Gaëtan Kerschen

• PhD in Aerospace Engineering
• Professor (Full) at University of Liège

This survey paper contains a review of the past and recent developments in system identification of nonlinear dynamical structures. The objective is to present some of the popular approaches that have been proposed in the technical literature, to illustrate them using numerical and experimental applications, to highlight their assets and limitation...

a b s t r a c t The concept of nonlinear normal modes (NNMs) is discussed in the present paper and its companion, Part II. Because there is virtually no application of the NNMs to large-scale engineering structures, these papers are an attempt to highlight several aspects that might drive their development in the future. Specifically, we support th...

The harmonic balance (HB) method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems. The objective of this paper is to exploit the method for bifurcation analysis, i.e., for the detection and tracking of bifurcations of nonlinear systems. To this end, an algorithm that combines the computation of t...

This paper is devoted to the probabilistic uncertainty quantification of orbital lifetime estimation of low-altitude satellites. Specifically, given a detailed characterization of the dominant sources of uncertainty, we map this input into a probabilistic characterization of the orbital lifetime through orbital propagation. Standard Monte Carlo pro...

The objective of this paper is to propose an exact closed-form solution to the H ∞ > optimization of piezoelectric materials shunted with inductive-resistive passive electrical circuits. Realizing that Den Hartogʼs method which imposes fixed points of equal height in the receptance transfer function is approximate, the parameters of the piezoelectr...

The objective of this Chapter is to demonstrate how the intentional utilization of nonlinearity can bring important benefits in the area of engineering design. Two different applications are considered, namely (i) the tailoring of the nonlinear normal modes of a structure through the addition of a nonlinearity and (ii) the management of the bifurca...

This work presents a novel vibration damping approach for bladed structures. Piezoelectric transducers bonded to a structure can be used simultaneously as actuators and sensors to mitigate the vibrations of their host. This can be achieved by connecting a transducer to a digital vibration absorber composed of a voltage sensor, a digital processing...

A novel piezoelectric shunt damping methodology for structures featuring different mode families with high modal density is introduced in this work. To this end, multiple digital vibration absorbers comprising each a voltage sensor, a digital processing unit and a current injector are connected to multiple piezoelectric transducers bonded to these...

By drawing a parallel between the controller parameters of a piezoelectric inductive-resistive (RL) shunt with a negative capacitance (NC) and the active control method positive position feedback (PPF), we prove that there exists an equivalence between the controller parameters and their receptance functions. Based on these findings, exact H∞ tunin...

Two of the most popular vibration testing methods for nonlinear structures are control-based continuation and phase-locked-loop testing. In this paper, they are directly compared on the same benchmark system, for the first time, to demonstrate their general capabilities and to discuss practical implementation aspects. The considered system, which i...

In this paper, a modified active tuned inerter damper concept which is more suitable for practical applications is proposed. The proposed device is composed of a pair of collocated reactive actuator and force sensor. A second-order low-pass filter and a proportional term are combined to form the controller. The equivalent mechanical model of the co...

A novel sequential tuning procedure for passive piezoelectric shunts targeting multiple structural modes is proposed in this work. The control authority on each targeted mode can be quantitatively chosen ab initio and is shown to be limited by passivity requirements, which highlights the fundamental limitations of multimodal piezoelectric shunts. B...

Although simple and effective, the performance of passive control techniques is limited in terms of authority and robustness. This paper studies the potential of using an active control system for mitigating the vibrations of a bladed rail assembly. First, we discuss the optimal placement of the piezoelectric patches. Although locating the patches...

In this paper, a new computational method for the purpose of multimodal vibration mitigation using multiple tuned mass dampers is proposed. Classically, the minimization of the maximum amplitude is carried out using direct H∞ optimization. However, as shall be shown in the paper, this approach is prone to being trapped in local minima, in view of t...

In this paper, an active nonlinear energy sink (ANES) based on force feedback is investigated. The proposed device is composed of a pair of collocated actuator and force sensor. The control law is implemented by feeding back the output of the force sensor, through one single integrator and one double integrator of its cube. Its working principle ca...

