Marie Curie Europe

TC5: Vibration Testing, Identification of Linear and Nonlinear Systems

TC5: Vibration Testing, Identification of Linear and Nonlinear Systems 

A training course coordinated by Jean-Claude Golinval, Gaëtan Kerschen, Guido De Roeck
time:   July 6-10, 2009
location: University of Liege, Belgium

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Aim of the course
The demand for enhanced and reliable performance of vibrating structures in terms of weight, comfort, safety, noise and durability is ever increasing while, at the same time, there is a demand for shorter design cycles, longer operating life, minimization of inspection and repair needs, and reduced costs. With continual interest to expand the performance envelope of structures at ever increasing speeds, there is the need for designing lighter, more flexible, and consequently, more nonlinear structural elements. It follows that the demand to utilize nonlinear (or even strongly nonlinear) structural components is increasingly present in engineering applications.


In the theory of mechanical vibrations, mathematical models -termed structural models- are helpful for the analysis and the understanding of the dynamic behavior of the structure being modeled. Experimental testing and system identification play a key role because they help the structural dynamicist to reconcile numerical predictions with experimental investigations. The term ‘system identification’ is sometimes used in a broader context in the technical literature and may also refer to the extraction of information about the structural behavior directly from experimental data, i.e., without necessarily requesting a model. Here, system identification refers to the development (or the improvement) of structural models from input and output measurements performed on the real structure using vibration sensing devices. 
Linear system identification is a discipline that has evolved considerably during the last 30 years. Modal parameter estimation -termed modal analysis- is indubitably the most popular approach to performing linear system identification in structural dynamics. The model of the system is known to be in the form of modal parameters, namely the natural frequencies, mode shapes and damping ratios. The popularity of modal analysis stems from its great generality; modal parameters can describe the behavior of a system for any input type and any range of the input. Numerous approaches for modal parameter identification have been described in details in TC3. 
The basic principles that apply to a linear system and that form the basis of modal analysis are no longer valid in the presence of nonlinearity. In addition, even weak nonlinear systems can exhibit extremely interesting and complex phenomena which linear systems cannot. These phenomena include jumps, bifurcations, saturation, subharmonic, superharmonic and internal resonances, resonance captures, limit cycles, modal interactions and chaos. 
The theory of nonlinear dynamics has received considerable attention in TC1, TC2 and TC4. The motivation behind TC5 is threefold. First, it is meant to provide a concise point of departure for researchers and practitioners alike wishing to assess the current state of the art in the identification of nonlinear structural models. Second, the course intends to review several methods that have been proposed in the technical literature and to highlight some of the reasons that prevent these techniques from being applied to complex structures.
The last goal of this training course is to identify future research needs which would help to ‘push the envelope’ in nonlinear system identification. Typical sources of nonlinearities which will be treated in TC5 are:
  • Geometric nonlinearity results when a structure undergoes large displacements and arises from the potential energy.
  • Inertia nonlinearity derives from nonlinear terms containing velocities and/or accelerations in the equations of motion, and takes its source in the kinetic energy of the system (e.g., convective acceleration terms in a continuum and Coriolis accelerations in motions of bodies moving relative to rotating frames).
  • A nonlinear material behavior may be observed when the constitutive law relating stresses and strains is nonlinear.
  • Damping dissipation is essentially a nonlinear and still not fully modeled and understood phenomenon. The modal damping assumption is not necessarily the most appropriate representation of the physical reality, and its widespread use is to be attributed to its mathematical convenience. Dry friction effects (bodies in contact, sliding with respect to each other) and hysteretic damping are examples of nonlinear damping.
  • Nonlinearity may also result due to boundary conditions (for example, free surfaces in fluids, vibro-impacts due to loose joints or contacts with rigid constraints, clearances, imperfectly bonded elastic bodies), or certain external nonlinear body forces (e.g., magnetoelastic, electrodynamic or hydrodynamic forces). 
All systems referenced in this training course are assumed to be time-invariant and deterministic, i.e., for given excitation conditions, the system response is always the same without any uncertainty. 
Course Program
Lecture 1 – Introduction 
Lecture 2 - Overview of linear system identification and model updating in structural dynamics a. Time domain and frequency domain methods for input-output and output-only system identification; examplesb. Robust global optimization/minimization strategies; case-studies of optimization techniques; propagation of uncertainty 
Lecture 3 - Vibration testing (Testing techniques for linear and nonlinear systems; experiments able to evidence nonlinear parameters, friction, modification of stiffness; restoring force identification, state surface, parametric model) 
Lecture 4 - Fundamentals of nonlinear dynamicsa. Dynamics of nonlinear oscillationsb. Complicated dynamics of a two degree-of-freedom nonlinear system: a nonlinear normal mode perspective (dynamics of the undamped system, dynamics of the weakly damped system). 
Lecture 5 - Nonlinear system identification in structural dynamics: a literature reviewa. By-passing nonlinearity: linearizationb. Time-domain methodsc. Frequency-domain methods d. Modal methodse. Time–frequency analysisf. Black-box modeling  
Lecture 6 - Detection of nonlinearity  
Lecture 7 - Characterization of nonlinearity (the location of the nonlinearity, the type of the nonlinearity, the functional form of the nonlinearity) 
Lecture 8 - Parameter estimation in the presence of nonlinearity: established methods a. The restoring force surface method b. Direct parameter estimation and restoring force surfacesc. NARMAX modeling d. The Hilbert transforme. The Volterra series and higher-order frequency response functionsf. The reverse path method. 
Lecture 9 - Parameter estimation in the presence of nonlinearity: recent methodsa. The conditioned reverse path methodb. The nonlinear identification through feedback of the output methodc. Frequency domain ARX modelsd. The nonlinear resonant decay method e. The modal methodsf. The wavelet method g. The subspace methodh. Structural model updating in the presence of nonlinearity 
Lecture 10 - Damage detection: response modification due to damage in linear and nonlinear systems 
Lecture 11 – Constructive utilization of nonlinearity a. Damage identificationb. The nonlinear energy sinkc. Micro-electromechanical systems: the electrostatically-actuated micro-beamd. Biological systems: the dynamics of the DNA molecule
Invited Lecturers

