The Hysteresis Bouc-Wen Model_ a Survey

Arch Comput Methods Eng (2009) 16: 161–188DOI 10.1007/s11831-009-9031-8

O R I G I NA L PA P E R

The Hysteresis Bouc-Wen Model, a Survey

Mohammed Ismail · Fayçal Ikhouane · José Rodellar

Received: 14 November 2008 / Accepted: 14 December 2008 / Published online: 20 February 2009© CIMNE, Barcelona, Spain 2009

Abstract Structural systems often show nonlinear behav-ior under severe excitations generated by natural hazards. Inthat condition, the restoring force becomes highly nonlinearshowing significant hysteresis. The hereditary nature of thisnonlinear restoring force indicates that the force cannot bedescribed as a function of the instantaneous displacementand velocity. Accordingly, many hysteretic restoring forcemodels were developed to include the time dependent natureusing a set of differential equations. This survey containsa review of the past, recent developments and implementa-tions of the Bouc-Wen model which is used extensively inmodeling the hysteresis phenomenon in the dynamically ex-cited nonlinear structures.

1 Introduction

This paper deals with the analysis, modeling and identifica-tion of a special class of systems with hysteresis. This non-linear behavior is encountered in a wide variety of processesin which the input-output dynamic relations between vari-ables involve memory effects. Examples are found in bi-

M. Ismail · J. Rodellar (�)Dept. of Applied Mathematics III, School of Civil Engineering,Technical University of Catalonia, Campus Nord, Module C-2,08034 Barcelona, Spaine-mail: [email protected]

M. Ismaile-mail: [email protected]

F. IkhouaneDept. of Applied Mathematics III, Technical School of IndustrialEngineering of Barcelona, Technical University of Catalonia,Comte d’Urgell, 187, 08036 Barcelona, Spaine-mail: [email protected]

ology, optics, electronics, ferroelectricity, magnetism, me-chanics, structures, among other areas. In mechanical andstructural systems, hysteresis appears as a natural mecha-nism of materials to supply restoring forces against move-ments and dissipate energy. In these systems, hysteresisrefers to the memory nature of inelastic behavior where therestoring force depends not only on the instantaneous defor-mation, but also on the history of the deformation.

The detailed modeling of these systems using the lawsof Physics is an arduous task, and the obtained models areoften too complex to be used in practical applications in-volving characterization of systems, identification or con-trol. For this reason, alternative models of these complexsystems have been proposed. These models do not come, ingeneral, from the detailed analysis of the physical behaviorof the systems with hysteresis. Instead, they combine somephysical understanding of the hysteretic system along withsome kind of black-box modeling. For this reason, some au-thors have called these models “semi-physical”.

Within this context, a hysteretic semi-physical model wasproposed initially by Bouc early in 1971 and subsequentlygeneralized by Wen in 1976. Since then, it was known as theBouc-Wen model and has been extensively used in the cur-rent literature to describe mathematically components anddevices with hysteretic behaviors, particularly within the ar-eas of civil and mechanical engineering. The model essen-tially consists in a first-order non-linear differential equationthat relates the input displacement to the output restoringforce in a hysteretic way. By choosing a set of parametersappropriately, it is possible to accommodate the response ofthe model to the real hysteresis loops. This is why the mainefforts reported in the literature have been devoted to thetuning of the parameters for specific applications.

The starting point of the so-called Bouc-Wen model isthe early paper by [38], where a functional that describes

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162 M. Ismail et al.

Fig. 1 Graph force versus displacement for a hysteresis functional

the hysteresis phenomenon was proposed. Consider Fig. 1,where F is a force and x a displacement. Four values of Fcorrespond to the single point x = x0, which means that Fis not a function. If we consider that x is a function of time,then the value of the force at the instant time t will dependnot only on the value of the displacement x at the time t ,but also on the past values of x. The following simplifyingassumption is made in [38].

Assumption 1 The graph of Fig. 1 remains the same for allincreasing function x(·) between 0 and x1, for all decreasingfunction x(·) between the values x1 and x2, etc.

Assumption 1 is what, in the current literature, is calledthe rate-independent property [318]. To precise the form ofthe functional F , [38] elaborates on previous works to pro-pose the following form:

dF

dt= g

(x,F , sign

(dx

dt

))dx

dt. (1)

Consider the equation

d2x

dt2+ F (t) = p(t) (2)

for some given input p(t) and initial conditions

dx

dt(t0), x(t0) and F (t0) (3)

at the initial time instant t0. Equations (1) and (2) describecompletely a hysteretic oscillator.

The paper [38] notes that it is difficult to give explicitlythe solution of (1) due to the nonlinearity of the function g.For this reason, the author proposes the use of a variant of

Fig. 2 Evolution of the Bouc-Wen model literature

the Stieltjes integral to define the functional F :

F (t) = μ2x(t) +∫ t

β

F(V t

s

)dx(s), (4)

where β ∈ [−∞,+∞) is the time instant after which thedisplacement and force are defined. The term V t

s is the to-tal variation of x in the time interval [s, t]. The function F

is chosen in such a way that it satisfies some mathemati-cal properties compatible with the hysteresis property. Thefollowing is an example of this choice given in [38] so thatthese mathematical properties are satisfied:

F(u) =N∑

i=1

Aie−αiu, with αi > 0. (5)

Equations (2)–(5) can then be written in the form

d2x

dt2+ μ2x +

N∑i=1

Zi = p(t), (6)

dZi

dt+ αi

∣∣∣∣dx

dt

∣∣∣∣Zi − Ai

dx

dt= 0; i = 1, . . . ,N. (7)

Equations (6)–(7) are what is now known as the Boucmodel. The derivation of these equations is detailed in [38].The objective here is not to enter in these details, but onlygive a short idea on the origin of the model.

Equation (7) has been extended in [325] to describerestoring forces with hysteresis in the following form:

z = −α|x|zn − βx|zn| + Ax for n odd, (8)

z = −α|x|zn−1|z| − βxzn + Ax for n even. (9)

Equations (8)–(9) constitute the earliest version of whatis called now the Bouc-Wen model. This paper presents anoverview of the literature related to this model. It can be seenin Fig. 2 that the last few years have witnessed an increasinginterest to this model with a book dedicated exclusively tothe Bouc-Wen model [140]. This literature encompasses awide range of issues ranging from identification to model-ing, control, analysis, etc. These issues are treated in detailin the following sections.

The Hysteresis Bouc-Wen Model, a Survey 163

2 Physical and Mathematical Consistencyof the Bouc-Wen Model

In the current literature, the Bouc-Wen model is mostly usedwithin the following black-box approach: given a set of ex-perimental input-output data, how to adjust the Bouc-Wenmodel parameters so that the output of the model matchesthe experimental data? The use of system identification tech-niques is one practical way to perform this task. Once anidentification method has been applied to tune the Bouc-Wen model parameters, the resulting model is consideredas a “good” approximation of the true hysteresis when theerror between the experimental data and the output of themodel is small enough. Then this model is used to study thebehavior of the true hysteresis under different excitations.

By doing this, it is important to consider the followingremark. It may happen that a Bouc-Wen model presents agood matching with the experimental real data for a spe-cific input, but does not necessarily keep significant physicalproperties which are inherent to the real data, independentlyof the exciting input. In this section we draw the attentionto this issue by considering physical properties like stability,passivity, thermodynamic consistency etc.

On the other hand, the Bouc-Wen model is a nonlineardifferential equation, and has to have some general math-ematic properties to be used properly. We consider in thissection the existence and uniqueness of the solution of themodel, along with the bijective relationship between the pa-rameters and the input/output behavior.

2.1 Physical Consistency of the Bouc-Wen Model

The following properties have been considered in the litera-ture:

– Bounded input-bounded output (BIBO) stability– Consistency with the asymptotic motion of physical sys-

tems– Passivity– Thermodynamic admissibility– Accordance with Drucker and Il’iushin stability postu-

lates– Consistency with the hysteresis property.

2.1.1 BIBO Stability

The BIBO stability property of the Bouc-Wen model hasbeen formulated in [144] as follows: Let us conceptualizea nonlinear hysteretic behavior as a map x(t) �→ �s(x)(t),where x represents the time history of an input variable and�s(x) describes the time history of the hysteretic outputvariable. For any bounded input x, the output of the truehysteresis �s(x) is bounded. This bounded BIBO stability

property stems from the fact that we are dealing with me-chanical and structural systems that are stable in open loop.

The Bouc-Wen model that approximates the true hystere-sis �s(x) is

�BW(x)(t) = αkx(t) + (1 − α)Dkz(t), (10)

z = D−1(Ax − β|x| |z|n−1z − γ x|z|n) (11)

where z denotes the time derivative, n > 1, D > 0, k > 0,0 < α < 1 and β + γ �= 0 (the limit cases n = 1, α = 0,α = 1, β + γ = 0 are treated separately in [144]).

This model is BIBO stable if and only if the following set� of initial conditions z(0) is nonempty:

� = {z(0) ∈ R such that �BW is BIBO stable for all C1

input signal x(t) with fixed values of the parameters

A,β,γ,n}. (12)

The following is proved in [144].