Control-Based Continuation uses feedback control to follow stable and unstable branches of periodic orbits of a nonlinear system without the need for advanced post-processing of experimental data. CBC relies on an iterative scheme to modify the harmonic content of the control reference and obtain a non-invasive control signal. This scheme currently...

In this study, a digital impedance is used to realize both a linear and a nonlinear piezoelectric tuned vibration absorber in order to mitigate the vibrations of a nonlinear structure. The digital processing unit enables the synthesis of impedances with arbitrary functional forms, thereby easing the implementation of nonlinear absorbers. The superi...

The tuning of a simplified current blocking shunt circuit able to mitigate the vibration amplitude of multiple structural resonances is addressed in this article. The proposed strategy exploits the two-port network formalism in combination with physically motivated approximations to tune sequentially the electrical elements of the different branche...

The objective of this study is to develop an optimization methodology to find a layout that traces a prescribed force–displacement curve through a topology optimization approach. To this end, we propose an objective function to minimize the difference between a prescribed force–displacement curve and the curve calculated at each iteration of the op...

This paper presents an adaptive smooth controller for a class of nonlinear dynamical systems in the presence of bounded uncertainties with unknown bounds. Motivated by the concept of sliding mode control, a continuous control law that drives the sliding variable to a user-specified small domain in a finite time is developed. Also, an adaptation law...

The objective of this works is to investigate the interest of analogous piezoelectric networks for vibration mitigation of multiple nonlinear resonances. An electrical network providing a similar dynamics as the considered structure is first designed according to linear considerations. Then, a nonlinear electrical component is added to the network...

This paper deals with the nonlinear system identification of structures exhibiting distributed nonlinearities, which has become of great interest recently, due to the continuous interest to improve the performance of structures. This brings the need for designing lighter and more flexible structural elements, which are usually characterized by mode...

Nonlinear system identification has become of great interest during the last decades. However, a common and shared framework is not present yet, and the identification may be challenging, especially when real engineering structures are considered with strong nonlinearities. Subspace methods have proved to be effective when dealing with local nonlin...

Piccolo tubes are parts of aircraft wings anti-icing system and consist of titanium pipes inserted into the internal structure of the slat. Due to differential thermal expansion, clearances between the tube and its support are unavoidable and cause the overall system to exhibit highly nonlinear behavior, resulting from impacts and friction. This pa...

We analyze isolated resonance curves (IRCs) in single-degree-of-freedom systems possessing nonlinear damping. Through the combination of singularity theory and the averaging method, the onset and merging of IRCs, which coincide to isola and simple bifurcation singularities, respectively, can be analytically predicted. Numerical simulations confirm...

The objective of this study is to develop the first fully passive nonlinear piezoelectric tuned vibration absorber (NPTVA). The NPTVA is designed to mitigate a specific resonance of a nonlinear host structure. To avoid the use of synthetic inductors which require external power, closed magnetic circuits in ferrite material realize the large inducta...

The objective of the present paper is to provide experimental evidence of isolated resonances in the frequency response of nonlinear mechanical systems. More specifically, this work explores the presence of isolas, which are periodic solutions detached from the main frequency response, in the case of a nonlinear set-up consisting of two masses slid...

The linear vibration absorber is a widely-used vibration mitigation device. However, when the absorber is tuned according to Den Hartog’s equal-peak method, the resulting narrow bandwidth may decrease its effectiveness, especially when the host structure is uncertain or in the presence of environmental variability. In this paper, a new tuning strat...

Nonlinear normal modes offer a solid theoretical framework for interpreting a wide class of nonlinear dynamic phenomena. However, their computation for large-scale models can be time consuming, particularly when nonlinearities are distributed across the degrees of freedom. In this paper, the nonlinear normal modes of systems featuring distributed g...

Important properties of linear systems, such as force-displacement proportionality and invariance of the resonant frequency, are not satisfied by nonlinear systems. The objective of this paper is to demonstrate that the intentional addition of properly tuned nonlinearities to a nonlinear system allows to retrieve those linear properties, enlarging...

