Past, present and future of nonlinear system identification in structural dynamics

doi:10.1016/J.YMSSP.2005.04.008

Journal Article•DOI•

Past, present and future of nonlinear system identification in structural dynamics

Gaëtan Kerschen¹, Keith Worden², Alexander F. Vakakis³, Alexander F. Vakakis⁴, Jean Claude Golinval¹ - Show less +1 more•Institutions (4)

University of Liège¹, University of Sheffield², University of Illinois at Urbana–Champaign³, National Technical University of Athens⁴

01 Apr 2006-Mechanical Systems and Signal Processing (Academic Press Ltd Elsevier Science Ltd)-Vol. 20, Iss: 3, pp 505-592

TL;DR: In this article, a review of the past and recent developments in system identification of nonlinear dynamical structures is presented, highlighting their assets and limitations and identifying future directions in this research area.

read less

About: This article is published in Mechanical Systems and Signal Processing.The article was published on 2006-04-01 and is currently open access. It has received 1000 citations till now. The article focuses on the topics: Nonlinear system identification & System identification.

...read moreread less

Summary (11 min read)

Jump to: [Review] – [2.1. Dynamics of nonlinear oscillations] – [2.2.1. Dynamics of the undamped system] – [Linear oscillator] – [2.2.2. Dynamics of the weakly damped system] – [3. Nonlinear system identification in structural dynamics: a literature review] – [3.1. By-passing nonlinearity: linearisation] – [3.2. Time-domain methods] – [3.3. Frequency-domain methods] – [3.4. Modal methods] – [3.5. Time-frequency analysis] – [3.6. Black-box modeling] – [3.7. Structural model updating] – [4.1. Literature review] – [4.2. An example of nonlinearity detection: the Hilbert transform 4.2.1. Theory] – [4.2.2. Assessment] – [5.1. Literature review] – [5.1.1. The location of the nonlinearity] – [5.1.2. The type of the nonlinearity] – [5.1.3. The functional form of the nonlinearity] – [5.2. An example of nonlinearity characterisation: the restoring force surface method] – [6. Parameter estimation in the presence of nonlinearity: established methods] – [6.1.2. Application example] – [6.2. Direct parameter estimation and restoring force surfaces] – [6.2.1. Theory] – [6.2.2. Assessment of the RFS and DPE methods] – [6.3. NARMAX modeling 6.3.1. Theory] – [6.3.2. Assessment] – [6.4.2. Assessment] – [6.5.1. Theory] – [6.5.2. Assessment] – [6.6.2. Assessment] – [7. Parameter estimation in the presence of nonlinearity: recent methods] – [7.1. The conditioned reverse path method] – [7.1.1. Theory] – [7.1.2. Application example] – [7.1.3. Assessment] – [7.2. The nonlinear identification through feedback of the output method] – [7.2.1. Theory] – [7.2.2. Assessment] – [7.3.2. Assessment] – [7.4.2. Application example] – [7.4.3. Assessment] and [8. Summary and future research needs]

Review

It follows that the demand to utilise nonlinear (or even strongly nonlinear) structural components is increasingly present in engineering applications.
The substantiation that a model within its domain of applicability possesses a satisfactory range of accuracy consistent with the intended application of the model (Schlesinger et al., 1979 [61] ).
The subject of nonlinear dynamics is extremely broad, and an extensive literature exists.

2.1. Dynamics of nonlinear oscillations

Even though at sufficiently small-amplitude motions nonlinearity may not 'perturb' significantly the linear dynamics, when the energy of the motion increases, stiffness, inertial and/or damping nonlinearities may introduce dynamical phenomena that are radically different than those predicted by linear theory.
What makes nonlinear dynamics challenging to analyse and model is the well-known result that the principle of linear superposition does not apply to nonlinear systems.
This alternative definition enables the direct extension of the concept of NNM to damped oscillators, although extension of 'undamped'.
In general, bifurcations of equilibrium positions or periodic orbits of nonlinear systems are the source of additional distinctively nonlinear features in the dynamics.
Considering the steady state responses of nonlinear systems to harmonic excitations, IRs or RCs lead to forced resonances, which, apart from the case of fundamental resonance (i.e., strong steady response at the frequency of the external excitation), have no counterparts in linear dynamics: subharmonic, superharmonic, combination, or autoparametric resonances.

