Nonlinear Frequency Response Functions Research Articles

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

There are two major contributions which are described in this paper. Firstly, a new graphical user interfaces interpretation tool for the non-linear frequency response function is proposed. The tool is developed by pv-wave package to provide better ways and user friendly of displaying, interpreting, and analysing of non-linear systems with significant non-linear effects. Secondly, the application of the non-linear auto-regressive moving average model with eXogenous input (NARMAX) identification methodology has been fully demonstrated and shown to have great potential in modelling real systems. To demonstrate the improved visualisation and interpretation of the multi-dimensional non-linear frequency response functions and the application of the NARMAX methodology, two cases have been analysed in detail as bench test to obtain the generalised higher-order frequency response functions for both continuous time non-linear differential equations and identified discrete time NARMAX models. The results obtained have shown that frequency response functions computed directly from the fitted models are very consistent with the actual frequency response functions.

Non-linear system identification and fault diagnosis using a new GUI interpretation tool

There are two major contributions which are described in this paper. Firstly, a new graphical user interfaces (GUI) interpretation tool for the non-linear frequency response functions (FRF) is proposed. The tool is developed by pv-wave package to provide better ways and user friendly of displaying, interpreting and analysing of non-linear systems with significant non-linear effects. Secondly, the application of this identification tool for fault detection, identifying in the civil engineering. To demonstrate the improved visualisation and interpretation of the multi-dimensional non-linear FRF and the application of the non-linear auto-regressive moving average model with exogenous input (NARMAX) methodology, two engineering applications have studied including a non-linear oscillator and fault diagnosis in civil engineering structures. The results obtained have shown that the interpretation of the higher-order FRF has been comprehensively studied and non-linear effects have been related back to the physical models of the systems.

Frequency response of a nonlinear acoustical resonator of arbitrary shape

The frequency response of a nonlinear acoustical resonator is investigated analytically and numerically. First, natural frequencies of a resonator whose shape is close to cylindrical are studied analytically by perturbation. An asymptotic formula is derived for the natural frequencies as a function of resonator shape. The solution at first order in the perturbation shows that each natural frequency can be shifted independently via appropriate spatial modulation of the resonator wall. Numerical calculations for resonators of different shapes establish the limits of the asymptotic formula. Second, the nonlinear interactions of modes in a nearly cylindrical resonator are investigated with Hamiltonian formalism as a basis for studying wave interaction in a resonator of arbitrary shape. A nonlinear frequency response function for the fundamental mode is derived by assuming that nonlinear interaction occurs only with the second harmonic. Bent tuning curves are predicted which lean up or down depending on whether the second-harmonic frequency is above or below the corresponding natural frequency, respectively. A fully nonlinear 1-D numerical code is used to verify the analytical result for the frequency response. [Work supported by ONR.]

Dynamic Wavelet and Equivalent Models

The representation of nonlinear dynamic wavelet models in the form of an equivalent global model which is valid over the operating range of the system is investigated. The results are used to analyse and interpret the nonlinear wavelet models using nonlinear frequency response functions.

Spectral analysis for nonlinear wave forces

The spectral analysis of nonlinear wave forces is investigated based on the recently introduced Dynamic Morison equation. This equation accounts for nonlinearities associated with history or vortex shedding effects accurately, in contrast to the standard Morison equation which has undesirable features in this respect. An expression for the response power spectrum is initially derived in terms of the higher-order nonlinear frequency response functions and the spectrum of the input velocity. It is then shown how this can be evaluated for the Dynamic Morison model to yield an expression for the output power spectrum in terms of the coefficients of the time-domain differential equations and properties of the input. A comparison of the output spectra computed using traditional methods and the new expression of the response spectrum over several data sets is included. An interpretation of the Morison equation response spectrum is finally given.

ANALYSIS OF NONLINEAR SYSTEM USING NARMA MODELS

The purpose of this paper is to develop a systematic method of time-domain nonlinear system identification using NARMA model. In the beginning the Hilbert transform test was used for nonlinearity, then an orthogonal parameter estimation algorithm is applied to a NARMA model which represents the nonlinear system model. Finally the nonlinear frequency response functions were computed. Application of the methodology to a combined Duffing and Van Der Pol nonlinear system and the seismic response data of Van Nuys building to Whittier and Northridge earthquakes are studied.

Spectral and bispectral analysis for single‐ and multiple‐input nonlinear phreatic aquifer systems

In general, stochastic partial differential equations and/or boundary conditions describing flow in phreatic aquifers are nonlinear. This paper uses second‐order perturbation to approximate the response of a phreatic aquifer system subject to multiple random inputs. The approach is equivalent to assuming quadratic nonlinearities in the original nonlinear phreatic aquifer system. Using frequency domain representations, the resulting model is decomposed into linear and nonlinear (quadratic) frequency response functions (FRF). For Gaussian input, the response of a nonlinear phreatic aquifer system is non‐Gaussian and is typically skewed. Thus the variance spectrum is insufficient to describe the response process. The bispectrum, defined as the Fourier transform of the third‐order cumulant sequence, represents the distribution of skewness with frequency and further serves as a measure of the nonlinearity of the system. Given input processes of aquifer recharge and stream stage fluctuations, the linear and quadratic FRF are used to construct the spectrum and bispectrum of the response process. The role of nonlinearity in a stream‐aquifer system is examined.

