Anti-Noise Performance of the FT Continuous Zoom Analysis Method for Discrete Spectrum

As a discrete spectrum correction method,the Fourier transform(FT) continuous zoom analysis method is widely used in vibration signal analysis,but little effort had been made on this method's anti-noise performance.It is widely believed that the analysis accuracy of the method can be substantially improved by increasing the zoom multiple,however,with the zoom multiple increases,the frequency estimation accuracy may decline sometimes in practices.Aiming at the problems above,this paper analyzes the sources of frequency estimation error when a harmonic signal mixed with and without noise is processed using the FT continuous zoom analysis.According to the characteristics that the local maximum of the zoom spectrum may be wrongly selected when the signal is corrupted with noise,the number of wrongly selected spectrum lines is deduced under different signal-to-noise ratio and local zoom multiple,and then the maximum frequency estimation error is given accordingly.The validity of the presented analysis is confirmed by simulations results.The frequency estimation accuracy of this method will not improve any more under the influence of noise,and there is a best zoom multiple,when the zoom multiple is larger than the best zoom multiple;the maximum frequency estimation error will fluctuate back and forth.The best zoom multiple curves under different signal-to-noise ratios given provide a theoretical basis for the choice of the appropriate zoom multiples of the FT continuous zoom analysis method in engineering applications.     

FFT+FT离散频谱校正法参数估计精度

研究用FFT谱连续细化傅里叶变换分析法进行离散频谱校正时的参数估计误差。分析无噪声情况下频率﹑相位﹑幅值的估计误差随细化倍数的变化规律,估计精度随细化倍数的增大而提高,当细化倍数大于40时,最大估计误差几乎可忽略不计。在高斯白噪声的影响下,细化后频谱序列最大值找错的概率随细化倍数的增加而增加,综合考虑频率分辨率对频率估计精度的影响及频谱序列最大值找错的概率,提出用归一化频率估计综合误差和归一化频率估计最大可能误差两个指标评价此校正法对频率的估计精度,并基于此给出不同信噪比条件下的最优细化倍数。采用非线性最小二乘拟合法对噪声影响下的FFT谱连续细化傅里叶变换分析校正法进行改进,通过仿真模拟验证改进后该校正方法具备更高的校正精度和抗噪能力。     

Design of two-channel filter bank using nature inspired optimization based fractional derivative constraints

In this article, a novel approach for 2-channel linear phase quadrature mirror filter (QMF) bank design based on a hybrid of gradient based optimization and optimization of fractional derivative constraints is introduced. For the purpose of this work, recently proposed nature inspired optimization techniques such as cuckoo search (CS), modified cuckoo search (MCS) and wind driven optimization (WDO) are explored for the design of QMF bank. 2-Channel QMF is also designed with particle swarm optimization (PSO) and artificial bee colony (ABC) nature inspired optimization techniques. The design problem is formulated in frequency domain as sum of L₂ norm of error in passband, stopband and transition band at quadrature frequency. The contribution of this work is the novel hybrid combination of gradient based optimization (Lagrange multiplier method) and nature inspired optimization (CS, MCS, WDO, PSO and ABC) and its usage for optimizing the design problem. Performance of the proposed method is evaluated by passband error (ϕ_p), stopband error (ϕ_s), transition band error (ϕ_t), peak reconstruction error (PRE), stopband attenuation (A_s) and computational time. The design examples illustrate the ingenuity of the proposed method. Results are also compared with the other existing algorithms, and it was found that the proposed method gives best result in terms of peak reconstruction error and transition band error while it is comparable in terms of passband and stopband error. Results show that the proposed method is successful for both lower and higher order 2-channel QMF bank design. A comparative study of various nature inspired optimization techniques is also presented, and the study singles out CS as a best QMF optimization technique.     