2.2.1. Dynamics of the undamped system

Through this example, the objective is to demonstrate that even nonlinear systems of very simple configuration can possess surprisingly complicated and rich dynamics.
A more detailed discussion of the dynamics of this system can be found in (Kerschen et al., 2005b [53] ; Lee et al., 2005 [97] ), where additional dynamical features such as passive energy transfer are studied.
This convention holds for every branch except S11AE, which, however, are particular branches because they form the basic backbone of the entire plot.
Four elements of the frequency-energy plot are described in what follows: (a) the backbone of the plot; (b) the branches of symmetric solutions; (c) the branches of unsymmetric solutions and (d) the special orbits.
There is a sequence of higher-and lower-frequency periodic solutions bifurcating or emanating from branches S11AE, which are denoted as tongues.

Linear oscillator

The two-DOF system with essential stiffness nonlinearity.
Similar results hold for the other S-branches.
Periodic motions on the U-tongues are not NNMs because non-trivial phases between the two oscillators are realised.
Close-up of S13AE branch in the frequency index-logarithm of energy plane; the special periodic orbit is represented by triple stars ð ÃÃÃ Þ; at certain points of the branch the corresponding motions in the configuration plane ðy; vÞ are depicted (Lee et al., 2005 [97] ). whereas motion on S-tongues corresponds to one-dimensional curves.
If the system initially at rest is forced impulsively, and if one of the stable, localised special orbits is excited, the major portion of the induced energy is channeled directly to the invariant manifold of that special orbit, and, hence, the motion is rapidly and passively transferred from the linear to the nonlinear oscillator.

2.2.2. Dynamics of the weakly damped system

This section intends to demonstrate that the intricate structure of NNMs of Fig. 9 can lead to complicated transient responses of the corresponding weakly damped system.
When viewed from such a perspective, one can systematically interpret the complex transitions between multi-frequency modes of the transient, weakly damped dynamics by relating them to the different branches of NNMs in the frequency-energy plot.
As energy decreases even further there occurs escape from RC, and the motion evolves along branches S13, S15, S17, etc.
The numerical evidence of these findings is given in Fig. 12e ,f which depicts the dominant harmonic components of the measured displacements computed using a wavelet transform.
The results of this example show that even low-dimensional nonlinear oscillators with relatively simple configuration may possess very complicated and rich dynamics.

3. Nonlinear system identification in structural dynamics: a literature review

Nonlinear structural dynamics has been studied for a relatively long time, but the first contribution to the identification of nonlinear structural models date back to the 1970s (Ibanez, 1973 [100] ; Masri and Caughey, 1979 [101] ).
Part of the reason for this shift in emphasis is the increasing attention that this research field has attracted, especially in recent years.
For clarity, the methods are classified according to seven categories, namely by-passing nonlinearity: linearisation, time and frequencydomain methods, modal methods, time-frequency analysis, black-box modeling and structural model updating.
The textbook (Worden and Tomlinson, 2001 [67] ) is a reference book for anyone conducting tests on nonlinear structures and then constructing a dynamic model of the system as discussed in (Singh, 2004 [107] ) (its companion is (Ewins, 2000 [11] ), dedicated to linear structures).

3.1. By-passing nonlinearity: linearisation

The curve-fitting algorithms used will associate a linear system with each FRFin some sense the linear system which explains it best.
As the nonlinear system FRF will usually change its shape as the level of excitation is changed, any linearisation is only valid for a given excitation level.
Hagedorn and Wallaschek (1987 [116] ) have developed an effective experimental procedure for doing precisely this.
Nonlinear system identification is carried out in (Rice, 1995 [118] ) by comparing an experimentally derived equivalent linear model to the one derived directly from the assumed equation of motion.
The harmonic balance method described in (Nayfeh and Mook, 1979 [28] ) can be employed for linearising nonlinear equations of motion with harmonic forcing.

