A Wavelet Based Approach for Model and Parameter Identification of Nonlinear Systems

*Corresponding author. Tel.: #1-410-516-7647; fax: #1-410-516-7473.E-mail addresses: [email protected] (R. Ghanem), [email protected] (F. Romeo).

International Journal of Non-Linear Mechanics 36 (2001) 835}859

A wavelet-based approach for model and parameteridenti"cation of non-linear systems

Roger Ghanem!,*, Francesco Romeo"

!The Johns Hopkins University, Baltimore, MD 21218, USA"Universita% di Roma **La Sapienza++, Rome, Italy

Abstract

A procedure is presented for identifying the mechanical parameters of zero-memory non-linear discrete structuralsystems. The procedure allows both the parameter estimation of a priori known dynamical models as well as theidenti"cation of classes of suitable non-linear models based on input}output data. The method relies on a wavelet-baseddiscretization of the non-linear governing di!erential equation of motion. Orthogonal Daubechies scaling functions areused in the analysis. The scaling functions localization properties permit the tracking of fast variations of the state of thedynamical system which may be associated with unmodeled dynamics of measurement noise. The method is based on theknowledge of measured state variables and excitations and applies to single and multi-degree-of-freedom systems undereither free or forced vibrations. ( 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Non-linear dynamics; System identi"cation; Time-varying systems; Wavelets; Galerkin projection

1. Introduction

Modeling non-linear structural systems consists of two main components, namely the characterization ofthe non-linearities and the parameter estimation. The detection of the presence of non-linearities, theirclassi"cation and, eventually, the determination of their location, are the main steps involved in thecharacterization phase. The generation of a working model through a parameter estimation techniquerepresents the next step towards developing a predictive model. The analysis of non-linear systems iscomplicated by the inaplicability of the principle of superposition. Thus, the response of the system dependsboth on the level and on the location of the excitation. Most linear analysis techniques are thus notapplicable. A number of alternative approaches have been proposed in the literature for the identi"cation ofnon-linear systems arising in structural applications, with a number of surveys available [1,29]. Among thetime-domain techniques, a direct parameter estimation method has been proposed [26]. Accordingly, bothlinear and non-linear elements are postulated at certain locations throughout a discrete model of the system.The model parameters are then estimated by curve "tting the data with the assumed model through

0020-7462/01/$ - see front matter ( 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 0 5 0 - 0

a least-squares technique. The most appropriate set of parameters to include in the model is obtainedthrough a statistical signi"cance test. The main advantage of this method is its simplicity, while somedrawbacks arise from the need to excite the system in the range where there are modal frequencies and fromthe necessary data "ltering. The latter step is needed to remove the integration errors introduced in order toobtain velocity and displacement data from the measured accelerations. The use of ARMA models for linearsystem identi"cation has been extended to deal with non-linear systems through the NARMAX models [6].The main idea here is to non-linearly regress the current response data point on the past response and inputdata points. The method can still be viewed as a direct parameter estimation based on the sampled dataobtained after substituting "nite di!erence approximations for the time derivatives in the equation of motion.The latter model can be regarded as a subgroup of the AVD models (acceleration, velocity and displacement)[7,8]. In fact, it has been shown that a more general form of time-series models of the original non-lineardi!erential equation is obtained by &regressing' a dominant variable (which can be the acceleration, thevelocity or the displacement) not only on the past sampled response but also on the past sampled velocitiesand accelerations. This technique has been shown to be e!ective and useful for non-linear modelling eventhough some practical limitations arise when the number of parameters is too large and correlated noiseterms are generated. Another interesting approach is based on the representation of the structural non-linearrestoring force as a set of parallel linear subsystems allowing the development of an identi"cation techniquebased on a multi-input}single-output (MISO) linear system [3,33,38]. This procedure begins by #ipping thesystem so that the force becomes the output and the response becomes the force. Linear and non-linear pathsare then used to connect, respectively, the linear and non-linear inputs (the actual response) to the output (theactual force). Once obtained the MISO linear system, standard frequency-domain techniques can be used tocharacterize and model the non-linear system. The representation of the input/output system behaviorthrough functional series models is the basic approach for a widely studied group of identi"cation techniques.A "rst functional representation of non-linear systems which is a generalization of the linear convolutionintegral was introduced by Volterra early in the 20th century and system identi"cation techniques based onthe Volterra/Wiener representation theory were later developed [5,15]. Other functional relationships havebeen also investigated by a number of authors. Surface "tting to the restoring forces represented over thephase plane by using an orthogonal polynomial series in the time domain has been presented [24]. Thismodel, also referred to as the force state mapping, was then extended for the identi"cation of multi-degree-of-freedom systems [25]. In developing this technique, it was assumed that the modal matrix of the systemwas known and it was used to transform the restoring force functions into generalized forces. This force statemapping method was applied to the identi"cation of hysteretic structures [4,30]. Models for identifying theparameters of non-linear equivalent lumped systems by using ordinary polynomial series have also beenproposed [2,37]. The main limitations of all the above mentioned procedures are their reliance on the modalmatrix of some equivalent linear system, their signi"cant computational requisites, and the lack of physicalsigni"cance to be attributed to many of the associated parameters.