Frequency estimation of discrete time signals based on fast iterative algorithm

A fast iterative algorithm for frequency estimation is developed in this paper to improve the frequency tracking performance. If the signal is transformed by a mathematical tool, the signal to noise ratio (SNR) should not be greatly reduced after the transformation. The analysis presented in this paper showed that the traditional method for frequency estimation causes large noise at high frequency range, therefore, the suitable estimation range of traditional method is only from 0 to fs/6 Hz (fs is the sample frequency). In order to overcome this limitation, a new structure of iterative algorithm is established to extend the upper bound frequency from fs/6 to fs/2 Hz. The experimental noisy sinusoid signal frequency estimation and chirp signal frequency tracking confirmed that the novel algorithm showed improved performance. Furthermore, the average estimation error was decreased over 30% (under SNR = 15 dB) when applying the novel iterative algorithm. The novel iterative algorithm will have broad applications in fields of signal processing and communication systems.     

非整周期采样信号频率估计的相频匹配方法

为提高非整周期采样信号的频率估计精度,提出一种非整周期采样信号频率估计的相频匹配方法。首先,为抑制信号非整周期采样对自相关的影响,对采样信号进行加窗自相关;其次,根据加窗自相关信号初相位为零的特点生成参考信号,实现参考信号与加窗自相关信号的相位匹配;最后,根据柯西不等式,利用参考信号和加窗自相关信号构造反映参考信号和加窗自相关信号频率匹配程度的误差函数,误差函数最小值对应的频率即为信号频率估计值。计算验证和LFMCW雷达测距实验表明该方法不受信号非整周期采样的影响,有效地提高了非整周期采样信号的频率估计精度,改善了LFMCW雷达的测距精度。     

变焦距镜头凸轮结构优化设计

变焦距镜头普遍采用机械补偿法来改变焦距,变焦凸轮是其实现连续变焦的关键。为了减小变焦距镜头的质量和体积,本文研究了用机械补偿方式改变变焦距镜头焦距的原理,采用NX/Nastran软件对变焦凸轮结构进行了仿真分析,并针对分析结果对变焦凸轮结构进行了优化设计。优化结果使变焦凸轮质量减小了88g,壁厚减小了1.5mm,外径尺寸减小了3mm。对优化后的变焦凸轮进行了模态分析和热分析。模态分析显示,凸轮主体一阶频率为50.3Hz;热分析显示,凸轮在90℃温差作用下变形大小为0.006mm。最后通过对光学传递函数的检测以及振动和高低温试验验证了仿真分析的正确性,表明对变焦凸轮的优化设计是成功的。     

Subsample interpolation bias error in time of flight estimation by direct correlation in digital domain

This work presents an investigation of the bias error introduced in time of flight estimation realized by subsample interpolation in digital domain. The time of flight estimation is accomplished based on the evaluation of the peak position of the cross correlation function. In order to cope with the discrete nature of the cross-correlation function, subsample estimation exploits three time domain interpolation techniques: parabolic, cosine, Gaussian and frequency domain interpolation using phase angle. An empirical equation relating the maximum value of the bias error to sampling frequency and signal parameters (center frequency and envelope bandwidth) has been derived. It is found that the maximum value of the bias error is in inverse cubic relation to sampling frequency and in quadratic relation envelope bandwidth for cosine interpolation. The maximum value of the bias error is in inverse cubic relation to sampling frequency and in quadratic relation to center frequency and envelope bandwidth for parabolic interpolation. The coefficients related to the approximation technique are given. Results can be applied for bias errors estimation or correction when fast subsample interpolation is used and application of phase domain interpolation is unacceptable due to processing speed limitations. The equations for minimum required sampling frequency are derived by balancing the interpolation error against Cramer–Rao lower bound.     

