knowledge or assumptions about its internal structure, parts, or model.Although some quasistatic models have been proposed and shown to describe the force-displacement relationship of the MR damper reasonably well, they have not been able to model its nonlinear force-velocity behavior.
More accurate dynamic models have been developed and can be divided into two categories: nonparametric and parametric.Nonparametric models are based solely on the performance of the device and typically require a large amount of experimental data to show the response of the fuid to various loads under diferent operating conditions.Tese proposed models are based on Chebyshev polynomials [19][20][21], neural networks [22][23][24], and neuro-fuzzy systems [25][26][27].Neural networks can accurately reproduce the nonlinear behavior of MR fuids.However, since there is no specifc mathematical expression for nonparametric models, the information obtained from the laboratory results must be examined in the form of complex cognitive methods to achieve acceptable accuracy with the neural 2 Structural Control and Health Monitoring network method.In order to obtain appropriate results, the data set must be subjected to strict conditions.In addition, part of the data is always used to learn the model.On the other hand, the sensitivity of nonparametric models to interference from laboratory methods and sensors is low and one can work with diferent data sources, including realworld data collection under less controlled conditions.Multilayer perceptron (MLP) networks are one of the most commonly used neural network types and have the distinction of using only a single nonlinear function.Tey have also been shown to model both simple and complex systems accurately.Neuro-fuzzy models are another example of nonparametric models that have been proposed to emulate the behavior of MR dampers.Because these devices are highly nonlinear, fuzzy logic has been proposed as an alternative to the computationally intensive models that are currently in use.Neural networks were subsequently used to ft the fuzzy logic parameters.Fuzzy logic incorporates human knowledge about the system into the controller using membership functions, which are quantities that defne imprecise or vague concepts, such as large, weak, hard, and moderate.Te desired output is determined based on fuzzy information of the inputs, similar to how the human brain makes decisions.Although several nonparametric models can efectively reproduce the behavior of MR dampers, their application is often hindered by their complexity and the large amount of experimental data required for training and model validation.Terefore, parametric models are more commonly employed in simulations and in the development of control algorithms.Te objective of this study is to compile and present the most successful parametric models for this type of damper.In this study, we delve into models that are typically the most ftting parametric types, having undergone revisions throughout various research and testing phases.Grasping the evolution process, the interrelated combinations, and the role of each component in delivering the damping force ofers a valuable guide.Tis insight can help propel research forward, culminating in more comprehensive relationships that align with the intricate and highly nonlinear behavior of this damper series.
Our research primarily concentrates on the potential and competence of the fnal model, especially its capability to yield an inverted model.Te signifcance of indicator models is underscored.While developing some of these models, the dependency of the model on a governing parameter, namely, its interaction with the device, was also taken into account and discussed.From this vantage point, this research proves invaluable for researchers aiming to choose an apt model for their control structure design.Notably, each model can be rationally associated with one of the standard variables (either current or voltage) used in the damper during the tool's identifcation process.Te precision and the researchers' emphasis on this aspect are accentuated in our study.
On the other hand, there are only a handful of models that rely on the frequency of stimulation during the identifcation phase.Te dependence of the damper model on the frequency of the external stimuli is an aspect that is complicated in practical application.Te vibrational properties of earthquakes themselves are very complex and extensive in terms of their variety and frequency content.Moreover, from a practical point of view, it is unsuitable to determine this content during an earthquake.So, there is no simple practical method to fnd out how much this frequency content infuences the results.Terefore, this dependence model is limited to a small number in this overview and there is no separate section for them.
To streamline access to these models, reference is made to Table 1.Tis table lists the model name, the relation number mentioned in the text, its reversibility, and the kind of governing parameters.As observed, there are select models that, in addition to meeting reversibility criteria, can replicate precise numerical results pivotal for devising a suitable control algorithm for structural applications.

Dynamic Models Based on the Bingham Model
Tis group of modelers used the stress-strain diagram of the Bingham viscoplastic model [28] to describe the behavior of MR fuids.Based on the viscoplastic model, when the stress is greater than the feld-dependent yield stress, the damper fuid can be represented by the Bingham equation as follows: where η is the viscosity of the fuid and _ c is the shear strain rate.Te material behaved viscoelastically when the stress was lower than the yield stress.Many researchers based their modifed version of the damper on this model.

Bingham Standard
Model.Spencer et al. [16] proposed an idealized mechanical model used by Stanway et al. [29] to express the behavior of an ER damper based on the model described in (1).Tis model consists of a Coulomb friction element parallel to the viscous segment, as shown in Figure 1.
In this rigid viscoplastic model, the force generated by the damper is given by the following equation: where _ x, c 0 , and f c denote the piston velocity, damping coefcient, and frictional force, respectively.f 0 is the ofset in the resulting damping force owing to the accumulator.Tis model describes only the force-displacement behavior of the MR damper; however, it cannot accurately represent the force-velocity behavior, particularly in the roll-of region.[30] proposed an extension to the Bingham model for ER fuids (Figure 2).Spencer et al. [16] used the viscoelastic-plastic model for MR dampers.Tis model consists of a classical Bingham model connected to the Kelvin-Voigt representation of a standard linear solid model, also known as the Zener model (with two springs and a dashpot).In the Zener

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where c 0 denotes the damping coefcient of the Bingham model.Parameters k 1 , k 2 , and c 1 are the stifness and damping coefcients of the Zener model, respectively.Although this model is suitable for predicting the behavior of a damper, its use in numerical problems is difcult.

Bingham Plastic Model
. By adding a yielding force F y to the linear damping model, Werely et al. [31][32][33] introduced the Bingham plastic model, as shown in Figure 3, which is similar to the Bingham standard model (equation ( 2)).Te equations describing the model are as follows: Te Bingham plastic model is expressed as follows: Tis model assumes that, in the preyield state, the materials are rigid and do not fow.Te shaft velocity is zero if F is lower than F y .Te fuid fowed when the applied force exceeded F y .Subsequently, the materials became Newtonian fuids with a nonzero yield stress.Although this model can accurately predict the force-displacement diagram and dissipated energy owing to the presence of the yield force, which is reminiscent of Coulomb friction, the force-velocity diagram of the MR damper is not correctly modeled.Te behavior of the damper is displayed rigidly in the preyield area, which is termed the roll-of area.[34] proposed the modifed Bingham plastic model shown in Figure 4. Tis model consists of a standard Bingham model with a stifened k 1 .Te equation of the model is as follows:

Modifed Bingham Model. Zhou
2.5.Improved Bingham Model.Based on an experiment performed by Occhiuzzi et al. [35], they found that the standard Bingham model overestimates the viscous component of the force-displacement cycle for i � 0 A and underestimates it for i � 2.5 A, where i is the current of the MR coil.Tey ofered a relation between the damping coefcient and the magnetic feld of the damper, which fnally led to a dependency on the current.Te resulting modifed Bingham model is as follows:  6 Structural Control and Health Monitoring between the viscous damping coefcient and the piston velocity of the damper, according to the following equation: where c(i) and α(i) are current-dependent parameters.Te fnal improved Bingham model can be obtained by integrating equations ( 7) and ( 8): Because their experimental data did not show any trace of roll-of behavior, this model could not handle such an infuence.
2.6.Nonlinear Bingham Hysteretic Model.Zhang and Huang [36] proposed a nonlinear hysteretic model for the Bingham model.At low velocities, this model can describe the hysteresis characteristics of the force-velocity diagram and the nonlinear behavior of the MR damper.Te nonlinear Bingham hysteretic model (Figure 5) is given by where F y denotes the yield force of the MR fuid; c 0 , c 1 , and c 2 are constant coefcients of the nonlinear damping term of the model; k denotes the equivalent stifness of the model spring element; f 0 is the deviation force owing to the accumulator; and x 1 is the elongation of the spring, which is given by (11).
Te model variables are dependent on the applied current.

