Nonlinear Output Frequency Response Functions Research Articles

Nonlinear frequency response function: experimental study on gearbox fault detection under step load

For rotating machinery, excitation signals are difficult to measure, making it impossible to detect fatigue damage using nonlinear frequency response function. The purpose of this study is to propose a fatigue damage detection method for gearboxes under stepped load conditions based on normalized nonlinear output frequency response functions (NOFRFs). That is, using the vibration signals from the gearbox during the step loading to calculate the cross power spectrum, and therefrom estimating the normalized NOFRFs. This method is used to detect gearboxes containing prefabricated damaged gears or bearings. The research result shows that gear and bearing damage can lead to an increase in the index value of normalized NOFRFs. The different type of damage lead to the significant diversification of higher-order normalized NOFRFs, which further demonstrates that the method proposed in this article is not only effective in gearbox damage detection, but also has certain potential in fault classification. Then, experimental research for accumulated fatigue damage of gear was conducted. The results show that the normalized NOFRFs index is a better indicator of fatigue damage than the kurtosis index.

Application of nonlinear system identification and feature extraction in fault diagnosis of the cracked cantilever beam

The cantilever beam structure is extensively applied in civil, transportation, and mechanical fields. It is generally prone to structural damage like cracks arising from material defects, operation conditions, and environmental factors. Cracks in a cantilever beam generally induce nonlinear behavior, potentially compromising structural durability and even posing threats to production safety. The present study primarily focuses on online and quantitative fault diagnosis of cracks in the cantilever beam. A simulation model is established for the cantilever beam with varying cracks, where responses corresponding to excitation signals are collected. A Nonlinear Autoregressive with Exogenous inputs (NARX) model for the cantilever beam is identified, and Nonlinear Output Frequency Response Functions weighted contribution rate (WNOFRFs) are evaluated from the established NARX model. A WNOFRFs-associated index is extracted to quantify the severity of breathing cracks. Simulation results indicate that this approach can online diagnose and quantify the severity of crack faults to a certain extent. The applicability of this approach is further verified through experiments on a cracked cantilever beam.

An estimation method of normalized nonlinear output frequency response functions for fatigue damage detection using stochastic response signal

AbstractNonlinear output frequency response functions (NOFRFs) and its modified version, normalized NOFRFs (norm‐NOFRFs), are composed of multiple spectra, which can sensitively indicate the occurrence of fatigue damage in machinery. However, their application is restricted to a certain extent. Currently, neither NOFRFs nor norm‐NOFRFs can be estimated with only a single response signal. Aimed at this problem, the nonlinear auto‐regressive (NAR) model was introduced, and an estimation method of norm‐NOFRFs is proposed in this paper. Only one stochastic response signal is required in the proposed method, which removes the restrictions on the number and intensity of signals to identify norm‐NOFRFs, and expands the range of scenarios for using norm‐NOFRFs in damage detection. The feasibility and robustness of the proposed method are verified by the nonlinear model in numerical simulation. The effectiveness of revealing and quantifying the fatigue damage is verified by experimental study for three kinds of mechanical parts.

Fast evaluation of generalized associated linear equations (GALEs) for nonlinear systems characterization and compensation

Nonlinear Output Frequency Response Functions (NOFRFs), which are a one-dimensional extension of the linear Frequency Response Function (FRF) to nonlinear cases, have been introduced for nonlinear system characterization and compensation. The evaluation of NOFRFs is accomplished by solving a series of linear difference or differential equations, known as Generalized Associated Linear Equations (GALEs). However, the derivation of GALEs often includes numerous recursive and symbolic calculations, posing significant challenges in both practical applications and computer programming. In the present study, a fast-computing algorithm, based on combinatorial calculations as opposed to recursive and symbolic calculations, is proposed for fast evaluation of GALEs. This allows for the evaluation of NOFRFs in a more streamlined and straightforward manner. The applications of the proposed fast-computing algorithm for nonlinear system characterization and compensation are demonstrated through several case studies including a High Power Amplifier (HPA) and a Duffing system. The results of the case studies show the effectiveness of applying the GALEs and NOFRFs for system design and safety control across a wide range of engineering applications.