One of the main feature of non-linear vibrations is an amplitude-dependent period of oscillations. Bytracking the changes of frequency with amplitude, the non-linearity in a system can be characterized andidenti"ed. The plot of amplitude versus frequency is the system characteristic typically obtained by usingHilbert transform time-domain techniques. Such transform can be applied to both frequency- and time-domain data leading to a representation of the system response in terms of two time functions, theinstantaneous envelope and frequency, from which the dissipative and restoring forces of the system can berecovered. Such an approach is e!ective only when the response is composed of just a single quasi-harmoniccomponent. In order to overcome such limitation, extending the applicability of the procedure to theidenti"cation of non-linear multi-degree-of-freedom systems, a preliminary decoupling of the response intoa series of quasi-harmonic components is required. The joint use of the Gabor and Hilbert transforms to thetransient response has been proposed in an attempt to address this issue [19,34]. Signi"cant e!orts arerequired for "ltering and smoothing in order to obtain clear backbone curves of amplitude versus frequency

836 R. Ghanem, F. Romeo / International Journal of Non-Linear Mechanics 36 (2001) 835}859

and closely spaced modes of vibration can limit the applicability of such approaches. The latter consider-ations lead [35] to take advantage of the time-scale properties of the wavelet functions in order to analyzenon-linear vibrations. The wavelet transform ridges and skeletons of the system impulse response are used toextract the decay envelope and frequency characteristics. The continuous wavelet transform using the Morletwavelet function provides the necessary mode decoupling thanks to the frequency localization of thewavelets. Di!erently, in [16] the key role is played by the wavelet time localization property. Indeed thehysteresis curves and damping coe$cients of structural systems are identi"ed by expressing the tangentsti!ness as a series of Daubechies (D4) scaling functions. The proposed method does not require priorassumptions of the system non-linearities and its performance is shown through numerical and experimentaltests on a shear-type multi-degree-of-freedom system.

A wavelet-based procedure for the identi"cation of linear systems with either time-invariant or time-varying parameters has been recently proposed by the authors [14]. The localization properties of the dualfunctions of the wavelets, namely the scaling functions, are relied upon to resolve fast variations in the state ofthe dynamical system that would have been averaged out using other techniques. This work illustratesfurther developments enabling to address the identi"cation of systems with zero-memory non-linearbehavior. The "rst section is devoted to investigate the possibilities o!ered by the identi"cation algorithm todeal with systems with known non-linearities. Numerical applications to single and two degree-of-freedomsystems characterized by cubic non-linearities and dry friction under either free or forced vibrations are thenpresented. Stick/slip phases in dry friction systems are taken into account. Next, the characterization ofnon-linearities is addressed by postulating a non-linear behavior that is a linear combination of knownnon-linear models. A selection criterion across the class of acceptable models is then implemented and thecoe$cients in the selected models estimated. In the latter case, the unavoidable presence of noise a!ecting themeasurements can be tackled through the wavelet-transform-based denoising strategy presented in [14].

2. Identi5cation of systems with known non-linearities

The wavelet-domain representation of the original non-linear di!erential problem is the core of theidenti"cation scheme developed in this section. The expression of the equation of motion for a genericmulti-degree-of-freedom non-linear system can be written in the form

MxK#g(x,x5 )"f, (1)

where M refers to the n]n diagonal mass matrix, n is the number of degrees of freedom, and the term g(x,x5 )represents a generic non-linear function of displacement and velocity. Starting with the single degree-of-freedom system (SDOF), the function g(x,x5 ) assumes di!erent expressions according to the nature of thesystem. For a linear system it takes the form

g-*/

(x, x5 )"cx5 #kx. (2)

Alternatively, one of the common forms of zero-memory non-linearities arising in structural dynamics, andthat will be used in this paper, involve a function g(x,x5 ) that represents sti!ness, damping and cross-couplingpolynomial non-linearities as given by the following equation:

g10-

(x, x5 )"M+

m/2

c(m)x5 m#M+

m/2

k(m)xm#M+

m/1

N+n/1

d(m,n)x5 mxn, (3)

where M, N represent the order of the non-linearity. Another common form of the function g(x,x5 ), that willalso serve to demonstrate the method developed in this paper, is associated with dry friction, such as the

R. Ghanem, F. Romeo / International Journal of Non-Linear Mechanics 36 (2001) 835}859 837

constant and linear Coulomb models, given, respectively, by the following two equations:

g#&

(x5 )"e#&

Dx5 Dx5

, g#&

(x, x5 )"(e#&#g

#&DxD)

Dx5 Dx5

. (4)

The Wavelet}Galerkin solution of the di!erential equation (1) begins by projecting both the solution x(t)and the excitation f(t) on the subspace <

jspanned by the Daubechies' scaling functions [10]

uj,k

(t)"2j@2u(2jt!k) at scale J. Such resolution level is assumed to be so high that no signi"cant variationswould occur in its detail component. Using the expression x

i(t)"2j+J

k/Ia6i,k

2j@2u(2jt!k) for the expansionof the response at the ith degree of freedom [14], and after substitution, the equation of motion can berewritten as

m22jJ+k/I

akuK (y!k)#c2j

J+k/I

aku5 (y!k)#k

J+k/I

aku(y!k)

# gAJ+k/I

aku(y!k),

J+k/I

aku5 (y!k)B"

J+k/I

bku(y!k). (5)

In the latter expression the bounds on the summations are I"k0!¸#2 and J"k

1!1 where k

0"2jt

0and k

1"2jt

fare integers with t

0"0, t

f"1 being the initial and "nal times, respectively; ¸"2N is the

number of non-zero coe$cients depending on the choice of Daubechies' scaling function. Taking the innerproduct of Eq. (5) with u(y!l) and omitting the bounds on the summations for notational clarity, we get

m22j+k

akP

2jq

0

uK (y!k)u(y!l) dy#c2j+k

akP

2jq

0

u5 (y!k)u(y!l) dy

# k+k

akP

2jq

0

u(y!k)u(y!l) dy#P2jq

0

gA+k

aku(y!k), +

k

aku5 (y!k)Bu(y!l) dy

"+k

bkP

2jq

0

u(y!k)u(y!l) dy, l"2!¸,2, 2j!1. (6)