Hysteresis-Regularized Bingham Model.
Using the standard Bingham model, Soltane et al. [37] proposed the hysteresis regularized Bingham model (HRB).Tey applied a regularization technique [38] to the discontinuous Bingham equation and converted it into a continuous relation as shown in Figure 6.Equation (12) shows the generated force of the MR damper, based on the following model: where _ x 0 , and together with the velocity dimension, is a regularization parameter that exponentially controls the growth of the damping force.Equation ( 12) is converted into (13) to consider the nonlinear hysteresis behavior of the MR damper.Structural Control and Health Monitoring where _ x h is the scale factor, which has a velocity dimension and defnes the width of the hysteresis loop.Dynamic Bingham models have been used to simulate the behavior of MR dampers in several studies.It is simple, efective, and numerically easy to handle.However, there is another group of models for these devices that is even more signifcant, which are described in the following sections.

Dynamic Models Based on the Bouc-Wen Hysteresis Operator
Te Bouc-Wen model is widely used for modeling hysteretic systems.Tis model was frst proposed by Bouc [39], which was subsequently generalized by Wen [18,40].It is extremely versatile and can represent a wide range of hysteresis behaviors in devices and materials.Terefore, they have been used to model MR dampers.Te force generated by this model for a nonlinear hysteretic system is Function g(x, _ x) is a nonhysteretic component that can include the elastic property and/or viscous behavior of the damper.x and _ x denote the displacement and velocity of the system, respectively.α is the scaling parameter for the hysteretic term of the model.z, the hysteretic component of the model that depends on the time-history of the displacement, is the core of the model and assumes the shape of the hysteresis loop.Te evolutionary variable was obtained using the following equation: where c, β, A, and n are key parameters that defne the shape of the hysteresis loop.

Standard Bouc-Wen Model.
Based on the Bouc-Wen model, Spencer et al. [16] proposed a new model consisting of three components, as shown in Figure 7. Equation ( 16) describes the force generated by the model.
where c 0 is a constant parameter for the viscosity term; k 0 is the stifness of the damper, which refects the elastic phenomenon of the MR fuid; and z is the hysteretic parameter of the model, given by (15).Tis parameter represents the dynamic behavior of the device.x 0 is the initial displacement 5: Nonlinear Bingham hysteretic model (Section 2.6).

Modifed Bouc-Wen (Phenomenological) Model.
Te standard Bouc-Wen model provides a good prediction of the force-displacement behavior of the damper; however, it does not perform well in describing the roll-of region of the force-velocity loop.To better predict the damper response in this region, a modifed version of the model was proposed by Spencer et al. [16] as shown in Figure 8. Te following equation describes the force generated by the model.
In ( 17), k 1 is the accumulator stifness; c 0 is the viscous damping parameter at high velocity; c 1 is the damping parameter of the dashpot added to the model to produce a roll-of efect at the low velocities; k 0 controls the overall stifness of the device in a high-velocity situation; and x 0 denotes the initial displacement of the damper and produces an ofset force when multiplied by k 1 .Tis model has been used as an MR-damper model in extensive articles (for example, [49][50][51][52][53]).It must be mentioned that this model is referred to as the phenomenological model in many studies.For the proposed model to correctly predict the alternating behavior of the MR fuid under magnetic feld changes, Spencer et al. [16] proposed a voltage-dependent linear relation for some of the parameters in the models as follows: In this relation, u is an intrinsic variable that relates the dependence of the parameters on the applied voltage and is calculated using the frst-order flter as where v is the voltage applied to the current driver and η refects the response time of the MR damper [16].Assuming that the parameters of ( 18) are polynomials of the third degree, Yang et al. [54] performed a process of parameter identifcation and proposed a model for a large-scale MR damper.Te suggested relations for the parameters are as follows: Spaggiari and Dragoni [55] studied the behavior of the damper at low frequencies.Tey modifed the model by interpolating the parameters α, c 0 , and c 1 based on the results of laboratory tests.Te proposed quadratic polynomial functions for these parameters are as follows:

Bouc-Wen
Model in the Shear-Mode MR Damper.Yi et al. [56] proposed a model for shear mode MR dampers established on the Bouc-Wen base model by Spencer et al. [16].Spencer team damper was tubular in which MR fuid was sealed in a cylinder with a movable piston (see Figure 9).
In shear-mode dampers, the MR fuid is confned to the parallel plates.Figure 10(a) shows the schematic of this type of damper.Te common models for these dampers consist of a viscous friction element and hysteretic component (see Figure 10(b)).
where z is the evolutionary variable given by (15).

Modifed Bouc-Wen Model with Mass
Element.Tis model, proposed by Yang et al. [60] for a large-scale MR damper, incorporates the MR fuid stiction phenomenon, as well as the efect of shear-thinning and fuid inertia.A where m is equivalent mass which represents the MR fuid stiction phenomenon and inertial efect; k is accumulator stifness and MR fuid compressibility; f 0 is the damper friction force due to seals and measurement bias; and c 0 ( _ x) is the postyield plastic damping coefcient, which is defned as a monodecreasing function with respect to absolute velocity _ x, to describe the MR fuid shear-thinning efect which results in the force roll-of of the damper resisting force in the low-velocity region.Te postyield damping coefcient is given by the following equation: where a 1 , a 2 , and p are positive constants.Lau and Liao [61,62] and Tsampardoukas et al. [63] used this model in train and truck suspensions, respectively.Additionally, Yang et al. [60] presented two more dynamic models based on the standard Bouc-Wen form and the modifed version updated with a mass element (Figure 12) [60].
According to Figure 7, the force generated by the damper in the standard Bouc-Wen model with the mass element is according to (25) and that for the modifed Bouc-Wen model with the mass element is according to the following equations [60]: Te variables are defned in equations ( 15) and (17).