In the last decade, the problem of detached resonant curves (DRCs) has received a growing attention. However, most of recent research works seem to ignore papers from the sixties and the seventies, which already investigated the subject. With the aim of recovering this “forgotten” literature, we briefly review the majority of the studies on the top...

The introduction of the frequency-domain nonlinear subspace identification (FNSI) method in 2013 constitutes one in a series of recent attempts toward developing a realistic, first-generation framework applicable to complex structures. If this method showed promising capabilities when applied to academic structures, it is still confronted with a nu...

The objective of this study is to develop a fully passive nonlinear piezoelectric vibration absorber that mitigates a specific resonance of a nonlinear system. To this end, the saturation of a passive inductor based on a magnetic circuit in ferrite material is exploited to realize a cubic nonlinearity in the absorber. The resulting variable inducta...

Piezoelectric coupling to an analogous electrical network has already proven its interest for vibration mitigation. As such a solution optimizes the energy transfers between the electrical and mechanical domains, an extension to energy harvesting is considered.

One of the major challenges in nonlinear system identification is the selection of appropriate mathematical functions to model the observed nonlinearities. In this context, piecewise polynomials, or splines, offer a simple and flexible representation basis requiring limited prior knowledge. The generally-adopted discretization for splines consists...

Aircraft are more than ever pushed to their limits for performance reasons. Consequently, they become increasingly nonlinear and they are more prone to undergo aeroelastic limit cycle oscillations. Structural nonlinearities affect aircraft such as the F-16, which can undergo store-induced limit cycle oscillations (LCOs). Furthermore, transonic buzz...

The influence of a nonlinear tuned vibration absorber (NLTVA) on the airfoil flutter is investigated. In particular, its effect on the instability threshold and the potential subcriticality of the bifurcation is analyzed. For that purpose, the airfoil is modeled using the classical pitch and plunge aeroelastic model together with a linear approach...

The paper offers the fully analytic solution to the motion of a satellite orbiting under the influence of the two major perturbations, due to the oblateness and the atmospheric drag. The solution is presented in a time-explicit form, and takes into account an exponential distribution of the atmospheric density, an assumption that is reasonably clos...

Modal testing of nonlinear engineering structures is currently a research area that attracts substantial attention. Recently, a nonlinear generalisation of phase separation testing has been proposed for identifying nonlinear normal modes (NNMs) based on input and output measurements. The nonlinear phase separation (NPS) method integrates identifica...

This paper presents a robust adaptive control methodology based on the concept of second-order sliding modes for satellite formation control in the presence of unknown, but bounded uncertainties. By introducing two sliding variables to achieve the concept of second-order sliding modes, a robust controller is designed so that it forces the first sli...

A passive vibration absorber, termed the nonlinear tuned vibration absorber (NLTVA), is designed for the suppression of chatter vibrations. Unlike most passive vibration absorbers proposed in the literature for suppressing machine tool vibrations, the NLTVA comprises both a linear and a nonlinear restoring force. Its linear characteristics are tune...

A passive vibration absorber, termed the nonlinear tuned vibration absorber (NLTVA), is designed for the suppression of chatter vibrations. Unlike most passive vibration absorbers proposed in the literature, the NLTVA comprises both a linear and a nonlinear restoring force. Its linear characteristics are tuned in order to optimize the stability pro...

The current drive for increased efficiency in aeronautic structures such as aircraft, wind-turbine blades, and helicopter blades often leads to weight reduction. A consequence of this tendency can be increased flexibility, which in turn can lead to unfavorable aeroelastic phenomena involving large-amplitude oscillations and nonlinear effects such a...

Uncertainties in the dynamical system affect all real-life applications in astrodynamics. This is particularly true for satellites orbiting about small bodies owing to their irregular shapes and to the difficulties in characterizing their physical properties at the early stages of a dedicated spacecraft mission. Hence, accommodating uncertainty in...

A passive vibration absorber, termed the nonlinear tuned vibration absorber (NLTVA), is designed for the suppression of chatter vibrations. Differently from most passive vibration absorbers proposed in the literature , the NLTVA comprises both a linear and a nonlinear restoring force. Its linear characteristics are tuned in order to optimize the st...