Fault diagnosis of rotor systems with breathing slant cracks based on nonlinear output frequency response functions

When it comes to practical engineering, there is a high likelihood of shafts developing slant cracks due to torque transmission. These cracks exhibit breathing characteristics, repeatedly opening and closing under gravity and radial forces. To diagnose rotor systems with breathing slant cracks, the nonlinear output frequency response functions (NOFRFs) method is introduced. Firstly, a dynamic model of rotor systems with breathing slant cracks is established, and the system's frequency spectra are obtained by solving the model with different crack angles, depths, and positions. Secondly, the NOFRFs values at different orders are derived based on the NOFRFs theory. The simulation results display increased nonlinear characteristics with crack angle and depth. The system's nonlinear characteristics are stronger when cracks are closer to the disk. The [Formula: see text] value for the first-order is sensitive to crack angles, the [Formula: see text] value for the second-order is sensitive to crack depths, and the [Formula: see text] value for the fourth-order is sensitive to crack positions. As a result, the NOFRFs response at different orders can detect variations in angles, depths, and positions of the breathing slant crack. Finally, experiments are conducted on a rotor system with breathing slant cracks, and the results match those of the simulation, indicating that the NOFRFs method is feasible for identifying breathing slant cracks.

Tool Condition Monitoring Based on Nonlinear Output Frequency Response Functions and Multivariate Control Chart

Tool condition monitoring (TCM) is a key technology for intelligent manufacturing. The objective is to monitor the tool operation status and detect tool breakage so that the tool can be changed in time to avoid significant damage to workpieces and reduce manufacturing costs. Recently, an innovative TCM approach based on sensor data modelling and model frequency analysis has been proposed. Different from traditional signal feature-based monitoring, the data from sensors are utilized to build a dynamic process model. Then the nonlinear output frequency response functions, a concept which extends the linear system frequency response function to the nonlinear case, over the frequency range of the tooth passing frequency of the machining process are extracted to reveal tool health conditions. In order to extend the novel sensor data modelling and model frequency analysis to unsupervised condition monitoring of cutting tools, in the present study, a multivariate control chart is proposed for TCM based on the frequency-domain properties of machining processes derived from the innovative sensor data modelling and model frequency analysis. The feature dimension is reduced by principal component analysis first. Then the moving average strategy is exploited to generate monitoring variables and overcome the effects of noises. The milling experiments of titanium alloys are conducted to verify the effectiveness of the proposed approach in detecting excessive flank wears of solid carbide end mills. The results demonstrate the advantages of the new approach over conventional TCM techniques and its potential in industrial applications.

Performance assessment of hydropower generator unit with the nonlinear transmissibility function method

The hydropower generator unit (HGU) is a complex nonlinear dynamic system, and how to utilize the monitoring data to achieve the performance assessment of the HGU is a challenge at present. In this paper, based on the linear transmissibility function and the nonlinear output frequency response function, the nonlinear transmissibility function (NTF) method is proposed to characterize the variation rule of the HGU performance, and the effectiveness of the proposed method is confirmed by the monitoring data from a hydropower station located in Guangxi, China. Firstly, the feature parameters related to the operation state of the unit are extracted from the monitoring data of the HGU. Secondly, the HGU system is divided into three subsystems according to the installation position of the sensors collecting these feature parameters, and the NTFs of them are estimated from multiple time scales. Finally, the performance of the HGU is comprehensively assessed utilizing the Fe indicator and the correlation transmissibility damage indicator. The analysis results demonstrate that the NTF method can reveal the performance variation trend and detect the early fault of the HGU, which is of great significance to the stable operation of the HGU.