Introducing the general de"nition of the 2-term connection coe$cient [20]

!(n)k"P

`=

~=

u(n)(y!k)u(y) dy, (7)

where a superscript (n) denotes nth-order derivative, the linear part of Eq. (6) can be simpli"ed using thetwo-term connection coe$cient according to the scheme developed for the linear-time-varying case [14]. Onthe other hand, and depending on the nature of the non-linear function g( ' ), higher-order connectioncoe$cients will be needed in order to reduce the non-linear functional. As an instance, consider a non-linearity of the form g

10-(x,x5 )"xx5 . The following scaling function expansion is obtained:

g(x,x5 )"d(1,1)xx5 "d(1,1)+i

aiu(y!i)+

k

aku5 (y!k) (8)

and the corresponding integral term in Eq. (6) is rewritten using the three-term connection coe$cient as

+i

+k

aiakP

2jq

0

u(y!i)u5 (y!k)u(y!l) dy"+i

+k

aiak)(1,0)

k~i,l~ifor l"2!¸,2, 2j!1. (9)


Therefore the bilinear non-linearity involves the 3-term connection coe$cient )(1,0)k~i,l~i

. Setting d(1,1)"d theequation of motion is rewritten as

m+k

ak!(2)

k~l#c+

k

ak!(1)k~l

#kal# d+

i

+k

aiak)(1,0)

k~i,l~i"b

lfor l"2!¸,2,2j!1, (10)

which can be also expressed as

+kCm!(2)

k~l#c!(1)

k~l#kd

kl#d

2j~1+

i/2~L

ai)(1,0)

k~i,l~iDak"b

lfor l"2!¸,2, 2j!1, (11)

which can "nally be written in a compact form as

C(a) 'a"b. (12)

Given that the governing equations are linear in the parameters c, k, and d, rearranging Eq. (12) yieldsa linear algebraic set of equations to be solved for these parameters c, k, d, given the scaling functioncoe$cients of the input and output signals. Two considerations are worth mentioning. First, as provided bythe Galerkin scheme, the latter equation represents the transformation of the original non-linear di!erentialproblem into a non-linear algebraic problem in the scaling functions domain. Second, the presence ofpolynomial-type non-linearities introduces n-term connection coe$cients where n depends on the order ofthe non-linearity. For other forms of non-linearity, such as those described by Eqs. (4), more involvedfunctionals need to be computed. Although in the formulation resulting in Eq. (11), and given the linearity ofthe governing equation with respect to m, c, and k, knowledge of the input f (t) and output x(t) avoids thesolution of a non-linear set of equations, higher-order connection coe$cients still need to be computed. At"rst look, this seems to be a serious drawback. Indeed, even though the two main algorithms available forevaluating the connection coe$cients [11,20] do not give any restriction on the number of terms involved inthe integrals, the corresponding numerical implementations of such algorithms [18,32] are capable ofperforming the inner products of up to four scaling functions and their derivatives. Moreover, non-linearitiesdi!erent from the polynomial ones seem to be unmanageable using this technique. An alternative strategy isdeveloped in order to overcome these limitations. Speci"cally, since the output signal, representing theresponse of the system, is measured, a signal associated with the non-linear functional, g( ' ), can be readilysynthesized and then expanded by means of scaling functions. For example, and considering once more theprevious example, where g(x,x5 )"xx5 , an alternative equation to (11) is obtained

m+k

ak!(2)

k~l(¸!1)#c+

k

ak!(1)

k~l(¸!1)#ka

l#dg

l"b

lfor l"2!¸,2, 2j!1, (13)

where the following scaling function expansion:

g(x,x5 )"dxx5 "d2j~1+

k/2~L

gku(y!k) (14)

is now used instead of Eq. (8). The resulting set of algebraic equations to be solved for the unknownparameters has the same structure as that of the linear case and is given by

CM h"b1 . (15)

In this equation the vector hT"Mc, k, dN contains both the linear and non-linear unknown parameters andthe generic lth row of the (2j#¸!2]3) matrix CM reads

!Ml"C2j+

k

aj,k

!(1)k~l

alglD. (16)


Table 1Estimated parameters of a non-linear oscillator based on Eqs. (11) and (13)

Eq. (11) Eq. (13)

COVc(

1.8E-3 1.2E-3COV

kK1.6E-4 1.6E-3

COVdK

2.9E-2 3.0E-3

The corresponding lth term of the vector b1 is given by

bMl"b

l!22jm+

k

aj,k

!(2)k~l

. (17)

A numerical test has been performed on a SDOF system characterized by the non-linear function of Eq. (8)in an e!ort to compare the results obtained from Eqs. (11) and (13). The values of the parameters arem"1.0, c"1.26, k"25.0, d"5.0 and the system is excited by a sinusoidal force with amplitude A"1.0and frequency l"2.0Hz. The comparison between the estimates is presented in Table 1 and it showsa similar level of accuracy for the two formulations. The results in the table refer to the coe$cients ofvariation of the estimates taken over a statistical population representing the successive time windows. Themean values of the estimates over such windows have been found, for both cases, to be coincident with theirtrue values.

Highlighting the generality of the above formulation and its applicability to any non-linearity, as long asits functional form is known, a straightforward extension to multi-degree-of-freedom systems is nextdeveloped following analogous arguments to those presented above for the SDOF case. A generic set ofn-coupled non-linear governing di!erential equations, is given by Eq. (1). For clarity of treatment, andconsistent with the numerical examples presented in a subsequent section, the 2 degree-of-freedom (2-DOF)case is used to demonstrate the general MDOF case. Referring to the model sketched in Fig. 3, the scalarexpression of the linear terms (2) will be replaced by the vectorial one with the sti!ness and damping matricesK and C.