Amplitude-Dependent Bouc-Wen Model. Ali and
Ramaswamy [64] proposed the amplitude-dependent Bouc-Wen model.In this model, the parameters c 0 , k 0 , and α expressed in (16) depend on the amplitude of excitation and the input current to the MR damper.Te relationship between these parameters and the input current, i c , changes linearly, and the efects of the motion amplitude are considered as a quadratic function of the stroke amplitude (x a ), as shown in the following equation: where the parameters c 1 , c 2 , . .., α 5 , and α 6 are constants obtained from the identifcation process.
3.6.Current-Dependent Bouc-Wen Model.Dominguez et al. [41] proposed a new Bouc-Wen model, in which the current is considered as a variable.In the standard Bouc-Wen model, the current is not considered a variable; therefore, the model parameters must be estimated for each current excitation.However, this can be computationally expensive.Upon entering the fow, ( 16) can be rewritten as follows: Te evolutionary variable (z) is described by the following frst-order diferential equation: If A(I) and β(I) are assumed to be unity and zero, respectively, z(I) is obtained as follows:

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In addition, owing to the experimental results, the relationship between the current and the parameters c and k 0 is linear, and the parameters c 0 , α, and F z0 change exponentially with the current.
where ω is the excitation frequency; x max is the excitation displacement amplitude; and d 1 ,. .., d 4 are constants.z is an evolutionary variable defned by the diferential equation in (30).A linear relationship exists between the current and parameters c and k 0 .For the parameters c 0 , α, and F z0 , this dependency is related to the values of the driven current.When the device current was low, these parameters changed linearly, and at the point of high current, an exponential relationship was chosen.A border value, I c , was defned to separate the low and high current values.Dominguez et al. [65,66] used this model in their study.
3.8.Asymmetric Sigmoid Modifed Bouc-Wen Model.Ma et al. [67] found that the Bouc-Wen model equation modifed in [16] can be further modifed.Spencer et al. [16] described the behavior of an MR damper linearly using current-dependent functions according to (20), whereas in the proposed model [67], the damper force can be described in a general form with two independent functions, as shown in the following equation: where f i (i) represents the current function and f h (x, _ x, € x) is a hysteresis function that can be the same as the modifed Bouc-Wen model introduced by Spencer et al. [16] according to (17).Te current function f i (i) is derived from the following equation where k 2 and a 2 are positive constants and I 0 is an arbitrary constant that represents a bias and is identifed from experimental or benchmark data.
3.9.Nonsymmetrical Bouc-Wen Model.Because of nonsymmetrical hysteresis behavior that can be seen in the force-velocity response of some MR dampers, Kwok et al. [68] proposed a nonsymmetrical model based on the standard Bouc-Wen model.Te evolutionary variable expressed in (15) can be rewritten as follows: 12 Structural Control and Health Monitoring Te nonsymmetrical Bouc-Wen model proposed by Kwok et al. [68] is obtained as (37) by adjusting the velocity value where μ is a scale factor for the adjustment and sgn(.) is a signum function.Te force generated by the MR damper according to Figure 13 is obtained from the following equation: where z is the asymmetric Bouc-Wen hysteresis operator expressed in (37).Note that the overall efect of the shift in the hysteresis switching is in the vicinity of zero velocity, whereas the form of hysteresis is maintained for the rest of the loop.
In this general form of the Bouc-Wen model, some parameters are redundant.To eliminate this redundancy, a normalized form of the Bouc-Wen model was proposed in [69][70][71][72][73] using a transition parameter w(t): Terefore, the Bouc-Wen model can be rewritten as where Te normalized form of the Bouc-Wen model is an equivalent representation of the original Bouc-Wen model.Tis model has fewer parameters; therefore, the overparameterization in the original Bouc-Wen model was removed.In this representation, the evolutionary variable is in the range Tsouroukdissian et al. [74] proposed a normalized Bouc-Wen model for small-scale MR dampers.Teir proposed model have the same form as (42), with one major diference: they replace x with _ x. Terefore, the force generated by the MR damper is obtained from the following equation: Parameter w(t) is calculated using (43).Equation ( 45) is more compatible with the force-velocity behavior of MR dampers.Parameters k x , σ, and n are constant values, and k w is considered a linear variable with voltage.In their research, two diferent forms were considered for ρ: a linear voltagedependent relation or a constant value.Te results showed that the accuracy of the frst model (voltage-dependent) was higher than that of the second.In 2009, Rodriguez et al. [75] used the same normalized Bouc-Wen model [47] for a largescale MR damper.
Bahar et al. [76] modifed the normalized Bouc-Wen model for a large-scale MR damper.To improve the model accuracy for large-scale dampers, they developed normalized Bouc-Wen model parameters, as shown in the following equation: Te frst term in this formula, k x x(t), represents the linear-elastic force added to (45).According to the parameter identifcation results, k x is constant and k _ In 2010, the same group [77] used their model as an MR damper in a benchmark structure and proposed a new successful hierarchical control algorithm.

Normalized Bouc-Wen
Model II.Te model proposed by Dominguez et al. [42] has several parameters.Tis redundancy makes the identifcation and modeling procedure of the MR damper complex.Dominguez et al. [43], based on the same models [41,42], presented a new model with fewer parameters and higher accuracy.If in (29), β is zero, then n is assumed to be equal to z and A � c.Te evolutionary variable (z) becomes normalized and has the simple form of the following equation: Parameter z was obtained from the solution of the above equation.
In the above equation, λ i is a constant originating from integration.As the MR damper hysteresis force is velocitydependent in practice, in (49), the velocity is replaced by the displacement (i.e., _ x is replaced by x), as shown in the following equation: Te frst part of (51) shows the upper curve of the hysteresis loop and the second part shows the lower curve.Te force generated by a damper in such a model is expressed as follows: Te parameter ρ is a new evolutionary parameter.It is used instead of the parameter α in the previous form of the Bouc-Wen model presented in (22).Te parameters ρ, c 0 , and k 0 can be approximated by the root-squared function weighted by the proper constants.Parameter λ did not change signifcantly with respect to the current variation.Terefore, it can be considered a constant parameter.
Another efectual decrease is that Dominguez et al. [65] assumed that the integration constant parameter λ could be approximated by − € x.With this change, ( 49) is written as the following equation: While the Bouc-Wen model is widely used, it exhibits two primary shortcomings when characterizing the hysteresis phenomenon.First, there is a signifcant discrepancy between the hysteresis curves derived from experimental data and the model when assessing constant characteristic parameters.Optimizing these parameters demands substantial computational efort.Second, the model's performance hinges on specifc excitation conditions.Typically, the hysteresis loop is formulated for a harmonic excitation characterized by a defned amplitude, frequency, and current excitation.Altering these conditions necessitates recalibrating the Bouc-Wen parameters.Recognizing these challenges, Dominguez et al. [43] introduced a normalized Bouc-Wen model that streamlines the number of constant characteristic parameters.Tey elucidated the impact of each parameter on the hysteresis loop and proposed an innovative method to extract these parameters from experimental data.Conclusively, they presented a model rooted in the normalized Bouc-Wen framework, wherein displacement, velocity, acceleration, and current excitation serve as inputs, with the hysteresis force as the output.
3.12.Normalized Phenomenological Model.Bai et al. [78] introduced a normalized phenomenological model by incorporating the concept of normalization into the modifed Bouc-Wen model.In the modifed Bouc-Wen model or phenomenological models shown in Figure 8, a dashpot denoted by c 1 is connected in series with the Bouc-Wen model to generate the roll-of force at low velocities.Such combinations increase the capacity of the model to describe the hysteresis behavior of the MR damper.However, the complexity of the model also increases because of the differential equation required to represent the dashpot.Te concept of normalizing the Bouc-Wen model was introduced by Ikhouane and Rodellar [70][71][72], as discussed in the previous section.By incorporating the normalized Bouc-Wen model into the phenomenological model, (17) can be written as Te above equation, when replaced with a similar relation as in (17), initiates the normalized phenomenological model.
Te intricacy of the phenomenological model somewhat restricts its applicability in real-time control systems.To address this, researchers simplifed the model, focusing on both its structural and expressive facets.One particular challenge is the inclusion of the dashpot in the phenomenological model.Its presence augments the complexity of the mathematical model, primarily due to the diferential equation that describes the dashpot, complicating the parameter identifcation process.However, by incorporating the concept of normalization within the phenomenological model, it is possible to decrease the number of parameters by one.