3.2. Time-domain methods

A method is said to be a time-domain method if the data considered during the identification process take the form of time series (e.g., force and acceleration).
The NARMAX structure is general enough to admit many forms of model including neural networks although the estimation problem becomes nonlinear and the orthogonal estimator will not work (Billings et al., 1991a [166] ).

3.3. Frequency-domain methods

A wide range of frequency-domain methods have been proposed in the technical literature during the last two decades.
The first major application in the field of structural dynamics occurred with the work of Gifford (1989 [190] ).
Higher-order spectra have also received some consideration for system identification (Bendat, 1998 [108] ).
Other early contributions to nonlinear system identification using frequency-domain data are those of Yasuda and co-authors (1988a,b [127, 128] ) in which the harmonic balance method is used in an inverse way to estimate parameters.
A second alternative referred to as the conditioned reverse path (CRP) method was presented in (Richards and Singh, 1998 [222] ) and is exposed in Section 7.1.

3.5. Time-frequency analysis

A typical feature of nonlinear vibrations is that the instantaneous natural frequency and damping coefficient of the system may become functions of time depending upon the type of nonlinearity.
Time-frequency analysis is also suitable for the analysis of nonlinear oscillations.
Quadratic representations which include the Wigner-Ville distribution and the Cohen-class of distributions have also received some attention (Feldman and Braun, 1995 [277] ; Wang et al., 2003a [278] ).
A different approach using the wavelet transform for nonlinear system identification consists in expanding the system response and excitation in terms of scaling functions (in (Kitada, 1998 [283] )), it is the tangent stiffness of the structural system which is expressed as a series expansion of wavelets).

3.6. Black-box modeling

One major difficulty of nonlinear system identification is that the functional S½ which maps the input xðtÞ to the output yðtÞ, yðtÞ ¼ S½xðtÞ, is generally unknown beforehand.
The most comprehensive programme of work to date is that of Billings and co-workers starting with (Billings et al., 1991a [166] ) for the multi-layer perceptron structure and (Chen et al., 1990b [292] ) for the radial basis function network.
In (Fan and Li, 2002 [303] ), a hybrid approach that embeds neural networks to represent unknown nonlinearities in a otherwise typical physical model is developed.
First and foremost, the identified model parameters do not provide physical information about the structure, which may limit the practical usefulness of the resulting model.

3.7. Structural model updating

For the investigation of more complex structures in a wider frequency range, resorting to models with many DOFs is inevitable.
Very often, the initial model is created using the finite element method (see, e.g., (Zienkiewicz, 1977 [308] )), and structural model updating is termed finite element model updating.
In (Kyprianou, 1999 [317] ; Kyprianou et al., 2001 [318] ), model updating is realised through the minimisation of an objective function based on the difference between the measured and predicted time series.
Several efforts have been made in order to define features (i.e., variables or quantities identified from the structural response that give useful insight into the dynamics of interest) that facilitate correlation.
This strategy aims at reducing the number of computer simulations required during optimisation while maintaining the pertinent characteristics of the problem.

4.1. Literature review

According to the scheme in Fig. 4 , the detection of structural nonlinearity is the first step toward establishing a structural model with a good predictive accuracy.
It is arguably the most often-used test, by virtue of the fact that almost all the commercial spectrum analysers allow its calculation; however, it does not distinguish between the cases of a nonlinear system and noisy signals.
A more sophisticated diagnostic tool introduced in (Simon and Tomlinson, 1984 [334] ) is provided by the Hilbert transform.
H½ which diagnoses nonlinearity on the basis of measured FRF data.
Recently developed techniques include the use of autocorrelation functions of residuals from overdetermined FRF calculations (Adams and Allemang, 2000b [345] ) and the use of multisine excitations (Vanhoenacker et al., 2001 [346] ; Verboven et al., 2005 [347] ).

