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

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.

A copula-based Gaussian mixture closure method for stochastic response of nonlinear dynamic systems

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.

A new GUI interpretation tool for the non-linear frequency response function

International audience

Stochastic bifurcation

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.

Equivalent linear modal parameter estimation for a nonlinear aeroelastic system

Among the irregular responses of nonlinear dynamic systems, chaotic responses of nonlinear systems are probably the most attractive phenomena along with the new observations in the last decades. A nonlinear deterministic system may behavior chaotically under regular such as periodic excitations. Regular motion of a system subjected to periodic exertions is usually periodic. In contrast with regular motions, final states of chaotic vibrations are extremely nonperiodic. This research is to analyzing the irregular behavior of dynamic systems with implementation of a newly developed criterion named Periodicity-Ratio. The development of a methodology for diagnosing the irregular motions from the regular motions of a dynamic system is presented. The Periodicity-Ratio describes the degree of periodicity of motion and can be conveniently used to distinguish a nonperiodic motion from a regular vibration or oscillation and to diagnose whether or not a motion is chaotic and the other irregular responses of the nonlinear dynamic systems, without plotting any figures. The analyses on the irregular behavior of nonlinear dynamic systems with the implementation of the Periodicity-Ratio will be demonstrated.

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

Direct integration variance prediction of random response of nonlinear systems

Inverse substructuring method has been recently proposed and applied for inverse analysis of the dynamical response of product transport system. The component-level frequency response functions (FRFs) and the coupling dynamic stiffness for facilitating the cushioning packaging design are all predicted from only the system-level FRFs. However, the system-level FRFs from coupling degree of freedoms may not be measured accurately because of the difficulties of vibration excitation and response measurement for the coupled interface between packaged product and vehicle within the limited accessible space. The aim of this paper is to develop a new FRF-based indirect inverse substructuring method for the analysis of the dynamic characteristics of a three-substructure coupled product transport system without measuring system-level FRFs at the coupling degree of freedoms. By enforcing the dynamic equilibrium conditions at the coupling coordinates and the displacement compatibility conditions, a closed-form analytical solution to inverse sub-structuring analysis of multi-substructure coupled product transport system is derived based on the relationship of easy-to-monitor component-level FRFs and the system-level FRFs at the coupling coordinates.. The proposed method is validated by a lumped mass-spring-damper model, and the predicted coupling dynamic stiffness is compared with the direct computation, showing exact agreement. Then, the FRF tests of a physical prototype of multi-substructure coupled product transport system are performed to further check the accuracy of the suggested method. The method developed offers an approach to predict the unknown coupling dynamic stiffness from measured FRFs purely. The proposed method may help to obtain the main controlling factors and contributions from the various structure-borne paths for product transport system. Copyright © 2014 John Wiley & Sons, Ltd.

In periodic forced response analysis of nonlinear structures, most of the time analytical solutions cannot be obtained due to the complex behavior of the nonlinearity and/or due to the number of nonlinear equations to be solved. Therefore, numerical methods are widely used. For periodic forced response analysis of nonlinear systems, generally Harmonic Balance Method (HBM) or Describing Function Method (DFM), which transform the nonlinear differential equations into a set of nonlinear algebraic equations, are used. In the literature, there exist several nonlinear algebraic equation solvers based on Newton’s method which have different convergence properties and computational expense. In this paper, comparison of computational performance of different nonlinear algebraic equation solvers are studied where, solvers with different convergence order are selected based on the number of Jacobian matrix and vector function evaluations. In order to compare the performance of these selected nonlinear solvers, a lumped parameter model with cubic stiffness nonlinearity is considered. Several case studies are performed and nonlinear solvers are compared to each other in terms of solution time based on error tolerance used.

Large degree of uncertainties may exist simultaneously in system parameters and external excitations of engineering structures. To capture the performance of such nonlinear multi-degree-of-freedom structures is still a great challenge in stochastic dynamics. In the present paper, the probability density evolution method is adopted and extended to reduce the dimension of parametric FPK equation of an uncertain-parameter structure subjected to additively white noise process. Numerical examples validate the proposed algorithm. Problems to be further studied are discussed.

This is the first part of a paper on an innovative approach of studying general nonlinear structural dynamic systems. In this part, the theoretical background and some numerical simulations are presented. Firstly, a frequency domain nonlinear compensation method is proposed. It can be used to recover the linear path of a general non-linear system as long as the Volterra expansion is held between the input and the output. Secondly, a new nonlinear element location scheme is developed based on the system linear path identification. Lastly, as a natural extension in the development of the contents of this paper, an in-situ nonlinear restoring force state mapping technique is proposed. The use of these new methods is demonstrated through several numerical examples of either single or multiple DOF systems incorporating some typical types of nonlinearities.