Nonlinear spectrum feature fusion diagnosis method for RV reducer of industrial robots

RV reducers of industrial robots may be possibly subjected to wear and vibration impact and invalidated for keeping the repeatability due to long-time dynamic load in their life cycle. Any failure of RV reducers may decrease the reliability and lead to incredible loss due to the unexpectedly shutdown of the manufacturing system. Failures can cause nonlinear interference when occurring in RV reducers and pose a challenge to RV reducer fault diagnosis. Therefore, it is important to take proper measures to diagnose the failure in RV reducers and reduce the influence of nonlinear interference. Nonlinear spectrum features fusion method is proposed for diagnosing the faults of RV reducers in industrial robots. For the purpose of extracting fault features more effectively, nonlinear output frequency response function is employed to obtain nonlinear frequency spectrum for exactly describing the nonlinear mechanism of faulty phenomenon. Moreover, enhanced evidence theory with similarity measure is proposed to compute the basic probability assignment functions of evidences according to the similarity between different modes for improving the identification accuracy of different faults. RV reducer experiments with different faults are performed to simulate robot operating conditions for obtaining normal and fault data under a variable speed working condition. Among the 180 sets of test data including normal conditions, the diagnostic accuracy of the proposed nonlinear feature fusion method is 92.22% and greatly preferable to that of the ordinary frequency spectrum method. Experimental results demonstrated that the nonlinear spectrum method can effectively extract nonlinear features and the introduction of similarity can effectively improve the diagnostic accuracy in the proposed feature fusion method.

A novel nonlinear spectrum estimation method and its application in on-line condition assessment of bearing-rotor system

In this paper, a novel approach for nonlinear spectrum estimation is proposed and combined with Support Vector Machine (SVM) classifier to assess the condition of bearing-rotor systems, including fault types and severity. The proposed approach involves extraction and reconstruction of target harmonics, dynamic process modeling of the reconstructed signal based on a data-driven method, and extraction of nonlinear spectrum features in the form of nonlinear output frequency response functions (NOFRFs). The focus is on the influence of model stability and accuracy evaluation on the robustness of NOFRFs. Then the robustness of the NOFRFs based on the proposed estimation approach is compared with the conventional estimation approach using simulation cases and experiments. SVM is used to classify the extracted nonlinear spectrum features. Finally, the classification accuracy of NOFRFs evaluated by the proposed approach is compared with time-domain and frequency-domain features. Neighborhood Component Analysis (NCA) is used to select sensitive features and improve classification accuracy. The proposed condition assessment strategy does not require the selection of sensitive features, ensuring the stability of assessment results, which is also a significant advantage of the proposed strategy.

A Novel Data-Driven Feature Extraction Strategy and Its Application in Looseness Detection of Rotor-Bearing System

During the service of rotating machinery, the pedestal bolts are prone to looseness due to the vibration environment, which affects the performance of rotating machinery and generate potential safety hazard. To monitor the occurrence and deterioration of the pedestal looseness in time, this paper proposes a data-driven diagnosis strategy for the rotor-bearing system. Firstly, the finite element model of a rotor-bearing system is established, which considers the piecewise nonlinear force caused by pedestal looseness. Taking the vibration response as output and periodic unbalanced force as input, the system’s NARX (Nonlinear Auto-Regressive with exogenous inputs) model is identified. Then GALEs (Generalized Associated Linear Equations) are used to evaluate NOFRFs (Nonlinear Output Frequency Response Functions) of the NARX model. Based on the first three-order NOFRFs, the analytical expression of the second-order optimal weighted contribution rate is derived and used as a new health indicator. The simulation results show that compared with the conventional NOFRFs-based health indicator, the new indicator is more sensitive to weak looseness. Finally, a rotor-bearing test rig was built, and the pedestal looseness was performed. The experiment results show that as the degree of looseness increases, the new indicator SRm shows a monotonous upward trend, increasing from 0.48 in no looseness to 0.90 in severe looseness, a rise of 89.7%. However, the traditional indicator Fe2 has no monotonicity, which further verifies the sensitivity of the first three-order NOFRFs-based second-order optimal weighted contribution rate and the effectiveness of the proposed data-driven feature extraction strategy.