4.2. An example of nonlinearity detection: the Hilbert transform 4.2.1. Theory

The Hilbert transform diagnoses nonlinearity on the basis of measured FRF data.
This is illustrated in the Nyquist plot of Fig. 17 in the case of a hardening cubic stiffness; the characteristic circle is rotated clockwise and elongated into a more elliptical form.
By making use of the parity of the real and imaginary parts of the FRF, the Hilbert transform can be recast in a slightly different form to that described above EQUATION EQUATION ).
An alternative approach establishes the position of the FRF poles in the complex planes and forms the decomposition (8) .
The data was truncated by removing data above and below the resonance leaving 151 spectral lines in the range 9.25-32:95 Hz.

4.2.2. Assessment

Computation is fairly straightforward for a baseband FRF, but complications can arise for zoomed FRFs.
An appealing feature of the Hilbert transform is that the form of the distortion observed for a nonlinear system FRF can give some insight into the qualitative form of the nonlinearity.
Perhaps the main limitation of the Hilbert transform is associated with all detection methods which look for distortion in a measured quantity; it is that there is currently no established technology to determine if the deviations observed in the FRF are statistically significant.
A further (probably minor) concern is that the Hilbert transform does not strictly detect nonlinearity, but non-causality.
It is not established beyond doubt that all nonlinear systems have noncausal FRFs (in the sense that their corresponding impulse responses have support for negative times) and this means that there may exist classes of nonlinear systems which the Hilbert transform would not detect.

5.1. Literature review

Since the paper mainly focuses on parameter estimation techniques, the relevant methods for nonlinearity characterisation will solely be cited or very briefly described.
In fact, this research topic could deserve its own survey paper.
A nonlinear system is said to be characterised when the location, type and functional form of all the nonlinearities throughout the system are determined.
Without a precise understanding of the nonlinear mechanisms involved, the identification process is bound to failure.
This is evidenced in (Malatkar and Nayfeh, 2003c [354] ) in which a simple cantilever plate may exhibit 2:1 and 3:1 internal resonances, external combination resonance, energy transfer between widely spaced modes, period-doubled motions and chaos.

5.1.1. The location of the nonlinearity

The spatial localisation of local nonlinearities is the first step in the characterisation process.
The literature on this topic is not so extensive because nonlinearities may often be located easily, at least for simple structures.
Some information may be gleaned by studying FRFs at various excitation levels and examining the deformation shapes of the modes which are most corrupted by the nonlinear response; nonlinearities may be assumed where the relative displacements of these mode shapes are the largest.

5.1.2. The type of the nonlinearity

Nonlinearity classification is useful for determining the type of the nonlinearity.
The form of the distortion introduced during a Hilbert transform of the FRF can also be characteristic of the type of nonlinearity (Simon and Tomlinson, 1984 [334] ; Worden and Tomlinson, 2001 [67] ).
The hardening or softening characteristic of the system can be easily deduced from the backbone curve which depicts the frequency as a function of the free vibration envelope.
For symmetrically distributed inputs and restoring forces with only odd terms, the output has a symmetric distribution and consequently a zero bispectrum.

5.1.3. The functional form of the nonlinearity

A priori knowledge and nonlinearity classification may help to select a reasonably accurate model of the nonlinearity.
The number of possible terms increases rapidly with the polynomial order, but most often not all terms in the expansion have a significant contribution to the restoring force.
The key advantage of the method is that a collection of potential models together with their posterior probability is obtained instead of the single best model; this allows for more flexibility in deciding the most appropriate model of the nonlinearity.

5.2. An example of nonlinearity characterisation: the restoring force surface method

The RFS procedure which is described in Section 6.1 has been applied to the characterisation and identification of automotive shock absorbers in a number of publications (see Section 3.2).
The data was obtained by FIAT engineers using the experimental facilities of the vehicle test group at Centro Ricerche FIAT, Torino.
Briefly, data was recorded from an absorber which was constrained to move in only one direction in order to justify the assumption of SDOF behaviour.
The base was then excited harmonically using a hydraulic actuator.
The surfaces from the tests at other frequencies showed qualitatively the same characteristics, i.e., a small linear stiffness and a bilinear damping.

