【作者】 李锐；

【导师】 杨叔子； 轩建平；

【作者基本信息】 华中科技大学 ， 机械电子工程， 2021， 博士

【摘要】 频率是一个重要的信号参数,它被广泛地应用于各行业中,如用于机械结构的故障诊断、声呐和雷达探测、光谱分析、地震定位和天体演变研究等应用中。在基于频率的梁结构故障诊断中,频率的估计越精确,裂纹定位和裂纹深度估计越精确。基于共振的信号激励方法可以有效地激励振动,提取的固有频率可以用于裂纹的位置定位和深度估计。受频谱泄漏效应的影响,现有的频率估计方法均无法精确地估计频率距离小于3 bins（bin为频率分辨率,1bin=采样频率/样本数）的两分量。傅立叶变换（FT）被广泛应用于信号参数分析。实际应用中,离散傅立叶变换（DFT）承载着FT的数字实现。DFT有常规DFT（ODFT）和对称DFT（SDFT）两种主要形式。目前,几乎所有的信号分析方法均基于ODFT。与ODFT相比,SDFT具备更多的FT性质,如对称性、积分性和插值等性质。经分析比较,本文得出SDFT比ODFT更适合作为FT的离散形式这一结论。为解决频率估计方法的不足,本文研究了SDFT。研究发现偶数SDFT不严格对称于零,本文对其进行了改进。为解决两近距离频率分量的估计问题,本文先后提出了两种频率估计方法。第一种是旁瓣法,当两等幅值频率分量的频率距离大于1.5bins时,两频率分量的最大估计误差约为0.07bins。第二种方法是基于SDFT的相位差法（也称为数值法）。仿真试验结果表明该方法显著提高了两近距离频率分量的估计精度,且具备较好的抗噪性能。当两等幅值频率分量的频率距离大于0.5 bins时,该方法的最大误差约为0.04 bins。模态分解可以将复杂信号分解为单音信号,它为提取裂纹梁的固有频率提供了一种新的途径。假设频率分辨率为1Hz,当待估频率不是特别小时,现有方法的最小估计误差可达10^-6Hz。实数单音信号有且只有两个频率分量（一个正分量和一个负分量）,因而上述数值法可以用于实数单音的频率估计。然而该数值法在瞬时频率计算时,存在计算时间长,偶尔失效,对直流敏感和不能计算瞬时幅值等缺点。本文基于SDFT的积分性质,构造了一种高精度的瞬时频率、幅值和相位的计算方法,该方法无数值法的四个缺点。采用该方法估计一个实单音信号的频率,估计误差约为10^-11Hz。采用该方法分析一个幅值调制和频率调制的信号时,瞬时频率和幅值均存在较大的波动,但是,瞬时相位的误差很小。通过微分瞬时相位可以重新估计瞬时频率,仿真试验表明重新估计的瞬时频率具有极高的精度。最后,为验证所提出的频率估计方法,本文结合视频运动放大方法,模态信号提取方法和频率估计方法,提取了张开裂纹梁的前三阶模态频率。结果显示该方法具备较高的可靠性,频率估计值与理论值的最大相对误差约为3.66%。更多还原

【Abstract】 Frequency is an important signal parameter,and it is widely used in various industries.For example,it is used in mechanical structure fault diagnosis,sonar and radar detection,spectroscopy,seismic location positioning,and celestial mechanics research.In the frequency-based fault diagnosis of beam structures,the more accurate the frequency estimation is,the more precise the crack location and depth estimation are.The signal excitation method based on resonance can effectively excite vibration.The extracted natural frequencies can be used for crack location and depth estimation.Affected by the spectrum leakage effect,none of the existing frequency estimation methods can accurately estimate the two components whose frequency distance is less than three bins.Fourier transform（FT）is widely used in signal parameter analysis.In practical application,discrete Fourier transform（DFT）carries the digital implementation of FT.DFT has two main forms:ordinary DFT（ODFT）and symmetric DFT（SDFT）.Currently,almost all signal analysis methods are based on ODFT.Compared with ODFT,SDFT has more FT properties,such as symmetry,integration,and interpolation.After analysis and comparison,this thesis concludes that SDFT is more suitable as the discrete form of FT than ODFT.In order to solve the shortcomings of frequency estimation,this thesis studies SDFT.It is found that even-number SDFT is not strictly symmetrical to zero,and this thesis improves it.In order to solve the estimation problem of two close specified frequency components,this thesis proposes two frequency estimation methods successively.The first one is the sidelobe method.When the frequency distance between two equal amplitude components is greater than 1.5 bins,the maximum estimation error is about 0.07 bins.The second method is the phase difference method based on SDFT（also called numerical method）.The simulation experiments show this method significantly improves the estimation accuracy of the two close specified frequency components,and this method has better anti-noise performance.When the frequency distance between two equal amplitude components is greater than 0.5 bins,the maximum estimation error is about 0.04 bins.Modal decomposition can decompose complex signals into single-tone signals,and it provides a new way to extract the natural frequencies of cracked beams.Assuming that the frequency resolution is 1Hz,when the frequency to be estimated is not very small,the minimum estimation error of the existing method reaches 10^-6Hz.A real single tone has two frequency components（a positive component and a negative component）,so the above numerical method can estimate the instantaneous frequency of a real single tone.However,the numerical method has disadvantages such as long calculation time,occasional failure,sensitivity to DC,and inability to calculate instantaneous amplitude.Based on the integral property of SDFT,a high precision calculation method of instantaneous frequency,amplitude,and phase is constructed.This method does not have the four shortcomings of the numerical method.When estimating the frequency of a real single tone with this method,the estimation error is about 10^-11Hz.When analyzing an AM and FM signal with this method,the instantaneous frequency and amplitude fluctuate heavily,but the instantaneous phase error is very small.The instantaneous frequency can be re-estimated by differentiating the instantaneous phase.Simulation experiments show that the re-estimated instantaneous frequency has extremely high accuracy.Finally,in order to verify the proposed frequency estimation method,this thesis combines the video motion magnification method,modal signal extraction method,and frequency estimation method to extract the first three mode frequencies of a beam with an open crack.The results show that this approach has high reliability,and the maximum relative error between the estimated frequency and the theoretical value is about 3.66%.更多还原