The evaluation of Nonlinear Output Frequency Response Functions based on tailored data-driven modelling for rotor condition monitoring

Poorly assembled or faulty rotors often behave nonlinearly. The accurate extraction of weak nonlinear features is a vital issue. This paper proposes to extract the partial dynamic properties of concern for nonlinear feature based rotor condition monitoring. The realization of this concept relies on a novel tailored data-driven NARX (Nonlinear Auto Regressive with eXogenous input) modelling approach. The tailored NARX model represents part of the system’s dynamic properties of concern. The details are first to use the harmonic product spectrum to determine the rotational speed of the inspected rotor system. After that, the least mean squares method is used to extract the harmonic components from the measured vibration signal. The low-order harmonics of concern are then reconstructed for the tailored NARX modelling. Then the Nonlinear Output Frequency Response Functions are evaluated from the tailored NARX model as features for rotor condition monitoring. Finally, the proposed method is validated by simulation and experiment cases. The results show that the obtained features based on the proposed tailored NARX modelling approach are more robust and more accurate than conventional methods when assessing rotor faults. The proposed approach provides a reference for the inspection of the assembly quality of components such as bearings and the on-line health monitoring of rotating machinery.

An effective approach based on nonlinear spectrum and improved convolution neural network for analog circuit fault diagnosis.

In this work, an effective approach based on a nonlinear output frequency response function (NOFRF) and improved convolution neural network is proposed for analog circuit fault diagnosis. First, the NOFRF spectra, rather than the output of the system, are adopted as the fault information of the analog circuit. Furthermore, to further improve the accuracy and efficiency of analog circuit fault diagnosis, the batch normalization layer and the convolutional block attention module (CBAM) are introduced into the convolution neural network (CNN) to propose a CBAM-CNN, which can automatically extract the fault features from NOFRF spectra, to realize the accurate diagnosis of the analog circuit. The fault diagnosis experiments are carried out on the simulated circuit of Sallen-Key. The results demonstrate that the proposed method can not only improve the accuracy of analog circuit fault diagnosis, but also has strong anti-noise ability.

Review on the Application of the Nonlinear Output Frequency Response Functions to Mechanical Fault Diagnosis

Key components of mechanical equipment are prone to defects or failures during operation, resulting in equipment performance degradation because of harsh working environments and abnormal working conditions. The faults may induce the mechanical structures to behave nonlinearity. Some linear methods cannot accurately describe systems’ nonlinear characteristics. Nonlinear output frequency response functions (NOFRFs) well reveal the generation mechanism of nonlinear phenomena. The main objective of this article is to summarize the development and state-of-the-art research of NOFRFs in mechanical fault diagnosis. The method of NOFRFs is briefly introduced. Associated indexes of NOFRFs are summarized and two case analyses on the performances of associated indexes are conducted. Recent applications of NOFRFs in mechanical fault diagnosis are analyzed from the following aspects: 1) direct diagnosis of NOFRFs; 2) diagnosis of NOFRFs-associated indexes; and 3) NOFRFs+ diagnosis. Finally, the limitation of NOFRFs in fault diagnosis is pointed out and future directions are given.