Fast algorithms for frequency, amplitude and phase evaluation of nonsinusoidal signals with noises

A fast and accurate algorithm for frequency, amplitude and phase estimation of the signals with white Gaussian noises is proposed in this paper. The proposed algorithm need two sample and computation process, one of which is used for frequency estimation in half cycle of the signal and another of which is used for amplitude and phase estimation in another half cycle. The proposed algorithm spends at most 1 cycle. Frequency estimation is based on numerical differentiation, and amplitude and phase estimation is based on fast Fourier Transform. With an initial sample frequency of 512 × 50 Hz, the signal is sampled and the frequency of the signal with white Gaussian noises is estimated at an error of 0.001% over a range of 1 Hz–1000 kHz. With another sample frequency based on the estimated frequency, the signal is once again sampled and the amplitude of the signal is estimated an error of 0.001% over a range of 1 V–320 V and the phase angle of the signal is estimated an accuracy of 0.001% over a range of 0–360. Using Matlab software, the simulation results of the test example are satisfactory.     

Estimation method for errors of an aerostatic planar XY stage based on measured profiles errors

This paper describes a method for estimating the geometric errors of an aerostatic planar XY stage. The method calculates two-dimensional (2D) position errors and flatness based on measured guideway profiles. Profile measurements, estimates of motion error, and geometric error models are considered to estimate the 2D position and flatness errors along the X- and Y-axes. A three-probe system is used to measure the guideway profiles for the X- and Y-axes. The motion errors, which are used as input data for geometric error models, are calculated from the force equilibrium of aerostatic bearings along the measured guideway profiles and compared with laser interferometer measurements. Geometric error models were derived to calculate the 2D position and flatness errors of the stage. The measured results and estimates for the 2D position error and flatness differed by 0.5 and 0.6 μm, respectively. Therefore, the proposed estimation method for 2D position and flatness errors of an aerostatic planar XY stage will be a useful tool during the machining and assembly of guideways.     

基于频谱细化和相位差校正的全息谱研究

针对全息谱在频谱密集情况下精度会降低的问题,提出了基于频谱细化和相位差校正的全息谱分析方法。该方法采用复解析带通滤波器选带细化方法对以阶次频率为中心的局部区间进行细化分析,通过提高频率分辨率来消除密集频谱对精确获取幅值、相位信息的影响,运用相位差法对细化区间内的最大谱峰进行频率、幅值、相位校正,准确地提取出各阶次的幅值、相位信息,最后合成全息谱图。仿真及对柔性转子试验台振动信号的分析结果表明,基于频谱细化和相位差校正的全息谱能有效提高其分析精度,更加精确有效地诊断旋转机械的故障。     

离散频谱能量重心法频率校正精度分析及改进

研究噪声对离散频谱能量重心法的频率校正精度的影响,推导了在高斯白噪声背景下用能量重心法对加对称窗的离散频谱进行校正的频率误差理论公式,分析找错和找对最大值谱线情况下的理论误差和某些情况下校正误差较大的原因,为了提高能量重心校正法的频率校正精度,提出用谱线间相位差为阈值作为选择用3条或4条谱线进行校正依据的改进措施。通过对加Hanning窗的离散频谱进行计算机仿真计算,结果表明在大噪声背景下改进的能量重心校正法有很高的频率校正精度,与理论推导十分吻合,验证了理论推导的正确性,表明改进后的能量重心法具有更高的抗噪性能,扩大了能量重心校正法的工程应用范围。     

基于FIR优化滤波的旋转高频信号注入法及其应用

针对传统旋转高频信号注入法中信号处理精度低、延时时间长及过程复杂等缺陷,提出一种基于有限冲激响应(finite impulse response,FIR)优化滤波的改进旋转高频信号注入法.该方法采用等纹波最佳逼近FIR滤波器提取高频电流信号,实现高频电流信号提取误差最小.通过对高频电流作外差处理,提取转子位置误差信号,省去旋转高频信号注入法中的同步轴系滤波单元,降低系统的复杂性.通过线性相位补偿,实现转子速度与位置估计最小延迟.构建无轴承永磁同步电机无速度传感器矢量控制平台,验证算法的有效性.仿真实验结果表明:通过离线优化设计FIR滤波器及线性相位补偿,该方法在全速范围内能够准确估计转子的位置与速度,与卡尔曼滤波相比,其估计精度更高,鲁棒性更强.     