The focus of this study is on the development of a robust controller with a simple gain adaptation for satellite formation control. The complete nonlinear dynamics of the motion of the follower satellite relative to the leader satellite is considered and the rigorous proof for the stability of the controlled formation system is given in the presenc...

The objective of this paper is to derive an approximate closed-form solution to the H ∞ optimization of piezoelectric materials shunted with inductive-resistive passive electrical circuits in the presence of damping in the primary structure. To this end, the homotopy perturbation method (HPM) is utilized in which the zero-order solution is the rece...

This paper explores the relationship that exists between nonlinear normal modes (NNMs) defined as invariant manifolds in phase space and the spectral expansion of the Koopman operator. Specifically, we demonstrate that NNMs correspond to the zero level sets of specific eigenfunctions of the Koopman operator. Thanks to this direct connection, a new,...

Most studies tackling hysteresis identification in the technical literature follow white-box approaches, i.e. they rely on the assumption that measured data obey a specific hysteretic model. Such an assumption may be a hard requirement to handle in real applications, since hysteresis is a highly individualistic nonlinear behaviour. The present pape...

The objective of this paper is to demonstrate that the addition of properly-tuned nonlinearities to a nonlinear system can increase the range over which a specific resonance responds linearly. Specifically, we seek to enforce two important properties of linear systems, namely the force-displacement proportionality and the invariance of resonance fr...

Bifurcation theory and continuation methods are well-established tools for the analysis of nonlinear mechanical systems subject to periodic forcing. We illustrate the added value and the complementary information provided by singularity theory with one distinguished parameter. While tracking bifurcations reveals the qualitative changes in the behav...

Bifurcation theory and continuation methods are well-established tools for the analysis of nonlinear mechanical systems subject to periodic forcing. We illustrate the added value and the complementary information provided by singularity theory with one distinguished parameter. While tracking bifurcations reveals the qualitative changes in the behav...

This paper develops a principle of similarity for the design of a nonlinear absorber, the nonlinear tuned vibration absorber (NLTVA), attached to a nonlinear primary system. Specifically, for effective vibration mitigation, we show that the NLTVA should feature a nonlinearity possessing the same mathematical form as that of the primary system. A co...

This paper presents a new adaptive methodology for sliding mode control of a nonlinear dynamical system in the presence of unknown, but bounded uncertainties. A continuous control law is first developed to compensate for the uncertainties and this Lyapunov-based approach eliminates chattering by replacing a discontinuous signum function with a cont...

Important properties of linear systems, such as force-displacement pro-portionality and invariance of the resonant frequency, are not satisfied by nonlinear systems. The objective of this paper is to demonstrate that the intentional addition of properly tuned nonlinearities to a nonlinear system allows to retrieve those linear properties, enlarging...

The objective of the present paper is to develop a two-step methodology integrating system identification and numerical continuation for the experimental extraction of nonlinear normal modes (NNMs) under broadband forcing. The first step processes acquired input and output data to derive an experimental state-space model of the structure. The secon...

Most studies tackling hysteresis identification in the technical literature follow white-box approaches, i.e. they rely on the assumption that measured data obey a specific hysteretic model. Such an assumption may be a hard requirement to handle in real applications, since hysteresis is a highly individualistic nonlinear behaviour. The present pape...

This paper develops a differential drag-based sliding mode controller for satellite rendezvous. It is chattering-free and avoids bang-bang type control to adjust the relative motion more efficiently. In spite of uncertain nonlinear perturbations and disturbances, it is shown that the in-plane relative motion between two satellites can be effectivel...

The objective of the present paper is to develop a rigorous identification methodology of nonlinear normal modes (NNMs) of engineering structures. This is achieved by processing experimental measurements collected under broadband forcing. The use of such a type of forcing signal allows to excite multiple NNMs simultaneously and, in turn, to save te...

