Estimation of Nonlinear Contributions in Human Controller Frequency Response Functions

Frequency response function estimation techniques and the corresponding coherence functions: A review and update

Measurement sciences counts numerous applications wherein one wishes to characterize the dynamics of a system. In case this characterization is intended non-parametrically the Frequency Response Function (FRF) is instrumental to achieve this goal. In the presence of nonlinearities of the system, the linear dynamics are accessible through the FRF which is known as the best linear approximation (BLA) of the nonlinear system. Once the BLA is determined and a periodic excitation was applied, one may describe the nonlinear distortion and the measurement noise characteristics. The description of the nonlinear distortion and measurement noise remains currently restricted to establishing its variance of both sources of error.In this paper, we go a step further in the characterization of the stochastic properties of the nonlinear distortion. Indeed, we show that the for a class of nonlinear systems, the nonlinear distortion acts as a stochastic gain contribution to the measured BLA depending on the input signal’s root mean square (RMS) value. Moreover we reveal that the random gain follows a log-normal distribution which allows improved uncertainty bounds on the FRF in the presence of nonlinear distortions.

Engineers and scientists want mathematical models of the observed system for understanding, design, and control. Many mechanical and civil structures are nonlinear. This paper illustrates a combined nonparametric and parametric system identification framework for modelling a nonlinear vibrating structure. First step of the process is the analysis: measurements are (semiautomatically) preprocessed, and a nonparametric best linear approximation (BLA) method is applied. The outcome of the BLA analysis results in nonparametric frequency response function, noise and nonlinear distortion estimates. Second, based on the information obtained from the BLA process, a linear parametric (state-space) model is built. Third, the parametric model is used to initialize a complex polynomial nonlinear state-space (PNLSS) model. The nonlinear part of a PNLSS model is manifested as a combination of high-dimensional multivariate polynomials. The last step in the proposed approach is the decoupling: transforming multivariate polynomials into a simplified, alternative basis, thereby significantly reducing the number of parameters. In this work a novel filtered canonical polyadic decomposition (CPD) is used. The proposed methodology is illustrated on, but of course not limited to, a ground vibration testing measurement of an F16 aircraft.KeywordsMIMO systemsNonlinearityDecouplingSystem identificationGround vibration testing

In many engineering applications the level of nonlinear distortions in frequency response function (FRF) measurements is quantified using specially designed periodic excitation signals called random phase multisines and periodic noise. The technique is based on the concept of the best linear approximation (BLA) and it allows one to check the validity of the linear framework with a simple experiment. Although the classical BLA theory can handle measurement noise only, in most applications the noise generated by the system -- called process noise -- is the dominant noise source. Therefore, there is a need to extend the existing BLA theory to the process noise case. In this paper we study in detail the impact of the process noise on the BLA of nonlinear continuous-time systems operating in a closed loop. It is shown that the existing nonparametric estimation methods for detecting and quantifying the level of nonlinear distortions in FRF measurements are still applicable in the presence of process noise. All results are also valid for discrete-time systems and systems operating in open loop.

In experimental modal analysis, the use of characterized excitation and the choice of window functions are fundamental to the estimation of system frequency response function (FRF) data. It greatly influences the quality of the resulting FRF measurements. Discussed are combination methods of several commonly-used excitation signals with window function, which is relative to lightly damped systems. Study shows that burst random signal with Rectangular works well for lightly damped structures, and pure random with Hanning will bring about higher error. A modal testing of white car body was done. Experimental parameters analysis are given to demonstrate the study result presented above. So it poses a practicable method for improving the accuracy of frequency response function (FRF) measurements of lightly damped structure by using proper excitation signal with appropriate window function.

Nonparametric frequency response function (FRF) estimators based on nonlinear averaging techniques are proposed. The suitability of such estimators for FFT based signal analyzers is studied. Under mild assumptions, it is shown that their bias is a function of the fourth-order moments of the perturbing errors on the input-output Fourier coefficients while the bias of the classical H/sub 1/ estimator is a function of the second-order moments. The performances of these estimators are analyzed in different situations: uncorrelated input-output perturbing errors (i.e., open-loop measurements), mutually correlated input-output disturbances (i.e., closed-loop measurements) and in presence of outliers. Analytical expressions are given for the bias when the input-output errors are zero-mean normal distributed (and mutually correlated). >