Restructured Model.
Figure 14 shows a schematic of the restructured model proposed by Bai et al. [78].Te restructured model was inspired by the phenomenological or modifed Bouc-Wen model [16].Tey removed the elastic element, k 0 , of the phenomenological model.Te position of segment c 0 moves and is located parallel to the springy component, k 0 .By incorporating the normalization concept, the restructured model can be written as follows: Te use of the restructured model is more straightforward than the modifed Bouc-Wen model [16] because of the lower dependency between the parameters.Trough the modifcation of the phenomenological model's structure and the integration of the normalization concept, a restructured model was introduced.Tis new model boasts a more streamlined and lucid structure, facilitating computer simulations and the parameter identifcation process.As a result, the foundational structure of the restructured model is an evolution from the original phenomenological model, with the added beneft of a reduced parameter count.

LuGre Models
Canudas et al. [79] proposed a dynamic model for friction that captures most of the experimentally observed friction behavior.Tis includes the Stribeck efect, hysteresis, springlike characteristics for stiction, and varying breakaway forces.Te model has the following form and is characterized by parameters a 0 , σ 0 , σ 1 , and σ 2 .

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Traditional models fall short in capturing the hysteretic behavior observed during the study of friction at transient velocities.Additionally, they fail to account for variations in the breakaway force based on experimental conditions and the minor displacements in the contact region during stiction.Addressing these gaps, Canudas et al. [79] introduced a dynamic friction model that melds the Dahl efect, representing stiction behavior, with arbitrary steady-state friction properties, potentially encompassing the Stribeck efect.
In 2002, Alvarez and Jiménez [80] modifed this model by considering the efect of the magnetic feld and assumed that the current that determines the intensity of the feld is proportional to the applied voltage.Te modifed model is Te novel modeling approach, grounded in the frstorder dynamic friction model of LuGre for MR dampers, boasts a more streamlined analytical structure.Impressively, it retains the capability to replicate force responses across various excitation signal types.A salient characteristic of this modeling framework is its facilitation of real-time parametric identifcation for the MR damper model parameters.Tis is possible because, with suitable parameter manipulation, the model can be linearized [80].
Sakai et al. [81] proposed a modifed model to produce a simple model which can create an inverse model and express the dynamic friction characteristics, as well as the hysteresis efect.Te modifed model is described as Sakai et al. [81] aimed to introduce a model capable of predicting the damping force using the MR damper's velocity, internal state, and input voltage.Beyond this primary capability, their proposed model can also generate an inverse dynamic model, thereby determining the necessary input voltage from a given desired damping force.Tis advancement was motivated by the limitations of the model presented by Canudas et al. [79], which, despite its fewer parameters and capacity to articulate the hysteresis function, fell short in efciently computing the optimal input voltage.
To manage the MR damper during the control process, Palka et al. [82] proposed an inversion of the LuGre model with thirteen parameters.

Biviscous Models
Biviscous models are another group of relations used to describe the nonlinear behavior of magnetorheological dampers.Generally, they are presented as piecewise lines or curves in a force-velocity diagram to illustrate the hysteresis behavior of MR dampers.

Nonlinear Biviscous Model.
Instead of assuming that the materials are rigid in preyield circumstances, Stanway et al. [83] assumed that the materials in both the preyield and postyield states behave plastically.Based on this assumption, Wereley et al. [31], Pang et al. [32], and Snyder et al. [33] have proposed nonlinear biviscous models.Tey assumed that the preyield damping C pre was much larger than the postyield damping C post (C pre > C post ).Terefore, the yield force can be determined by an extension of the postyield force line on the force-velocity curve and its intersection with the force axis, as shown in Figure 15.Te force generated by this model is expressed as follows: where _ x y denotes the yield velocity.16

Nonlinear Hysteretic Biviscous
Structural Control and Health Monitoring Wereley et al. [31] suggested some current-dependent relations for the four main parameters of their model: C pre , C post , _ x 0 , and F y .Based on the results obtained from the experimental data, second-order polynomials are needed for C pre , C post , and F y , whereas fourth-order polynomials are required for _ x 0 as functions of the applied current.

Nonlinear Hysteretic Arctangent Model. Wang and Liao
[86] introduced a model from Ang et al. [87] that describes the force-velocity relation of the hysteresis loop according to the following equation: In (62), α is the magnifcation factor and β is the rotation factor.Tese parameters must be detected from the shape of the resulting force-velocity curve.Te proposed model Structural Control and Health Monitoring produces a symmetric hysteresis loop around the zero point of the coordinate system.

Lumped-Parameter Model of Fluid Flow. Sims et al. [88]
introduced a lumped-parameter model for fuid fow within MR dampers.Teir design conceptualizes fuid motion through a lumped-parameter model intrinsically tied to the device's geometry, as depicted in Figure 17.Tey contended that such models should not only predict the damper's performance in isolation but also its behavior within a more intricate vibrating structure.Tis perspective diverged from earlier modeling priorities, which predominantly emphasized the damper's standalone performance, often characterized under sinusoidal excitation and open-loop scenarios.In response to these evolving needs, the authors advanced a modeling technique that balances these distinct prerequisites while retaining the paramount physical relevance of core parameters.Additionally, their approach facilitates system identifcation or model revision, ensuring that the model faithfully mirrors observed behaviors.Figure 18 shows a schematic representation of the model with lumped parameters.Te equations of motion for this object are as follows: Here, the quasisteady valve fow is represented by a nonlinear ψ function, which is a function of the quasisteady velocity x 1 and management signal H.For an MR damper, this signal is typically an electrical current (I) that generates a magnetic feld in the valve.Fluid compressibility is indicated by a spring with stifness k.Fluid inertia is denoted by m 1 .F is the force applied to the damper, which causes piston movement (x 2 ).An additional mass (m 2 ) is used to consider the mass of the piston parts and accessories.Te physical signifcance of the model is its ability to defne its parameters based on constitutive relationships using fuid properties and device.It is not necessary to observe the behavior of the actual appliance.Tis implies that the model can be developed before the device is built; thus, it is a suitable tool for the prototype design of a damper.
Te quasisteady behavior described by ψ can be described in a biviscous function format that follows: Te relationship between C pre , C post , and F y with the current applied to the damper (I) can be expressed as (65) using the hyperbolic tangent function.