Generation of sinusoidal pulsating flows in the channels of experimental setups

A new construction of the pulsator for generating sinusoidal pulsating flows of a continuous medium in the channels of experimental setups have been performed. The pulsator is based on operating experience of existing types of latter used for the frequency range 0.03 < f < 0.8 Hz and is differed from them due to the use of a profiled flap. The flow structure using the pulsator is studied based on the theoretical analysis and experimental verification via the hot-wire anemometry and SIV method. It is shown that the profiled flap allows more than an order increase in the frequency of the flap rotation f ˜ 25 Hz, which maintains an almost sinusoidal law of flow rate. A criterion for estimating the efficiency of pulsator has been presented and an analytical expression has been obtained for estimating the impact of inertial forces on the law of variation of average velocity over the phase angle. The deviation from the harmonic law realized by the submitted pulsator at f = 25 Hz is 3 times smaller than the deviation characteristic for a pulsator with a ball flap and 6 times smaller for one with a butterfly flap at f = 0.04 Hz.     

Wavelength-modulation Fourier interferometry with elimination of DC phase error

Wavelength-modulation Fourier interferometry has been employed to measure the thickness of a glass plate. However, nonlinear phase-modulation error causes a DC error in the evaluated phase distribution. In this study, a 5 N–2 phase-extraction algorithm was derived for the simultaneous measurement of the thickness and surface shape of a glass plate with eliminating the DC phase error. First, the condition for the error elimination on a complex plane was derived and a new algorithm polynomial was proposed to satisfy this condition. Next, using this condition and 5 N–4 algorithm, 5 N–2 algorithm was derived. The newly developed 5 N–2 algorithm was visualized in the frequency domain using the Fourier transform method. The DC error elimination ability of the 5 N–2 algorithm was confirmed through numerical error analysis. Finally, the surface shape and thickness of the glass plate were simultaneously measured using a large-aperture Fizeau interferometer and the 5 N–2 algorithm. The standard deviations of the thickness and surface shape profiling were 3.113 nm and 10.689 nm, respectively, which were smaller than those obtained using other phase-extraction algorithms.     

基于HWPT-ZFFT的二维全息谱计算方法

精确的频率、相位和幅值识别是进行全息谱计算的必备条件,针对含3个及以上的密集频谱成分,提出一种基于谐波小波包变换的频谱细化方法(harmonic wavelet packets transform-zoom fast Fourier transform,简称HWPT-ZFFT),较传统的复调制细化傅里叶变换所利用的低通及带通滤波器相比,其盒型频谱特性可将感兴趣频段的信号正交,无冗余、无泄漏地提取出,提高了识别精度。首先,利用谐波小波包对密集频谱成分进行滤波;然后,频移进而重采样,进行傅里叶变换得到细化的频率、幅值及相位;最后,计算密集频率下二维全息谱,进行双盘转子全息谱计算,考虑高次分倍频,得到更丰富的故障特征。仿真及双盘转子实验结果表明所提出方法的有效性。     

Prediction of hydraulic jump characteristics in a closed conduit using numerical and analytical methods

The hydraulic jump is an economical alternative to dissipate energy in the conduit and to reduce erosion at the culvert outlet. In the literature, very limited studies have been reported on the performance of hydraulic jump in a closed conduit. The innovation of this research is to employ a numerical method for the estimation of the hydraulic jump characteristics in a closed conduit with different positive slopes (S₀). The analytical method was used to develop several equations for hydraulic jump and the provided results were compared with the numerical method. The results indicate that the numerical method predicts the flow depth ratio after conduit with higher accuracy (error less than 5%) in comparison to the analytical method (error less than 10%). Furthermore, in the slope of 0.00, the energy loss increases by 16% with increasing the Froude number from 4.617 to 5.562 while this value is 23% and 22% for slopes of 0.01 and 0.02, respectively. Finally, several equations were developed for the prediction of hydraulic jump characteristics in terms of Fr₁, S₀, and conduit depth (D).     