2.1. Systems with cubic non-linearities

The "rst example used to demonstrate the proposed procedure consists of a cubic non-linearity in thesti!ness elements. In this case, the vector-valued function g(x,x5 ) has the following form:

g1(x

1, x

2)"k(3)

1x31#k(3)

2(x

1!x

2)3, g

2(x

1, x

2)"k(3)

2(x

2!x

1)3, (18)

where the indices 1 and 2 refer, respectively, to the "rst and second degree of freedom. As presented in Eq.(14), the scaling function expansion of the non-linearities in (18) results in

x31"

2j~1+

k/2~L

g1,k

u(y!k), (x1!x

2)3"

2j~1+

k/2~L

g2,k

u(y!k), (x2!x

1)3"

2j~1+

k/2~L

g3,k

u(y!k). (19)

Proceeding similarly to the SDOF case, an augmented matrix CM , with dimensions (n(2j#¸!2)]3n), isobtained. Assuming the masses to be known, two equations result from each projection of the two governing


Table 2Parameters adopted for the non-linear oscillators

m c k c(3) k(3) F0

)(0) )(¹f) ¹

f/¹ *t/¹

Du$ng 1.0 0.04 1.0 * 0.003 4 0.5 1.5 49.0 0.05Van der Pol 1.0 !0.2 1.0 0.2 * 10.0 0.5 1.5 10.0 0.01

equations of motion on one scaling function, u(y!l). These equations have the following general form:

C+J

I!(1)k~l

a1,k

a1,l

+JI!(1)k~l

(a1,k

!a2,k

) a1,l

!a2,l

g1,l

g2,l

0 0 +JI!(1)k~l

(a2,k

!a1,k

) a2,l

!a1,l

0 g3,lD C

c1,l

k1,l

c2,l

k2,l

k(3)11,l

k(3)12,l

D"C

b1,l

!m1+J

I!(2)k~l

a1,k

b2,l

!m2+J

I!(2)k~l

a2,kD . (20)

The resulting set of algebraic equations can be solved either by considering sequential groups of 2n equationsor by some minimization technique. In the following several numerical investigations addressing theidenti"cation of systems characterized by cubic non-linearities are presented. All the simulations have beencarried out using the scaling function with N"4 and the resolution level j"9. The di!erential equations arealso solved using accurate standard numerical techniques in order to generate the signals employed in theidenti"cation procedure. Starting with the SDOF case, a damped Du$ng oscillator and a Van der Poloscillator are analyzed. The selected parameters of these models have been used in the literature [24]. Bothsystems are excited by a swept-sine excitation

F(t)"F0

sin[)(t)]t, (21)

where the variable frequency )(t) changes linearly between t"0 and ¹f

according to the relationship

)(t)"at#b, (22)

where

a"u

2!u

12¹

f

, b"u1, u

1")(0), u

2"2)(¹

f)!)(0). (23)

The restoring forces of the two oscillators are given by

g(x,x5 )"cx5 #kx#k(3)x3 (24)

for the Du$ng, and

g(x,x5 )"cx5 #c(3)x2x5 #kx (25)

for the Van der Pol oscillator. The values of the parameters in the above models are summarized in Table 2.The results of the analysis are given in Tables 3 and 4 where the estimated parameters are placed side

by side with their corresponding coe$cients of variation (COV). The estimates are obtained through a


Table 3Forced vibrations: estimated parameters of the non-linear oscillators

Du$ng Van der Pol

c( kK kK (3) c( kK c( (3)

Estimates 0.04 1.0 0.003 !0.2 1.0 0.2COV 1.1E-7 2.3E-6 2.4E-6 2.2E-6 !3.8E-5 1.5E-6

Table 4Forced vibrations: estimated parameters of the Van der Pol oscillator

c( kK c( (3)

Estimates !0.2 1.0 0.2COV 2.2E-6 !3.8E-5 1.5E-6

least-squares solution over sequential sets of 32 equations. The COV index is provided in order to check thedispersion of the estimates throughout the duration of the analysis. In the analysis, *t is taken equal to 0.02and 0.05 for, respectively, the Van der Pol and the Du$ng analyzed cases.

Figs. 1 and 2 show the state-variable plots and the evolution of the parameter estimates of the twosimulations. The performance of the algorithm with respect to free-vibrations data for the two examples ofnon-linear oscillators with cubic non-linearities is next investigated. It must be emphasized that thepossibility of deducing the mechanical parameters of the system from the free response is a remarkablefeature for any method intended to identify non-linear systems. Indeed, such a possibility would permit theidenti"cation under any type of excitation. The results shown in Tables 5 and 6 refer, respectively, to theSDOF Du$ng and Van der Pol oscillators characterized by the same set of parameters as the forcedvibrations case. The initial conditions considered for both systems are x"1.0, 0.0.

The 2-DOF system is considered next. Fig. 3 shows a schematic of the non-linear links characterizing themodel. The numerical investigation is carried out considering the same non-linear relationships used for theSDOF analysis with the parameters shown in Table 7. In particular, the simulation concerning cubicsti!nesses assumes hardening and softening behaviors for the "rst and second element, respectively. The samesinusoidal excitation is considered to act on each mass and is given by

F(t)"F0[sin(u

1t)#sin(u

2t)]

with

F0"10 000, u

1"25.0, u

2"35.0. (26)

By adopting the same least-squares scheme as before with a time step *t"0.001 the results given in Tables8 and 9 are obtained. The restoring forces of each element of the system and the estimated parameters timeevolution are shown in Figs. 4 and 5.