Viscoelastic-Plastic Models
Weiss et al. [89] and Jolly et al. [90] indicated that MR fuid behaves as a viscoelastic fuid in the preyield region and as a viscoplastic fuid in the postyield region.Based on this knowledge and perception, some viscoelastic-plastic models have been developed.Subsequently, it extended it to the MR dampers [32,33].

Nonlinear Viscoelastic
Based on the force-velocity hysteresis loop of the MR dampers, there are two distinct rheological regions: the preyield and postyield regions.Te preyield area demonstrates strong hysteresis, which is regular for the viscoelastic behavior of such materials.Te postyield region changes with a nonzero yield force, in which the yield force is a function of the applied current (e.g., changes in the magnetic feld).Te structure of the nonlinear viscoelastic-plastic model introduced by Pang et al. [32] is based on the block diagram in Figure 19.

Preyield Mechanism. Te Kelvin element in
Figure 20(a) represents the mechanical analogy of the viscoelastic damper's behavior in the preyield region.Equation ( 66) expresses this mechanism in the time domain.
18 Structural Control and Health Monitoring where F υe is the viscoelastic element of the damper force.Te nonlinear shape function S υe is a preyield switching function.Tis is parallel to the postyield switching function.S υi undergoes a smooth transition from the preyield to postyield phase.Te S υe function depends on the yield velocity (υ g ) and is established during the identifcation process.S υe is obtained from the following equation: In the above equation, υ(t) is the instantaneous velocity, υ y is the yield velocity, and ε y is a smoothing parameter.Te force component is obtained as a derivative of the preyield mechanism.

Postyield Mechanism.
In the postyield stage, the damper behavior is similar to a viscous damping device with a nonzero yield force.Te postyield mechanical simulation, F υi , is shown in Figure 20(b) as a viscous element.Te force is expressed as follows: where S υi is a shape function similar to S υe .Tis function fulflls as a switching operation to trigger a postyield viscous mechanism when the damping force exceeds the yield force.Tis is obtained using the following equation: Terefore, the force component of the postyield mechanism is as follows: 6.1.3.Yield Force.Te yield force F c is a function of the applied domain and is a domain-dependent parameter that provides the damper with its semiactive capabilities.Te damper behavior shows some forms of the Coulomb force efect at low velocity.Tis efect is included in the force parameter F c and the shape function S c .Te shape function is given by the following equation:

Structural Control and Health Monitoring
In the above equation, v(t) is the velocity magnitude and v c is a smoothing factor that ensures a smooth transition from a negative velocity to a positive value and vice versa.
6.1.4.Final Form of the Viscoelastic-Plastic Model.Equation ( 74) shows the fnal form of the damping force of the viscoelastic-plastic model.
Te assumption behind this equation is that the damping force is a combination of functions, each of which is a linear mechanism with nonlinear shape functions.Wereley.In 1980, Wereley et al. [93,94] used the previously described preyield and postyield mechanisms and proposed a viscoelasticplastic model by changing the transition function from the preyield to the postyield regions.Te researchers observed that prevailing dynamic models heavily lean on experimental data.While this reliance is somewhat inescapable given the intricate material behaviors at play, the absence of a phenomenological approach in modeling ER fuid often amplifes the volume of experimental data needed for parameter estimation.Notably, models of this nature typically encompass a limited set of parameters.As a solution, they introduced a nonlinear viscoelastic-plastic model, embodying a novel approach to characterize ER fuid.A schematic representation of this model can be seen in Figure 21.

Viscoelastic-Plastic Model Proposed by
As shown in Figure 21, in this viscoelastic-plastic model and unlike in Figure 19, there is no yield force, and its descriptive relation is given by the following equation: where S υe and S υi are the nonlinear preyield and postyield shape functions, respectively, obtained from equations ( 69) and (65).F υe and F υi are the pre and postyield damping force components obtained from equations ( 61) and ( 64), respectively.

Viscoelastic-Plastic Model Proposed by Li.
Li et al. [20] started using a mechanism similar to that of the viscoelasticplastic modeling introduced by Wereley et al. [93,94] to model the behavior of the MR damper, whereafter they modifed the preyield and postyield mechanisms of the viscoelastic-plastic model.As shown in Figure 22, they developed a new model for the MR damper by integrating a rigid three-parameter model, similar to the mechanism used for viscoelastic behavior in the preyield region (Figure 22(a)).Tis model consists of a spring (k 2 ) connected in series with a rigid Kelvin-Voigt model, represented by k 1 and c 1 .
In addition to the viscoelastic force F υe , the stiction efect (F s ) resulting from the piston seal enters the damping force.Terefore, the damping force in the preyield region is given by the following equation: When the damper force F is higher than the yield force of the damper F c , the MR damper operates in the postyield phase.Both the inertial components and the fuid viscous residence contributed to the yield force.Terefore, the damper's postyield force (F p ) can be expressed as follows: In ( 77), c v is the viscous coefcient and R is the equivalent inertial mass.By combining these two phases, the resulting equation for the model proposed by Li et al. [20] was obtained as follows:

Viscoelastic-Plastic Continuous Physical Model (VEP).
To design an intelligent suspension system for high-speed trains, Li et al. [95] 23.Equation (80) shows the resulting equations for this model.
where k is the stifness and C pre is the viscous damping coefcient of the preyield phase.Te viscous-plastic behavior in the postyield phase was described using the Bingham model (Figure 24).Te damping force in the postyield region is expressed as follows: 20 Structural Control and Health Monitoring where A is the yield force and C post is the viscous damping coefcient in the postyield region.Li et al. [95] used a new hyperbolic function to describe the transition path from the preyield to the postyield regions, as shown in Figure 25.A smooth curve of the yielding process can be obtained by merging C pre _ y and A sgn( _ y) as a new function.Tus, both the Maxwell and Bingham models were merged into the viscoelastic-plastic model, as shown in Figure 26.Te fnal equation can be rewritten as follows: where F denotes the damping force of the damper, k denotes the stifness of the damper in the preyield region, c denotes the viscous damping coefcient in the postyield region, A denotes the yield force, and α denotes the restoring factor.Structural Control and Health Monitoring