Theorem 1 Let x(t), t ∈ [0,∞) be a C1 input signal and

z0 � n

√A

β + γand z1 � n

√A

γ − β. (13)

Then, Table 1 holds.

It is shown in [144] that class V corresponds to a linearbehavior so that it is irrelevant for the modeling of hysteresisbehavior.

2.1.2 Consistency with the Asymptotic Motion of PhysicalSystems

Reference [144] considers a structural base-isolation devicemodeled as a SDOF system with mass m > 0 and viscousdamping c > 0 plus a restoring force � characterizing thehysteretic behavior of the hysteretic material. The free mo-tion of the system is described by the second order differen-tial equation

mx + cx + �(x)(t) = 0 (14)

with initial conditions x(0) and x(0), and the restoringforce is assumed to be described by the Bouc-Wen model(10)–(11). The following is proved in [144].

Theorem 2 For every initial conditions x(0) ∈ R, x(0) ∈ R

and z(0) ∈ � �= ∅, the following holds:

(a) For all classes I–IV in Table 1, the signals x(t), x(t) andz(t) are bounded and C1.


Table 1 Classification of BIBOstable Bouc-Wen models Case � Upper Bound on |z(t)| Class

A > 0 β + γ > 0 and β − γ ≥ 0 R max(|z(0)|, z0) I

β − γ < 0 and β ≥ 0 [−z1, z1] max(|z(0)|, z0) II

A < 0 β − γ > 0 and β + γ ≥ 0 R max(|z(0)|, z0) III

β + γ < 0 and β ≥ 0 [−z0, z0] max(|z(0)|, z0) IV

A = 0 β + γ > 0 and β − γ ≥ 0 R |z(0)| V

All Other Classes ∅

(b) Assume that the Bouc-Wen model belongs to the classesI or II, then there exist constants x∞ and z∞ which de-pend on the Bouc-Wen model parameters (α,D,k,A,β,

γ,n), the system parameters (m, c) and the initial con-ditions (x(0), x(0) and z(0)), and there exists a con-stant c that depends on the parameters m,k,A,α,β, γ

such that for all c ≥ c, we have:

limt→∞x(t) = x∞, (15)

limt→∞ z(t) = z∞, (16)

αx∞ + (1 − α)Dz∞ = 0. (17)

Furthermore, we have:

x ∈ L1([0,∞)

)and lim

t→∞ x(t) = 0. (18)

Accordingly, Theorem 2 shows that, for the classes Iand II, the displacement x goes to a constant value asymptot-ically and that the velocity x goes to zero. This is compatiblewith experimental observations for base-isolation deviceswhich means that both classes are good candidates for thedescription of the real physical behavior of a base-isolationsystem. Based on numerical simulations, classes III and IVare shown not to behave in accordance with experimentalobservations.

2.1.3 Passivity

Passivity is related to the energy dissipation and means thatthe system does not generate energy. In [144], the model(10)–(11) is written as:

x = u,

z = D−1(Au − β|u||z|n−1z − γ u|z|n

), (19)

y = αkx + (1 − α)Dkz

where u is the input of the model and y is its output. Denot-ing

l1 = (1 − α)D2k

2A> 0, (20)

l2 = αk

2> 0, (21)

W(x, z) = l1z2 + l2z

2 (22)

it is shown that the Bouc-Wen model is passive with respectto the storage function W .

2.1.4 Thermodynamic Admissibility

The thermodynamic admissibility is investigated in [99] us-ing the endochronic theory (a theory of viscoplasticity with-out a yield surface, proposed in [313]). The Bouc-Wen typemodels that are considered are univariate and tensorial. TheBouc model [38] is univariate and is defined as

w(t) = A0u(t) + z(t), (23)

z(t) =∫ ϑ(t)

0μ(ϑ(t) − ϑ ′) du

dϑ ′ dϑ ′ (24)

where μ = μ(ϑ(t)) is the hereditary kernel and A0 ≥ 0. Theinput u is a relative displacement between two structural el-ements, while the output w is a structural restoring force.The time function ϑ is positive and non-decreasing whichdirectly depends on the strain and/or the stress tensors, andit is named internal or intrinsic time. One of the definitionsof ϑ proposed by Bouc is the total variation of u:

ϑ(t) =∫ t

0

∣∣∣∣du

dτ

∣∣∣∣dτ, or, equivalently,

dϑ = |du|, with ϑ(0) = 0. (25)

In this case (24) becomes

dz = Adu − βz|du|. (26)

A more general formulation of (26) was also proposedin [37]:

dz = Adu − βz|du| − γ |z|du with γ < β (27)

while [325] suggested a further modification, introducingthe positive exponent n:

dz = Adu − (β sign(z du) + γ ) |z|n du. (28)


Reference [18] introduced the stiffness and strengthdegradation effects in the Bouc-Wen model (28). Only thestrength degradation case is considered in [99]:

dz = Adu − ν (β sign(z du) + γ ) |z|n du (29)

where ν is a positive and increasing function of the energydissipated by the system. A tensorial generalization of (28)is given in [160] for isotropic materials with elastic hydro-static behavior:

σd = A0εd + z, (30)

dz = Adεd − βz‖z‖n−2|z : dεd | − γ z‖z‖n−2(z : dεd) (31)

where εd and σd are the deviatoric part of the small straintensor and of the Cauchy stress tensor, respectively; z is thetensor defining the hysteretic part of the stress, while ‖ · ‖ isthe standard L2-norm.

The following has been proved in [99].

Theorem 3 The Bouc-Wen models of (23)–(31) fulfill thesecond principle of Thermodynamics if and only if the fol-lowing holds:

n > 0, (32)

β > 0, (33)

−β ≤ γ ≤ β. (34)

2.1.5 Accordance with Drucker and Il’iushin StabilityPostulates

Drucker’s postulate [27] implies that the hysteresis systemshould not produce a negative energy dissipation when theunloading-reloading process occurs without load reversal.For the Bouc-Wen model, it has been noted in many ref-erences that this may not be the case, although the effect ofthis violation on the expected results may be minor [48, 131,272, 306, 328].

An attempt to reduce the violation is presented in [47] bymodifying the Bouc-Wen model. On the other hand, [49, 50]show that for n = 1, β +γ > 0 and γ −β ≤ 0 (β being posi-tive by assumption), the Bouc-Wen model verifies Drucker’spostulate. The more general result for n arbitrary is obtainedin Theorem 3 of [99], as the thermodynamic consistency forthe Bouc-Wen model implies that it verifies Drucker’s pos-tulate.

2.1.6 Consistency with the Hysteresis Property

It is shown in [136, 138] that, to be consistent with the hys-teresis property, the Bouc-Wen model (10)–(11) has to ver-ify

maxt≥0

(x(t)) ≤ (1 − α)Dz0

α(35)

where max(x(t)) is the maximal value of the input x(t) fort ≥ 0. The hysteresis property means that the output dependson the sign of the derivative of the input.

2.1.7 Conclusion

As a conclusion of the above results, for univariate Bouc-Wen models, the class I of Table 1 is BIBO stable, consistentwith the motion of physical systems, passive, and consistentwith the second law of Thermodynamics. Furthermore, con-dition (35) is needed for the consistency with the hystere-sis property. For the multivariate Bouc-Wen model, the onlyconsistency result available in the literature is that of [99]stated in Theorem 3.

2.2 Mathematical Consistency of the Bouc-Wen Model

This section presents the results available in the literaturewith regard to

1. The existence and uniqueness of the solution of the Bouc-Wen model.

2. The uniqueness of the description of the Bouc-Wenmodel.

2.2.1 Existence and Uniqueness of Solutions

It is shown in [144] that the differential equations (10)–(11)has a unique solution if n ≥ 1.

2.2.2 Uniqueness of the Description

It is shown in [137] that there exists an infinite number ofdifferent sets of Bouc-Wen model parameters that lead to thesame input/output behavior of the model. This means thatsome parameters of the model are redundant. To eliminatethis redundancy, a normalized form of the Bouc-Wen modelis introduced in [136, 138] by using the transformation

w(t) = z(t)

z0(36)

so that the model (10)–(11) can be written in the form

�BW(x)(t) = κxx(t) + κww(t), (37)

w(t) = ρ(x − σ |x(t)| |w(t)|n−1w(t)

+ (σ − 1)x(t)|w(t)|n) (38)

where

ρ = A

Dz0> 0,

σ = β

β + γ≥ 1

2,

κx = αk > 0,

κw = (1 − α)Dkz0 > 0.

(39)


This form is defined only for the class I as it is con-sistent with the physical properties considered before.The following inequality is also obtained: maxt≥0 w(t) =max(|w(0)|,1).

2.2.3 Conclusion

The mathematical and physical consistency conditions showthat the Bouc-Wen model has to be used under the nor-malized form (37)–(38), along with the inequalities n ≥ 1,ρ > 0, σ ≥ 1/2, κx > 0, κw > 0 and maxt≥0(x(t)) ≤ κw

κx. In

this case, we also have maxt≥0 w(t) = max(|w(0)|,1).