The course will be delivered by recognized professors in the field of structural dynamics:

  • Douglas E. Adams - Purdue University, West Lafayette, IN, USA
  • Pierre Argoul - Ecole Nationale des Ponts et Chaussées, France
  • Guido De Roeck - Katholieke Universiteit Leuven, Belgium
  • Grigorios Dimitriadis - University of Liege, Belgium
  • Luigi Garibaldi - University of Torino, Italy
  • Jean-Claude Golinval - University of Liege, Belgium
  • Gaëtan Kerschen - University of Liege, Belgium
  • Alexander Vakakis - National Technical University of Athens, Greece
  • Fabrizio Vestroni - University of Rome « La Sapienza »
  • Keith Worden - University of Sheffield, UK
Preliminary Readings
  • G. Kerschen, K. Worden, A.F. Vakakis, J.C. Golinval, Past, present and future of nonlinear system identification in structural dynamics, Mechanical System and Signal Processing 20, 505-592, 2006 
  • K. Worden, G. R. Tomlinson, Nonlinearity in Structural Dynamics: Detection, Identification and Modelling, Institute of Physics Publishing Ltd 2001, ISBN 0 7503 0356 5

Hosting Institution

Department of Aerospace and Mechanical Engineering, University of Liege

For further information please contact:

Vincenzo Gattulli - The SICON General Secretary -
Piazzale E. Pontieri 2 - 67040 Monteluco di Roio (Italy)
tel. +39 0862 434511, fax +39 0862 434548 , e-mail: mailto: This e-mail address is being protected from spam bots, you need JavaScript enabled to view it


Event Time Table









July 6

July 7

July 8

July 9

July 10

9:00 ­ 9:45


Lecture 5a-c - GK 

Lecture 9a-b - DA

Lecture 9f - PA

Lecture 11a - FV


Lecture 2a - GDR

Lecture 5d-f - GK

Lecture 9b-c - DA

Lecture 9f-g - PA/LG

Lecture 11a - FV

Coffee Break


Lecture 2b - GDR

Lecture 6 - KW

Lecture 9d - GD

Lecture 9g - LG

Lecture 11b - AV


 Case studies - GDR

Lecture 7 - KW

Lecture 9d - GD 

Lecture 9h - JCG

Lecture 11b - AV



Lecture 3 - GDR

Lecture 8 - KW

Lecture 9e - AV

Lecture 10 - DA

Lecture 11c - JCG


Lecture 4a - GDR

Lecture 8 - KW

Lecture 9e - AV

Lecture 10 - DA

Lecture 11d - GK

Coffee Break


Lecture 4a-b - AV

Lecture 8 - KW

Lecture 9e - GK

Lecture 10 - DA



Lecture 4b - AV

Lecture 8 - KW

Lecture 9e - GK

Lecture 10 - DA




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