Nonlinear Stifness Viscoelastic-Plastic Model (nkVEP).
Li et al. [95] observed a sudden decline in the slope of the force-displacement and force-velocity curves derived from MR damper behavior, particularly within the preyield region.Tis abrupt deviation is largely attributable to the damper's inadequate internal pressure.While bolstering the pressure in the reservoir can mitigate this efect, it cannot be entirely eradicated.Earlier viscoelastic-plastic models fell short in capturing this distinct decline in force-velocity and force-displacement relationships.Consequently, there was a palpable need to formulate a model that aptly represents this phenomenon.Based on this fnding, they proposed a viscoelastic-plastic model with nonlinear stifness (nkVEP), as shown in Figure 27.Te equation for this model is in the form of the following equation: By ftting the nonlinear stifness data, the function k(x, y) can be rewritten as where ρ is the amplifcation coefcient, d opf is the position of the sudden change point in the stifness, and k 0 is the reference value of the stifness.In addition, the main parameters of the nkVEP model were presented for the generalized fuctuating current (i) according to the following equations: 6.6.Stifness-Viscosity-Elasto-Slide (SVES) Models.Te nonlinear SVES model includes a linear combination of nonlinear mechanisms.Figure 28 shows a schematic of the SVES model proposed by Wereley et al. [93] to describe the hysteresis behavior of the MR dampers.Teir model has three parallel amplitude-dependent elements: linear stifness, linear viscous dashpot, and a nonlinear elasto-slide element consisting of a stifness and Coulomb element in series.Figure 29 shows the efects of each of these mechanisms on the hysteresis loop of the damper.Te linear stifness and dashpot parameters provide the necessary slope and damping characteristics for the hysteresis loop.Te elasto-slide element conveys stifness in the region where the velocity of the damper changes its sign, and the displacement from the extreme position is less than a specifc value (2x s ).
Te elasto-slide element is equivalent to a Coulomb element for the memory phenomenon of the hysteresis loop.
Te force predicted by this model is where F s (t) is the stifness element force, F c (t) is the viscous dashpot element force, and F es (t) is the elasto-slide element force (four legs as a parallelogram), as shown in the following equation: 22 Structural Control and Health Monitoring In ( 87), the sliding force (N) is given by the yielding displacement (x s ) multiplied by k 2 and x is the amplitude over the cycle.

Algebraic Models
Given that the models raised for MR dampers are complex and computationally heavy, researchers have sought to provide simpler models.Earlier models predominantly stemmed from theoretical examinations of the device's behavior or that of its specifc fuid.While these studies provided valuable insights, they often proved time-intensive and necessitated specialized expertise.In contrast, certain research groups prioritize operational and practical applications, striving to construct models that are as streamlined as possible, yet retain the fundamental and efective properties.Tese models serve a dual purpose: they facilitate the numerical simulation of devices in structural control and guide the management of the devices.However, MR dampers have high hysteresis and nonlinear behavior; therefore, the introduced models should show this phenomenon clearly and have sufcient accuracy.Algebraic  models are another possibility proposed by researchers.In this group of models, the hysteresis behavior of the damper is mainly expressed simple algebraic expressions.Te accuracy of these models is desirable to a large extent.
7.1.Weng Model.Weng et al. [96] presented a model employing an algebraic expression based on the arctangent function to simulate the hysteresis behavior of an MR damper.In this model, the employment of both the hyperbolic tangent function and the sign function successfully replicates the damper's behavior without resorting to differential relations, such as those found in the Bouc-Wen relation.Te model introduced by them is given by the following equation: where F d is the restoring force of the MR damper, f 0 is the preload value owing to the nitrogen accumulator, c b represents the viscous damping coefcients, and f y is the yield force.Parameter k is a specifc shape coefcient, and _ x 0 is the hysteretic velocity._ x and € x are the velocity and acceleration of the damper piston, respectively.[97] proposed a modifed version of the Weng et al. [96] model.Tey found that Weng's model was in good agreement with the experimental data, except for the lower current inputs of the highest excitation velocity.Tis was possibly due to the fuid inertial force, which became more critical at lower input currents than the induced yield stress because the excitation acceleration increased.Terefore, to improve the model, they added an inertial force term to the following equation:

Cesmeci and Engin Model. Cesmeci and Engin
Parameter m in (89) represents the virtual mass to be determined based on experimental data.

Metered Model.
To enhance Weng's model, Metered et al. [98] incorporated amplitude and excitation frequency parameters.While these parameters increase the model's accuracy in predicting the damper's response to external excitation, they also constrain its application in structural control.Tis limitation arises because buildings are exposed to inherently random vibrations, making it challenging to reliably predict the excitation's amplitude and frequency.Teir proposed model is given by the following equation: where a is the amplitude and f is the excitation frequency.Parameters A 1 , A 2 , F 1 , and F 2 are constants obtained from the identifcation and curve-ftting processes.

Balamurgan Model.
Based on their experimental results, Balamurgan et al. [99] found that an MR damper exhibited voltage-dependent nonlinear hysteresis behavior.However, any change in the input voltage has a visual efect on the postyield saturation peak force and magnitude of the hysteresis loop.Tey suggested that the resulting damping value could be expressed in a general form consisting of two independent functions: the control voltage of the device and hysteretic force.
In the above equation, F i is a function of the voltage, which is shown by the parameter υ, and F h is a hysteretic function.Function F i is a monotonous nonlinear incremental function of the voltage and can be used as a gain function.Te nonlinear gradual behavior of F i can be represented by an asymmetric sigmoid function that has a bias with the axis [100].Equation (92) shows their proposed damper force.
24 Structural Control and Health Monitoring where k 2 and a 2 are positive constant values, I 0 is an arbitrary constant value bias, and v 0 is a constant.Tese parameters must be derived from measured data.

