Order Frequency Response Functions Research Articles

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.

Novel vibration structural health monitoring technology for deep foundation piles by non‐stationary higher order frequency response function

Novel vibration structural health monitoring technology for deep foundation piles by non‐stationary higher order frequency response function

Nonlinear parameter estimation in damping with volterra series through harmonic probing

In this work nonlinear system identification procedure, based on Volterra series, is presented to distinguish a damping nonlinearity from stiffness nonlinearity, using measured first and third harmonic amplitude characteristics. First and higher order volterra kernel synthesis formulations and frequency response functions (FRFs) have been developed in damping and stiffness nonlinearity. The characteristics of first and higher order harmonic amplitudes at various excitation level is studied for the both the nonlinearity. This paper shows comparison study between the Volterra series, Runge-Kutta fourth order in analysis of damping and stiffness nonlinearity in the mechanical systems. Nonlinearity can lead to various system behaviour, like jump phenomenon, stable and unstable region, super harmonic resonances. Using first and higher order harmonic amplitudes, formulated from Volterra series response representation, nonlinear parameters are estimated through the recursive iteration method.

Frequency Domain Updating of Thin-Walled Workpiece Dynamics Using Reduced Order Substructuring Method in Machining

The structural dynamics of thin-walled parts vary as the material is removed during machining. This paper presents a new, computationally efficient reduced order dynamic substructuring method to predict the frequency response function (FRF) of the workpiece as the material is removed along the toolpath. The contribution of the removed mass to the dynamics of the workpiece is canceled by adding a fictitious substructure having the opposite dynamics of the removed material. The equations of motion of the workpiece are updated, and workpiece FRFs are evaluated by solving the hybrid set of assembled equations of motion in frequency domain as the tool removes the material between two consecutive dynamics update steps. The orders of the initial workpiece structure and the removed substructures are reduced using a model order reduction method with a newly introduced automatic master set selection criterion. The reduced order FRF update model is validated with peripheral milling tests and FRF measurements on a plate-shaped workpiece. It is shown that the proposed model provides ∼20 times faster FRF predictions than the full order finite element (FE) model.

Nonlinear frequency response analysis of forced periodic operation of non-isothermal CSTR using single input modulations. Part II: Modulation of inlet temperature or temperature of the cooling/heating fluid

Periodic operation of a non-isothermal CSTR subject to a single input modulation is analyzed considering a nth order reaction and using the nonlinear frequency response (NFR) method, introduced in our previous publications. The method is based on deriving the asymmetrical second order frequency response function (FRF) and analyzing its sign. In Part I of this paper, the periodic modulations of inlet concentration or flow-rate of the reaction stream were investigated. In this Part II, periodic operations are analyzed for modulations of the temperature of the inlet reaction stream or the temperature of the heating/cooling fluid. Conditions that need to be fulfilled in order to achieve process improvement through periodic operation are identified. The method is applied for the same numerical example taken from literature which was used in Part I. The results obtained by the NFR method are compared with the results of numerical simulations. Good agreement is obtained, except for forcing frequencies close to the resonant frequency. In these cases, even the sign of the DC component was not predicted correctly.

Nonlinear frequency response analysis of forced periodic operation of non-isothermal CSTR using single input modulations. Part I: Modulation of inlet concentration or flow-rate

Periodic operations of a non-isothermal CSTR with n-th order reaction, subject to a single input modulation, is analysed using the nonlinear frequency response (NFR) method, introduced in our previous publications. The method is based on deriving the asymmetrical second order frequency response function (FRF) and analysing its sign. In Part I of this paper, periodic operation with modulation of the inlet concentration or flow-rate of the reaction stream is analysed. As a result, conditions regarding the reaction order, process parameters and frequency of the input modulation are identified that need to be fulfilled in order to achieve process improvement through the periodic operation compared to conventional steady state operation. The method is applied for a numerical example from literature and the results obtained by the NFR method are compared with the results of numerical simulation. Good agreement is obtained, except for imposed forcing frequencies close to the resonant frequency and high forcing amplitudes.

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.

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.

Nonlinear frequency response analysis of dehydration phenomena in polymer electrolyte membrane fuel cells

Dehydration phenomena in a PEM fuel cell were investigated by nonlinear frequency response analysis (NFRA) in a differential H2/H2 cell. The linear H1,0 spectra, which are equal to classic EIS spectra, showed not only an increase of the membrane resistance but also an increase of the anode reaction resistance, caused by dehydration leading to the decrease of the protonic conductivity of the polymer network in the catalyst layer. With this, active sites with long protonic pathes to the membrane become inactive. In order to further clarify this effect, modelling work was used. Therefore, proton transport was incorporated into an existing model of a differential H2/H2 cell. Finally, the key features of NFRA spectra under dehydration and CO poisoning are compared in order to discuss the suitability of NFRA for unambiguous diagnosis of PEMFC. It can be seen that while the linear spectrum is not sufficient to distinguish between both cases, the second order frequency response functions can be used for discrimination.

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.

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.

Nonlinear frequency response analysis of PEM fuel cells for diagnosis of dehydration, flooding and CO-poisoning

Membrane dehydration, fuel cell flooding and anode catalyst poisoning by carbon monoxide are diagnosed in a single polymer electrolyte fuel cell (PEMFC) using nonlinear frequency response analysis (NFRA). A sinusoidal perturbation of high amplitude is applied to the fuel cell current and the resulting voltage is analysed using the concept of higher order frequency response functions. It is shown that the linear part of the system response corresponds to classical Electrochemical Impedance Spectra (EIS), which are not sufficient to clearly distinguish between different fuel cell failures. Therefore, the nonlinear behaviour is additionally taken into account in the form of the second order frequency response function. With this, it is possible to distinguish unambiguously between the three analysed PEM fuel cell failures.