THE ITD MODE-SHAPE COHERENCE AND CONFIDENCE FACTOR AND ITS APPLICATION TO SEPARATING EIGENVALUE POSITIONS IN THEZ -PLANE

The properties of a newly defined mode-shape coherence and confidence factor are discussed in detail in connection with the Ibrahim time domain method. It has been found that this factor is able to distinguish three kinds of modes, true physical modes within the frequency range (0∼f_π) (corresponding to an angular range (0∼π) of eigenvalue positions in the Z -plane), frequency folded and/or overlapped modes and computational (noise) ones. Making use of this ability gives at least two benefits; (a) it releases the limitation on the user's constant Δt₁and increases the flexibility in use of all the user's constant combinations, and (b) it extends the angular range of eigenvalue positions to a multiple of 2 π by using a large value of Δt₁and makes the identification of eigenvalue positions in the Z -plane more accurate by a sort of zoom process. Digital simulation and measurements on a gearbox were chosen as application examples to illustrate the above advantages.     

Bispectrum feature extraction of gearbox faults based on nonnegative Tucker3 decomposition with 3D calculations

Nonnegative Tucker3 decomposition（NTD） has attracted lots of attentions for its good performance in 3D data array analysis. However, further research is still necessary to solve the problems of overfitting and slow convergence under the anharmonic vibration circumstance occurred in the field of mechanical fault diagnosis. To decompose a large-scale tensor and extract available bispectrum feature, a method of conjugating Choi-Williams kernel function with Gauss-Newton Cartesian product based on nonnegative Tucker3 decomposition（NTD_EDF） is investigated. The complexity of the proposed method is reduced from o（nNlgn） in 3D spaces to o（RiR2nlgn） in 1D vectors due to its low rank form of the Tucker-product convolution. Meanwhile, a simultaneously updating algorithm is given to overcome the overfitting, slow convergence and low efficiency existing in the conventional one-by-one updating algorithm. Furthermore, the technique of spectral phase analysis for quadratic coupling estimation is used to explain the feature spectrum extracted from the gearbox fault data by the proposed method in detail. The simulated and experimental results show that the sparser and more inerratic feature distribution of basis images can be obtained with core tensor by the NTD EDF method compared with the one by the other methods in bispectrum feature extraction, and a legible fault expression can also be performed by power spectral density（PSD） function. Besides, the deviations of successive relative error（DSRE） of NTD_EDF achieves 81.66 dB against 15.17 dB by beta-divergences based on NTD（NTD_Beta） and the time-cost of NTD EDF is only 129.3 s, which is far less than 1 747.9 s by hierarchical alternative least square based on NTD （NTD_HALS）. The NTD_EDF method proposed not only avoids the data overfitting and improves the computation efficiency but also can be used to extract more inerratic and sparser bispectrum features of the gearbox fault.     

Performance evaluation of two different neural network and particle swarm optimization methods for prediction of discharge capacity of modified triangular side weirs

Technical design of side weirs needs high accuracy in predicting discharge coefficient. In this study, discharge coefficient prediction performance of multi-layer perceptron neural network (MLPNN) and radial basis neural network (RBNN) were compared with linear and nonlinear particle swarm optimization (PSO) based equations. Performance evaluation of the model was done by using root mean squared error (RMSE), coefficient of determination (R²), mean absolute error (MAE), average absolute deviation (δ) and mean absolute relative error (MARE). Comparison of the results showed that both neural networks and PSO based equations could determine discharge coefficient of modified triangular side weirs with high accuracy. The RBNN with RMSE of 0.037 in test data was found to be better than MLPNN with RMSE of 0.044 and multiple linear and nonlinear PSO based equations (ML-PSO and MNL-PSO) with RMSE of 0.043 and 0.041, respectively. However, due to their simplicity, PSO based equations can be sufficient for use in practical cases.