Free-vibrations analyses have also been carried out for the 2-DOF systems. The initial conditionsconsidered are the same for the both cases of non-linearity, namely x

1"0.1,x

2"0.1, x5

1"0, x5

2"0.

Unlike the forced vibrations case, the chosen sampling rate for the free-vibrating Du$ng oscillator is*t"0.003. The results of the latter analyses are shown in Tables 10 and 11.


Fig. 1. Du$ng oscillator; left: state-variable plots; right: estimated parameters.


Fig. 2. Van der Pol oscillator; left: state-variable plots; right: estimated parameters.


Table 5Free vibrations: estimated parameters of the non-linear oscillators

Du$ng Van der Pol

c( kK kK (3) c( kK c( (3)

Estimates 0.04 1.0 0.003 !0.2 1.0 0.2COV 2.2E-6 1.02E-7 5.9E-5 3.48E-6 1.4E-5 6.7E-6

Table 6Free vibrations: estimated parameters of the Van der Pol oscillator

c( kK c( (3)

Estimates !0.2 1.0 0.2COV 3.48E-6 1.4E-5 6.7E-6

Fig. 3. 2-DOF system with cubic non-linearities.

2.2. Systems with dry friction

The applications addressed in the present section deal with systems whose motion is resisted by friction.Such class of systems is of considerable interest in several engineering contexts. In particular, for machinerycomponents in relation to the contact problem and in earthquake protection of structures by sliding isolationsystems. A number of models have been proposed in the literature to describe the behavior of such systems.


Table 7Parameters of the 2-DOF non-linear system

m1

m2

c1

c2

k1

k2

c(3)1

c(3)2

k(3)1

k(3)2

Du$ng 12.0 12.0 21.0 60.0 7241 14900 * * 72.14 !223.0Van der Pol 12.0 12.0 21.0 60.0 7241 14900 50.0 150.0 * *

Table 8Forced vibrations: estimated parameters of the 2-DOF Du$ng system

c(1

c(2

kK1

kK2

kK (3)1

kK (3)2

Estimates 21.0 60.0 7241 14 900 72.14 !223.0COV 2.3E-5 2.2E-5 2.3E-6 3.1E-6 8.2E-6 !6.9E-5

Table 9Forced vibrations: estimated parameters of the 2-DOF Van der Pol system

c(1

c(2

kK1

kK2

c( (3)1

c( (3)2

Estimates 21.0 60.0 7241 14 900 50.0 150.0COV 4.6E-4 7.6E-5 2.1E-5 9.0E-6 2.9E-5 1.3E-4

In the following, attention will be focused on SDOF and 2-DOF systems with either constant Coulomb,linear/Coulomb and velocity-dependent friction. Analytical expressions for the steady-state solution toconstant Coulomb frictional oscillators under harmonic excitation is available in the literature [12]. Thatanalytical solution applies to the response characterized by at most two stops per cycle. Subsequent studieshave led to extended solutions allowing multiple stops per cycle [31] and transient response [22]. Morerecently techniques have been developed to replace the discontinuous friction force}sliding velocity relationwith continuous functions [28]. Such representation of the Coulomb friction force eliminates the need tokeep track of the stick}slip phases and their transitions. Moreover, the proposed continuous functions can beselected to be arbitrarily close to the exact discontinuous friction force. The identi"cation of systems with dryfriction has also received considerable attention. Parametric identi"cation of systems with constant dryfriction has been carried out through a time-series model [8]. It was accordingly demonstrated that AVD(acceleration, velocity and displacement) models can accurately predict the time response of both SDOF andMDOF systems using high-level excitation in order to avoid the occurrence of stick/slip motion. Anothermethod, based on free-vibration decrements was also proposed to identify Coulomb and viscous damping[13,21]. This method is based on the evolution of decaying peaks and valleys in the oscillatory responsewhich allows to isolate the viscous e!ect and then extract the Coulomb e!ect. The algorithm is applicable toSDOF systems and cannot account for other frictional characteristics than the constant Coulomb one.Aiming at identifying the linear viscous damping, the linear restoring force and the dry friction, the numericalinvestigations that are carried out in the following sections refer to systems experiencing both continuous andstick/slip motion.


Fig. 4. Du$ng oscillator; left: "rst element; right: second element.


Fig. 5. Van der Pol oscillator; left: "rst element; right: second element.


Table 10Free vibrations: estimated parameters of the 2-DOF Du$ng system

c(1

c(2

kK1

kK2

kK (3)1

kK (3)2

Estimates 21.0 60.0 7241 14900 72.12 !222.3COV 2.1E-6 1.13E-6 1.59E-7 1.46E-7 4.1E-3 !3.89E-2

Table 11Free vibrations: estimated parameters of the 2-DOF Van der Pol system

c(1

c(2

kK1

kK2

c( (3)1

c( (3)2

Estimates 21.0 60.0 7240.99 14 900 50.08 149.2COV 4.56E-5 1.99E-5 6.92E-6 2.12E-6 1.69E-2 6.25E-2

2.2.1. Constant Coulomb frictionThe equation of motion of the SDOF mass}spring}dashpot system with constant Coulomb friction

considered in this section is given by

mxK#cx5 #kx#kmg sign(x5 )"P, (27)

where m, c and k represent the mass, the viscous damping constant and the spring constant, respectively; thefrictional force is F

f"kmg sign(x5 ) where k is the sliding coe$cient of friction; P represent a generic

excitation which, for the harmonic case, takes the form P"P0

sin(ut#/). The Signum function is de"nedas

sign"G#1 for x5 (t)'0,

0 for x5 (t)"0,

!1 for x5 (t)(0.

