Equivalent linear modal parameter estimation for a nonlinear aeroelastic system

3R12. Nonlinearity in Structural Dynamics: Detection, Identification and Modeling. - K Worden and GR Tomlinson (Univ of Sheffield, UK). Inst Phys Publ, Bristol, UK. 2001. 659 pp. ISBN 0-7503-0356-5. $130.00.Reviewed by K Yagasaki (Dept of Mech and Syst Eng, Gifu Univ, 1-1 Yanagido, Gifu, 501-1193, Japan).This book provides backgrounds in some techniques for analyzing nonlinear structural dynamical systems and is intended for engineers and scientists in the fields of structural dynamics and nonlinear systems. It is also suitable for postgraduate and senior undergraduate students in these disciplines as a textbook and for expert structural dynamicists as a survey. Many engineering examples are given. As for the mathematics of the reader, calculation of vectors and matrices, and basic knowledge of linear differential equations and Fourier analysis are almost sufficient. The most enthusiastic topics in modern nonlinear dynamics are chaos and bifurcation. There have also been remarkable advances in the field of nonlinear control. However, these subjects are not treated in this book at all. Instead, it explains useful techniques, which include the frequency response functions (FRFs) and Hilbert transformations, for detection of nonlinearity and system identification (ie, estimation of the governing equations) when the systems do not exhibit very different motions, like chaos, from ones of linear systems. The techniques are especially important from a point of view of structural engineering because really nonlinear behavior like chaos is thought to be rather uncommon in the problems of that field. The book begins by describing the relevant backgrounds including the frequency response functions (FRFs) in linear dynamics for discrete- and continuous-time systems in Chapter 1. The discussion is first given for single-degree-of-freedom (SDOF) systems and finally generalized to multi-degree-of-freedom (MDOF) systems with an outline of modal analysis. Chapter 2 gives fundamental results and classical approaches to nonlinear systems in structural dynamics. An idea of FRF distortion is also stated there. Chapter 3 discusses FRFs for nonlinear systems in detail and describes how they are used to obtain information about nonlinearity.In Chapters 4 and 5, the Hilbert transformation, which can not only detect nonlinearity but also solve system identification problems, is explained. A mathematical exposition of the method by complex analysis is also presented. In Chapters 6 and 7, several techniques of system identification for discrete- and continuous-time equations are discussed. In Chapter 8, the concept of FRFs is generalized to higher-order FRFs with the assistance of the Volterra series. The generalization presents a method of system identification for MDOF systems. Finally, three experimental examples of a built-in beam rig, automotive shock absorber, and bilinear beam rig (the last of which is motivated from constructing a system with localized damage in the benchmark of fault detection algorithms) are given, and the techniques of the earlier chapters are applied to demonstrate their effectiveness in Chapter 9. This chapter is especially interesting for the reader from a practical point of view. A substantial set of appendices is also valuable for not only the beginners, but also ordinary researchers in the field. In summary, this reviewer recommends Nonlinearity in Structural Dynamics: Detection, Identification and Modeling for students and researchers in structural dynamics who want to study techniques for detection of nonlinearity and system identification in realistic problems.

The results of an analytical investigation are presented for the aeroelastic response of a two-dimensional, structurally nonlinear airfoil subject to a forced excitation. The system is modeled as a two-dimensional, rigid airfoil section free to move in both the bending and pitching directions and possessing a rigid flap. The airfoil is mounted by torsional and translational springs attached at the elastic axis, and the flap motion is used to provide the forcing input to the system. The airfoil is immersed in an aerodynamic flow environment, modeled using incompressible thin airfoil theory for unsteady oscillatory motion. The equations of motion for the aeroelastic system are solved using a fourth-order Runge-Kutta numerical integration technique to provide time-history solutions of the response of the airfoil in the pitch and plunge directions. The time-histories are analysed using Fourier transform-based techniques to obtain frequency-domain response and transfer functions. Results show that the nonlinear response of the aeroelastic system contains frequencies other than the forcing frequency. When modal frequencies and damping values are calculated using standard Fourier-based techniques, it is shown that the super- and sub-harmonic frequency content in the nonlinear response can contribute to errors when results are compared to those obtained for the equivalent linear system. This paper describes an investigation of a method of analysis that, while based on the Fourier transform, has been modified to recognize and accommodate the nonlinear contribution to the system response. The method, developed by Bendat [1], uses a band-limited random input and separates the linear and nonlinear components of the response within the frequency domain. Results are given for the application of this method to the specific case of the structurally nonlinear aeroelastic system. It is shown that the method may be used to successfully recover the linear frequency response function using the input and output data for the nonlinear system.