Hyperbolic Tangent-Function Models
Researchers have used the hyperbolic tangent function to express the hysteresis behavior of MR damper.However, exact classifcation of these models is difcult.Some of these models have been described in previous sections.In this section, the most efective hyperbolic model is presented.
8.1.Bass Model.Te Bass hyperbolic tangent mode [102] consists of two sets of spring-dashpot elements connected by a mass element, as shown in Figure 30.Te inertial-mass element resists motion through Coulomb friction.Te displacement and velocity of the mass relative to the fxed base are denoted by x 0 and _ x 0 , respectively.Te displacement and velocity of the damping piston relative to the mass are represented by x 1 and _ x 1 , respectively, and the general movement parameters of the damper are represented by x and _ x, respectively, which are the result of adding these parameters with the correct sign.
Moreover, the parameters k 1 and c 1 model the preyield viscoelastic behavior model.Te postyield viscoelastic behavior is represented by a spring (k 0 ) and a dashpot (c 0 ).m 0 denotes the inertial mass of both the fuid and piston, and f 0 is the yield force.V ref is the reference velocity that afects the overall shape of the transition curve from the elastic to the plastic region of the function.Te damper output force and dynamic characteristics of the system are presented as equations ( 87) and ( 88) in the state-space form.
Te model has unknown parameters in the form of frstand second-order polynomial functions.Tese parameters are current-dependent input operations, which are obtained from the identifcation of a specifc MR damper device.8.2.Kwok Model.Kwok et al. [101] proposed a model in which the hyperbolic tangent function was used to represent the MR damper hysteresis loop.Linear functions were used to show the conventional viscous damping and spring stifness.Te Kwok model can be expressed as follows: In these equations, c and k are the viscous and stifness coefcients, respectively; α is the hysteresis scale factor; z is the hysteresis variable given by the hyperbolic tangent function; and f 0 is the damper force ofset.Te components constituting the hysteresis curve are shown in Figure 31, [101].Tese parameters describe the force-velocity response of the damper.Te viscosity term (c _ x) forms an inclined line that depicts the relationship between the velocity and the dissipating force of the damper in the postyield regions (at both ends of the hysteresis loop).k is responsible for opening the horizontal ellipse composed of kx, from the vicinity of the zero velocity.Te parameter β forms the fundamental form of the hysteresis curve.Tis coefcient is the dampingvelocity scale coefcient that defnes the slope of the hysteresis loop.δ is a scale coefcient, and the sign of the displacement determines the width of the hysteresis loop using the δ sgn(x) term.Te overall hysteresis loop was scaled by the α coefcient, which determines the height of the hysteresis loop.In the original study, the unknown parameters of the model were identifed using the PSO optimization algorithm [103].
Models employing the hyperbolic tangent function to articulate the internal dynamic behavior of the damper typically lack an intrinsic inverse model.Nevertheless, certain models can be streamlined based on the integration of the hyperbolic function within the model's fnal equation.In such cases, the hyperbolic tangent function is substituted with the sign function, facilitating the formulation of inverse function relationships.Te Kwok model is a notable example within this category.8.3.Yang Models.Yang et al. [104] worked on MR dampers that have a friction lagging efect in their force-velocity diagram.Figure 32 shows the force-velocity and forcedisplacement diagrams of the damper.Tey proposed a nonlinear algebraic hysteretic model for dampers, based on a hyperbolic function.Te following equation represents this relationship: In these formulas, F is the damping force; _ x and x are the velocity and displacement of the piston, respectively; c is the viscous damping coefcient; k is the stifness coefcient; and α is the hysteretic factor.Parameter z represents the internal dynamic efect of the device from the Coulomb damping force, which describes the hysteresis behavior of the force- Te parameters used in the above relation are known and accepted symbols in this feld.Here, F is the damping force, _ x is the velocity, x is the displacement, c 0 is the viscous damping coefcient, and k is the stifness coefcient.f b , α, β, and c are the parameters describing the hysteresis characteristics of the device.Operator sgn(.) is the sign function, and f 0 is the ofset of the damping force.Te model parameters are linear and quadratic equations relative to the damper input-management current, which is derived from the identifcation process.
8.4.Cheng Models.In 2020, Cheng et al. [106] proposed a parametric model for MR dampers that considered the amplitude and frequency of excitation.Te proposed model was a modifed hyperbolic tangent model.Te results show that the proposed model has higher ftting accuracy and can be well adapted to changing conditions.According to the type and size of the damper used and verifcation of parameters' sensitivity, the proposed model is as follows: where c(I, A, f) and β(I, A, f) are functions of the current I and the excitation amplitude and frequency are denoted as A and f, respectively.α(I) is a function of the current.Furthermore, the proposed model is invertible, allowing for tracking of the damping force.Te authors endeavored to enhance result accuracy by elucidating the relationship between the damping force and the frequency of external excitation.Yet, as previously noted, the suitability of such a model for structural control felds, characterized by their randomness and nonharmonic nature, remains ambiguous.

Magic Formula
Te magic formula ofers an efective approximation for nonlinear curves [107].Comprising a combination of various trigonometric functions, it is extensively utilized to estimate vehicle tire performance in mechanical simulations.Indeed, this model represents a specifc application of the hyperbolic tangent function in terms that delineate and epitomize the internal dynamic performance of the damping device.Te common magic formulation is where β, C, D, and E are the parameters that control the shape of the curve.α is an independent variable that can be selected from the main movement parameters of the device, such as displacement, velocity, and defection angle, based on its actual responses.Pan et al. [107] showed that the acquired tire force-defection angle curve is similar to the plastic-fuid curve of the MR damper, with some diferences in describing the hysteresis characteristics.Te magic formula representing the MR damper is given by In this equation, x represents the displacement and _ x represents the velocity for both damper pistons.Te other parameters of the model are defned as follows: A is the hysteretic factor that determines the half-width of the MR damper hysteresis curve and is generally positive; β is generally a positive stifness factor; C is a shape factor that determines the "S" curve shape and is approximately equal to 1; D is used as the peak factor and determines the damping-force saturation of the damper, which is generally a positive value; E is the curvature factor that determines the yield characteristics, which is typically less than 1; k c is the viscous damping coefcient; k is the MR-damper stifness coefcient; and f 0 is the bias force.
Te parameters in this context can be chosen as constant, linear, or quadratic polynomials, depending on their dependency on the damper management current, a dependency that becomes evident postidentifcation.For parameters A and β, it is also possible to have a dependency on the amplitude and excitation frequency.Tis model can be inverted with the introduction of certain simplifcations.Te precision of the resultant inverse model is contingent upon the identifed parameters and their relation to either the current or the amplitude and frequency of the external excitation.

Dynamic Models Based on the Dahl Hysteresis Operator
Te Dahl model [108,109] belongs to the frst generation of dynamic-friction models.In addition to MR dampers, this model has been used in standard simulation models in the aerospace industry [110,111].Te starting point of the model is as follows: Tis form of the Dahl model was further developed and became a well-known equation, characterized by the following relations for better numerical implementation: Te parameter defnitions are defned as follows.F is the friction force; ž is the state variable interpreted as the elastic deformation of surface ruggedness of the adjacent bodies; _ x is the sliding velocity; x is the body displacement; σ 0 is the Structural Control and Health Monitoring rugged stifness; i is the hysteresis loop-shape coefcient; and F c is the Coulomb friction.
Te most explicit form of the Dahl model is 10.1.Modifed Dahl Model. Figure 33 shows a modifed Dahl model proposed by Zhou et al. [112] based on the experiments of Zhou [34], and Zhou et al. [113] performed on MRdamper devices.Tey found that the results could be represented in the form of a model based on the Dahl operator, which simulates the Coulomb friction force observed in the results.
where k 0 is the stifness coefcient, c 0 is the viscous damping coefcient, f d is the Coulomb force regulated by the applied magnetic feld, z is the dimensionless hysteresis variable given by ( 103), and f 0 is the force exerted by the seals to measure the bias.
where the constants k x and k w are voltage dependent and w is obtained from (43). Figure 34 illustrates this model schematically.Tis model was used in several studies [74,75,114,115].
10.3.Asymmetric Dahl Model.Garcia et al. [116] tested a prototype of an MR damper whose force-displacement response curves and the force-velocity obtained were asymmetric.Hence, they proposed modifcations to the Dahl viscous model to consider the efects of asymmetry on the behavior of the MR dampers.Te force produced by the damper is as follows (105), and its viscosity and friction parameters are obtained from the following relationships: All the parameters (k i ) were obtained from the identifcation process.