6. Parameter estimation in the presence of nonlinearity: established methods

An important assumption which conditions the success of parameter estimation is that all the nonlinearities throughout the system have been properly characterised.
Several established methods for parameter estimation in the presence of nonlinearity are described.
The basic procedures were introduced by Masri and Caughey (1979 [101] ), although the approach described here resembles more the variant developed independently by Crawley and Aubert (1986 [133] ) and Crawley and O'Donnell (1986 [132] ) and referred to as force-state mapping.
In the first case, direct sampling of the displacement, velocity and acceleration data requires considerable instrumentation.

6.1.2. Application example

The acceleration shown in Fig. 20b was measured at the beam tip, and the displacement and velocity were deduced from this signal using integration and filtering procedures.
By looking for the minimum value of the normalised mean-square error (NMSE (Allen, 1971 [390] )) between the predicted and measured restoring force for variations of the clearance d, this parameter could have been identified; the smallest NMSE was 1.80%, which is the sign of an accurate identification.

6.2. Direct parameter estimation and restoring force surfaces

The procedure therefore required a priori estimates of the modal matrix and mass matrix.
Al-Hadid and Wright showed that unless a time-consuming iterative version of the procedure was adopted, any model parameters would be biased.
Research continued, and in Al-Hadid and Wright (1989, [138] ) a useful form of the identification procedure was obtained by utilising a physical coordinate representation for the nonlinear forces while retaining a modal coordinate approach to the underlying linear system.

6.2.1. Theory

If the masses are displaced and released, they are restored to equilibrium by internal forces in the links.
This means that coefficients for both equations are known up to the same scale m 2 .
If this were not true the system would fall into two or more disjoint pieces.
Yang and Ibrahim (1985 [135] ) observed that if the unforced equations of motion are considered, the required overall scale can be fixed by a knowledge of the total system mass; i.e., all system parameters can be obtained from measurements of the free oscillations.
The DPE scheme has also been implemented for distributed systems in (Liang and Cooper, 1992 [396] ).

6.2.2. Assessment of the RFS and DPE methods

They are essentially appealing because of their simplicity and efficiency for identification of SDOF systems or nonlinearity localised between two DOFs.
These methods offer a convenient means of determining the functional form of the nonlinearity through the visualisation of the RFS as shown in Section 5.2.
A characterisation of the elastic and dissipative forces can be obtained by taking a cross section of this three-dimensional plot along the axes where either the velocity or the displacement is equal to zero, respectively.
A difficulty lies in the need for numerical differentiation or integration which may introduce errors in the estimation of signals; careful signal processing is required.

6.3. NARMAX modeling 6.3.1. Theory

Suppose one is interested in the SDOF linear system EQUATION.
One can pass to the discrete-time representation EQUATION ) Model ( 42) is now termed a NARX (Nonlinear ARX) model.
Suppose the measured output has the form EQUATION where y c ðtÞ is the 'clean' output from the system.
As above, the estimated outputs must be compared with the measured outputs, with good agreement a necessary condition for accepting the model.

6.3.2. Assessment

The NARMAX modeling technique is very versatile.
If the method has a disadvantage, it is that the models do not directly give insight into the physics of the system being modeled, although it is possible to pass to a continuous-time model from the NARMAX model by using the HOFRFs.
Further, the transform commutes with differentiation, so EQUATION Adding (59) and iÂ (69) yields a differential equation for the analytic signal Y, i.e., EQUATION or, the quasi-linear form EQUATION Now, the derivatives € Y and _ Y are known functions of A and o by ( 67) and (68) .
The inverse of the latter mapping AðoÞ is sometimes referred to as the backbone curve of the system.