Nonlinear single track model analysis using Volterra series approach

In this paper, an original approach based on the Volterra series theory is applied in order to analyse a nonlinear single track model, which is considered to describe the vehicle dynamics behaviour in the nonlinear domain. This model is based on a polynomial approximation up to the third order of the Pacejka formula that describes the full tyre behaviour. The analysis of the model is carried out using a truncated form of the Volterra series; this allows the extraction of an analytical formulation of the nonlinear response characteristics. The analysis is focused on the extraction of the first order frequency response function expression and the understeer angle curve vs lateral acceleration, which characterises the vehicle typology and stability. The resulting equations and illustrations in both the cases are presented.

Estimation of single solute adsorption isotherms applying the nonlinear frequency response method using non-optimal frequencies

In order to estimate single solute adsorption isotherms, the nonlinear frequency response (FR) of a chromatographic column is analyzed experimentally and evaluated using the concept of higher order frequency response functions (FRFs) based on the Volterra series and generalized Fourier transform. In this case study, it has been investigated the adsorption of ethyl benzoate on octadecyl silica from a mixture of methanol and water (60:40) as a solvent. Experiments are performed using a standard gradient HPLC unit. For estimation of adsorption isotherms by the nonlinear FR method the column inlet concentration is changed in a nearly sine waveform around several steady-state concentrations. Using this method the first three local derivatives of a single solute adsorption isotherm are estimated from the low frequency asymptotes of the corresponding functions, i.e. the phase and first order derivative of the FRFs. For an accurate estimation of isotherm coefficients periodical experiments should be preformed for frequencies below a certain critical frequency. This is the frequency needed for approaching the low frequency asymptotic behaviour of the corresponding functions close enough, so that errors due to the non-feasibility of experiments with zero frequency can be neglected. Unfortunately, depending on the properties of the system, it can happen (as for the system investigated here) that experiments for the critical frequency would be too long and cannot be realized. In order to study the loss of accuracy of the nonlinear FR method, when it is applied for non-optimal frequencies, experiments are performed for frequencies approximately one order of magnitude higher than the critical frequency required to evaluate the FRF phases. The obtained isotherm model coefficients are compared with the ones estimated using conventional frontal analysis as a reference method. The isotherms determined by two methods are similar, however a closer look reveals that peaks predicted under overloading conditions differ.

Evaluation of the potential of periodically operated reactors based on the second order frequency response function

A new, fast and easy method for analysing the potential for improving reactor performance by replacing steady state by forced periodic operation is presented. The method is based on Volterra series, generalized Fourier transform and the concept of higher-order frequency response functions (FRFs). The second order frequency response function, which corresponds to the dominant term of the non-periodic (DC) component, G 2( ω, − ω), is mainly responsible for the average performance of the periodically operated processes. Based on that, in order to evaluate the potential of periodic reactor operation, it is enough to derive and analyze G 2( ω, − ω). The sign of this function defines the sign of the DC component and reveals whether a performance improvement by cycling is possible compared to optimal steady state process. The method is used to analyze the periodic performance of a continuous stirred tank reactor (CSTR), a plug flow tubular reactor (PFTR) and a dispersive flow tubular reactor (DFTR), after introducing periodic changes of the input concentrations. A homogeneous, n-th order reaction is studied under isothermal conditions.

Theoretical investigation of the adsorption of a binary mixture in a chromatographic column using the nonlinear frequency response technique

The nonlinear frequency response of a chromatographic column for the adsorption of two dissolved components is analyzed using the concept of higher order frequency response functions (FRFs) which is based on the Volterra series and generalized Fourier transform. By applying this concept a nonlinear model of a system is replaced by an infinite series of the FRFs of the first, second, etc. order. The FRFs up to the third order are derived theoretically starting from the equilibrium-dispersive model, which is used for description of a chromatographic column, and applying the harmonic probing method. The functions that relate outlet concentration changes of each component to the corresponding inlet concentration changes are derived. At the inlet of a chromatographic column, it is considered: (a) the concentration change of one of the components keeping the concentration of the other component constant and (b) the concentration change of both components keeping their ratio constant. The FRFs are calculated numerically for different steady-state concentrations and relative mixture compositions. It has been found that, despite certain differences in initial conditions, the FRFs exhibit similar behavior. For higher frequencies, the amplitudes of the FRFs tend to zero and phases to −∞. In the low frequency range, which is of interest for investigation of equilibrium parameters, these functions have similar behavior, but tend to different asymptotic values. Correlations between coefficients of competitive adsorption isotherms, i.e. partial isotherm derivatives, and the derived FRFs are established. This theoretical result offers the potential to use the analysis of the nonlinear frequency response of a chromatographic column for estimation of competitive adsorption isotherms.

Analysis of orthotropic bearing non-linearity using non-linear FRFs

Among other critical conditions in rotor systems the large non-linear vibration excited by bearing non-linearity causes the rotor failure. For reducing this catastrophic failure and predictive detection of this phenomenon the analysis of orthotropic bearing non-linearity in rotor system using higher order frequency response functions (HFRFs) is conducted and is shown to be theoretically feasible as that of non-rotating structures. The complex HFRFs based on the Volterra series are newly developed for the process and investigated their features by using the simple forms of the FRFs associated with the forward and the backward modes.