General Dynamic Models
Tis section presents models that rely on the potential of identifcation methods developed in recent years for faster and more accurate identifcation of complex problems with increased unknowns.Shortly, a signifcant increase in this branch of research will occur to reproduce the numerical version of MR dampers.
In 2018, Zhao et al. [117] presented a dynamic hysteretic model to describe the force-velocity properties of MR dampers with various current excitations.It can be used to simulate the nonlinear properties of MR dampers more accurately.Te model was presented using nonlinear state equations.
where U is the model input, η is a state variable, Φ(U) is the model output, a is the feedforward gain, k i is an integration gain, and Δ(η) is the output of the dead-zone operator.b is the bias.x 0 and k i can afect the width and amplitude of the hysteresis loop.Because of its dynamic nature, the model can represent the hysteresis of MR dampers.It provides a simple building block for developing various dynamic hysteretic models for MR dampers and represents a general framework.Te structure of the model is shown in Figure 35.
In 2018, Zhao and Xu [118] proposed a model based on the sigmoid function that was based on a comparison of experimental data.Te data used were predicted by the original sigmoid model and considered the Stribeck efect (Figure 36).
Te model was designed to overcome the shortcomings of the original sigmoid model considering the properties of the MR damper, including the viscosity, elasticity, force hysteresis, and force mutation in the low-velocity region.Tis is formulated as follows: Te parameters were identifed using genetic algorithms.Te results show that their model can well predict the Stribeck efect in the low-velocity region of small-scale dampers.
In 2018, Yuan et al. [119] proposed a model by combining the hyperbolic tangent and exponential functions.
where the constants S1 and S2 depend on the geometric and physical properties of the fuid MR, respectively.However, the model still requires scale parameters to match the numerical force-velocity curve with the corresponding curve results from laboratory tests.Yu et al. [120] proposed a new model based on the hysteresis division method to describe the complex nonlinear behavior of an MR damper.Tis method divides the force-velocity hysteresis loop into a branch and backbone curve (Figure 37).
where _ x denotes the damper velocity and a, b, c, d, e, and f are the parameters that must be identifed in association with the excitation.Tis model accurately describes the nonlinear hysteresis characteristics of small MR dampers.Moreover, it has a time-saving and efcient identifcation process and can be integrated into semiactive-vibration-control strategy.However, the performance of the model against nonharmonic excitation and its efciency were not considered in their study.
In 2019, He et al. [121] proposed a mechanism-based unifed MR-damper model based on a neuro-fuzzy technique to represent the direct dynamics of MR dampers.Based on kinetics and rheology, the damping force of an MR damper consists of the friction, elasticity, viscosity, inertia, and shear terms.Te model parameters have unique physical meanings, and the model can be conveniently used for semiactive control in vehicle applications.Bui et al. [122] proposed a novel parametric dynamic model based on a quasistatic model (QS) and hysteresis multiplier of the magic formula (Figure 39).Teir model could efectively and accurately capture the strong nonlinear hysteresis phenomenon of MR dampers.Te quasistaticmagic-formula (QSMF) model formulates both the shear and fow modes of MR dampers, and the asymmetric hysteresis responses of the damper are more accurate than those of the Spencer and Pan models.
Additionally, the QSMF-model parameters are easier to identify, and the model is feasible and applicable to semiactive control systems based on small-scale MR dampers.

Concluding Remarks
A successful control algorithm is based on a good sense of the seismic behavior of the structure and sufcient knowledge of control device operation.Much efort has been made to identify MR dampers as efcient but complicated tool; however, their complicated operation has also been noted.Various relationships have been proposed to describe the highly nonlinear behavior of these devices.An algorithm is more successful if the control device has clear management rules (inverse model) to act as a control and a regulative narrative.Terefore, proper identifcation of these devices alongside using the correct inverse model is essential.Te selection of an appropriate mathematical model describing the device is important during the identifcation process.It is important to note that the frst step in selecting an appropriate mathematical model always begins with laboratory samples.Te response of a damper to changes in kinetic parameters, such as displacement and velocity, and changes in the damping force with changes in control factors, such as voltage or current, are essential.
Te models discussed in this review are characterized by parametric relations.Researchers have employed various specifc functions to depict the damper's behavior, resulting in a subset of models adept at accurately simulating the device's operation.Tese models can be broadly categorized into two distinct groups.Te frst encompasses models that depict the nonlinear internal dynamics of the damper, where a parameter is expressed through a diferential formulation, exemplifed by the Dahl or Bouc-Wen models.Despite their efcacy in simulating damper behavior, these models exhibit constraints in facilitating continuous damper control, making them more suited for bang-bang or clippedcontrol algorithms.Conversely, the second group leverages geometric and explicit functions, excelling in crafting inverted models.Tis permits designers to implement continuous control algorithms, achieving  Structural Control and Health Monitoring a damping force closely aligned with the desired control force.However, the damper's response to variations in the amplitude and frequency of external stimuli warrants deeper investigation.
Te research showcased here represents only a fraction of the extensive eforts made over the past decades to decipher the nonlinear behavior of MR dampers.Tese dampers stand out as one of the most efective and practical tools to enhance the seismic performance of structures.Our fndings highlight several vital considerations: Undoubtedly, what has been done thus far has been efective but not sufcient.Te strong dependence of device segments on their physical properties prevents the provision of empirical and simple mathematical formulas.Terefore, continuous eforts are being made to identify the factors that afect the damping behavior of the device and to provide the most appropriate mathematical models that can express the complexity of the device response, particularly in areas where rheological fuid yields.At the frst glance, the use of complex functions appears to be an efective way to identify MR dampers.However, it should be noted that the numerical reproduction of the damper response is only half that is required by the designer to design a successful and efcient control algorithm.Te second is the management of the devices to increase the efciency of the control confguration.Terefore, the proposed model should be simple, accurate, and efcient.Tis simplicity plays a crucial role in the implementation of the inverse model, which plays a signifcant role in controlling the device and generating the dissipating force closest to the required control force.Te combination of these two parts is not simple in practice, and this is an issue that requires further research.
Health Monitoring model, the Kelvin representation is preferred over the Maxwell representation to demonstrate linear viscoelastic behavior in the preyield region.Te following equations are associated with this model:
x changes Structural Control and Health Monitoring linearly with voltage.Te parameters n, ρ, and σ change exponentially.Te identifed k w has a more complicated relationship with the voltage.

Figure 23 :
Figure 23: Mechanical analogy of the viscoelastic phenomena in the preyield phase (Section 6.4).

Structural
Control and Health Monitoring velocity loop.β and δ are the coefcients of the slope and the width of the hysteresis loop, Tese unknown parameters of the model are obtained from the identifcation process, which are linear or quadratic relationships of the damper input current.In another study on MR dampers, Yang et al. [105] proposed a model in which a componentwise additive strategy, including viscous damping, spring, and a hysteretic component, was used.Te damping force produced by this new model is expressed in the following equation:
(i) Te model should be simple with a restrained number of parameters.(ii) It must be capable of replicating the force-velocity cycle and the hysteresis behavior of the damper.(iii) Te performance of the proposed model, particularly in the transitional region around the yield stress, should be both accurate and closely approximated.(iv) Ensuring the model's invertibility is paramount.(v) Te model parameters be sufciently fexible yet straightforward in illustrating the dependency of results on control factors, such as current or voltage.