Generalized Frequency Response Functions Research Articles

A Buffeting-Net for buffeting response prediction of full-scale bridges

In order to predict power spectral density (PSD) and root of mean squares (RMS) of buffeting responses of full-scale bridges in real operation conditions, a neural network-based model in frequency domain (Buffeting-Net) is proposed. Inspired by the mechanism of buffeting, the model is composed of two sub-neural networks that output the buffeting force and the generalized frequency response function. These two sub-neural networks are connected according to the model of buffeting response in the frequency domain to output the predicted spectrum of response. Sub-neural networks are constructed by bi-directional gated recurrent unit neural networks for their capacity of modeling complex mappings among sequences. Monitoring data of two long-span bridges are used for model validation, and the accuracy of prediction is evaluated by the cosine similarity of the shape of power spectrum and the relative error of RMS responses. The prediction results of Buffeting-Net are also compared with the traditional FEM-based buffeting method. The results indicate that trained models can make much higher accurate predictions for the bridges in real operational conditions including typhoon and season wind than regular finite element model-based buffeting analysis method. The generalization of the Buffeting-Net is validated, the result indicates the Buffeting-Net can predict the buffeting response with high prediction accuracy even when the wind speed is much greater than that in training.

Auxiliary harmonic excitation generalized method for higher-order analysis of linear structure under multiple non-stationary non-Gaussian excitations

In the classical random vibration analysis, the frequency-domain analysis method, especially the time-frequency domain analysis method based on the frequency domain analysis method, is the most commonly analysis method. However, the time-frequency domain analysis method is challenging when dealing with multiple non-stationary non-Gaussian excitations. In this paper, an accurate and efficient auxiliary harmonic excitation generalized method for higher-order analysis of linear structures under multiple non-stationary non-Gaussian excitations is proposed. Firstly, the expression for the higher-order moment spectrum of response is theoretically deduced by the mode superposition method, in which the time-varying generalized impulse response function and the time-varying generalized frequency response function are introduced. Secondly, according to the relationship between the response of harmonic excitation and the time-varying generalized frequency response function, the expression for the high-order moment spectrum of response is reconstructed in terms of the response of harmonic excitation and the higher-order moment spectrum matrix of the excitations, in which the response of the time-varying auxiliary harmonic excitation can be evaluated efficiently based on the time-domain explicit formulation method. Finally, two examples are investigated to verify the accuracy and efficiency of the proposed method.

Auxiliary harmonic excitation generalized method for random vibration analysis of linear structures under non-stationary Gaussian excitation

Frequency-domain methods are the most fundamental and effective methods for non-stationary random vibration analysis. However, the existing frequency-domain methods often involve truncation for degree of mode or decomposition of the power spectrum in multi-correlation conditions, which may have impact on the computational accuracy and efficiency of the methods. To avoid the truncation for degree of mode and the decomposition of the power spectrum, an accurate and efficient auxiliary harmonic excitation generalized method is proposed for random vibration analysis of linear structures under non-stationary Gaussian excitation. First, several existing frequency-domain analysis methods are investigated and discussed. Second, the generalized impulse response function, the generalized frequency response function, and the evolutionary generalized frequency response function are proposed. Meanwhile, the physical meaning of the evolutionary frequency response function is re-investigation, in which the response for the evolutionary harmonic excitation and the evolutionary frequency response function proves equivalent. Subsequently, according to the physical meaning, a novel and easy-to-implement algorithm of the response PSD is proposed. The response for the evolutionary auxiliary harmonic excitation can be quickly evaluated based on the time-domain explicit formulation method. Finally, two examples are investigated to verify the accuracy and efficiency of the proposed method.

Application of Generalized Frequency Response Functions and Improved Convolutional Neural Network to Fault Diagnosis of Heavy-duty Industrial Robot

The combination of nonlinear spectrum and convolutional neural network (CNN) is efficient for fault diagnosis of nonlinear system . However, in traditional method, the nonlinear spectrum calculation was accomplished by identification algorithm outside the CNN, which reduced the diagnosis efficiency. To solve this problem, a novel CNN with the function of spectrum calculation and fault diagnosis is designed, in which the spectrum calculation network and the fault diagnosis network are connected in series. By extracting the optimized parameters of network, the nonlinear spectrum based on generalized frequency response function (GFRF) is obtained in the former network. Then, the GFRF spectrum is automatically put into the latter network for feature extraction and diagnosis. Hence, after determining the structure of the CNN, only by system input and output, the fault diagnosis can be realized, which avoids the complex process in traditional method. What's more, a new error cost function model is designed to guide the network parameters optimization in the direction of feature classification , which is conductive to improve the diagnosis accuracy. The proposed network model is applied to the heavy-duty industrial robot system, and the best performance is demonstrated by several experiments.