3 Description of the Hysteresis Loop

When the input displacement is periodic with a loading-unloading shape, it is observed by means of numerical sim-ulations that the output Bouc-Wen restoring force is also pe-riodic asymptotically with the same period as the input. Thisfact has been demonstrated analytically only recently due tothe nonlinearity of the model [137]. The hysteresis loop isthe cycle obtained asymptotically when the hysteresis out-put is plotted versus the displacement input. Two types ofproblems have been discussed in the literature:

– Consider that the input is a periodic displacement and theoutput is the Bouc-Wen restoring force.

– Consider that the input is an external periodic force, andthe output is the displacement of a second-order systemthat includes a Bouc-Wen hysteresis.

For the first case, the periodic input displacement sig-nal with a loading-unloading shape is defined formally in[137]: x(t) is continuous over the time interval [0,∞) andis periodic with a period T > 0. Furthermore, there exists ascalar T + with 0 < T + < T such that x(t) is a C1 increasingfunction of time on the interval (0, T +) and a C1 decreas-ing function of time on the interval (T +, T ). The quantities

Xmin = x(0) and Xmax = x(T +) > Xmin denote the minimaland the maximal values of the input signal, respectively. Thesignal x(t) is called wave T -periodic and it is illustrated inFig. 3.

The description of the hysteresis loop uses the followinginstrumental functions:

ϕ−σ,n(μ) =

∫ μ

0

du

1 + σ |u|n−1u + (σ − 1)|u|n (40)

is well defined, C∞ and strictly increasing on (−1,1], sothat it is invertible, with inverse ψ−

σ,n.

ϕ+σ,n(μ) =

∫ μ

0

du

1 − σ |u|n−1u + (σ − 1)|u|n (41)

is well defined, C∞ and strictly increasing on [−1,1), sothat it is invertible, with inverse ψ+

σ,n.

ϕσ,n(μ) = ϕ−σ,n(μ) + ϕσ,n(μ)+ (42)

is well defined, C∞ and strictly increasing on (−1,1), sothat it is invertible with inverse ψσ,n.

The analytical description of the hysteresis loop is givenin the following.

Theorem 4 Define the functions ωm and φm for any non-negative integer m as

ωm(τ) = w(mT + τ) for τ ∈ [0, T ], (43)

φm(τ) = κxx(τ) + κwωm(τ) for τ ∈ [0, T ]. (44)

Then, the sequence of functions {φm}m≥0 (resp. {ωm}m≥0)

converges uniformly on the interval [0, T ] to a continuousfunction �BW (resp. w) defined as:

�BW(τ ) = κxx(τ) + κww(τ ) for τ ∈ [0, T ], (45)

w(τ ) = ψ+σ,n

(ϕ+

σ,n

[−ψσ,n(ρ(Xmax − Xmin))]

+ ρ(x(τ) − Xmin)) for τ ∈ [0, T +], (46)

Fig. 3 The T -periodic inputsignals


w(τ ) = −ψ+σ,n

(ϕ+

σ,n

[−ψσ,n(ρ(Xmax − Xmin))]

− ρ(x(τ) − Xmax)) for τ ∈ [T +, T ]. (47)

Define the time function φBW as φBW(t) = �BW(τ )

where the time t ∈ [0,∞) is written as t = mT + τ for allintegers m = 0,1,2, . . . and all real numbers 0 ≤ τ < T .Theorem 4 means that the time function hysteretic output�BW(x)(t) of the Bouc-Wen model approaches asymptot-ically the T -periodic function φBW(t). The limit cycle isthe graph (x(τ ), �BW(τ )) parameterized by the variable0 ≤ τ ≤ T . Equations (45) and (46) correspond to the load-ing, that is to an increasing input x(τ) with 0 ≤ τ ≤ T +,while (45) and (47) correspond to the unloading, that is to adecreasing input x(τ) with T + ≤ τ ≤ T .

In [241] it is assumed that the exponent n = 1. The hys-teretic part of system is then governed by

z = Ax − (β|x|z + γ x|z|) . (48)

The input signal x(t) is assumed wave T -periodic andthe output is also assumed asymptotically wave T -periodic.Depending on the signs of velocity x and hysteretic restoringforce z, the hysteretic loop is divided into four regions. Ineach region, (48) is integrated leading to

z =⎧⎨⎩

−A−B1 e(β−γ )x

β−γ(β − γ �= 0),

Ax + D1 (β − γ = 0),for x ≤ 0, z ≥ 0,

(49)

z =⎧⎨⎩

−A−B2 e(β+γ )x

β+γ(β + γ �= 0),

Ax + D2 (β + γ = 0),for x ≤ 0, z ≤ 0,

(50)

z =⎧⎨⎩

A−B3 e−(β−γ )x

β−γ(β − γ �= 0),

Ax + D3 (β − γ = 0),for x ≥ 0, z ≤ 0, (51)

z =⎧⎨⎩

A−B4 e−(β+γ )x

β+γ(β + γ �= 0),

Ax + D4 (β + γ = 0),for x ≥ 0, z ≥ 0, (52)

where Bi and Di are integration constants. To obtain thevalues of these constants, it is assumed that the hysteresisloop is continuous, and the continuity condition leads to fourequations for the integration constants. Due to the nonlinear-ity of the equations, these constants are not determined ex-actly. Instead, it is assumed that β = εβ and γ = εγ whereε is a small positive constant. An expansion in power se-ries of ε allows to obtain an approximate description of thehysteresis loop by neglecting the ε3 and higher powers.

In [88] (11) is integrated for D = 1 and n = 2. It isclaimed that

z =√

AB

Btan(√

AB(x + c))

for B < 0, (53)

z =√

AB

Btanh

(√AB(x + c)

)for B > 0, (54)

where B is the coefficient that multiplies z in (11) and c anintegration constant (the expression of c is not given analyt-ically and is supposed to be determined from experiments).This description is incorrect as

√AB cannot be a real num-

ber in (53) since A > 0 and B < 0.In [276], equivalent linearization is used to approximate

the system response under sinusoidal base excitation. Thesystem is described by

mxr + kxr + f (xr , xr ) = −my, (55)

f (xr , xr ) = kdxr + cd xr + αz, (56)

z = Axr − γ |xr zn−1|z − βxr |z|n (57)

where xr is the relative displacement and y = y0 cos(ωt).Equivalent linearization assumes that (55)–(57) are equiva-lent to the single equation

xr + 2ξωnexr + ω2nekxr = −y, (58)

where ξ and ωne are the equivalent system parametersand are determined as a function of the frequency ω. Thismethod assumes that the solution of (55)–(57) is a sine wave.Its phase and amplitude are given as a function of the equiv-alent parameters. Two main disadvantages of this methodare that (1) it is not applicable when the input signal y isnot a sine wave and (2) no formal upper bound on the er-ror between the approximated solution and the exact one isobtained. The size of the error is checked by means of ex-periments.

Also in [276], an averaging method is used to approxi-mate the response to a harmonic excitation. This method as-sumes a sine wave steady-state solution for the variables xr

and z, the amplitude a and phase φ of which is consideredtime-varying. There is a mistake in [276, (35)–(37)] where(35) is obtained assuming a and φ constant, while they aretime-varying in (37).

4 Variation of the Hysteresis Loop with the ModelParameters

Due to the nonlinearity of the Bouc-Wen model, most stud-ies on the influence of the model parameters on the hystere-sis loop shape have been based on numerical simulations.This is the subject of Sect. 4.2. The analytical description ofthe hysteresis loop in [137] allowed a systematic analysis ofthe Bouc-Wen hysteresis cycle in [143]. This is the subjectof Sect. 4.1.


Fig. 4 Methodologies of theanalysis of the variation of w(x)

4.1 Analytical Study of the Influence of the ModelParameters on the Hysteresis Loop

In [143], it is considered that the input displacement signalx(t) is wave T -periodic with Xmin = −Xmax. This allows todefine a normalized input

x(t) = x(t)

Xmax. (59)

The hysteresis part of the limit cycle is derived for load-ing as

w(x) = ψ+σ,n

[ϕ+

σ,n(−ψσ,n(δ)) + δ

2(x + 1)

], (60)

where δ = 2ρXmax. Unloading is not considered in [143] asit is shown that loading and unloading are symmetric.

To analyze the influence of the normalized set of parame-ters (σ , δ and n) on the shape of the limit cycle defined bythe graph (x, w(x)), [143] considers three optics:

1. Analyzing the variation of w(x) as a function of eachparameter (σ , δ and n) separately. This corresponds tostudying the variation of the point Q along the w-axis inFig. 4.

2. Studying the variation of the point S along the x-axis inFig. 4 (that is the width of the hysteresis loop).

3. Defining four regions of the graph (x, w(x)) and studyingthe evolution of the points defining each region. Theseregions are:(a) linear region Rl = [Psl,Plt ],(b) plastic region Rp = [Ptp,Pp],(c) transition region Rt = [Plt ,Ptp],(d) transition region Rs = [Ps,Psl].