Fault mechanism analysis and diagnosis for closed-loop drive system of industrial robot based on nonlinear spectrum

To solve the problem of nonlinear characteristics neglecting and fault mechanism analysis lacking in fault diagnosis research, a new method of fault mechanism analysis and diagnosis based on nonlinear spectrum is proposed. Firstly, based on the Permanent Magnet Synchronous Motor (PMSM) model of robot, the first 4-order spectrums based on nonlinear output frequency response function (NOFRF) in different states are obtained by batch calculation method. Secondly, the high-frequency spectrum distribution rule of NOFRF spectrum in different states are analyzed. Finally, in the closed-loop simulation environment of robot, the identification method based on data-driven is adopted for NOFRF spectrum calculation to verify power loss fault of PMSM. Meanwhile, the fault diagnosis experiment is also carried out. The experimental results indicate that the key characteristics distribution rule of NOFRF spectrums in the real environment is consistent with the theoretical analysis results, and compared with the traditional fault feature extraction methods by output signal, the diagnosis with fault feature of NOFRF spectrum for industrial robot closed-loop drive system has the highest accuracy, which verifies the validity of NOFRF spectrum as the fault feature.

Weak Fault Feature Extraction Based on Weighted Contribution Rate of Nonlinear Output Frequency Response Functions and Stochastic Resonance for Rub-Impact Fault Diagnosis

Weak Fault Feature Extraction Based on Weighted Contribution Rate of Nonlinear Output Frequency Response Functions and Stochastic Resonance for Rub-Impact Fault Diagnosis

A parametric frequency domain approach to analysis and design of critical design parameters of nonlinear energy harvesting systems: Parametric output spectrum and power generation functions

Nonlinear energy harvesting systems have been extensively studied which usually involve quite a few critical design parameters to determine. However, a straightforward parametric design method for such a purpose is still far from developed. Traditional methods with harmonic balance or numerical simulation usually cannot reveal a straightforward relationship between the energy harvesting performance and system parameters, and in most cases face relatively large computational costs during parametric studies. To this aim, a novel Volterra series-based frequency domain method is established in this study with respect to a typical nonlinear harvesting system. A concise parametric characteristic approach, namely the linear and nonlinear characteristic output spectrum (lnCOS) method, is thus proposed to achieve parametric characteristic expression for the nonlinear output frequency response and power generation function (PGF) of the system, referred to as the lnCOS function and the Parametric-PGF respectively. Compared with the harmonic balance method, the vibration responses/power generation can be obtained in a more straightforward way without solving complex algebraic equations, and the lnCOS method directly links the power outputs to critical system parameters in a concise parametric way. It is for the first time to provide explicit expressions for vibration responses and power generations with respect to structural parameters of a nonlinear energy harvesting system, which cannot be achieved in a traditional way. Based on the obtained explicit expressions of the lnCOS function/Parametric-PGF, the sensitivity analysis of vibration response and power generation with respect to structural parameters can be directly performed. Our work provides an efficient way to characterize both the linear and nonlinear effects on a nonlinear energy harvesting system and demonstrates an efficient approach to the analysis and design of nonlinear energy harvesting systems.

Analysis of nonlinear vibrations and health assessment of a bearing-rotor with rub-impact based on a data-driven approach

Affected by manufacture, assembly, and operating conditions, the rotor is prone to rub-impact fault, resulting in instability of a system. The nonlinearity of a system response includes instability and fault characteristics. Based on this, this paper proposes a novel health assessment approach for bearing-rotor system that combines data-driven and NOFRFs (nonlinear output frequency response functions), revealing the intrinsic relationship between NOFRFs-based feature and rub-impact fault. Before utilizing the approach, the vibration and stability of the bearing-rotor with rub-impact are simulated and analyzed. Based on Hertz contact theory, the nonlinear bearing forces are obtained. Then the dynamic model of the rolling bearing-rotor system considering unbalance and rub-impact is established. The influence of rub-impact on the nonlinear vibration and stability of the bearing-rotor system is analyzed by combining the orbit, the Poincaré map, and the bifurcation diagram. A rolling bearing-rotor test rig is built, and the rub-impact experiment was completed. The data-driven model of the system is identified by the filtered vibration displacement. The NOFRF-based feature of the data-driven model is extracted. With the increase of rub-impact severity, the dispersion of feature gradually increases, and the stability of the system decreases. The standard deviation of the feature is solved as a new health indicator. The result shows it is more effective for the evaluation of rub-impact than the conventional indicators.