Structural identifiability analysis of nonlinear time delayed systems with generalized frequency response functions

Structural identifiability analysis of nonlinear time delayed systems with generalized frequency response functions

Power spectral density analysis for nonlinear systems based on Volterra series

A consequence of nonlinearities is a multi-harmonic response via a mono-harmonic excitation. A similar phenomenon also exists in random vibration. The power spectral density (PSD) analysis of random vibration for nonlinear systems is studied in this paper. The analytical formulation of output PSD subject to the zero-mean Gaussian random load is deduced by using the Volterra series expansion and the conception of generalized frequency response function (GFRF). For a class of nonlinear systems, the growing exponential method is used to determine the first 3rd-order GFRFs. The proposed approach is used to achieve the nonlinear system’s output PSD under a narrow-band stationary random input. The relationship between the peak of PSD and the parameters of the nonlinear system is discussed. By using the proposed method, the nonlinear characteristics of multi-band output via single-band input can be well predicted. The results reveal that changing nonlinear system parameters gives a one-of-a-kind change of the system’s output PSD. This paper provides a method for the research of random vibration prediction and control in real-world nonlinear systems.

Axial stiffness identification of a ball screw feed system using generalized frequency response functions

Contact of balls in screw nuts and bearings causes the nonlinearity of a ball screw feed system. The identification of nonlinear stiffness helps to understand and control the feed system. The axial vibration of the ball screw feed system can be descripted by a single-degree-of-freedom (SDOF) system with polynomial nonlinear terms. Since generalized frequency response functions (GFRFs) can be expressed in terms of linear and nonlinear system parameters, one can estimate these parameters based on the measured GFRFs. In this effort, both single-tone and multitone harmonic inputs are employed to measure GFRFs, and a simple search algorithm is proposed to find the suitable frequency set of the multitone input. Parameter identification based on these two methods has been compared by numerical simulation and experiment. Since the response spectrum measured from vibration experiment of a ball screw system has even and odd harmonics, it is suitable to simplify the restoring force with square and cubic nonlinearity form. Static experiment is also conducted to verify the identified parameters, and the results show that the method based on single-tone inputs performs effectively in identifying axial stiffness of a ball screw feed system, but the method based on multitone inputs gets the incorrect results in practical application

Fault Diagnosis of Nonlinear Analog Circuit Based on Generalized Frequency Response Function and LSSVM Classifier Fusion

For fault diagnosis of nonlinear analog circuit, a novel method based on generalized frequency response function (GFRF) and least square support vector machine (LSSVM) classifier fusion is presented. The sinusoidal signal is used as the input of analog circuit, and then, the generalized frequency response functions are estimated directly by the time-domain formulations. The discrete Fourier transform of measurement data is avoided. After obtaining the generalized frequency response functions, the amplitudes of the GFRFs are chosen as the fault feature parameters. A classifier fusion algorithm based on least square support vector machine (LSSVM) is used for fault identification. Two LSSVM multifault classifiers with different kernel functions are constructed as subclassifiers. Fault diagnosis experiments of resistor-capacitance (RC) circuit and Sallen Key filter are carried out, respectively. The results show that the estimated GFRFs of the circuit are accurate, and the fault diagnosis method can get high recognition rate.