Table 2 Variation of the maximal hysteretic output with the normal-ized Bouc-Wen model parameters

ψσ,n(δ) 0 ↑ 1

δ 0 +∞ψσ,n(δ) ψ 1

2 ,n(δ) ↑ ψ+

n (δ)

σ 12 +∞

ψσ,n(δ) ψσ,1(δ) ↑{

δ2 if δ ∈ (0,2],1 if δ > 2

n with σ ≤ 1 1 +∞

ψσ,n(δ) ψσ,1(δ){

δ2 if δ ∈ (0,2],1 if δ > 2

n with σ > 1 1 +∞

The obtained results are summarized in tables where onlythe results obtained analytically are reported. The maximalvalue of the hysteretic output is shown to be ψσ,n(δ). Thevariation of this term with respect to each of the three pa-rameters σ , δ and n is given in Table 2. For example, thesecond row of the table is to be read as follows: the firstcolumn says that the term ψσ,n(δ) is analyzed with respectto the parameter σ . The second column says that the termψσ,n(δ) is increasing (↑) from the value ψ 1

2 ,n(δ) that cor-

responds to σ = 12 to the value ψ+

n (δ) that corresponds toσ = +∞.

Table 3 summarizes the results of the variation of the zeroof the hysteretic output with respect to the model parame-ters, δ, σ and n. The variation of the hysteretic output withthe Bou-Wen normalized model parameters is given in Ta-ble 4.


A detailed analytical study of the four regions of theBouc-Wen model hysteresis loop is also presented in [143].

4.2 Study of the Variation of the Hysteresis Loopwith the Bouc-Wen Model Parameters by Meansof Numerical Simulations

As explained in Sect. 3, [276] assumes a harmonic approx-imation of the periodic response of a second-order systemcoupled with a Bouc-Wen hysteresis. Based on this approxi-mation, and using numerical simulations, the conclusions ofthe study are the following:

1. When the amplitude of the excitation is small, the fre-quency response of the system is quasilinear hystereticsystem, while the system exhibits a large multivaluedfrequency regions as the amplitude of the excitation in-creases.

2. As the parameter γ varies from negative value to positiveone, the system frequency response curves gradually vary

Table 3 Variation of the hysteretic zero x◦ with the normalized Bouc-Wen model parameters

x◦ 0 ↓ −1

δ 0 +∞x◦ 2

δψ 1

2 ,n(δ) − 1 ↓ −1

σ 12 +∞

x◦ 2δϕ−

σ,1(ψσ,1(δ)) − 1 ↑{

0 if δ ∈ (0,2],2δ

− 1 if δ > 2

n 1 +∞

from hardening character to a quasilinear character andthen to soft character.

3. When β is increased, the response amplitude is de-creased.

4. When A is increased, the system natural frequency is in-creased.

5. When α is increased, the damping force is increased, thehysteresis force loop will become slim and the dampingforce will be more similar to viscous damping force, inaddition, large α will make the system more like linearsystem.

The study in [333] reports that there are five kinds of hys-teresis loops with physical meaning depending on the re-spective signs of the quantities β + γ and β − γ . Hardeningand softening behaviors are shown to be related to the signof β + γ . It is also noted that the parameter A controls theslope of the hysteresis loop at z = 0. Using numerical simu-lations it is shown that the parameter n governs the smooth-ness of the transition from linear to nonlinear range. It isalso observed that an increase of the parameter A increasesthe resonant frequency and reduces the resonant peak. More-over, it is observed that the increase of the parameters A andn makes the hysteresis loop narrower.

5 Interpretation of the Model Parameters

In this section, the interpretation of the standard and normal-ized parameters of the Bouc-Wen model is discussed.

Table 4 Variation of the hysteretic output w(x) with the normalized Bouc-Wen model parameters

w(x) with x ≥ 0 0 ↑ 1

δ 0 +∞w(x) with −1 < x < 0 0 ↓ w(x)δ=δ∗ ↑ 1

δ 0 δ∗ = ϕσ,n

(n

√−x

σ (x+1)−x

)+∞

w(x) with x = −1 0 ↓ −1

δ 0 +∞w(x) with x ≥ 2

δψ 1

2 ,n(δ) − 1 w(x)

σ= 12

↑ ψ+n

(δ2 (x + 1)

)σ 1

2 +∞w(x) with x < 2

δψ 1

2 ,n(δ) − 1 w(x)

σ= 12

ψ+n

(δ2 (x + 1)

)σ 1

2 +∞

w(x) w(x)n=1

⎧⎪⎪⎨⎪⎪⎩

if δ ∈ (0,2] δ2 x

if δ > 2

{−1 + δ

2 (x + 1) for − 1 ≤ x < 4δ

− 1

1 for 4δ

− 1 ≤ x ≤ 1

n 1 +∞


5.1 Interpretation of the Standard Bouc-Wen ModelParameters

The parameters α, A, β , γ and n play the role of governingand controlling the scale and general shape of the hystere-sis loop. Due to the lack of an analytical expression of thehysteresis loop, most works addressing this issue have usednumerical simulation to understand the influence of theseparameters [47, 187, 202, 241, 276, 333]. The way thesesimulations have been done is by fixing four parameters andvarying one (the free parameter). As shown in Sect. 4.1, thismethodology is hampered by the following facts:

1. The variation of the hysteresis loop with the free parame-ter depends on the precise values of the fixed parameters.

2. Any given quantity related to the hysteresis loop (likemaximum value, width, etc.) does not depend on a sin-gle parameter but rather on the whole set of parameters.

As a consequence, most of the results obtained in the lit-erature are either incomplete or incorrect. A more analyticapproach is given in [143] using the normalized parametersand its results are given in the following section.

5.2 Interpretation of the Normalized Bouc-Wen ModelParameters

The parameters of the normalized form of the model are in-terpreted as follows. ρ is the slope of the linear zone forXmax = 1, or the inverse of the apparent yield point of thenonlinear component of the Bouc-Wen model. The Para-meter δ can be seen as a measure for the ductility of themodel, while the parameter σ distinguishes the softeningor the hardening behaviors of the hysteretic component andtherefore the consistency with the laws of Thermodynam-ics. The parameter n characterizes the transition from linearto plastic behavior along the axis of ordinates in the map(x, w(x)).

6 The Bouc-Wen Model Parameter Identification

Identifying the Bouc-Wen model parameters consists inproposing a signal input (or several signal inputs) and anidentification algorithm that uses the measured output ofthe model along with that input to determine the unknownmodel parameters. This problem has stirred a lot of re-search effort due to its difficulty being nonlinear and non-differentiable. Some identification methods have been pro-posed with a rigorous analysis of the convergence of the pa-rameters to their true values, while others relied on numeri-cal simulations and experimentation. In the next, we give anoverview of these methods.

6.1 Least-Squares Based Identification

Reference [200] presents a three-stage identification algo-rithm that is a combination of sequential regression analysis,least-squares analysis and/or Gauss-Newton method alongwith the extended Kalman filtering technique.

In the first stage of identification, assuming equivalentlinear system at each time interval and by using the sequen-tial regression analysis, the system stiffness and dampingare identified by means of the following recursive formu-lation:

ak+1 = ak +PkHTk+1(1+Hk+1PkHT

k+1)−1 · (qk+1 −Hk+1a)

(61)

where qk+1 = the actual (k+1) measurement, P−1k = HT

k Hk

and Hk+1ak is the estimate of this measurement, while k =time step; a = [c1, c2, . . . , cN , k1, k2, . . . , kN ]T and N = de-grees of freedom.

In the second identification stage, a fixed n = 1 is as-sumed, and the model parameters (A, β , γ ) are identifiedusing the least-squares method proposed in [302] for dif-ferent α values and through minimizing the following errorfunction:

E =m∑

i=1

[zi − AI1i + βI2i + γ I3i

]2 where

I1i = x, I2i = |x||zi |n−1zi and I3i = |x||zi |n. (62)

In the third stage of identification, the extended Kalmanfilter technique is used to obtain better identification resultsby using the results from the second stage as an initial guessto the third stage to speed convergence.

Paper [173] describes an iterative least squares procedurebased on a modified Gauss-Newton approach to perform theparameter identification of an extended version of the Bouc-Wen model that accounts for strength and stiffness degrada-tion by [18].

References [55, 289] present an on-line identificationmethod based on a least-squares adaptive law with the for-getting factor [147]. For an unknown mass m, a Bouc-Wenhysteresis element model with additional polynomial-typenonlinear terms, is used to investigate the effects of persis-tence of excitation and of under- and overparameterization.This model form is given as

z = mx + kx + cx + dx3

−∫ t

0(1/η)

[ν(β|x||z|n−1z − γ x|z|n)]dt (63)

where d is the cubic term parameter and c the linear damp-ing parameter, while the stiffness parameter k replaces


(1/η)A in the common model degradation form. In this case,x and z are supposed to be available to measurement whilex and x are obtained by integration.