(28)

The initial simulations address the SDOF free vibrations (P"0). In order to assess the accuracy of theestimated parameters, a comparison is carried out with the free-vibrations decrement method proposed by[21]. The numerical values considered have been chosen identical to an example reported in the above-mentioned work. Namely, m"1, c"0.2, k"10.0 and e"0.2, where e"kmg; the duration of the responseis ¹"12 s with *t"0.023 s. The initial conditions are x(0)"1.0, x5 "0.0. The reference numerical solutionsare based on a "fth-order Runge}Kutta method. The error analysis developed for the free-vibrationslogarithmic decrement method shows that the estimation error, which is directly proportional to themeasurement error, can be minimized by a clever choice of the number of extreme excursions m accumulatedfor the estimation process. Analogous reasoning can be applied to the current wavelet-based identi"cationalgorithm. Indeed, by considering that the number of equations l entering the least-squares solution governsthe number of time lags during which the di!erential equation approximation error is minimized, an analogybetween m and l can be found. Thus, the results shown in Table 12 are given for a choice of the parametersl (wavelet) and m (log-decrement). Fig. 6 shows the total restoring force of the identi"ed oscillator. The resultsof the simulation, while demonstrating the reliability of the log-decrement estimates, show a satisfactoryaccuracy of the wavelet-based estimates. The sti!ness parameter k appears only among the wavelet-basedidenti"cation results since the decrement method does not allow for the estimation of this parameter. Indeed,


Table 12Free vibrations: estimated parameters of the SDOF constant Coulomb oscillator

WAVELET-GALERKIN LOG-DECREMENT

l c( kK e( m c( e(

32 0.193 10.00 0.206 5 0.2 0.199

Fig. 6. Total restoring force of the SDOF oscillator with constant Coulomb friction under free vibration.

the Coulomb friction estimates (e( ) of the log-decrement method are obtained through the assumed knownsti!ness parameter. Before addressing cases characterized by the occurrence of multiple stops per cycle, somemodeling considerations are needed. As previously mentioned, the occurrence of stick/slip motion presentssigni"cant modeling di$culties. As a matter of fact, whenever a sticking phase occurs, the time evolution ofthe parameters estimates provided by the proposed identi"cation algorithm is characterized by suddenjumps. This phenomenon is due to the loss of uniqueness of the solution corresponding to the zero-velocityintervals. During the slipping phase, on the other hand, accurate estimates of the time evolution are attained.In order to overcome the above di$culties an alternative mathematical representations of Coulomb frictionis used [28]. In particular, the discontinuous Signum function in Eq. (28) can be substituted by any of thefollowing four continuous functions:

f1(a

1,x5 )"Erf(a

1x5 ), f

2(a

2,x5 )"Tanh(a

2x5 ),

f3(a

3,x5 )"

2

pArcTan(a

3x5 ), f

4(a

4, x5 )"

a4x5

1#a4Dx5 D

. (29)

The Signum function can be represented to any desired level of accuracy by increasing the value of theparameters a

i, i"1, 2, 3, 4. Fig. 7 shows the comparison between the function sign(x) and the function

f2(a

2, x) for increasing values of the parameter a

2. A test on the norm of the error between the true solution of

Eq. (27) and the approximated one obtained by using the expressions (30), for the same value of the factors ai,

has been carried out. The minimum error has been obtained by using the function f2(a

2, x5 ), which will thus be

used throughout the following simulations with values of a2

chosen according to the problem at hand.


Fig. 7. Signum function representation.

The next investigation is devoted to assess the reliability of the proposed identi"cation method for a SDOFsystem with constant Coulomb friction subjected to harmonic excitation. The case studied refers to anexample of motion with two stops per cycle [22]. In particular, the values of the parameters are:m"1, c"0.5, k"25.0 and e"0.2. The harmonic excitation is given by P(t)"P

0sin(ut) with P

0"1.0 and

u"0.5. The coe$cient of the function f2(a

2,x5 ) representing the Signum function is a

2"1500; Figs. 8 and

9 show the accuracy of the alternative representation in terms of, respectively, total restoring force and statevariables. The estimated parameters are given in Table 13.

A 2-DOF system with viscous damping and constant Coulomb friction is studied next. The systemconsidered has been taken from the work of [8] and is shown in Fig. 10. The equations of motion of thesystem are given by

m11

xK1#c

11x51#e

1sign(x5

1)#k

12x2"p

1(t),

m22

xK2#c

22x52#e

2sign(x5

2)#k

21x1"p

2(t), (30)

where p1(t)"p

01sin(u

1t) and p

2(t)"p

02sin(u

2t) are the forces applied at the "rst and second mass,

respectively, while

m11

"m1"1.0N s2/4, m

22"m

2"1.0N s2/4,

k11

"k1#k

2"20 000 N/m, k

12"!k

2"!10 000N/m,

k22

"k2#k

3"20 000 N/m, k

21"!k

2,

c11

"c1"10.0N s/m, c

22"c

2"c

1,

e1"e

2"1.0N. (31)

In the cases studied the same level of excitation adopted in the above cited reference have been used. Namely,p01

"p02

"7.5, u1"105.0 and u

2"95.0. The results are illustrated in Table 14.


Fig. 8. Restoring force obtained by replacing the Signum function with f2(a

2, x5 ).

Fig. 9. Displacement and velocity responses obtained by replacing the Signum function with f2(a

2, x5 ).


