Higher-order Frequency Response Functions Research Articles

On the dynamic properties of statistically-independent nonlinear normal modes

Much attention has been given to the study of nonlinear normal modes (NNMs), a nonlinear extension to the eminently useful framework for the analysis of linear dynamics provided by linear modal analysis (LMA). In the literature, several approaches have gained traction, with each able to preserve a subset of the useful properties of LMA. A recently-proposed framework (Worden and Green, 2017) casts nonlinear modal analysis as a problem in machine learning, viewing the NNM as directions in a latent modal coordinate space within which the modal dynamics are statistically uncorrelated. Thus far, the performance of this framework has been measured in a largely qualitative way. This paper presents, for the first time, an exploration into the underlying dynamics of the statistically-independent NNMs using techniques from nonlinear system identification (NLSI) and higher-order frequency-response functions (HFRFs).In this work, the statistically-uncorrelated NNMs are found for two simulated nonlinear cubic-stiffness systems using a recently-proposed neural-network based approach. NLSI models are fitted to both physical and modal displacements and the HFRFs of these models are compared to theoretical values. In particular, it is found for both systems that the modal decompositions here permit an independent single-input single-output (SISO) representation that can be projected back onto the original displacements with low error. It is also shown via the HFRFs that the underlying linear natural frequencies of the modal dynamics lie very close to the underlying linear natural frequencies of the nonlinear systems, indicating that a true nonlinear decomposition has been identified.

Single Degree-of-Freedom Modeling of the Nonlinear Vibration Response of a Machining Robot

AbstractHighly dynamic machining forces can cause excessive and unstable vibrations when industrial robots are used to perform high-force operations such as milling and drilling. Implementing appropriate optimization and control strategies to suppress vibrations during robotic machining requires accurate models of the robot’s vibration response to the machining forces generated at its tool center point (TCP). The existing models of machining vibrations assume the linearity of the structural dynamics of the robotic arm. This assumption, considering the inherent nonlinearities in the robot’s revolute joints, may cause considerable inaccuracies in predicting the extent and stability of vibrations during the process. In this article, a single degree-of-freedom (SDOF) system with the nonlinear restoring force is used to model the vibration response of a KUKA machining robot at its TCP (i.e., machining tool-tip). The experimental identification of the restoring force shows that its damping and stiffness components can be approximated using cubic models. Subsequently, the higher-order frequency response functions (HFRFs) of the SDOF system are estimated experimentally, and the parameters of the SDOF system are identified by curve fitting the resulting HFRFs. The accuracy of the presented SDOF modeling approach in capturing the nonlinearity of the TCP vibration response is verified experimentally. It is shown that the identified models accurately predict the variation of the receptance of the nonlinear system in the vicinity of well-separated peaks, but nonlinear coupling around closely spaced peaks may cause inaccuracies in the prediction of system dynamics.

Bayesian model identification of higher-order frequency response functions for structures assembled by bolted joints

This paper proposes a procedure to identify a stochastic Bouc-Wen model for describing the dynamics of a structure assembled by bolted joints considering vibration data. The proposed identification approach is expressed into a Bayesian framework to take into account the data fluctuations related to uncertainties in the measurement process. The calibration of the model parameters uses the analytical expressions of the higher-order frequency response functions (FRFs) for approximating experimental measurements. The Metropolis-Hastings algorithm is employed for approximating posterior distributions. Once calibrated, the applicability of the probabilistic Bouc-Wen model is evaluated, and its dynamical behavior is compared with experimental measurements from the bolted structure. The results show that the stochastic version of the Bouc-Wen model can predict with adequate agreement, including hysteretic effects, the output of the jointed structure considering several excitation amplitudes.