The filtered signals z in the identification process aregiven by

z = θ∗T � (64)

where

θ∗ = [m,k, c, d,−(1/η)a1νβ, (1/η)a1νγ,

− (1/η)a2νβ, . . .]T (65)

and

� =[

sx

(s + α),

sx

(s + α),

sx

(s + α)

sx3

(s + α),

|x||z|0z(s + α)

,x|z|

(s + α)

|x||z|z(s + α)

,|x||z|2(s + α)

, . . .

]T

. (66)

To account for variable gain, a modified least-squaresadaptive law with forgetting factor [147] is used.

Paper [196] presents an adaptive on-line identificationmethodology with a variable trace method to adjust theadaptation gain matrix.

In [202], a linear parameterized estimator is proposed forthe on-line estimation of the hysteretic Bouc-Wen modelwith unknown coefficients (including the parameter n) writ-ten as

z = Ax +N∑

i=1

an

[β|x||z|i−1z + γ x|z|i], (67)

where N is a known upper bound on n. A discrete-time formof (67) is defined as follows:

z(k) = z(k − 1) + �tAx(k − 1)

+ �t

N∑i=1

ai

[β|x(k − 1)||z(k − 1)|i−1z(k − 1)

+ γ x(k − 1)|z(k − 1)|i], (68)

where the coefficients ai are 0 or 1. This discrete time modelgives rise to the following discrete-time linearly parameter-ized estimator:

z(k) = z(k − 1) + θ0(k)x(k − 1)

+N∑

i=1

[θ2i−1(k)|x(k − 1)||z(k − 1)|i−1z(k − 1)

+ θ2i (k)x(k − 1)|z(k − 1)|i], (69)

where the coefficients θi , i = 0, . . . ,2N are estimates ateach time tk of the corresponding coefficients shown in (68).Equation (69) indicates a linear parameterized form with re-spect to θi , i = 0, . . . ,2N . Now set

θ = [θ0, θ1, θ2, . . . , θ2N ]T (70)

where θi is a function of �t and time-dependent model pa-rameters. Also define

φ(k − 1) = [x(k − 1)|x(k − 1)||z(k − 1)|0z(k − 1)

× x(k − 1)|z(k − 1)|1|x(k − 1)||z(k − 1)|1× z(k − 1)

× x(k − 1)|z(k − 1)|2 · · · |x(k − 1)|× |z(k − 1)|N−1z(k − 1)x(k − 1)|z(k − 1)|N ].

(71)

Then the restoring force at time t can be expressed as

z(k) = z(k − 1) + φT (k − 1)θ. (72)

Based on the data collected from performance test, least-square method is used to estimate the model parameters, θi ,i = 0, . . . ,2N , for each discrete frequency where N = 5 waschosen.

In [278], the mechanical properties of low yield strengthsteel are studied to develop a new device for added damp-ing, stiffness and seismic resistance of rhombic low yieldstrength steel plate. The Bouc-Wen model is used in thefollowing form to approximate that mechanical behav-ior:

z = Ax − η(β|x||z|n−1 + γ x|z|n), (73)

η =(

�y

�y0

)−n

, (74)

�y =(

A

β + γ

)1/n

, (75)

where �y and �y0 are the present and the nominal yieldingdisplacements, respectively.

A discrete form of (73) is rearranged as

�zi = Ay1i − βy2i − γy3i , where (76)

y1i = �xi, and (77)

y2i = (β sgn (xi)|zi |n−1zi

)�xi, and (78)

y3i = (γ |zi |n

)�xi. (79)

Then, the optimal coefficients of A, β and γ can be acquiredby the fitting method of least squares through solving the


following equation⎡⎢⎣∑

y21i −∑

y1i y2i −∑y1i y3i∑

y22i

∑y2i y3i

sym.∑

y23i

⎤⎥⎦⎧⎨⎩

A

β

γ

⎫⎬⎭

=

⎧⎪⎨⎪⎩

∑y1i �zi

−∑y2i �zi

−∑y3i �zi

⎫⎪⎬⎪⎭ . (80)

6.2 Kalman Filter Based Identification

In [195] it is assumed that n = 1 and an extended Kalmanfilter is used to identify the rest of the parameters. A pro-cedure for nonlinear system identification based on the ex-tended Kalman filter is applied to soils under strong motionrecords [191]. The ground is modeled as 3DOFs hystereticstructure and the Bouc-Wen model is used in characterizingthe nonlinear backbone curve of soils. Considering a soil tobe purely hysteretic, the Bouc-Wen model of soil is writtenas

τ = Aγs − β|γs ||τ |n−1τ − γ γs |τ |n, (81)

where τ = shear stress; γs = shear strain; A = initial shearstress modulus. For non-degrading soils, using β = γ en-sures that the backbone curve is convex and symmetric aboutthe origin in this study. With β = γ the time-rate form of thebackbone curve is obtained from (81) as

τ = Aγs − 2βγs |τ |n, (82)

where A, β and n are the parameters to be identified.Identification is first carried out based on soil hysteresis

loops, using a weighted global iteration scheme for parame-ter identification consisting of the following steps:

1. Start the first run with estimates of the initial conditionX(0|0) and its error covariance matrix, P(0|0).

2. Carry out the extended Kalman filtering of the data andobtain at the end of the analysis the updated estimates ofX(n|n) and P(n|n).

3. For the next iteration, use the obtained parameter,Xp(n|n), as the new initial estimate, and scale the sub-matrix, Ppp(n|n), with a weighting factor w. The esti-mates of the initial conditions Xs(0|0) and Pss(0|0) arenot modified in the iteration. For instance, after i itera-tions, the new initial estimates for the i + 1 iteration is asfollows:

Xi+1(0|0) =[

X0s (0|0)

Xip(n|n)

], (83)

Pi+1(0|0) =[

P0ss(0|0) 0

0 w · Pipp(n|n)

]. (84)

4. Continue the analysis until the parameters can no longerbe improved or a local minimum identification error isreached.

Reference [361] presents three algorithms based uponthe simplex [24], extended Kalman filter [217], and gen-eralized reduced gradient methods. The objective is to es-timate the parameters of hysteresis for different classes ofinelastic structures using the generalized Bouc-Wen modelthat accounts for degradation and pinching [17]. This modelform contains 13 parameters to be identified and in whichthe restoring force is expressed as

Fr = αkx + (1 − α)kz, (85)

z = h(z)

{Ax − ν(β|x||z|n−1z + γ x|z|n)

η

}, (86)

in which A, β , γ , n are simple loop parameters while ν,η are functions containing additional loop parameters as afunction of the energy dissipated through hysteresis.

In [192], an adaptive on-line identification algorithm isproposed for parametric and non-parametric identificationof structural models, and is applied to a generalized Bouc-Wen model. The proposed identification methodology, a re-cursive least-square (Kalman filter) based algorithm, up-grades the adaptation gain matrix using an adaptive forget-ting factor that is expressed as the ratio between the mini-mum value of the diagonal elements of the adaptation gainmatrix and a set of pre-defined threshold values. This ap-proach requires only acceleration measurements.

6.3 Genetic Algorithm Based Identification

In [120] a genetic based identification algorithm is proposed.The reproduction procedure adopts the roulette wheel se-lection and the method of crossover and uniform mutation[3, 125].

A modification to the standard Bouc-Wen model is pro-posed in [175] to account for non-symmetrical hysteresis ex-hibited by an MR damper

f = cx + kx + αz − f0, (87)

z = (A − (γ + β sgn(zx)|z|n))(x − μ sgn(x)), (88)

where f0 is the initial damper displacement contributing tothe force offset and μ is the scale factor for the velocity ad-justment.

Then, the model parameters are identified by a geneticalgorithm that is improved here by

1. Removing the selection stage (it is absorbed into thecrossover and mutation operations).

2. Imposing a termination criterion on the basis of statisti-cal tests which guarantees the quality of a near-optimalsolution.


The parameters to be defined are c, k, f0, α, A, β , γ , n

and, μ. Hence, the chromosome becomes

Ci = {ci, ki, f0,i , αi,Ai, βi, γi, ni,μi},i = 1, . . . ,N, N = 50. (89)

The fitness function is designed from the averaged sumof normalized square error between the simulated and ex-perimental data. That is

ei = 1

n

n∑i=1

|vi |(

f simi − f

expi

�f

)2

(90)

where n = the number of data points, vi ∈ [−1,1] is thenormalized velocity and �f = fmax − fmin is the differencein experimental force data.

Reference [176] proposes an identification method basedon differential evolution using simulated noise-free data andexperimental data obtained from a nuclear power plant. TheBouc-Wen model parameter n is kept constant to the value 2.The used objective function is the mean square error (MSE)which is cast in the discrete normalized form as

MSE = 100

nσ 2x

n∑i=n

(x(t) − x(t |P )

)2, (91)

where σ 2x = the variance of the measured output and n =

the number of points in the measured output; x and x = themeasured and predicted time history, respectively; and P =the parameter vector.