Table 13Forced vibrations: estimated parameters of the SDOF oscillator with constant Coulomb friction

c( kK e(

Estimates 0.507 25.0 0.2COV 2.6E-1 9.45E-5 8.24E-4

Fig. 10. 2-DOF system with viscous and constant Coulomb friction.

Table 14Forced vibrations: estimated parameters of the 2-DOF system with constant Coulomb friction

c(11

kK11

c(22

kK22

kK12

e(1

e(2

Estimates 10.14 1995.8 10.09 20 018.7 !1012.0 0.97 0.964COV 8.26E-2 1.26E-2 7.75E-2 1.34E-2 !2.46E-2 8.65E-2 7.6E-2Error (%) 1.3 0.2 0.89 0.09 0.12 3.0 3.6

2.2.2. Linear/Coulomb and velocity-dependent frictionRecent research on the isolation of structures against earthquake motions has motivated the development

of other mathematical models of friction than the constant Coulomb model. For instance, an energyabsorbing base isolation system (displacement proportional friction) providing increasing frictional resist-ance has been investigated [36]. Such resistance is created by the tightening together of a set of friction platesby compressive forces proportionally increasing with the relative separation of a building and its foundation.In order to model the above-described systems, a linear/Coulomb oscillator with additional viscous dampingundergoing a periodic motion was considered. This type of oscillator was further investigated and analyticalsolutions were derived for its transient response [22]. The motion of a practical oscillator characterized byvelocity-dependent friction for the cases of free and harmonic forced vibration was also analyzed [23]. Themodel investigated in the latter reference provides for a coe$cient of friction increasing linearly with velocityup to a critical value beyond which it remains constant. This simpli"ed bilinear model derives from researchon sliding isolation systems utilizing Te#on}steel interfaces [9,27], where it was indicated that Te#on}steelinterfaces cannot be modeled by constant Coulomb friction, but rather by a coe$cient of friction dependingon both bearing pressure and velocity of sliding. A model was proposed that is characterized by a sliding


Fig. 11. Linear/Coulomb oscillator; approximation introduced by replacing the Signum function with f2(a

2,x5 ).

coe$cient of friction dependent exponentially on the velocity of sliding and varying between a minimum anda maximum value, the latter value being reached at large velocities of sliding. In what follows severalnumerical investigations addressing oscillators featuring the above-described frictional properties are pre-sented. As far as the linear/Coulomb model is concerned, the frictional force entering the equation of motion(27) can be expressed as [22]

Ff"(F#fkDxD) sign(x5 ), (32)

where k(x)"F#fkDxD represents the displacement-dependent sliding coe$cient of friction, fk is the slope ofthe increasing frictional force and F represents the portion of load carried by the device. Based on theknowledge of both the response, in terms of displacement x and velocity x5 , and the excitation, thewavelet-based identi"cation algorithm is applied in order to estimate the viscous damping c, the sti!nessk and frictional force parameters F and fk. The values of these parameters considered in the simulations are,m"1.0, c"0.5, k"25.0, the amplitude of the sinusoidal excitation is P

0"1.0 and e"F/P

0"0.2.

Denoting the ratio between the driving frequency u and the natural frequency u0

by b, two cases are studiedcorresponding to b"1.2 and 0.6. These two values give rise, respectively, to a continuous motion and toa motion with multiple stops per cycle of the considered oscillator. Moreover, for each value of b, twodi!erent slopes of the frictional force are studied, namely fk"5.0 (f"0.2) and 10.0 (f"0.4). In the modelreferring to the motion with multiple stops per cycle, the Signum function is replaced by Tanh(a

2x5 ) with

a2"200. In Fig. 11 the displacement and velocity time histories obtained by adopting the two functions are

shown. The estimated parameters are shown in Table 15. Next, Fig. 11 shows a comparison between themodels frictional forces and the estimated ones as functions of the displacement.


Table 15Forced vibrations: estimated parameters of the SDOF linear/Coulomb oscillator

b"1.2 b"0.6

f"0.2 f"0.4 f"0.2 f"0.4

c( 0.525 0.581 0.438 0.447COV 0.220 0.483 0.152 0.42kK 25.015 25.119 25.033 25.030COV 6.6E-3 1.26E-2 2.69E-3 9.87E-3e( 0.189 0.172 0.218 0.218COV 0.277 0.598 0.095 0.331fkK 5.101 10.337 4.778 9.787COV 0.106 0.129 0.058 0.097

The last numerical investigations concerning systems with dry friction refer to the case of a coe$cient offriction obeying the following equation [9],

k(x5 )"f.!9

!Df exp(!aDx5 D). (33)

In the latter expression f.!9

represents the coe$cient of friction at large velocity of sliding, Df is the di!erencebetween the maximum and minimum values of k and a is constant for a given bearing pressure and conditionof interface. The resulting equation of motion, assuming the Coulomb model, is given by

mxK#cx5 #kx#k(x5 )mg sign(x5 )"P0

sin(ut#/). (34)

As already pointed out, the latter equation has been adopted to model structures equipped with a slidingisolation system utilizing Te#on}steel interfaces. Consequently, the identi"cation refers to an oscillator withfriction corresponding to Te#on}steel interfaces under a known condition of pressure. In particular, thepressure considered is 6.9MPa and the associated value of the parameter a is equal to 0.0236 s/mm. Based onthe notation so far adopted, the following parameters have been selected for the numerical investigationwhose results are shown in Table 16: m"0.77, c"0.0, k"9.870, P

0"1.0, e"f

.!9g/P

0"1.17,

d"!Dfg, u0"n, b"0.5. As illustrated in Fig. 12, the response shows sticking intervals, thus an

approximated model with the function Tanh(a2x5 ) with a

2"1000 has been adopted. The same "gure

demonstrates the agreement between the responses obtained using the approximated continuous functionand the discontinuous one. The frictional force}displacement loops of both the actual model and theestimated one are also presented in Fig. 13.