Identification and parameter estimation of cubic nonlinear damping using harmonic probing and volterra series

Dynamic systems, such as vibration isolators, rotor-bearing systems, are inherently nonlinear and their dynamic behaviour often cannot be sufficiently explained or predicted by simple linear models. Presence of nonlinearity leads to certain characteristic behaviours in the response such as jump phenomenon, limit cycle, super-harmonic resonances and such behaviours can be accurately predicted only if the nonlinearity structure and related parameters are properly known. This emphasises the recently growing importance of nonlinear system identification. A majority of the identification works is based on a-priori knowledge of nonlinearity structure and most of them consider only stiffness nonlinearities, such as Duffing’s oscillator and bilinear oscillator. Not much work has been reported on nonlinearity structure identification for systems with damping nonlinearities. This paper, first discusses a systematic classification of nonlinearity structures based on first, second and third harmonic response amplitudes under harmonic excitation. Characteristics response for individual nonlinearity class is explained by Volterra series response formulation with higher order Frequency Response Functions. In the second part, a typical cubic damping nonlinearity is identified from cubic stiffness nonlinearity and an algorithm for estimating the nonlinear and linear damping parameters is developed. A new term called measurability ratio is introduced to show how it can help in deciding the most appropriate excitation frequency. Effect of truncating the Volterra series response on parameter estimation error is also studied for different excitation frequencies and varying excitation levels. It is shown that, with recursive iteration in computation of third harmonic amplitude, estimation accuracy can be further improved.

Nonlinear Frequency Response Analysis as a Tool for Identification of Adsorption Kinetics: Case Study—Pore-Surface Diffusion Control

In the present paper, the Nonlinear Frequency Response (NFR) analysis is applied for theoretical study of kinetics of adsorption governed by pore-surface diffusion. The concept of higher-order frequency response functions (FRFs) is used. Based on a nonlinear mathematical model for adsorption of pure gas and spherical adsorbent particles, the theoretical first- and second-order FRFs, which relate the adsorbate concentration in the particle to the surrounding pressure (F1(ω) and F2(ω,ω)), have been derived. The obtained FRFs have been simulated for different steady-state pressures and different ratios (between zero and one) of surface to pore diffusion coefficients. The results show that, unlike F1(ω), F2(ω,ω) exhibits features which unambiguously distinguish the pore-surface diffusion model from pure pore diffusion and micropore diffusion. Based on the characteristic features of F1(ω) and F2(ω,ω), a new methodology for direct estimation of the separate values of the pore and surface diffusion coefficients has been established.

Applications of higher-order frequency response functions to the detection and damage assessment of general structural systems with breathing cracks

Two types of cracks are often encountered in engineering structural systems and these are the open cracks and the breathing cracks. Existence of open cracks often leads to the loss of physical stiffness, resulting in a mostly linear structure with reduced load bearing capacity and vibration frequencies. The development of breathing cracks not only reduces structural stiffness, but tends to render the otherwise linear structure to become nonlinear, due to their bilinear stiffness characteristics associated with open and closed states. The nonlinear structural vibration responses can then be investigated and potentially employed to detect and identify breathing cracks within a structural system. In the present study, breathing cracks are modeled based on fracture mechanics from which bilinear stiffness values are obtained. These values are then incorporated into finite element models to compute the first- and second-order frequency response functions (FRFs) based on a proposed correlation technique which is both very accurate and resilient against measurement uncertainties. The existence of well-defined second-order FRFs has been firmly established for the bilinear oscillator, as well as a cantilevered beam with breathing cracks. By expressing the bilinear restoring force as a polynomial series, analytical derivation of second-order FRFs of general structural systems such as the GARTEUR AG11 structure with breathing cracks has been established for the first time. Further, a method of identification has been developed to identify the physical parameters of breathing cracks using second-order FRFs. With these new developments that are presented in this paper, a solid foundation has been laid for the potential applications of higher-order FRFs to damage identification and assessment of general real structural systems.

A new method for the accurate measurement of higher-order frequency response functions of nonlinear structural systems

Higher-order frequency response functions (FRFs) are important to the analysis and identification of structural nonlinearities. Though much research effort has been devoted recently to their potential applications, practical issues concerning the difficulty and accuracy of higher-order FRF measurement have not been rigorously assessed to date. This paper presents a new method for the accurate measurement of higher-order FRFs. The method is developed based on sinusoidal input, which is ideal for exciting a nonlinear structure into desired regimes with flexible control, and the correlation technique, which is a novel signal processing method capable of extracting accurate frequency components present in general nonlinear responses. The correlation technique adopted is a major improvement over Fourier transform based existing methods since it eliminates leakage and aliasing errors altogether and proves to be extremely robust in the presence of measurement noise. Extensive numerical case studies have been carried out to critically assess the capability and accuracy of the proposed method and the results achieved are indeed very promising. Interesting nonlinear behavior such as frequency shift and jump have been observed in first-, second- and third-order FRFs, as well as solitary islands which have been identified over which higher-order FRFs virtually do not change as input force amplitude varies. Higher-order FRFs over such solitary islands are essentially their theoretical counterparts of Volterra transfer functions which can be measured with very low input force and can be profitably employed for the identification of physical parameters of structural nonlinearities. Subsequently, a nonlinear parameter identification method has also been developed using measured higher-order FRFs and results are presented and discussed.