This paper investigates the effects of nonlinearities on the dynamics of flat panels in supersonic flow and presents new bifurcation results for a fairly general nonlinear aeroelastic system. The method of normal forms, bifurcation techniques, and various other dynamical systems tools are used to obtain these results. This paper deals with modeling of unsteady aerodynamic forces and moments acting on flat panels undergoing arbitrary motion in supersonic flow and the structural nonlinearities inherent in panels. While deriving the equations of motion, many in the past have neglected some nonlinear aerodynamic terms as insignificant from the modeling point of view. However, inclusion of these terms in the analysis may completely change the bifurcation behavior. In aeroelastic systems it is the nonlinear dissipative terms that can change the behavior, and it is essential to carefully model these terms in the physical problems. In the absence of dissipative terms, the nonlinear system is reversible, which provides the governing equations with near integrable structure. Thus certain analytical methods are used to study the bifurcation behavior of the system near critical points. For general nonlinear flat panels in supersonic flow, we present these effects as various bifurcation results due to symmetry-breaking imperfections. For aeroelastic systems, the reversible symmetry is broken by the addition of unsteady aerodynamic terms.

Non-intrusive frequency response analysis of nonlinear systems with interval uncertainty: A comparative study

Computational modal analysis performed using finite element analysis is an established method for estimating the modal parameters of complex structures. This work aims to provide insight into the techniques necessary to accurately model structural components within an aircraft by performing computational modal analysis on a rear-mounted pump support structure. The results of this analysis will be used to further aid in the understanding of the vibratory transmission path from the support structure to the aircraft cabin. The frequency response functions were used to validate the finite element model via a visual and analytical comparison against experimental data. A modal frequency response analysis was used to estimate the dynamic response at a discrete set of points on the structure. A model validation study showed an excellent correlation between the experimental and computational results for frequencies between 100–2000 Hz with an average percent difference of the centre frequencies of 9.4%. The agreement between the two sets of results across the 20–4000 Hz bandwidth deteriorated slightly to 11.7 %. Modifications were made to the webbing of the pump support yoke as well as the in-board and out-board isolator plates. These tests confirmed that differences between the frequency responses of the original and modified finite element models can be justified using modal analysis theory. It was found that these particular structural design changes did not significantly influence the modal response of the structure. The methodology presented in this work provides an outline for future modelling and analysis of aircraft structural designs.

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

An alternative approach to measure similarity between two deterministic transient signals

One of the most versatile and powerful approaches to the identification of nonlinear dynamical systems is the NARMAX (Nonlinear Auto_regressive Moving Average with eXogenous inputs) method. The model represents the current output of a system by a nonlinear regression on past inputs and outputs and can also incorporate a nonlinear noise model in the most general case. Although the NARMAX model is most often given a polynomial form, this is not a restriction of the method and other formulations have been proposed based on nonparametric machine learning paradigms, for example. All of these forms of the NARMAX model allow the computation of Higher-order Frequency Response Functions (HFRFs) which encode the model in the frequency domain and allow a direct interpretation of how frequencies interact in the nonlinear system under study. Recently, a NARX (no noise model) formulation based on Gaussian Process (GP) regression has been developed. One advantage of the GP NARX form is that confidence intervals are a natural part of the prediction process. The objective of the current paper is to discuss the GP formulation and show how to compute the HFRFS corresponding to GP NARX. Examples will be given based on simulated data.KeywordsNonlinear system identificationNARMAX modelsHigher-order Frequency Response Functions (HFRFs)Gaussian processes

Convergent systems constitute a practically important class of nonlinear systems that extends the class of asymptotically stable LTI systems. In this paper we extend frequency response functions defined for linear systems to non-linear convergent systems. Such nonlinear frequency response functions for convergent systems give rise to nonlinear Bode plots, which serve as a graphical tool for performance analysis of nonlinear convergent systems in the frequency domain. The results are illustrated with an example.