The estimation method of normalized Nonlinear Output Frequency Response Functions with only response signals under stochastic excitation

The nonlinearity degree of the system can be sensitively revealed by the methods based on Nonlinear Output Frequency Response Functions (NOFRFs). However, there is a lack of methods based on NOFRFs that work with only response signals under stochastic excitation, which limits the application of NOFRFs in practical engineering. In this study, a new concept of normalized NOFRFs is proposed, and a corresponding estimation method that employs only stochastic output response is established. Firstly, a method of estimating NOFRFs by Power Spectral Density (PSD) is established, and therefrom the normalized NOFRFs and a corresponding nonlinearity index are proposed. Then, by taking the PSD of the excitation signal as the input and the PSD of the response signal as the output, a general estimation method of normalized NOFRFs is built, and an output measured only estimation method of normalized NOFRFs is further deduced and established. In these two estimation methods, the assumption of white noise excitation is not required, and the estimated results are insensitive to the change of excitation intensity. Finally, the experiments of fatigue damage detection of three-point bending fatigue steel plate specimens and reducer box cases under stochastic excitation are carried out. The results demonstrate that, compared with the NOFRFs, the degree of fatigue damage can be effectively revealed by the normalized NOFRFs with output.

The Improved WNOFRFs Feature Extraction Method and Its Application to Quantitative Diagnosis for Cracked Rotor Systems.

During its operation, a rotor system can be exposed to multiple faults, such as rub-impact, misalignment, cracks and unbalancing. When a crack fault occurs on the rotor shaft, the vibration response signals contain some nonlinear components that are considerably tougher to be extracted through some linear diagnosis methods. By combining the Nonlinear Output Frequency Response Functions weighted contribution rate (WNOFRFs) and Kullback–Leibler (KL) divergence, a novel fault diagnosis method of improved WNOFRFs is proposed. In this method, an index, improved optimal WNOFRFs (IOW), is defined to represent the nonlinearity of the faulty rotor system. This method has been tested through the finite element model of a cracked rotor system and then verified experimentally at the shaft crack detection test bench. The results from the simulation and experiment verified that the proposed method is applicable and effective for cracked rotor systems. The IOW indicator shows high sensitivity to crack faults and can comprehensively represent the nonlinear properties of the system. It can also quantitatively detect the crack fault, and the relationship between the values of IOW and the relative depth of the crack is approximately positively proportional. The proposed method can precisely and quantitatively diagnose crack faults in a rotor system.

An improved NOFRFs-based fault feature extraction method and its application to quantitative diagnosis in rotor rub-impact

In this paper, the Jeffcott rotor model has been set up to extract and describe the nonlinear characteristics caused by the local rub-impact. During a mechanical fault diagnosis, the detection methods frequently lead to misjudgment only based on the output behaviors. Hence, nonlinear output frequency response functions (NOFRFs) have gradually developed. Like the linear transfer function, NOFRFs can reflect the relationship between the spectrums and the nonlinear system parameters. However, in the fault diagnosis of the rotor system, the current fault feature extraction methods based on NOFRFs can describe the damage severity tend qualitatively, but diagnostic results cannot be guaranteed by only analyzing the vibration response under a single rotational speed. In the present study, a definite amplification coefficient αn is clarified to keep the same magnification ratio of the high-order NOFRFs. It achieves the purpose that describes the amplified fault characteristics quantitatively. Furthermore, a novel fault diagnosis criterion—weighted contribution ergodic index (TR(n)), is proposed to consider system responses in a wide range of harmonic excitations, so a more comprehensive system characteristic can be obtained to make the diagnosis results more reliable. Simulation results have demonstrated that the new fault diagnosis method can enhance the sensitivity and stability to identify the rub-impact fault. Simultaneously, the experiments based on the rotor bench have verified the superiority of the new method in practical application.