Identification of Weakly Nonlinear Systems Using Describing Function Inversion

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.

ABSTRACTVibration-based structural damage detection has been the focus of attention by many researchers over the last few decades. However, most methods proposed for this purpose utilize extracted modal parameters or some indices constructed based on these parameters. A literature review revealed that few papers have employed Frequency Response Functions (FRFs) for detecting structural damage. In this paper, a technique is presented for damage detection which is based on measured FRFs. Proper Orthogonal Decomposition (POD) has been implemented on spatiotemporal responses in each frequency in order to reduce the dimension of the data. This is based on the concept that the forced harmonic response of a linear vibrating system can be fully captured utilizing a single basis vector. A different approach is also presented in this paper in which POD is applied to the frequency domain data. Operational Deflection Shapes (ODSs) have been decomposed using POD to localize the damage. The efficiency of the method is demonstrated through some numerical and experimental case studies.

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

Time–frequency characterization of nonlinear normal modes and challenges in nonlinearity identification of dynamical systems

Equivalent linear modal parameter estimation for a nonlinear aeroelastic system

With the growing structural complexity and growing demands on structural reliability, nonlinear parameters identification is an efficient approach to provide better understanding of dynamic behaviors of the nonlinear system and contribute significantly to improve system performance. However, the dynamic response at nonlinear location, which cannot always be measured by the sensor, is the basis for most of these identification algorithms, and the clearance nonlinearity, which always exists to degrade the dynamic performance of mechanical structures, is rarely identified in previous studies. In this paper, based on the thought of output feedback which the nonlinear force is viewed as the internal feedback force of the nonlinear system acting on the underlying linear model, a frequency-domain nonlinear response reconstruction method is proposed to reconstruct the dynamic response at the nonlinear location from the arbitrary location where the sensor can be installed. For the clearance nonlinear system, the force graph method which is based on the reconstructed displacement response and nonlinear force is presented to identify the clearance value. The feasibility of the reconstruction method and identification method is verified by simulation data from a cantilever beam model with clearance nonlinearity. A clearance test-bed, which is a continuum structure with adjustable clearance nonlinearity, is designed to verify the effectiveness of proposed methods. The experimental results show that the reconstruction method can precisely reconstruct the displacement response at the clearance location from measured responses at reference locations, and based on the reconstructed response, the force graph method can also precisely identify the clearance parameter.

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

An efficient algorithm based on group theory and the Woodbury formula for the dynamic responses of periodic structures

A system of linear interval equations is called solvable if each system of linear equations contained therein is solvable. In the main result of this paper it is proved that solvability of a general rectangular system of linear interval equations can be characterized in terms of nonnegative solvability of a finite number of systems of linear equations which, however, is exponential in matrix size; the problem is proved to be NP-hard. It is shown that three earlier published results are consequences of the main theorem, which is compared with its counterpart valid for linear interval inequalities that turn out to be much less difficult to solve.

Clearance turns up in a large number of engineering structures because of the errors during assembling, manufacturing, and wearing. The presence of clearance in engineering structures changes the normal dynamic response and will result in low precision and short lifetime. The clearance parameter identification of such nonlinear system is the prerequisite to control and eliminate the effect of clearance nonlinearity. In this paper, a derivative plot of probability density function (DPPDF) for displacement response is proposed to precisely identify the clearance value of continuous system, and the nonlinear subspace identification (NSI) method is modified to recognize the related contact stiffness based on the frequency response function (FRF) equations of continuous system. The DPPDF method is carried out by analyzing the distribution characteristic of displacement response, and the clearance value is derived through inspecting the probability density function (PDF) plot and the second derivative plot of the PDF. Based on the identified clearance, the clearance nonlinearity is regarded as external force, and the relationship between the dynamic responses and the external forces in frequency domain can be expressed as the form of FRF equations. Based on the FRF equations, the contact stiffness in continuous system is obtained with modified NSI method. This combined identification process is verified by a single-degree-of-freedom (SDOF) system and a cantilever beam system with clearances, and some influence factors of this identification process, including noise, transfer error, and force level, are discussed in detail. In the end, an experiment device with changeable clearance and contact stiffness was designed to conduct identification experiments, and the results show that the proposed methods perform effectively in identifying the clearance parameters.

A novel strategy to characterize and identify structural nonlinearities in MDOF systems based on reconstructing constant response tests from constant excitation tests is developed in this paper. Constant displacement frequency response functions (FRFs) can be measured by a stepped sine test where the displacement is controlled at every frequency of interest. In these FRFs, the nonlinear restoring force is effectively linearized and natural frequencies can be estimated by linear modal analysis. Using a series of constant displacement tests, the relationship of equivalent stiffness versus displacement can be established by curve fitting, and hence, the nonlinear stiffness characterized. This paper proposes a method to reconstruct the constant displacement FRFs from stepped sine tests with constant excitation; this avoids the requirement to control either the response or force amplitude leads to a faster and more stable testing program. Similarly, damping nonlinearities in structures can be characterized and identified by constant velocity tests reconstructed in a similar way. This approach of FRF reconstruction is mathematically simple and suitable for structures with weak nonlinearities. The method is demonstrated on a framed structure with unknown weak nonlinearities, and the nonlinear stiffness and damping parameters of the structure are identified and validated. The results demonstrate the feasibility and effectiveness of the approach and also show the potential for practical applications in engineering.

Iterative refinement of solution with biparameter for solving ill-conditioned systems of linear algebraic equations

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.

The paper presents a method for the identification of nonlinear system parameters by using an improved Genetic Algorithm and Grey Function. The paper firstly outlines several commonly used nonlinear identification methods such as RLS, RIV and COR and also their drawbacks. Then, a method based on the Genetic Algorithm and Grey Function is proposed and given in detail in the paper. Finally, a simulation experiment to TV set production data of an electronic factory was carried out. The simulations show that the method can gain good results and is also simple and effective.

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.