Higher-order FRFs and their applications to the identifications of continuous structural systems with discrete localized nonlinearities

Abstract Many modern structural systems are found to contain localized areas, often around structural joints and boundaries, where the actual dynamic behavior is far from linear. Such nonlinearities need to be properly identified so that they can be incorporated into the improved mathematical model used for design and operation. One approach to this task is to use higher-order frequency response functions (FRFs). Identification of structural nonlinearities using higher-order FRFs is currently an active and growing emerging research area and to date, much research has been conducted. However, most existing research seems to be confined to very simple nonlinear system models such as SDOF mass-spring-damper models or to the most, MDOF mass-spring chain models. For more general continuous nonlinear structures with discrete localized nonlinearities however, analytical derivation of higher-order FRFs remains unknown and as a result, identification of nonlinearities of such systems using measured higher-order FRFs becomes impossible to achieve. This missing link between higher-order FRFs and physical parameters of nonlinearities of continuous structures has to be established before any real progress can be possibly made. In this paper, a new novel method is developed which can be used to derive analytical higher-order FRFs of continuous structural systems with discrete localized nonlinearities. The method is generally applicable and theoretically exact, and it serves exactly as that missing link. Having established analytical higher-order FRFs which serve as the theoretical foundation for any subsequent identification, method of parameter identification of nonlinearity is then further developed. Important characteristics of higher-order FRFs are discussed, some of which are revealed the first time since there has never been higher-order FRFs derived from nonlinear continuous structures. Various numerical aspects on how to improve accuracy of identified nonlinear system parameters are discussed.

On the confidence bounds of Gaussian process NARX models and their higher-order frequency response functions

Abstract One of the most powerful and versatile system identification frameworks of the last three decades is the NARMAX/NARX 1 approach, which is based on a nonlinear discrete-time representation. Recent advances in machine learning have motivated new functional forms for the NARX model, including one based on Gaussian processes (GPs), which is the focus of this paper. Because of their nonparametric form, NARX models can only provide physical insight through their frequency-domain connection to Higher-order Frequency Response Functions (HFRFS). Because of the desirable properties of the GP-NARX form (no structure detection needed, natural confidence intervals), the analytical derivation of the HFRFs for the model is presented here for the first time. Furthermore, an algorithm for propagating uncertainty from the GP into the HFRF estimates is presented. A valuable by-product of the latter algorithm is a new test for nonlinearity, capable of detecting the presence of odd and even system nonlinearities. The new results are illustrated via two case studies; the first is based on simulation of an asymmetric Duffing oscillator. The second case study presents a validation of the new theory in the area of wave force prediction on offshore structures. This problem is one that has been considered by some of the authors before; the current paper takes the opportunity to highlight and correct a number of weaknesses of the original study in the light of modern best practice in machine learning.

Uncertainty Bounds on Higher-Order FRFs from Gaussian Process NARX Models

One of the most versatile and powerful algorithms for the identification of nonlinear dynamical systems is the NARMAX (Nonlinear Auto-regressive Moving Average with eXogenous inputs) approach. 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. In recent papers, one of the authors introduced a NARX (no noise model) formulation based on Gaussian Process (GP) regression and derived the corresponding expressions for Higher-order Frequency Response Functions (HFRFs). This paper extends the theory for the GP-NARX framework by providing a means of converting the GP prediction bounds in the time domain into bounds on the HFRFs. The approach is demonstrated on the Duffing oscillator.

