Relationship between harmonic balance method and non-linear output frequency response function approach

The concept of Nonlinear Output Frequency Response Functions (NOFRFs) was introduced in [1] for the analysis of nonlinear systems in the frequency domain where a least-squares (LS) method was proposed for the evaluation of the NOFRFs using system input-output data. However, this leastsquares (LS) based approach may experience numerical errors due to matrix inversion and consequently cannot produce accurate NOFRFs sometimes. In order to mitigate this problem, in the present study, a modified LS method (M-LS) is proposed to make sure that the information matrix to be inverted for the LS solution is appropriately constructed so as to improve the condition number of the information matrix and reduce the numerical errors. Simulation results have demonstrated the effectiveness of the newly proposed method and indicated that the new method will enable a more accurate evaluation of the NOFRFs and facilitate the frequency analysis of nonlinear systems directly using system input-output data.

The analysis of multidegree-of-freedom (MDOF) nonlinear systems is studied using the concept of nonlinear output frequency response functions (NOFRFs), which are a series of one-dimensional functions of frequency recently proposed by the authors to facilitate the analysis of nonlinear systems in the frequency domain. By inspecting the relationships between the NOFRFs of two consecutive masses in the locally nonlinear MDOF systems, a series of properties about the NOFRFs of the nonlinear systems are obtained. These properties clearly reveal how the system linear characteristic parameters govern the distribution of nonlinear effects induced by the system nonlinear component within the whole systems. The results obtained in the present work are of significance to the frequency response analysis for nonlinear systems. One important potential application of these results is the detection and location of damages in engineering structures, which make the structures behave nonlinearly.

It is well-known that if one or a few components in a structure are of nonlinear properties, the whole structure will behave nonlinearly, and the nonlinear component is often the component where a fault or an abnormal condition occurs. Therefore it is of great significance to detect the position of nonlinear components in structures. Nonlinear output frequency response functions (NOFRFs) are a new concept proposed by the authors for the analysis of nonlinear systems in the frequency domain. The present study is concerned with investigating the NOFRFs of nonlinear one-dimensional chain type systems, which have been widely used to model many real life structures. A series of important properties of the NOFRFs of locally nonlinear one-dimensional chain type structures are revealed. These properties clearly describe the relationships between the NOFRFs of different masses in a one-dimensional chain type system, and allow effective methods to be developed for detecting the position of a nonlinear component in the system. The results are an extension of the authors’ previous research studies to a more general and practical case, and have considerable significance in fault diagnosis and location in engineering systems and structures.

Non-linear output frequency response functions of MDOF systems with multiple non-linear components

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

Transmissibility of non-linear output frequency response functions with application in detection and location of damage in MDOF structural systems

The nonlinear output frequency response function and its application to fault detection

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.

摘要： 基于非线性输出频率响应函数(NOFRF)的独特优势，将非线性输出频率响应函数引入到转子不对中-碰摩耦合故障中，提出了基于非线性输出频率响应函数的转子不对中-碰摩耦合故障诊断方法。利用提出的方法辨识得到了不同对中度及其角度的转子系统的NOFRF值，对比分析了不对中度及其角度对转子系统各阶NOFRF值的影响。仿真结果表明随着不对中度和角度的增加，NOFRF中二次谐波二阶非线性输出频率响应函数和二次谐波四阶非线性输出频率响应函数有着较为明显的增幅；一次谐波三阶非线性输出频率响应函数随不对中角度的增加而减小。因此，可以依据二次谐波二阶非线性输出频率响应函数、二次谐波四阶非线性输出频率响应函数、一次谐波三阶非线性输出频率响应函数值的变化规律识别转子系统的不对中程度。最后通过试验结果验证了仿真结果的正确性，研究成果为具有不对中-碰摩耦合故障的转子系统的故障诊断提供了重要的依据。