Nonlinear aeroelastic phenomena such as store-induced LCOs, transonic buzz and stall flutter are the burden or modern aircraft: they reduce the performance and can even limit the flight envelope in both civil and military cases. Several nonlinear setups were studied experimentally in the last decades by the scientific community but most of them hav...

Nonlinear Normal Modes (NNMs) have a clear conceptual relation to the classical linear normal modes (LNMs), yet they offer a solid theoretical framework for interpreting a wide class of non-linear dynamical phenomena with no linear counterpart. The main difficulty associated with NNMs is that their calculation for large-scale models is expensive, p...

Engineering structures are designed to be lighter and more flexible, hence reducing the extent of application of linear dynamic models. Concurrently, vibration mitigation is required for enhancing the performance, comfort or safety in real-life applications. Passive linear vibration absorbers are purpose-built, often designed using Den Hartog’s equ...

The performance of a linear tuned vibration absorber (LTVA) and a nonlinear energy sink (NES) for the vibrationcert mitigation of an unain linear primary system is investigated. An analytic tuning rule for the LTVA when the primary system contains uncertainty is derived. The behavior of the linear system coupled to the NES is analyzed theoretically...

The current trend of developing more slender structures is increasing the importance of nonlinearities in engineering design, which, in turn, gives rise to complicated dynamical phenomena. In this study, we evidence the somewhat paradoxical result that adding purposefully nonlinearity to an already nonlinear structure renders the behavior more line...

The objective of the paper is to obtain the frequency response curves of nonlinear mechanical systems from broadband testing. The proposed approach consists in coupling an identification method with a continuation method. Specifically, the frequency-domain nonlinear subspace identification (FNSI) method is first used to derive an experimental model...

In the present paper, the observation and characterization of isolated response curves (IRCs) are experimentally reported in the case of a nonlinear system consisting of two masses sliding on an horizontal guide. Transverse springs are attached to one mass to provide the nonlinear restoring force, and a harmonic motion of the complete system is imp...

Because nonlinearity is now a frequent occurrence in real-life applications, the practitioner should understand the resulting dynamical phenomena and account for them in the design process. This tutorial focuses on nonlinear system identification, which extracts relevant information about nonlinearity directly from experimental measurements. Specif...

The current drive for increased efficiency in aeronautic structures such as aircraft, wind turbine blades and helicopter blades often leads to weight reduction. A consequence of this tendency can be increased flexibility, which in turn can lead to unfavourable aeroelastic phenomena involving large amplitude oscillations and nonlinear effects such a...

Nonlinear system identification is a vast research field, today attracting a great deal of attention in the structural dynamics community. Ten years ago, an MSSP paper reviewing the progress achieved until then [1] concluded that the identification of simple continuous structures with localised nonlinearities was within reach. The past decade witne...

In the present paper, a flexible and parsimonious model of the vibrations of nonlinear mechanical systems is introduced in the form of state-space equations. It is shown that the nonlinear model terms can be formed using a limited number of output measurements. A twostep identification procedure is derived for this grey-box model, integrating nonli...

This paper reviews the recent advances in computational methods for nonlinear normal modes (NNMs). Different algorithms for the computation of undamped and damped NNMs are presented, and their respective advantages and limitations are discussed. The methods are illustrated using various applications ranging from low-dimensional weakly nonlinear sys...

This paper addresses the computation of radiative exchange factors through Monte Carlo ray tracing with the aim of reducing their computation time when dealing with the finite element method. Both direction and surface samplings are studied. The recently-introduced isocell method for partitioning the unit disk is applied to the direction sampling a...

The objective of the present study is to explore the connection between the nonlinear normal modes of an undamped and unforced nonlinear system and the isolated resonance curves that may appear in the damped response of the forced system. To this end, an energy balance technique is used to predict the amplitude of the harmonic forcing that is neces...

In the present paper, isolated response curves in a nonlinear system consisting of two masses sliding on a horizontal guide are examined. Transverse springs are attached to one mass to provide the nonlinear restoring force, and a harmonic motion of the complete system is imposed by prescribing the displacement of their supports. Numerical simulatio...