Evaluation of periodic processes with two modulated inputs based on nonlinear frequency response analysis. Case study: CSTR with modulation of the inlet concentration and flow-rate

In our previous work, a new, fast and easy, nonlinear frequency response method for analysing potential improvements of reactor performance by forced periodic operation was presented. This method, which is based on Volterra series, generalized Fourier transform and the concept of higher-order frequency response functions (FRFs), gives an approximate value of the average process performance directly, without numerical simulation of the complete process. It was shown that the asymmetrical second order frequency response function, (G2(ω,−ω)) corresponds to the dominant term of the non-periodic (DC) component of the periodic steady-state response and determines the average performance of the periodic process. Thus, in order to evaluate the potential of a periodic reactor operation, it is enough to derive and analyse this function.In this work this method is extended to evaluating periodic operations with forced oscillations of two modulated inputs. In this case the nonlinear system has to be defined by three sets of frequency response functions, two of them correlating the output to each of the inputs and one set of cross-FRFs. The general methodology for this case is developed. It is further used to analyse the time-average performance of an isothermal continuous stirred tank reactor (CSTR) with forced periodic modulation of the inlet concentration and flow-rate, for a simple nth order homogeneous reaction. The analysis is performed for cases when the inlet concentration and flow-rate are modulated simultaneously. The optimal choice of the phase shift between the two inputs is discussed in detail.

A Volterra series approach to the frequency domain analysis of non-linear viscous Burgers’ equation

In this paper, higher order frequency response functions, based on the Volterra series, are employed to characterise the input-output behaviour of the non-linear viscous Burgers’ equation subject to sinusoidal excitation. First, a formal Volterra series representation for each spatial location is derived for the solution of Burgers’ equation with a boundary condition as the input to the system. Then a systematic method is presented to obtain the higher order frequency kernels of the Volterra series at each spatial location by solving a series of ordinary differential equations. It is shown that the convergence region of the individual harmonics with respect to the magnitude of the input excitation can be estimated by using these higher order kernels. The frequency characteristics of Burgers’ equation is investigated and compared with numerical simulation.

Analysis of Reciprocity Breakdown in Nonlinear Systems

The breakdown of the principle of reciprocity is a well-known phenomenon of nonlinear systems. A structure or system is said to exhibit reciprocity when the response at some point j to an input at some point i is identical to the response at point i when the same input is applied at point j. This paper seeks to explain this phenomenon by adopting a functional series representation which describes the input-output relationship. The frequency-domain Volterra Series representation utilises Higher-Order Frequency Response Functions (HFRFs) (generalisations of the linear FRF) to explain the behaviour of nonlinear systems. This breakdown in reciprocity may be observed through a breakdown in symmetry of HFRFs.

Efficient computation of higher order frequency response functions for nonlinear systems with, and without, a constant term

The behaviour of a nonlinear system can be profoundly affected by the presence of a constant or dc term in the system governing equation. These changes are reflected in the nonlinear frequency response characteristics of the system which provide a powerful insight into the system's dynamics. In this article, a new and efficient algorithm is presented for computing the higher order Volterra frequency response functions from nonlinear time-domain models that may contain a constant term. A comparison with previous methods is included to demonstrate the significant gains in computational efficiency that are achieved using the new method. The algorithm is applicable to systems modelled by nonlinear differential, or difference, equations and is easily automated. Several examples are used to illustrate the method, and to highlight the importance of dc terms in nonlinear system analysis.

On the use of non-linear vibrations and the anti-resonances of Higher-Order Frequency Response Functions for crack detection in pipeline beam

The identification of new scientific challenges, as well as the increasing high-performance computing support, indicates that the benefits of applying novel nonlinear techniques for crack detection will continue to grow. So, significant effort has been invested in recent years to develop effective techniques to detect crack in mechanical structures. The objective of this paper is to discuss and propose a robust diagnostic of damage based on non-linear vibrational measurements with particular regard to the Higher-Order Frequency Response Functions. An important observation is that the appearances of the non-linear harmonic components and the emerging anti-resonances in Higher-Order Frequency Response Functions can provide useful information on the presence of cracks and may be used on an on-line crack monitoring system for small levels of damage. Efficiency of the proposed methodology is illustrated through numerical examples for a pipeline beam including a breathing crack.

Nonlinear Dynamics and Damage Assessment of a Cantilever Beam With Breathing Edge Crack

Failures in structures and machine elements can be prevented through early detection of fatigue cracks using various nondestructive testing methods. Vibration testing forms one of the most effective and recent one among these methods. There are mainly two approaches to crack detection through vibration testing: open crack model and breathing crack model. The present study is based on breathing crack model, in which the nonlinear vibration response under harmonic excitation is formulated through Volterra series and higher order frequency response functions. Bilinear stiffness characteristic of a cracked cantilever beam is approximated by a truncated polynomial series and response amplitudes of various harmonics are investigated for both qualitative and quantitative characterization. A new procedure is suggested whereby the presence of a breathing crack in a structure can be first identified and then the severity of the damage can be estimated through harmonic probing.