Comparison of Stimuli for Nonlinear System Response Classification

As part of the development of an automated virtual design classification approach for nonlinear structural dynamics, alternative excitation functions are evaluated with respect to their overall performance and efficiency in feature-based response analysis. Robust design of nonlinear structures requires analysis of extensive parameter variations. Both the character of the stimulus and feature metrics used are central to the performance of a response classification approach. The main purpose of this study is to compare stimulus candidates with respect to their efficiency in response classification. A deterministic multilevel, multifrequency stepped-sine periodic test function is used as a baseline. Order-wise differences between generalized and linearized system frequency response functions are evaluated by a selected feature metric to allow categorization into primary, sub and super harmonic responses, as well as odd and even order response distortions. An alternative excitation function type is the pseudorandom phase multi-sine. Its robust variant estimates the best linear approximation of the generalized frequency response function and related nonlinear and noise variances, which can be used for response classification. The fast variant of this method further detects and classifies occurring even and odd order nonlinear responses using a hypothesis test. This article describes the application of these three methods to a virtually running two-piece rotor shaft model. Time response signals from simulated test parameter variations are used to calculate selected nonlinear feature metric values. The total simulation and measurement time, as well as the predictive performance in a few typical nonlinear response cases are evaluated.

A novel method of combining generalized frequency response function and convolutional neural network for complex system fault diagnosis.

To solve the problem of low accuracy in traditional fault diagnosis methods, a novel method of combining generalized frequency response function(GFRF) and convolutional neural network(CNN) is proposed. In order to accurately characterize system state information, this paper proposed a variable step size least mean square (VSSLMS) adaptive algorithm to calculate the second-order GFRF spectrum values under normal and fault states; In order to improve the ability of fault feature extraction, a convolution neural network (CNN) with gradient descent learning rate and alternate convolution layer and pooling layer is designed to extract the fault features from GFRF spectrum. In the proposed method, the second-order GFRF spectrum of each state of Permanent Magnet Synchronous Motor (PMSM) is obtained by VSSLMS; Then, the two-dimension GFRF spectrum, which is regarded as the gray value of the image,will be further transformed into image. Finally, the CNN is trained with learning rate by gradient descent way to realize the fault diagnosis of PMSM. Experimental results indicate that the accuracy of proposed method is 98.75%, which verifies the reliability of the proposed method in application of PMSM fault diagnosis.

A novel method of combining nonlinear frequency spectrum and deep learning for complex system fault diagnosis

A novel fault diagnosis method is proposed for complex systems by combining nonlinear frequency spectrum and stacked denoising auto-encoders (SDAE). In order to solve the problem of large calculation amount of generalized frequency response functions (GFRF), one-dimensional nonlinear output frequency response functions (NOFRF) are used to obtain nonlinear frequency spectrum. In order to overcome the problem of weak ability of fault features extraction, stacked denoising auto-encoders (SDAE) neural network is adopted to extract the fault features from nonlinear frequency spectrum. In this novel method, four orders nonlinear frequency spectrum of each state of Permanent Magnet Synchronous Motor (PMSM) are obtained by identification algorithm; Then, choosing suitable sampling points from four orders frequency spectrum to construct high-dimensional data; Finally, stacked denoising auto-encoders (SDAE) neural network is designed to realize the output of fault classification. Simulations indicate that the proposed method has good real-time performance and high diagnosis accuracy.

Gaussian process regression for the estimation of generalized frequency response functions

Bayesian learning techniques have recently garnered significant attention in the system identification community. Originally introduced for low variance estimation of linear impulse response models, the concept has since been extended to the nonlinear setting for Volterra series estimation in the time domain. In this paper, we approach the estimation of nonlinear systems from a frequency domain perspective, where the Volterra series has a representation comprised of Generalized Frequency Response Functions (GFRFs). Inspired by techniques developed for the linear frequency domain case, the GFRFs are modelled as real/complex Gaussian processes with prior covariances related to the time domain characteristics of the corresponding Volterra series. A Gaussian process regression method is developed for the case of periodic excitations, and numerical examples demonstrate the efficacy of the proposed method, as well as its advantage over time domain methods in the case of band-limited excitations.