6.4.2. Assessment

The method described above is one of the most successful approach to tracking the varying nature of vibration of a large class of nonlinear systems.
It is only truly suitable for monocomponent signals, i.e., those with a single frequency dominant.
The authors mention that a method for the decomposition of signals with multiple components into a collection of monocomponents signals is proposed in Huang et al. (1998 [176] ), which may extend the applicability of the method.
5. The Volterra series and higher-order frequency response functions.

6.5.1. Theory

For a general linear system, the input-output map can be expressed by Duhamel's integral EQUATION Eq. ( 79) is manifestly linear and therefore cannot hold for arbitrary nonlinear systems.
The use of the Volterra series in dynamics stems from the seminal paper of Barrett (1963 [405] ) in which the series was applied to nonlinear differential equations for the first time.
The method of harmonic probing was introduced by Bedrosian and Rice (1971 [168] ) specifically for systems with continuous-time equations of motion.
The HOFRFs are calculated and plotted using the FREP package of Tsang and Billings (1988 [418] ).
As an aside, the first-order FRF has been used in system identification for some time; the well-established technique of modal analysis (Ewins, 2000 [11] ) is based on the extraction of linear system parameters by curve-fitting to the FRF.

6.5.2. Assessment

The kernel transforms-the HOFRFs-provide an attractive means of identifying and interpreting interactions between input frequencies and give a visualisation equivalent to the Bode plot for a linear system.
One disadvantage of this visualisation is that the HOFRFs are objects with higher dimension than 3 for kernel orders higher than 3 and therefore one can only inspect lower-dimensional projections.
A related problem is that the radius of the convergence of the series may be restricted or that low-order truncations may not be accurate.
H where S yy ðoÞ, S xx ðoÞ and S yx ðoÞ contain the PSD of the response (e.g., acceleration signal), the PSD of the applied force and the cross-PSD between the response and the applied force, respectively.
Reverse path spectral methods were therefore introduced to accommodate the presence of nonlinearity.

6.6.2. Assessment

The reverse path formulation can solve nonlinear problems that can be modeled by means of a nonlinear differential or integrodifferential equation of motion in many fields because it is valid for random data with arbitrary probability, correlation and spectral properties (Bendat, 1998 [108] ).
In addition, identification is carried out through simple mathematical operations, and the results are in a form that is convenient to interpret.
A possible drawback of the method for MDOF systems (which has been removed in the CRP method) is that the excitation must be applied at the location of the nonlinearity in order to identify its coefficient.

7. Parameter estimation in the presence of nonlinearity: recent methods

Several recent methods for parameter estimation in the presence of nonlinearity are described.
These methods show promise for identification of more complex nonlinear structures because they are inherently capable of dealing with MDOF systems.
Numerical and/or experimental examples are presented to illustrate their fundamental concepts but also their assets and limitations.

7.1. The conditioned reverse path method

To address the problem of the location of the external force inherent in the reverse path method, the CRP method developed in (Richards and Singh, 1998 [222] ) employs spectral conditioning techniques to remove the effects of nonlinearities before computing the FRFs of the underlying linear system contained in the dynamic compliance matrix HðoÞ ¼ B À1 ðoÞ.
The key idea of the formulation is the separation of the nonlinear part of the system response from the linear part and the construction of uncorrelated response components in the frequency domain.
The nonlinear coefficients are estimated during the second phase of the method.

7.1.1. Theory

It follows from Eq. ( 120) that the dynamic compliance matrix H which contains the FRFs of the underlying linear system takes the form EQUATION ) Estimation of the nonlinear coefficients:.
To address this problem, the ordinary coherence function has been superseded by the cumulative coherence function g.
It indicates the contribution from the linear spectral component of the response of the ith signal.

7.1.2. Application example

The identification was carried within the range 0-500 Hz in which three structural modes exist.
This structure was also investigated within the framework of the European COST Action F3 (Golinval et al., 2003 [104] ).
As expected, the imaginary part of the coefficient is two orders of magnitude below the real part and can be safely neglected.