Conventional Volterra series model is hardly applied to engineering practice due to its parametric complexity and estimation difficulty. To solve this problem, nonlinear system identification using reduced complexity Volterra models is proposed. Since the nonlinear components often play a secondary role compared to the dominant, linear component of the system, they spend the most of identification cost. So it is worth establishing a balance between identification cost and model accuracy by reducing the complexity of nonlinear components. Refer to the idea of nonlinear output frequency response function, conventional Volterra model is simplified. And then a minimum mean square error criterion based method to identify the simplified model is proposed. The distinguishing feature of this method is high accuracy, good robustness, and significant reduction in the computational requirements compare to the identification of conventional Volterra models. The simulation show that the proposed method is effective, and the reduced complexity Volterra model is of good generalization ability in general. So this nonlinear system identification approach is quite applicable to engineering practice.

Experimental Study on Structural Damage Detection of Cracked Beams Based on Nonlinear Output Frequency Response Functions

Fault diagnosis of multivariable dynamic systems is studied by combining nonlinear spectrum feature with support vector machine. In order to resolve the problem of large calculated amount of solving nonlinear spectrum, a frequency domain variable step size normalized LMS adaptive algorithm is proposed based on the one-dimensional nonlinear output frequency response function (NOFRF). The step size is updated in real time according to the spectrum estimation error and the previous step size. After obtaining nonlinear spectrum data, kernel principal component analysis is used to compress data and extract spectrum feature. In order to improve fault recognition precision, a multi-feature fusion SVM fault classifier is established based on different frequency domain scales. Every sub-classifier is constructed by the spectrum feature of each order, and the diagnosis result can be obtained by weighed fusion of all sub-classifiers. Consider the difference of classification reliability for input features, sub-classifier weight is obtained using the distance between input and SVM separating hyperplane. Simulation experiments indicate that the proposed fault diagnosis method has good real-time performance and high recognition rate, so it can meet the requirements of online diagnosis of multivariable dynamic system.

The pillar porcelain insulator is an important protection device that is related to the safety of the entire power grid. Small damage of it may even cause a disaster. Nonlinear output frequency response functions (NOFRFs) can well reflect the nonlinear characteristics of early damage in the system. This paper uses the NOFRF entropy-based harmonic excitation detection method to detect the damage of the structure. Its effect has been verified by using metal specimens. Then, the pulse hammer detection and harmonic detection are carried out to detect the early-stage damage of the pillar porcelain insulator, and they achieve a good result, which verifies that the harmonic detection method can detect the early-stage damage in the pillar porcelain insulator as well. In addition, the orthogonal test of the simulated breathing crack model is used to search the greatest influence of the parameters of crack on the detection index. Through the orthogonal analysis, the results show that among the lengths, positions, and angles of the crack, the length of crack is the main factor that affects the detection index.

Improvement of the semi-analytical method, based on Hamilton's principle and spectral analysis, for determination of the geometrically non-linear response of thin straight structures. Part III: steady state periodic forced response of rectangular plates

Understanding the nonlinearity of pendulum tuned mass dampers (PTMDs) under large-amplitude vibration is important to enhance control performance. Here, the nonlinear effects of PTMDs on vibration control of a single-degree-of-freedom (SDOF) system under harmonic excitation are explored. The nonlinear governing equations of the linear SDOF structure coupled with the PTMD are formulated. The results obtained via the method of harmonic balance under various excitation amplitudes are compared with those obtained employing the linearized system. It is shown that the effects of PTMD nonlinearity can be ignored when the PTMD rotation angle is below about 9°. Upon increasing the excitation amplitude, remarkable differences are observed between the nonlinear and linear frequency response functions (FRFs). Large mass ratios of PTMD are found to lead to reduced PTMD motion and attenuated nonlinear effects. A parametric study is performed to find the optimal parameters of the nonlinear PTMD under various excitation amplitudes. The results indicate that the optimal tuning frequency ratio increases significantly with excitation amplitudes, whereas the optimal damping is less sensitive to excitation amplitudes. Finally, a 61-m-high steel chimney structure subjected to harmonic and random wind loadings is investigated to verify the proposed PTMD design method.