When controlling an aircraft the pilot perceives the motion kinematics in different ways. For instance, he sees the motion due to changes in the outside visual scene or a display and he feels the motion with his vestibular organ in the inner ear or from sensors in his skin. In this paper the pilot’s use of these different cues in a manual aircraft control task is studied using both single-loop and multi-loop identification methods. The task of the pilot was to control the roll-axis of an aircraft with double integrator dynamics. Two kinds of control tasks were analyzed: the disturbance task, where the pilot had to compensate for a disturbance working on the aircraft, and the target-following task, where the pilot had to minimize the difference of the roll-angle with a reference. The cues presented to the pilot were the error at a central display and the actual roll-angle on a peripheral display. In the single-loop experiments the total, lumped, frequency response function is identified. In the multi-loop method two frequency response functions are identified separately, representing the feedback of the central visual and the peripheral visual cues. The experiment showed that due to the addition of peripheral visual the response to the central visual decreases. The central visual also generates more phase in case of the target task. The estimation of the peripheral frequency response function with the multiloop method was difficult, which could be due to the choice of the forcing functions. Pilot behavior did not change when applying the multi-loop identification method, a result that paves the way for employing this method in other multi-sensory environments.

A ‘Sub-PolyMAX’ method is proposed in this paper not only for estimating modal parameters, but also for identifying structural flexibility by processing the impact test data in narrow frequency bands. The traditional PolyMAX method obtains denominator polynomial coefficients by minimizing the least square (LS) errors of frequency response function (FRF) estimates over the whole frequency range, but FRF peaks in different structural modes may have different levels of magnitude, which leads to the modal parameters identified for the modes with small FRF peaks being inaccurate. In contrast, the proposed Sub-PolyMAX method implements the LS solver in each subspace of the whole frequency range separately; thus the results identified from a narrow frequency band are not affected by FRF data in other frequency bands. In performing structural identification in narrow frequency bands, not in the whole frequency space, the proposed method has the following merits: (1) it produces accurate modal parameters, even for the modes with very small FRF peaks; (2) it significantly reduces computation cost by reducing the number of frequency lines and the model order in each LS implementation; (3) it accurately identifies structural flexibility from impact test data, from which structural deflection under any static load can be predicted. Numerical and laboratory examples are investigated to verify the effectiveness of the proposed method.

Frequency response function (FRF) estimation from measured data is an essential step in the design, control, and analysis of complex dynamical systems, including thermal and motion systems. Especially for systems that require long measurement time, missing samples in the data record, e.g., due to measurement interruptions, often occur. The aim of this article is to achieve accurate identification of nonparametric FRF models of periodically excited systems from noisy output measurements with missing samples. An identification framework is established that exploits a wavelet-based transform to separate the effect of the missing samples in the time domain from the system characteristics in tre frequency domain. The framework encompasses both a time-invariant and a time-varying wavelet-based estimator, which provides different mechanisms to address the missing samples. Experimental results from a thermodynamical system confirm that the estimators enable accurate identification.

Flexible multibody system simulation allows for fast and adequate investigation of the dynamics of mechanical systems. But in case of a system response with large deformations the time response does not uncover the causes, i. e. the resonances of the system. The identification of the systems eigenfrequencies gives more insight in resonance phenomena, but in case of periodic time-variant systems the often used snap-shot-eigenfrequencies do not reveal the real system dynamics, which has to be described by more than only one frequency response function. Based on the formulation of a flexible multibody system and the theory of ordinary linear periodic differential equations, partial frequency response functions, describing the real characteristics of a periodic system, are calculated and compared to the snapshot-frequency response functions.

In modal testing, the measured Frequency Response Functions (FRFs) are often affected by transducer mass effects. Especially in Multi-Input-Multi-Output (MIMO) modal test, multiple sensors are employed and the effects of additional masses are more significant. This paper deal with removing masses effects of multiple sensors from the measured FRFs. The proposed method offers some advantages over the available techniques in that extra FRFs measurements with different configurations are not required during the correction process. First, a general correction formulation is derived theoretically. Then, validations of the proposed methods are demonstrated using simulated data. It demonstrates a good performance in numerical simulation. To investigate the performance of the proposed method in practical application, further simulations are presented by employing noise-polluted data. And it is shown that the accuracy of correction results will be affected to some extent by the noise. It is thus suggested that the measured FRFs be preprocessed using the curve-fitting procedure or noise reduction processing before applying the proposed method. Since the FRFs measurements required for corrections refer to different excitation points, it requires moving the exciter to different locations. This is usually easy to achieve in hammer impact testing. However, moving the exciter is usually inconvenient in shaker modal testing. Further work is necessary to extend this proposed method so as to make it also applicable for shaker modal test case.