The optimization of the objective function is quoted from[263] and stated as obtaining the parameter vector P whichminimizes the MSE. The parameter vector is subjected to theconstraint

P min ≤ P ≤ P max. (92)

The so-called direct search optimization methods do notusually provide a mechanism to restrict the parameters in therange defined by inequality (92); neither does the differentialevolution. However, an unconstrained optimization methodcan be transformed to a constrained one using the conceptof the penalty function. This function determines a penaltyto be added to the value of the MSE any time any parameterexceeds the range limits. The penalty function selected forthis case is given by

Penalty(Pi) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

20((Pi(min) − Pi)/P(min))2

for Pi < Pi(min),

0

for Pi(min) < Pi < Pi(max),

20((Pi(max) − Pi)2/Pi(max))

2

for Pi > Pi(max).

(93)

In [209], a population-search algorithm is proposed toestimate the 13 parameters of the generalized Bouc-Wenmodel. With a given load-displacement trace as input, theoptimization problem can be stated as the determination ofthe parameter vector

p = (A,α,β, γ,n, δν, δη, ζs, q,p,ψ, δψ,λ) (94)

such that the objective function

gN(p) = 1

N

N∑j=1

[x(tj ) − x(tj |p)]2 (95)

is minimized, where N = the number of data points used.Minimization of the objective function is subject to the con-straint that all parameters in p with the exception of γ arepositive.

6.4 Gauss-Newton Iterative Based Identification

In [346], a method of estimating the parameters of hystereticBouc-Wen model on the basis of possibly noise corruptedinput-output data is proposed. The system model is

x + ax + bx + z = f, (96)

z = Ax − β|x||z|n−1z − γ x|z|n. (97)

The observations are assumed to be of the form

y = x + ε (98)

where (i) ε is a zero mean Gaussian white noise with vari-ance σ 2

e , (ii) the input f is zero mean Gaussian white noiseand can be measured noise free, and further the input f isindependent of the observation noise, (iii) the initial condi-tions are known and are set to zero for simplicity. Hence,the integrated mean squared error is selected for minimiza-tion as

Q = 1

T

∫ T

0ε2dt = 1

T

∫ T

0

[y − x

]2dt (99)

with respect to the parameters a, b, A, β , γ , n. Equation (99)is differentiated w.r.t. each parameter to get a system of nor-mal equations

Q = 1

T

∫ T

0[y − x](∂x/∂θi)dt = 0, i = 1,2, . . . ,6 (100)

that is solved using the Gauss Newton iterative procedure,where θ = [a b A β γ n].

In [187], Gauss-Newton iterations are used as a methodof estimating the parameters of hysteretic system with slipon the basis of input-output data. The model used is called


slip-lock [17] as an extended version of the Bouc-Wenmodel, and describes the pinching of hysteresis loops:

x + ax + bx + z = p(t), (101)

z = Ax1 − β|x1||z|n−1z − γ x1|z|n, (102)

x1 =√

2

π

s

σexp

[− z2

2σ 2

]z, (103)

x = x1 + x2, (104)

s = δsE(t), (105)

where s = δsE is the slip, δs is a constant, E is the systemenergy dissipation; and σ = the Gaussian density function.Then, the parameters to be identified are a, b, A, β , γ , n, δs

and σ the elements of the column vector θ .The basic philosophy of the identification method begins

with the selection of the integrated mean squared error be-tween the actual observation of acceleration ˆx(t) and themodel response x(t) as a cost function for minimization, i.e.,

minimize Q = 1

T

∫ T

0

[ ˆx(t) − x(t)]2

dt (106)

with the parameters a, b, A, β , γ , n, δs and σ , where T =the sampling time.

Since x(t) is non-linear function in all the parameters,(106) can be solved iteratively. The Gauss-Newton iterativeprocedure is used as:

θm+1 = θm − H−1m ∇Qm, (107)

where Hm = the Hessian matrix of Q, but only the first orderderivative is used. ∇Qm = the Jacobian vector of Q, and thesubscript m denotes the computation of the matrix Hm or thevector ∇Qm with respect to the estimate of the parametervector θ at the mth iteration.

A three-stage procedure is suggested for carrying out theiteration:

– In the first stage, the parameters n and σ are kept fixedand Q is minimized over the six parameters a, b, A, β , γ ,and δs .

– In the second stage, the parameter σ is kept fixed and Q

is minimized over the seven parameters a, b, A, β , γ , n,and δs .

– In the third stage, Q is minimized over all the eight para-meters.

6.5 Bootstrap Filter Based Identification

In [186], a parametric identification method is proposed foran extended form of the Bouc-Wen model that accounts forstick-slip phenomenon [17]. The method uses the bootstrapfilter, a filtering method based on Bayesian state estimation

and Monte Carlo method. Also, a method to decide the ini-tial estimates of the parameters is suggested to obtain stablesolutions as well as their fast convergence to the optimal val-ues.

6.6 Identification Using Periodic Signals

Reference [235] proposes a frequency domain parametricidentification method of non-linear hysteretic isolators de-scribed by

mx(t) + r(t) = F(t), (108)

r(t) = kx(t) + z(t), (109)

z = Ax − β|x||z|n−1z − γ x|z|n. (110)

Assuming a known mass m, the hysteretic restoring forcegoverned by (109) is identified by taking measurements ofboth the external excitation F(t) and the displacement re-sponse x(t) (or the acceleration x alternatively). The vectorof model parameters to be identified is {y} = {k α β γ n}T .

The signals F(t) and x(t) are periodic and are expressedas

F(t) = F0

2+

N∑j=1

Fj cos jωt +N∑

j=1

F ∗j sin jωt, (111)

x(t) = a0

2+

N∑j=1

aj cos jωt +N∑

j=1

a∗j sin jωt, (112)

where

{F} = {F0 F1 F2 · · · FN F ∗1 F ∗

2 · · · F ∗N }T , (113)

{a} = {a0 a1 a2 · · · aN a∗1 a∗

2 · · · a∗N }T , (114)

ω is the vibration frequency and N is the order number ofharmonics truncated. Applying the Galerkin (harmonic bal-ance) method into the time domain determining function

D(t) = F − m...x − x{k + A − [β sgn(x)

× sgn(F − kx − mx) + y]|F − kx − mx|n} (115)

achieves

d(y) = 0 (116)

where {d(y)} is the residual of the equations and a mini-mization problem in terms of non-linear least squares arisesas

ming(y) = ‖d(y)‖2 = dT d. (117)

This non-linear least squares optimal problem is solvediteratively by the Levenberg-Marquardt algorithm.


In [136, 138], an identification method for the normal-ized Bouc-Wen model is proposed. Using the analytical de-scription of the hysteresis loop developed in [137], an algo-rithm is proposed along with its analytical proof. It consistsin exciting the Bouc-Wen model with two periodic signalswith a loading-unloading shape (wave periodic) which givesrise asymptotically to a hysteretic periodic response. The ob-tained two limit cycles are then used as input to determineexactly the unknown parameters. The identification method-ology is summarized in the following steps:

– STEP 1. Excite the Bouc-Wen model with a wave peri-odic signal x(t). After a transient, the output �BW(t) willhave a steady state as �BW(t). Since both the input andthe output are accessible to measurements, the relation(x, �BW(x)) is known.

– STEP 2. Choose a nonzero constant q and excite theBouc-Wen model with the input x1(t) = x(t) + q . Af-ter a transient, the output �BW,1(t) will have a steadystate �BW,1(t). Since both the input and the output areaccessible to measurements, the relation (x1, �BW,1(x1))

is known.– STEP 3. Compute the coefficient κx as

κx = �BW,1(x + q) − �BW(x)

q. (118)

– STEP 4. Compute the function θ(x) as

κww(x) = �BW(x) − κxx � θ(x). (119)

– STEP 5. Determine the unique zero of the function θ(x),that is the quantity x∗ such that θ(x∗) = 0.

– STEP 6. Compute the parameter a as

a =(

dθ(x)

dx

)x=x∗

. (120)

– STEP 7. Choose two design constants x∗1 and x∗2 suchthat x∗2 > x∗1 > x∗. Then compute parameters n and b

using the following equations

n =log(

(dθ(x)dx

)x=x∗2−a

(dθ(x)dx

)x=x∗1−a)

log(θ(x∗2)θ(x∗1)

), (121)

where log(.) denotes the natural logarithm

b = a − (dθ(x)dx

)x=x∗2

θ(x∗2)n. (122)

– STEP 8. Compute the parameters κw and ρ as

κw = n

√a

b, (123)

ρ = a

κw

. (124)

– STEP 9. Compute the function w(x) using

w(x) = θ(x)

κw

. (125)

– STEP 10. Choose a design constant x∗3 such that x∗3 <

x∗. Then compute the parameter σ as follows:

σ = 1

2

( (dw(x)

dx)x=x∗3ρ

− 1

(−w(x∗3))n+ 1

). (126)

6.7 Simplex Method Based Identification

Reference [262] presents a two-step system identificationapproach that does not require the semiactive device to betested apart from the structure, but rather mounted into it. Itconsists in (i) identification of a model for the primary struc-ture without the semiactive damper attached; (ii) installationof the semiactive damper in the structure and simultaneousidentification of the remaining parameters for the primarystructure and of a model for the semiactive control device.The simplex algorithm is employed to optimize the dynam-ical parameters.