Volterra-series-based equivalent nonlinear system method for subharmonic vibration systems

ABSTRACTSubharmonics generation in the nonlinear system, directly using the traditional finite Volterra series, cannot generally represent the aimed system. In this paper, a new approach is presented, which is an extension of single finite Volterra series for representation and analysis of the subharmonic vibration system based on equivalent nonlinear system. The equivalent nonlinear system, which is constructed by pre-compensating the subharmonic vibration system with the super-harmonic nonlinear model, yields the input–output relation between the virtual source and the response of the aimed nonlinear system. Orthogonal least square method is employed to identify the truncated order of Volterra series and predominant Volterra kernels of the equivalent nonlinear system. The MGFRFs (modified generalised frequency response functions) of the equivalent nonlinear system is obtained from the data of the virtual source and response, and verified by comparing the response estimated by the MGFRFs with its true value. Therefore, the aimed subharmonic vibration system can be analysed by taking advantage of a truncated Volterra series based on the equivalent nonlinear system. Numerical simulations were carried out, whose results have shown that the proposed method is valid and feasible, and suitable to apply on representation and analysis of subharmonic vibration systems.

Inverse modeling approach for parametric frequency domain analysis of an electrohydraulic system

Abstract This paper presents parametric frequency domain analys.isf an electrohydraulic system, in particular, with respect to the hydraulic supply pressure. The supply pressure is a design parameter of significance such that it affects control bandwidth, load capability and system durability. However, intensive analysis on this parameter has not been reported yet due to the nonlinear dynamic features of an electrohydraulic system. In this paper, the inverse generalized frequency response functions are derived and used to conduct multi-dimensional Fourier analysis of the control input and to investigate its relation to the supply pressure. In addition, the hydraulic energy consumption is analyzed with respect to the supply pressure. The analysis reveals that the supply pressure is a critical design parameter that determines the degree of nonlinearity and trade-off between the control bandwidth and energy consumption.

Research on Fault Evolution Feature Extraction and Identification of Sun Gear Cracks in Planetary Gearbox Based on Volterra High-Order Kernel Generalized Frequency Response Graphic Analysis

In this paper, relying on the Volterra series nonlinear system model and the high-order kernel Hilbert’s reconstructed kernel fast solved algorithm, a fault feature frequency domain identification method based on Volterra high-order kernel generalized frequency response graph analysis is proposed. Firstly, the method uses the system input and output vibration signals to determine the Volterra model. Then, the Volterra high-order kernel function is solved quickly by reproducing kernel Hilbert space method, and the generalized frequency response function is used to identify the model. Finally, multidimensional high-order spectral pattern analysis is used to separate and extract the fault and degree characteristic information implied by frequency and phase coupling in the third-order kernel function. Following the theoretical approach, in the experimental part, this paper uses the planetary gearbox fault loading test rig to complete the data collection and establishes the Volterra experimental model through the measured data. The generalized frequency responses of each order kernel function are compared and analyzed and the capability of distinguishing and the adaptability of different order kernel functions for the degree of crack failure are discussed. The effects of changing the memory length of the Volterra model and the order of the kernel function on the recognition result are verified. The final experimental results show that the use of reproducing kernel Hilbert space can effectively avoid the dimension disaster problem that occurs in the high-order kernel solution process. Moreover, the third-order kernel can describe more intuitively the nonlinear system model under multifactor coupling than the second-order kernel. Finally, Volterra series model the third-order kernel’s generalized frequency response can effectively distinguish between nondefective and faulty gears, and its resolution is enough to distinguish the degree of failure of gear cracks.

Estimation of Multi‐Order Spectra for Nonlinear Closed‐Loop Systems

AbstractIn this paper, a multi‐order spectra estimation method is proposed for a nonlinear closed‐loop system based on the Volterra series. Owing to the correlation between the noise and the input, the estimation accuracy is poor when the nonlinear spectra of the plant is obtained using the traditional estimation method. In order to overcome this problem, a two‐step scheme is used to estimate the multi‐order spectrum of a nonlinear system operating in closed‐loop. Firstly, the generalized frequency response functions (GFRFs) from the reference signal to the input of the plant are estimated, and they are used to simulate the noise‐free input spectra of the plant. Secondly, the GFRFs of the controlled plant are estimated using the noise‐free input spectra and the output spectra. Because the GFRFs are multidimensional functions, the required amount of calculation for the estimation is very large. To reduce computational complexity, a simplified GFRF model is adopted to estimate the multi‐order nonlinear spectrum of the plant. In this model, the GFRF is transformed to a one‐dimensional function. Two simulation experiments are provided to illustrate the proposed approach.

Nonlinear Frequency-Domain Analysis of the Transformation of Cortical Inputs by a Motoneuron Pool-Muscle Complex.