The use of block overlapping and time windowing in the estimation of frequency response functions of linear systems is considered. The systems are subjected to stationary white random excitation and there is no external noise. The sources of errors for the bias and variance of the frequency response are identified to be due to the Fourier transforms of the transient terms or end effects that arise due to non-periodic signals. Using a pure time delay as an example for the system, it is shown that there is a limit to the improvement achievable in reducing the variance as the block overlap is increased.

Experimental modal analysis (EMA) with oblique excitation (i.e. oblique impact testing) is useful in improving the long test time problem of conventional EMA with normal excitation (i.e. triaxial normal impact testing), in order to extract all the important dynamic characteristics of a 3D complex structure. In this study, a new methodology involving a vector projection method is introduced to find the driving point frequency response function (FRF) in the oblique direction, without the need for a special fixture with an obliquely oriented impedance head. Hence, this presents a low cost and practical solution for scaling the mode shape, as compared to the traditional approach. Moreover, the concurrent forces characteristic of oblique excitation are used in the development of the theoretical relationship between FRFs with oblique excitation and those with normal excitation. This is important for the validation of the oblique impact testing result, such as the FRF and modal parameter estimations. The experimental results show that oblique impact testing has reliable and effective results, as compared with the triaxial normal impact testing, in terms of the FRF correlation, natural frequency discrepancy, modal damping ratio error and modal assurance criterion of the unit modal mass mode shape.

During transportation, protective packaging is subjected to random dynamic compressive loads that arise from random vibrations generated by the vehicle. The ability of the protective packaging to withstand these dynamic compressive loads depends on the environmental vibration levels, the nominal stresses and the material's characteristics. Previous research has shown that cumulative damage, in the packaging system under random dynamic compression, will result in a change in the overall stiffness of the system. This change is manifested as a shift in the system's fundamental resonant frequency. Natural frequency estimates are often extracted using a least squares regression curve fit applied to an estimate of the system's frequency response function. Frequency response function estimates are generally obtained using the Fourier transform with a single input/single output (SISO). This approach is suitable for many applications; however, it is not well suited to non-linear systems subjected to non-stationary excitation where the vibration level (overall root-mean-square value) can vary. This paper investigates the use of an optimised reverse multiple input/single output algorithm for reliably tracking variations in the condition of packaging elements subjected to excitation with varying magnitude (root-mean-square). Results are presented from the analysis of physical experiments performed on expanded polystyrene cushions as well as empty corrugated paperboard containers. The experiments performed using the polystyrene samples were designed to limit natural variation in the system's natural frequency; whereas the paperboard samples were allowed to naturally damage under dynamic loading. Copyright © 2015 John Wiley & Sons, Ltd.

As an alternative to operational modal analysis and classical experimental modal analysis (EMA), a novel method was introduced previously, namely impact-synchronous modal analysis (ISMA). The effectiveness ISMA on rotor and structural dynamic systems has been proven in previous literature. More recently, an automated impact device (AID) was introduced which utilized tachometer pulse as initiation signal and its effectiveness on ISMA was proven. An attempt to further enhance this device in term of equipment and cost is then proposed by replacing the tachometer with the in-use tri-axial accelerometer through utilizing the filtered response of cyclic load component as an initiation signal to control the impact device, which is also the primary aim for this study. Prior to modal testing, accuracy of this device is illustrated at desired phase angles of 0 deg, 90 deg, 180 deg, and 270 deg. Subsequently, frequency response function (FRF) estimations obtained for ISMA using enhanced AID has demonstrated the suppression capabilities of this device on disturbances, i.e., reduction of 93.58% at 30 Hz and 57.78% at 60 Hz, resulting in a high correlation for signature assurance criterion (SAC) and cross signature assurance criterion (CSAC). Modal parameters extracted from the EMA and ISMA using impact device are presented and compared, for the first three natural modes of the test rig. It is found that natural frequencies are deviating by less than 6%, whereas modal assurance criterion (MAC) values between the mode shapes of the two tests are found to be above 0.9.