In [115], the simplex and Levenberg-Marquardt opti-mization methods are used to fit experimental data withcurves given by a Bouc-Wen model in a dynamic suspen-sion modeling problem.

6.8 Support Vector Regression Based Identification

Reference [363] proposes a non-linear structural identifica-tion scheme to identify the Bouc-Wen type structures. It pro-duces the unknown power parameter n of the model by themodel selection strategy, transforms the non-linear differ-ential equation into a linear problem through the high or-der Adam Moulton implicit equation [275], and utilizes thesupport vector regression data processing technique to solvenon-linear structural parameters.

6.9 Constrained Nonlinear Optimization BasedIdentification

Paper [273] uses the Bouc-Wen model to describe mag-netorheological dampers, and proposes a methodology ofidentification to determine the model parameters. The valuen = 2 is supposed and the model estimation problem is re-duced to an optimization problem where the performanceindex is a classical normalized L2-norm of the output fittingerror.

In [320, 336] and [349] a constrained nonlinear optimiza-tion is used for identification purpose.


6.10 Non-Parametric Identification

In [220], the Bouc-Wen nonlinear hysteresis term is approx-imated by a power series expansion of suitable basis func-tions, then the coefficients of the functions are determinedusing standard least-squares methods.

In [199], a method relying on deconvolution to estimatethe non-linear hysteretic force z from experimental recordsis used.

7 Modeling Using the Bouc-Wen Model

The Bouc-Wen model has been extensively adopted in manyengineering fields to represent the hysteresis behavior ofsome nonlinear components. In this section, we present anoverview of the use of this model.

7.1 Magnetorheological Dampers

Magnetorheological (MR) dampers are hysteretic devicesthat employ rheological fluids to modify their mechanicalproperties. The stiffness and damping characteristics of theMR damper change when the rheological fluid is exposedto a magnetic field. The Bouc-Wen model has been used todescribe the hysteresis behavior of these devices.

Reference [90] proposes a non-linear model for MRdamper which incorporates the current (I ), amplitude andfrequency excitation (ω) as input variables. In this sense, themodified Bouc-Wen model is reformulated as follows:

F(x(τ), x(τ ), I,ω, x,0 ≤ τ ≤ t; t)= (

d1ωd2)(

d3ωd4max

)[c0(I )x + k0(I )x + α(I)z], (127a)

z(I ) = A(I)x + β(I)|x||z|n−1z − γ x|z|n, (127b)

c0(I ) = c1 + c2(1 − e−c3(I−Ic)

)for I > Ic, (127c)

c0(I ) = c4 + c4 − c1

Ic

I for I ≤ Ic, (127d)

k0(I ) = k1 + k2 I, (127e)

α(I) = α1 + α2(1 − e−α3(I−Ic)

)for I > Ic, (127f)

α0(I ) = α1 + α4 − α1

Ic

I for I ≤ Ic, (127g)

β(I) = β1 − β2 I, (127h)

Fz0(I ) = Fz01 + Fz02(1 − e−Fz03(I−Ic)

)for I > Ic, (127i)

Fz0(I ) = Fz04 + Fz04 − Fz01

Ic

I for I ≤ Ic, (127j)

where A and γ assumed to be one and zero, respectively,and the 16 constant parameters c1, c2, c3, c4, k1, k2, α1, α2,α3, α4, γ1, γ2, Fz01, Fz02, Fz03 and Fz04 relate the character-istic shape parameters to current excitation. Ic is the critical

current in which the characteristic parameters change theirlinear behavior in low velocity to exponential behavior inhigh velocity.

Paper [240] presents a simple mechanical model consist-ing of a Bouc-Wen element in parallel with a viscous damper(with damping coefficient c0). It was used and verified to ac-curately predict the behavior of a prototype shear-mode MRdamper over a wide of range of inputs by. The force f ex-erted by this model is

f = c0 x + α z (128)

where z is given by (139). Device model parameters α andc0 are considered dependent on the control voltage V as fol-lows:

α = α(V ) = αa + αb V, (129)

c0 = c0(V ) = c0a + c0b V . (130)

A Bouc-Wen hysteretic voltage-dependent model is de-veloped from performance tests of a 3 kN MR damper de-vice that is incorporated to an isolated structure subjectedto earthquake excitations in order to reduce its responseby [202]. The restoring force F(t) is expressed as

F(t) = C(V )x + z, (131)

z = Ax + β|x||z|n−1z + γ x|z|n (132)

where C(V ) is a voltage-dependent parameter.In [299], an extension to the standard Bouc-Wen model is

proposed to account for the rolloff that appears in the force-displacement relationship at small velocities. In this modi-fied model, the force is given by

F = αz + c0(x − y) + k0(x − y) + k1(x − x0)

= c1y + k1(x − x0) (133)

where z and y are governed by

z = −β|x − y| z |z|n−1 − γ (x − y)|z|n + A(x − y), (134)

y = 1

c0 + c1

(αz + c0x + k0(x − y)

)(135)

in which k1 = accumulator stiffness; c0 = viscous dampingat large velocities; c1 = viscous damping for force rolloff atlow velocities; k0 = stiffness at large velocities; and x0 =initial displacement of spring k1. This modification is usedby [343] to reproduce the hysteresis loop of a large scale MRdamper.

In [175], a non-symmetrical Bouc-Wen model for MRfluid dampers is presented. The adopted strategy is to adjustthe velocity value in calculating the hysteretic variable but


retain the general expression for the Bouc-Wen model. Theresulting non-symmetrical Bouc-Wen model is

z = (A − (

γ + β sgn[z(x − μ sgn(x))])|z|n)× (x − μ sgn(x)) (136)

where μ is the scale factor for the adjustment, provided that

μ sgn(x) → 0∣∣x≈0. (137)

The overall effect is to shift the hysteresis switching inthe vicinity of zero velocity while maintaining the hystereticshape in the rest of the hysteresis loop.

Paper [344] presents a phenomenological dynamic modelbased on the Bouc-Wen model to estimate large-scale MRdamper behavior under dynamic loading. This model ac-commodates the MR fluid stiction phenomenon, as well asfluid inertial and shear thinning effects. Moreover, the pro-posed model is supposed to be more effective in describingthe force rolloff in the low velocity region, force overshootswhen velocity changes in sign, and two clockwise hysteresisloops at the velocity extremes. The damper force is given by

f − f0 = mx + c(x)x + kx + αz, (138)

z = Ax − β|x||z|n−1z − γ x|z|n. (139)

In this model, m = equivalent mass which representsthe MR fluid stiction phenomenon and inertial effect; k =accumulator stiffness and MR fluid compressibility; f0 =damper friction force due to seals and measurement bias;and c(x) = post-yield plastic damping coefficient which isdefined as a monodecreasing function with respect to ab-solute velocity |x| to describe the MR fluid shear thinningeffect which results in the force rolloff of the damper resist-ing force in the low velocity region. The post-yield dampingcoefficient is assumed to have the form

c(x) = a1 e−(a2|x|)p (140)

where a1, a2 and p = positive constants.In [349] and [240], a Bouc-Wen based phenomenological

model of a shear-mode MR damper is presented while [135]proposes a simplified version of the Bouc-Wen model forMR dampers called the Dahl model.

7.2 Structural Elements

The Bouc-Wen model has been used to simulate a variety ofhysteretic structural elements behaviors. For example, andbased on the modification of the Bouc-Wen-Baber-Noorimodel, general features of the hysteretic behavior of woodjoints (with yielding plates or yielding nails or yieldingbolts) and structural systems were characterized by [104].

The model accounts for degradation and pinching in theform:

z = h(z)

{Ax − ν(β|x||z|n−1z + γ x|z|n)

η

}(141)

where ν and η are the strength and stiffness degradation re-spectively.

In [243], the standard form of the model is incorporatesinto ABACUS to investigate the dynamic frictional contactof a bolted joint under harmonic loading.

Reference [321] investigates the effects of connectionfailure on structural response of steel buildings under earth-quakes, a smooth connection-fracture hysteresis modelbased on the Bouc-Wen model. The form of the model thataccounts for degradation is extended to represent asymmet-ric smooth hysteresis in which the ultimate positive and neg-ative restoring forces are not equal. This extensions is for-mulated as

z = x

η

(A − ν|z|n[β sgn(xz) + γ + φ(sgn(z) + sgn(x))

])

(142)

in which φ is an additional parameter accounting for theasymmetric yielding behavior. The hysteresis loops shiftdownward or upward depending on the sign (positive or neg-ative) of the parameter φ.

The inelastic behavior of connections is described in[190], taking into account capacity uncertainties of connec-tions.

The work in [23] utilizes the Bouc-Wen model to de-scribe hysteretic dampers that interconnect two adjacentstructures subjected to seismic excitation.

Paper [19] studies the nonlinear response of single pilesunder lateral inertial and seismic loads using (144) as amacroscopic model. This model consists of distributed non-linear springs (described by the Bouc-Wen model), com-bined with a distributed viscous (frequency dependent)dashpots, placed in parallel, and describes the lateral soilreaction. The variable y represents the value of pile deflec-tion that indicates yielding in the spring.