3. Non-linearity characterization

The identi"cation scheme developed so far has been applied to systems whose behavior was assumed to beappropriately described by a known model. However such information about a given system of interest isseldom available. It is therefore important to develop identi"cation procedures that permit the characteriza-tion of a given system under conditions of incomplete knowledge. In this section the problem of selecting themost appropriate model with assumed non-linear terms among a list of over and under-speci"ed models isaddressed. The general structure of the algorithm developed in the previous sections does not restrict theform of the non-linearity. In particular, the functional form of the non-linearity could be a linear combinationof a number of forms that are likely to occur in real structures. Therefore the characterization task can betackled by introducing a criterion capable of discerning the most likely model among a set of plausible ones.


Table 16Forced vibrations: estimated parameters of the SDOF system with sliding friction exponential law

c( kK e( dK

Estimates 2.48E-3 9.87 1.17 0.909COV 1.6E-2! 3.82E-4 1.23E-3 2.07E-3

!Standard deviation p.

Fig. 12. Velocity-dependent friction; approximation introduced by replacing the Signum function with f2(a

2, x5 ).

A normalized mean-square error (MSE) can be used as a measure of `the goodness of "ta between theestimated acceleration response and the simulated or experimental one either under the same excitation orduring free vibrations. This error criterion is de"ned as [6,17,26]

MSEi"C

+Nj/1

(xKij!xK(

ij)2

Np2xK i

D]100 for i"1,2, M, (35)

where the M and N are, respectively, the number of degrees of freedom and the number of data points.Several numerical investigations have been carried out in order to assess the reliability of such an errorcriterion when intentionally wrong models are used. For each type of non-linear model indicated in theleftmost column of Tables 17 and 18 are reported the MSE values obtained for SDOF and 2-DOF systems,


Fig. 13. Comparison between estimated and model frictional force.

Table 17MSE for the SDOF non-linear oscillators

True Linear 7 Terms

(1) Du$ng 2.2E-10 4.66 1.31E-8(2) Van der Pol 2.89E-7 87.13 0.610Coulomb 1.18E-3 2.747 1.1553(1)#(2) 5.63E-8 76.75 0.057

Table 18MSE for the 2-DOF non-linear oscillators

True Linear 10 Terms! 10 Terms"

1 2 1 2 1 2 1 2

(1) Du$ng 4.8E-10 1.6E-10 7.5 1.22 3.43E-8 1.21E-8 3.7E-7 1.4E-7(2) Van der Pol 3.6E-9 1.8E-9 25.5 6.07 5.45E-7 1.8E-7 3.09E-4 1.7E-4Coulomb 1.56E-6 7.07E-7 8.27 1.74 1.27E-2 7.55E-3 * *

(1)#(2) 1.1E-9 2.17E-10 8.44 1.37 6.45E-7 1.31E-7 2.9E-7 1.6E-7

!Terms including Coulomb non-linearities."Terms including square-law non-linearities.


respectively. The MSE indices reported under the columns `truea have been obtained by comparing theaccelerations of a reference model that actually generated the data and the accelerations of the estimatedmodels with the non-linear terms matching those of the reference model. The error associated withunder-speci"ed models is next presented under the column `lineara. In this case the MSE values refer to thecomparison between the actual non-linear models and estimated models in which only the linear terms wereretained. Di!erent over-speci"ed models have also been investigated. For the SDOF models in Table 17a comparison has been carried out between the actual non-linear models and estimated models characterizedby the following terms (`7 Termsa): k, c, k(2), c(2), k(3), c(3) and e (see Eqs. (3) and (4)). For the 2-DOF cases,two sets of non-linearities combinations over-specifying the estimated models have been considered. The `10Terms*a heading in Table 18 refer to the set of terms: k

1, c

1, k

2, c

2, k(3)

1, c(3)

1, k(3)

2, c(3)

2, e

1and e

2. The `10

Terms**a refer to: k1, c

1, k

2, c

2, k(2)

1, k(2)

2, k(3)

1, c(3)

1, k(3)

2and c(3)

2. The last rows in both tables refer to

a reference model with both Du$ng and Van der Pol non-linearities. The indices 1 and 2 appearing in each2-DOF estimated model case refer to the "rst and second degree of freedom, respectively (see Fig. 3). Theresulting MSE values show that the MSE minima always occur when the right model is chosen. The leastdi!erence between the MSE indices for true and over-speci"ed models is of two orders of magnitude. Thelarge values obtained for the under-speci"ed cases are clear signs of the presence of non-linearities. Moreover,the computed MSE indices are a clear indication as to the relative in#uence of the considered non-linearities.

4. Conclusions

The parametric identi"cation of zero-memory non-linear systems has been addressed in this work.A procedure based on the Wavelet}Galerkin solution of the governing equations of motion were imple-mented to estimate the parameters of models of non-linear systems from input}output data. The procedure iscapable of handling arbitrary forms of non-linearities. In the absence of prior information on the functionalform of the non-linear model, a selection criterion is provided to characterize the non-linearity from amonga class of plausible models. Systems with cubic non-linearities and systems whose motion is resisted byfriction, experiencing both continuous and stick/slip motion, have been addressed. Accurate parametersestimation has been achieved under both free and forced vibrations. The encouraging results presented havebeen obtained through ideal numerical investigations based on noiseless data.

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A Wavelet Based Approach for Model and Parameter Identification of Nonlinear Systems

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