This paper investigates the implementation of model predictive control strategies in order to control the pitch response of a nonlinear aeroelastic system. Model predictive control, also known as receding horizon control, entails computing an optimal sequence of control inputs over a finite time horizon in order to minimize a cost functional subject to constraints. This approach requires a dynamic model for the system in order to predict the system response to candidate control input sequences. In this paper, a simulated nonlinear aeroelastic system is modeled in terms of third-order Volterra series, and these Volterra models are implemented in a model predictive control algorithm to control the pitch response of the system. The Volterra model is obtained from simulated input – output data using a multiresolution-based kernel identification algorithm that has previously been developed by the author and used to generate Volterra models from both simulated and flight test data from several aeroelastic systems. This paper investigates the feasibility of using Volterra-based model predictive control strategies to control a simulated nonlinear aeroelastic system corresponding to a model of the Texas A&M Nonlinear Aeroelastic Testbed. The model predictive control algorithm is used to compute trailing edge flap deflection commands in order to control the pitch response of the system at different free stream velocities, including a flight condition for which the underlying linear aeroelastic system is in an unstable flutter condition.

High-speed digital image correlation is a camera-based displacement measurement technique that has experienced increasing popularity for modal analysis. It is due to a high spatial density of measurement that provides an outstanding interpretation of mode shapes or operational deflection shapes. However, its sensitivity is typically lower than other techniques and transducers such as accelerometers or laser vibrometry. With these measurements, the estimation of any transfer function in the frequency domain is noisier in valley regions of the response, including anti-resonances. Therefore, an accurate analytical model of the response should be used for modal identification. Modal analysis algorithms perform an identification of the modal parameters and a reconstruction of the response based on the transfer function between force excitation and displacement response. This is called the frequency response function. However, force measurements are not available in many experiments, where the motion excitation is recorded instead. This is the case when the vibration is applied to the specimen through a motion in its base. This transfer function, also known as the transmissibility function, relates the motion of excitation and response, whose shape differs from the frequency response function. This difference in the response function is relevant in a relatively noisy measurement like using digital image correlation. Hence, in this work, a transformation of the transmissibility function into equivalent frequency response function, based on theoretical models, is performed prior to modal identification in base motion tests. More suitable experimental response functions are obtained, increasing the accurateness of the modal parameter estimation.KeywordsModal analysisTransmissibility functionsFull-field measurement

Frequency-Domain Gray-Box Identification of Industrial Robots

The receptance method has been proved to be effective for active vibration control of linear systems, theoretically and experimentally. A remarkable advantage of this method is that it does not require any prior knowledge of the system matrices, which are typically obtained from the finite element method. In this paper, a modified receptance method is proposed for obtaining the time-varying receptances of a general nonlinear system; the method is then applied to a structurally nonlinear aeroelastic system for active flutter suppression. A two-dimensional aeroelastic model with a polynomial nonlinearity in the pitching stiffness is considered. The Runge-Kutta method is used for numerical simulations. The poles of the closed-loop system are assigned to improve the system stability; both fixed and varied poles cases are studied. The effects of poles values, weight coefficients in indicator function, and control limitations are discussed. Results show that the aforementioned parameters affect both the system response and the input of the control surface, and the modified receptance method is effective for active flutter suppression of the nonlinear aeroelastic system by rational pole assignments.

Bayesian model selection for nonlinear aeroelastic systems using wind-tunnel data

Shaking table is a nonlinear system, which is a more nonlinear system with payload. System can be usually as linear system nearby working point in control strategy. 1 H -estimator or 2 H -estimator is used for identifying the Frequency Response Function (FRF) of the system. 1 H -estimator is a lower-estimator and 2 H -estimator is an over-estimator, both have large estimating errors. In this paper, a new estimator, m H -estimator, is used for the identification of the shaking table system’s FRF, and whose parameters are estimated by differential evolution (DE) which makes m H closed to the true FRF H . This control strategy can reduce the steps of iterative learning control (ILC) of shaking table system, and the affection of payload characteristic.