7.3 Base Isolation Devices

The base isolation devices aim to absorb or reflect the earth-quake input energy to the structure to keep linear structuralvibration. Many devices are strongly nonlinear showing dif-ferent hysteretic behaviors. The Bouc-Wen model has beenintroduced for its intrinsic ability in reproducing a widerange of real devices behavior.

A combined energy dissipation system was developed by[204]. In this system lead rubber dampers and their paral-lel connection with oil dampers are used in the braces of


a structural frame. The restoring force characteristics of thelead rubber damper is simulated by the Bouc-Wen hystereticmodel having the following form:

F = αFy

dy

x + (1 − α)Fyz, (143)

z = 1

dy

(Ax − β|x||z|n−1z − γ x|z|n) (144)

where Fy and dy are the yielding force and yielding dis-placement, respectively.

The same form of (144) was also used to control the evo-lutionary parameter z by [76] to model the friction of Teflon-steel interfaces in sliding-based seismic isolation devices.The frictional force is given by

Ff = μs W z (145)

where μs is the coefficient of sliding friction, W is the struc-tural weight, z takes values ±1 during sliding (yielding) andduring sticking (elastic behavior), the absolute value of z

is less than unity. Equation (144) was also used by [83] tomodel the dissipated energy by the wire rope isolators forbuilding equipments, which have found numerous applica-tion in the shock and vibration isolation in military and in-dustrial fields.

Because of its versatility and suitability for direct closedform stochastic linearization, the Bouc-Wen model was em-ployed within a method of random vibration analysis of baseisolated shear structures with hysteretic dampers by [75] and[214] to simulate the load deformation path of the hystereticdevices.

In [216], a parametric stochastic analysis of an isolatedbridge is proposed with the aim to assess isolation perfor-mance and to investigate effects of energetic influence onprotection efficiency. Isolated bridge (piers and seismic iso-lators) is described by a simple two degree of freedom Bouc-Wen hysteretic model. Engineering mechanical quantitieswere related to the analytical parameters of the model in or-der to model real structural elements. Accordingly, the evo-lutionary parameter z is given by

z = x

[1 − 1

2

(z

xy

)n

(1 + sgn(zx))

], (146)

xy =(

1

β + γ

)1/n

, the limit elastic displacement (147)

assuming A = 1. It was reported that, by adopting the Bouc-Wen and by using appropriate parameters for this model, itbecomes able to suitably characterize an ordinary buildingand the considered base isolators.

7.4 Mechanical Systems

The Bouc-Wen model provides useful predictions of the re-sponses of suspension seats of the off-road machines to tran-sient inputs [115]. It is concluded that the Bouc-Wen modelcan provide a useful simulation of an existing seat and as-sist the optimization of an individual component in the seat,without measuring the dynamic properties of components inthe seat except those of the component being optimized. Theused form of the model is

FBW = (k − ks)x − β|x|FBW − γ x|FBW | (148)

in which k and ks are positive stiffnesses, x is the relativevertical displacement, while β and γ give the effect of hys-teresis, and FBW is the Bouc-Wen force that combines all thenon-linear effects due to the different seat components.

Reference [293] investigates the interaction effect be-tween electrical substation equipment (an important ele-ment within the power transmission lifeline) items con-nected by nonlinear rigid bus conductors. The symmetrichysteretic behaviors are described by the original Bouc-Wenmodel while the components having asymmetric hysteresisare modeled by a modified version of the Bouc-Wen modelfor highly asymmetric hysteresis loops with constant para-meters. The auxiliary differential equation of this modifiedmodel is given as

z = x[A − |z|nψ(x, x, z)

], (149)

ψ = β1 sgn(xz) + β2 sgn(xx) + β3 sgn(xz)

+ β4 sgn(x) + β5 sgn(z) + β6 sgn(x) (150)

in which βi, i = 1,2, . . . ,6, are parameters controlling theshape of the hysteresis loop and x is the relative displace-ment.

To predict correctly the lateral vibration of an elevatorin motion by [324], it was necessary to deal with a rate-dependent stick-slip phenomenon which may occur at arubber-to-metal interface which is observed at the interfacebetween the guide roller and the rail of a typical elevatorin motion. To consider the rate-dependent nonlinearity, themth-power velocity damping model c|x|m sgn(x) is incor-porated to the Bouc-Wen model, so the variable z is ex-pressed as

zm = c|x|m sgn(x) + z, (151)

where z is given by (139).Without introducing any modifications to the standard

Bouc-Wen model, an investigation of the dynamics of asmall non-linear oscillator having softening hysteretic char-acteristic and weakly coupled with a linear oscillator wasperformed by [180].


7.5 Piezoelectric Actuators

The Bouc-Wen model was experimentally modified by [203]to describe the hysteresis phenomenon of the piezoelectricactuator and used later by [119] and [194] to simulate thehysteresis of a piezoelectric element. The modified form is

z = α d33V − β|V |z − γ V |z|, (152)

where d33 is the piezoelectric coefficient and V denotes theinput voltage. α, β and γ control the shape of the hystere-sis loop. The same form of the Bouc-Wen model was alsoemployed to describe the frictional hysteresis force of animpact drive mechanism by [118].

7.6 Soil Behavior

In a study of the grain-crushing-induced landslides, [111]used the hysteretic stress-strain Bouc-Wen-type constitutivemodel in conjunction with a Mohr-Column friction law andTerzaghi’s effective stress principle to model the hystereticstress-displacement relationship of the soil inside the shearband τ where,

τ = τy z, (153)

in which τy = the ultimate shear strength and z is given by:

z = 1

xy

(1 − |z|n[b + (1 − b) sgn(xz)

])(154)

where x is the lateral velocity, xy is a parameter accountingfor the elasto-plastic slip tolerance and the parameter b con-trols the shape of the unloading-reloading curve. Its range ofvalues is between 0 and 1.

Another extension to the Bouc-Wen hysteresis model isperformed in [254] to provide a better representation of theactual shearing stress-strain behavior of soils under constantand variable amplitude cyclic loading. In particular, it is de-sired to obtain a better representation of the soil responseunder small amplitude non-symmetric and non-zero meanloading reversals. Simultaneously, it is also desired to ob-tain a better representation of the equivalent viscous damp-ing ratio at high strain levels. The smooth-hysteretic modelcan be obtained if a finite number p of nonlinear-hystereticsprings described by the standard Bouc-Wen model are usedin parallel, then the shearing stress τ is given by

τ = (1 − α)Gm γs + (1 − α)

M∑p=1

Gmp zp, (155)

zp = Ap γs − βp|γs ||zp|np−1zp − γp γs |zp|np (156)

with the following conditions:

M∑p=1

Gmp = Gm, (157)

M∑p=1

τmp = τm (158)

where

τmp = Gmp

(Ap

βp + γsp

)1/np

(159)

in which Gm is the (small strains) shear modulus; γs is theshearing strain; and τmp is the maximum shear stress.

A discrete version of the Bouc-Wen model to describestrong ground motion in Iceland is used in [242].

7.7 Energy Dissipation Systems

In [277], a modification of the Bouc-Wen model was pro-posed by adding the isotropic hardening mechanism to re-produce the nonlinear strain hardening in seismic resistanceof rhombic Low Yield Strength Steel under practical recip-rocating loading test. The modified form is

z = Ax − η(β|x||z|n−1z + γ x|z|n), (160)

η =(

�y

�y0

)−n

, (161)

�y ={

�y0 if �max ≤ �y0,

�y0 + μ(�max − �y0) if �max ≥ �y0,(162)

where �y0 is the initial yielding displacement; �max is theabsolute value of the maximum displacement in the loadinghistory; μ = α−ν; ν is the rate of kinematic transformation;and η ranges from 0 to 1 and follows the transformation ofdeformation history, which is decreasing.

Based on the extension of the Bouc-Wen model thataccounts for degradation, (141) with h(z) = 1, a frictionenergy dissipating system attached to reinforced concretepanel is studied. The extension of the Bouc-Wen model isemployed to model the flexibility of the connection betweenthe steel elements and the concrete panel, and to model theimpact of these bolts as well as the expansion of the con-crete holes as a result of the impact. It is concluded that,when modeling friction using elements that are based on theBouc-Wen model, the violation of the model to Drucker andIliushin postulates increases the displacement by an insignif-icant amount over some paths and has negligible effect onthe predicted behavior, [272].


8 Conclusion

This paper surveyed the literature related to the Bouc-Wenmodel. It has been organized into sections that address spe-cific issues like modeling, control, identification, etc. Eachsection presented what, in the author’s point of view, arethe main contributions relative to that specific issue. It hasnot been possible to present every single work relative tothe Bouc-Wen model as issues discussed in some papers arevery peculiar and do not fall within any general category.Nevertheless, all the papers that the authors are aware ofhave been cited in the reference part of this survey.

Acknowledgements The first author acknowledges the support ofthe Generalitat de Catalunya through the FI fellowships programs.This research has been funded by projects DPI2008-06463-C02-01 andDPI2005-08668-C03-01 of the Spanish Ministry of Science and Inno-vation.

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