7.1.3. Assessment

It turns out that the CRP method is a very appealing and accurate method for parameter estimation of nonlinear structural models.
In addition, the cumulative coherence is a valuable tool for the characterisation of the nonlinearity.
The formulation of the method is such that it targets identification of MDOF systems, which enabled the identification of a numerical model with 240 DOFs and two localised nonlinearities (Kerschen and Golinval, 2005a [223] ).
A finite element model of the underlying linear structure is built from the knowledge of the geometrical and mechanical properties of the structure and is updated using linear model updating techniques based upon FRFs (Arruda and Santos, 1993 [422] ; Balmes, 1993 [423] ; Lin and Ewins, 1994 [424] ).

7.2. The nonlinear identification through feedback of the output method

The NIFO formulation (Adams and Allemang, 2000a [232] ) is a recent spectral approach for identification of MDOF nonlinear systems.
As for the CRP method, the central issue is to eliminate the distortions caused by the presence of nonlinearities in FRFs.
It exploits the spatial information and treats the nonlinear forces as internal feedback forces in the underlying linear model of the system.

7.2.1. Theory

Let us write the equation of motion in the frequency domain (116) in the form EQUATION.
The nonlinear forces may be considered as internal feedback forces and may be evaluated from the measured outputs.
An estimation of the FRFs of the underlying linear system HðoÞ and the nonlinear coefficients A i ðoÞ is therefore available.
A 'PSD version' of this equation is preferred, which is obtained by using the same procedure as the one used for obtaining Eq. ( 120) from Eq. ( 119).
The use of PSDs reduces the degree to which linearly correlated terms corrupt the numerical conditioning of the data matrices.

7.2.2. Assessment

The NIFO technique is similar to the CRP method in several aspects; it is, therefore, an attractive method for nonlinear system identification of MDOF structures.
Unlike the CRP method, it is simple to implement, and the estimation of the linear and nonlinear coefficients is carried out in a single step.
Modes of type (i) may be identified using classical curve-fitting methods.
There is only a modal force for mode 4 but now mode 2 responds due to the nonlinear coupling.

7.3.2. Assessment

The authors believe that it paves the way for the analysis of practical systems with high modal density.
One may also account for non-proportional damping, which is another interesting feature of the method.
The comparison between experimental features f i and predicted features fi should be preferred.
In linear dynamics, natural frequencies and mode shapes provide a sound basis for ascertaining whether the prediction of the model will adequately represent the overall dynamic response of the structure.
The correction of the model begins with the selection of the updating parameters.

7.4.2. Application example

This structure was also investigated within the framework of the European COST Action F3 (Golinval et al., 2003 [104] ).
A structural model was created using the finite element method, and the effect of the geometric nonlinearity was modeled with a grounded spring at the connection between the cantilever beam and the short beam.
The correlation study was performed by comparing experimental and predicted POMs.
There is also a good agreement between the experimental and numerical results in Fig. 38 , which confirms that the updated model has a good predictive accuracy.

7.4.3. Assessment

Structural model updating has the inherent ability to provide reliable models of more complex nonlinear structures.
Several crucial issues remain largely unresolved, and there is much research to be done:.
There are no universal features applicable to all types of nonlinearities; test-analysis correlation is still a difficult process.
In addition, the initial model cannot be assumed to be close to the 'actual' model because a priori knowledge about nonlinearity is often limited; the starting point of the optimisation may be far away from the sought minimum.

8. Summary and future research needs

There is a substantial body of literature on nonlinear system identification; it should be recalled that the paper is inevitably biased toward those areas the authors are most familiar with.
Besides rendering parameter estimation tractable, other important issues must be addressed adequately to progress toward the development of accurate, robust, reliable and predictive models of large, threedimensional structures with multiple components and strong nonlinearities.
The following discussion presents some of the key aspects that, the authors believe, will drive the development of nonlinear system identification in the years to come.

Did you find this useful? Give us your feedback