Large flexural vibrations of thermally stressed layered shallow shells

A dynamic nonlinear theory for layered shallow shells is derived by means of the von Karman-Tsien theory, modified by the generalized Berger-approximation. Moderately thick shells with polygonal planform composed of multiple perfectly bonded layers are considered. The shell edges are assumed to be prevented from in-plane motions and are simply supported. A distributed lateral force loading is applied to the structure, and additionally, the influence of a static thermal prestress, corresponding to a spatial distribution of cross-sectional mean temperature, is taken into account. In the special case of laminated shells made of transversely isotropic layers with physical properties symmetrically distributed about the middle surface, a correspondence to moderately thick homogeneous shells is found. Application of a multi-mode expansion in the Galerkin procedure to the governing differential equation, where the eigenfunctions of the corresponding linear plate problem are used as space variables, renders a coupled set of ordinary time differential equations for the generalized coordinates with cubic as well as quadratic nonlinearities. The nonlinear steady-state response of shallow shells subjected to a time-harmonic lateral excitation is investigated and the phenomenon of primary resonance is studied by means of the “perturbation method of multiple scales’. A unifying non-dimensional representation of the nonlinear frequency response function is presented that is independent of the special shell planform.

Reconstruction of linear and non-linear continuous time models from discrete time sampled-data systems

A new approach for identifying continuous time models from discrete time sampled-data records is presented. The proposed method involves estimating and validating a discrete time model, linear or non-linear, based on sampled data records, evaluating the discrete time linear and non-linear frequency response functions and then curve fitting to the frequency response data to yield a continuous time model. No numerical differentiation and integration is involved and hence higher derivatives of input and output data records are avoided. Errors which would be introduced by the numerical approximation of differentiation and integration are therefore eliminated. The orthogonal estimator which is introduced to curve fit to the complex frequency response functions provides information on the model structure and the unknown parameter values for linear and non-linear continuous time models. The advantage of this approach is that non-linear differential equation models which can be related to the physical behaviour of the system can be readily computed from discrete time data.

Nonlinear flexural vibrations of layered plates

By means of a higher-order plate-bending theory developed by Whitney and Pagano along with Berger's hypothesis, large amplitude forced vibrations of moderately thick laminated specially orthotropic plates are investigated. The theory includes shear deformation and rotatory inertia in the same manner as Mindlin's theory for isotropic homogeneous plates. The in-plane forces due to large deflections are assumed to be constant within the plate domain. Considering time-harmonic forcing a Kantorovich-Galerkin procedure provides the formulation of this problem in the lower band of the frequency domain. In the case of laminates made of isotropic layers an analogy to thin homogeneous plates is given, which is complete in the case of polygonal planforms and hard hinged supports. Furthermore, the plate deflection is determined by the solution of two (second-order) Helmholtz Klein Gordon boundary value problems. Inserting these results into a proper domain integral leads to Berger's normal force. This problem-oriented strategy renders the nonlinear frequency response functions of deflection of the undamped layered plate.

Interpretation of non-linear frequency response functions

Analytical and graphical methods of interpreting generalised frequency response functions for non-linear systems are derived. It is shown that non-linear phenomena can be classified into intra-kernel and inter-kernel interference and that worst-case responses can be computed. The results are illustrated using several discrete- and continuous-time non-linear systems.

Spectral analysis for non-linear systems, Part I: Parametric non-linear spectral analysis

A new theory of spectral analysis for non-linear systems is introduced. The method consists of estimating the parameters in a NARMAX model description of the system and then computing the generalised frequency response functions directly from the estimated model. The paper is divided into three parts. Part I introduces a methodology for estimating the models and computing the frequency response functions. Part II concentrates on the interpretation of the non-linear frequency response functions and Part III which will be published in the next issue describes a series of case study examples and illustrates in detail how to apply the algorithms to real systems.

Spectral analysis for non-linear systems, Part II: Interpretation of non-linear frequency response functions

In Part I of this series new methods of parametric spectral analysis for a wide class of non-linear systems were introduced. In this, the second part, the interpretation and properties of the non-linear frequency response functions are discussed and illustrated by example.

Rolling Element Bearing Vibration Transfer Characteristics: Effect of Stiffness

Vibration transmission through rolling element bearings was investigated in order to aid signal interpretation for use in machinery condition monitoring studies. The relationship between bearing nonlinear stiffness and frequency response function was derived for bearings of types commonly used in rotating machinery. By considering the contact deformation of individual elements the stiffness characteristic for a complete bearing was evaluated for the case of a radially applied load. Variation in stiffness of the complete bearing as elements moved through the load zone was calculated for bearings of a specified type and size. These values were also used to dynamically model typical rolling element bearing nonlinear behavior. Derived nonlinear stiffness coefficients were substituted into the equations of motion for a model bearing. Solution was obtained by digital integration, and the bearing frequency response function was then evaluated for discrete increments of static loading. Rolling element bearing frequency response was found to be very dependent on load. For operation under light load the bearings had especially strong nonlinear vibration transfer characteristics. At this operating condition a small increase in radial load produced a significant change in the bearing transfer characteristic (due to an increase in resonance frequency).