Parameter estimation of Duffing oscillator using Volterra series and multi-tone excitation

Use of Volterra series in nonlinear system identification is well established now. The series represents response of a nonlinear system in a functional series form consisting of convolution integrals involving higher-order impulse response functions known as Volterra kernels. Multi-dimensional Fourier transforms of these Volterra kernels give the higher-order frequency response functions (FRFs). The measurement of these FRFs under harmonic excitation and their relationship with the first-order FRFs provide a basis for estimation of the nonlinear parameters. However, most of the methods employ single-tone excitation, which provide limited FRF measurement in a single experiment. In the present study, a novel procedure based on multi-tone excitation is presented for a typical Duffing oscillator and it is demonstrated that accurate estimation of both nonlinear and linear parameters is possible with fewer number of experiments.

Nonlinear Frequency Response of Electrochemical Methanol Oxidation Kinetics: A Theoretical Analysis

In this theoretical contribution, nonlinear frequency response analysis was applied for the investigation of electrochemical methanol oxidation. This technique expresses the input–output behavior of any weakly nonlinear system with the help of the Volterra series expansion and generalized Fourier transform into so-called higher order frequency response functions. These functions contain the system’s nonlinear fingerprint. They can be derived analytically from a nonlinear model. These functions can be obtained experimentally from the measurement of higher harmonics induced by a high amplitude sinusoidal perturbation of the system of interest. Frequency response functions up to the second order have been derived analytically for four different model varieties describing the kinetics of the electrochemical methanol oxidation. The first-order frequency response function corresponds to the reciprocal value of the well-known electrochemical impedance, which represents the linear part of the frequency response. This function does not contain sufficient information for discrimination between the different kinetic models. In contrast, the symmetrical second-order frequency response functions show differences in shape, which substantiate the availability of the theoretical prerequisites for model discrimination. A detailed parametric study for all four model variants has been performed. The results show that the basic features of the shapes of the amplitude spectra corresponding to the four models remain unique. The ubiquitousness of the qualitative differences between the competing models, for the whole set of parameters chosen for our analysis, suggests that the aforementioned amplitude spectra contain sufficient information for an unequivocal model discrimination.

Structural damage assessment in a cantilever beam with a breathing crack using higher order frequency response functions

Vibration measurements offer an effective, inexpensive and fast means of non-destructive testing of structures and various engineering components. There are mainly two approaches to crack detection through vibration testing; open crack model with emphasis on changes in modal parameters and secondly, the breathing crack model focusing on nonlinear response characteristics. The open crack model based on linear response characteristics can identify the crack only at an advanced stage. Researchers have shown that a structure with a breathing crack behaves more like a nonlinear system, similar to that of a bilinear oscillator and the nonlinear response characteristics can very well be investigated to identify the presence of the crack. In the present study, the bilinear restoring force is approximated by a polynomial series and a nonlinear dynamic model of the cracked structure is developed using higher order frequency response functions. The effect of crack severity on the response harmonic amplitudes are investigated and a new procedure is suggested whereby the crack severity can be estimated through measurement of the first and second harmonic amplitudes.

Fast evaluation of periodic operation of a heterogeneous reactor based on nonlinear frequency response analysis

The concept of higher-order frequency response functions (FRFs), which is based on Volterra series representation of nonlinear systems, is used to analyse the time-average performance of a perfectly mixed reactor subject to periodic modulation of the inlet concentration, for a simple n-th order heterogeneous catalytic reaction. The second order frequency response function G 2( ω,− ω), which corresponds to the dominant term of the non-periodic (DC) component, essentially determines the average performance of the periodic process. Thus, in order to evaluate the potential of a periodic operation, it is sufficient to derive and analyse the G 2( ω,− ω) function. The sign of this function defines the sign of the DC component and reveals whether the periodic operation is favourable compared to conventional steady state operation, or not. It will be shown that, for the case investigated, the sign of this function depends both on the reaction order and on the shape of the adsorption isotherm.