Corticomotor coherence in the beta and/or gamma bands has been described in different motor tasks, but the role of descending brain oscillations on force control has been elusive. Large-scale computational models of a motoneuron pool and the muscle it innervates have been used as tools to advance the knowledge of how neural elements may influence force control. Here, we present a frequency domain analysis of a NARX model fitted to a large-scale neuromuscular model by the means of generalized frequency response functions (GFRF). The results of such procedures indicated that the computational neuromuscular model was capable of transforming an oscillatory synaptic input (e.g., at 20 Hz) into a constant mean muscle force output. The nonlinearity uncovered by the GFRFs of the NARX model was responsible for the demodulation of an oscillatory input (e.g., a beta band oscillation coming from the brain and forming the input to the motoneuron pool). This suggests a manner by which brain rhythms descending as command signals to the spinal cord and acting on a motoneuron pool can regulate a maintained muscle force. In addition to the scientific aspects of these results, they provide new interpretations that may further neural engineering applications associated with quantitative neurological diagnoses and robotic systems for artificial limbs.

A Fast Computational Framework for Seismic Random Response of Multi-support Structures

This paper presents a fast computational framework for seismic random response of multi-support structures, of which computational efficiency handers its application to practical engineering. First of all, after structural modelling based on the finite element method, the large mass method is used to exert the load on the supports in the form of seismic acceleration history, which ensures that the equations of motion of the system are solved directly. Secondly, the impulsive load is exerted on the supports and corresponding dynamic analysis is carried out. On the basis of the impulsive response, a generalized frequency response function (FRF) can be obtained via the Fourier transform. Thirdly, according to classic formulation, auto power spectrum density (PSD) functions of structural response may be calculated, which involves the generalized FRFs and the PSD function matrix of the seismic accelerations at the supports. The proposed framework is verified by comparing its computational results with the analytical solution of a single DOF system. A 5-span bridge during earthquakes is taken as a case study. The results show that three factors influencing spatial variation of seismic ground motion should be taken into consideration to get safer random response of a multi-support structure.

Spectral Analysis of Electrohydraulic System

This paper presents spectral analysis of an electrohydraulic system. For a linear system, spectral analysis using a frequency response function (FRF) offers great insight into system dynamics and controls. The objective of this paper is to extend such benefits to the nonlinear electrohydraulic system. To achieve the objective, generalized frequency response functions (GFRFs) and output spectra of the electrohydraulic system are analyzed in frequency domain. In this paper, two different approaches are proposed to derive the GFRFs. In the first approach, the analytic GFRFs are derived from physical dynamics of the electrohydraulic system. Thus, the dynamic features of the electrohydraulic system can be explored with respect to the physical parameters explicitly in frequency domain. In the second approach, the experimental GFRFs are identified from frequency response data. Although the explicit relationship with the physical parameters is not available, they can predict the output spectrum without a priori knowledge of the electrohydraulic system. The proposed approaches are applied to derive the GFRFs analytically and experimentally for spectral analysis of an electrohydraulic system. Spectral analysis reveals the critical dynamic features of the electrohydraulic system in frequency domain, and it turns out to be crucial for system design, identification, and controls of the electrohydraulic system.

Nonlinear modeling and identification of proton exchange membrane fuel cell (PEMFC)

The nonlinear autoregressive moving average model with exogenous inputs (NARMAX) approach is employed to analyze time-frequency domains for the proton exchange membrane fuel cell (PEMFC) system. The time-domain nonlinear NARMAX model of PEMFC system is modeled from measured experimental input and output data. The outputs obtained by using the time-domain NARMAX model were confirmed by model validation tests which compare the model's predictive outputs with actual system outputs, in an interactive environment. This provides a potentially nondestructive tool for the modeling of a nonlinear PEMFC system. Then, the generalized frequency response function (GFRF) is employed in the frequency-domain, which is computed directly from the time-domain NARMAX model, to provide information about the types of nonlinearities in a PEMFC. The analysis revealed that the time-domain model provides better parameter estimation and a good forecasting ability. The frequency-domain response function is consistent with actual reaction mechanisms. The results illustrate that the time-frequency domain NARMAX approach can be effectively employed in modeling and identifying of PEMFC system.