Nonlinear normal modes of vibration have been the focus of many studies during the past years and different characterizations of them have been proposed. The present work focuses on damped systems, and considers nonlinear normal mode motions as trajectories lying on an invariant manifold, following the geometric approach of Shaw and Pierre. We prov...

With continual interest in expanding the performance envelope of engineering systems, nonlinear components are increasingly utilized in real-world applications. This causes the failure of well-established techniques to mitigate resonant vibrations. In particular, this holds for the linear tuned vibration absorber (LTVA), which requires an accurate...

Isolated resonance curves are separate from the main nonlinear forced-response branch, so they can easily be missed by a continuation algorithm and the resonant response might be underpredicted. The present work explores the connection between these isolated resonances and the nonlinear normal modes of the system and adapts an energy balance criter...

The harmonic balance (HB) method has been widely used in the past few years, as a numerical tool for the study of nonlinear models. However, in its classical formulation the HB method is limited to the approximation of periodic solutions. The present paper proposes to extend the method to the detection and tracking of bifurcations in the codimensio...

Although they are generally modelled as linear systems, aircraft structures are known to be prone to nonlinear phenomena. A specific challenge encountered with fighter aircraft, besides aeroelastic nonlinearity, is the modelling of the wing-to-payload mounting interfaces. For large amplitudes of vibration, friction and gaps may be triggered in thes...

In the present paper, an averaging perturbation technique leads to the determination of a time-explicit analytic approximate solution for the motion of a low-Earth-orbiting satellite . The two dominant perturbations are taken into account: the Earth oblateness and the atmospheric drag. The proposed orbit propagation algorithm comprises the Brouwer–...

This paper proposes a piezoelectric vibration absorber, termed the nonlinear piezoelectric tuned vibration absorber (NPTVA), for the mitigation of nonlinear resonances of mechanical systems. The new feature of the NPTVA is that its nonlinear restoring force is designed according to a principle of similarity, i.e., the NPTVA should be an electrical...

The objective of the present paper is to develop a two-step methodology integrating system identification and numerical continuation for the experimental extraction of nonlinear normal modes (NNMs) under broadband forcing. The first step processes acquired input and output data to derive an experimental state-space model of the structure. The secon...

The objective of this study is to mitigate, or even completely eliminate, the limit cycle oscillations in mechanical systems using a passive nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA). An unconventional aspect of the NLTVA is that the mathematical form of its restoring force is not imposed a priori, as it is the case...

Optimization of fuel consumption is a key driver in the design of spacecraft maneuvers. For this reason, growing interest in propellant-free maneuvers is observed in the literature. Because it allows us to turn the often-undesired drag perturbation into a control force for relative motion, differential drag is among the most promising pro- pellantl...

The nonlinear tuned vibration absorber (NLTVA) is a recently developed nonlinear absorber which generalizes Den Hartog׳s equal peak method to nonlinear systems. If the purposeful introduction of nonlinearity can enhance system performance, it can also give rise to adverse dynamical phenomena, including detached resonance curves and quasiperiodic re...

Aircraft structures are known to be prone to nonlinear phenomena, especially as they constantly become lighter and hence more flexible. One specific challenge that is regularly encountered is the modeling of the mounting interfaces between aircraft subcomponents. Indeed, for large amplitudes of vibration, such interfaces may loosen and, in turn, tr...

Limit cycle oscillations (LCOs) occur in many mechanical systems and they are often a source of danger. The addition of a relatively small mass to the host system, attached through a linear spring and a damper (linear tuned vibration absorber, LTVA), significantly improves its stability. The use of a purely nonlinear spring in the absorber (nonline...

The paper presents the solution to the problem of the relative motion between two satellites orbiting Earth under the influence of the oblateness and atmospheric drag perturbations. Starting from the analytic solution to the problem of the absolute motion, the closed-form equations of motion are obtained. No simplifying assumptions are made on the...

Most identification methods in non-linear structural dynamics assume in advance a mathematical model of the non-linearities. This is however possible in specific situations only, since non-linear effects may be caused by numerous phenomena and a priori knowledge is generally limited. The present paper investigates the usefulness of piecewise third-...