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This article proposes a time-domain procedure for a non-Gaussian stationary random vibration test with prescribed power spectral densities. Previous procedures for generating non-Gaussianity suffered from certain defects. For example, zero-memory nonlinear transformation, an algorithm frequently applied to transform Gaussian signals into non-Gaussi...
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... can realize this aim by adopting control algorithms. For the non-Gaussian situation, the control objectives also include the kurtosis and skewness. The flowchart of the time-domain procedure for a Gaussian test is presented in Fig. 2. ...
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... this section, we propose a novel ZMNL transformation process, including a new ZMNL transformation for MIMO situation, a rescaling method and a FIR filter control strategy, to generate non-Gaussian reference signals with specified power spectra densities. The process is combined with the original Gaussian test procedure shown in Fig. 2 to form a non-Gaussian test ...
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... non-Gaussian time-domain procedure, a numerical test is conducted on a cantilever beam model. As can be seen from Fig. 21, a we can see the distortions still exist and the power-spectral densities increase over the whole frequency band. The power-spectral densities controlled by Smallwood algorithm demonstrate an obvious control instability in Fig. 24. The characteristic of the control instability of Smallwood algorithm is that the instability always starts at the frequencies where the exceeding power spectral lines are unable to be supressed. As a result, we draw the conclusion that the distortion caused by ZMNL transformation cannot be controlled by control algorithms and can even ...
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... into reference signals, that is the reason why traditional control algorithms are ineffective in controlling power spectra distortion. The proposed FIR filter is designed to directly correct the power spectra of reference signals. The power spectra of the cantilever beam model at point 1 and point 2 with the application of FIR filter are shown in Fig. 25. We can see that there are no power spectra distortion. The matrix power control algorithm is employed to control the power spectral densities shown in Fig. 25. The resulting data presented in Fig. 26 shows that ASD at both points are well controlled within their tolerances. At this time, no control instability occurs. From the ...
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... designed to directly correct the power spectra of reference signals. The power spectra of the cantilever beam model at point 1 and point 2 with the application of FIR filter are shown in Fig. 25. We can see that there are no power spectra distortion. The matrix power control algorithm is employed to control the power spectral densities shown in Fig. 25. The resulting data presented in Fig. 26 shows that ASD at both points are well controlled within their tolerances. At this time, no control instability occurs. From the example, we can see that the proposed FIR filter can eliminate the errors that ZMNL induces and is helpful to stabilize the control process. Table 5, Table 6, ...
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... spectra of reference signals. The power spectra of the cantilever beam model at point 1 and point 2 with the application of FIR filter are shown in Fig. 25. We can see that there are no power spectra distortion. The matrix power control algorithm is employed to control the power spectral densities shown in Fig. 25. The resulting data presented in Fig. 26 shows that ASD at both points are well controlled within their tolerances. At this time, no control instability occurs. From the example, we can see that the proposed FIR filter can eliminate the errors that ZMNL induces and is helpful to stabilize the control process. Table 5, Table 6, respectively. The test procedure was coded with ...
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... Fig. 28, we observe that the harmonics of the table response spectra are obviously added in the low frequency band. Correspondingly, the phase and coherence also deviate from their references. Another experiment was conducted using a FIR filter to improve the quality of the table's response signals. The response spectra and kurtoses are shown ...
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... Fig. 28, we observe that the harmonics of the table response spectra are obviously added in the low frequency band. Correspondingly, the phase and coherence also deviate from their references. Another experiment was conducted using a FIR filter to improve the quality of the table's response signals. The response spectra and kurtoses are shown in Fig. 29 and Fig. 30. As observed in Fig. 29, the shapes of the PSD are corrected. The coherence between point 1 and point 2 is improved compared with that in Fig. 28. However, we can observe that the improvement of the CSD is not obvious with the application of the FIR filter, the shape distortion of CSD can still be observed, and this issue ...
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... of the table response spectra are obviously added in the low frequency band. Correspondingly, the phase and coherence also deviate from their references. Another experiment was conducted using a FIR filter to improve the quality of the table's response signals. The response spectra and kurtoses are shown in Fig. 29 and Fig. 30. As observed in Fig. 29, the shapes of the PSD are corrected. The coherence between point 1 and point 2 is improved compared with that in Fig. 28. However, we can observe that the improvement of the CSD is not obvious with the application of the FIR filter, the shape distortion of CSD can still be observed, and this issue is worth further study in the future. ...
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... also deviate from their references. Another experiment was conducted using a FIR filter to improve the quality of the table's response signals. The response spectra and kurtoses are shown in Fig. 29 and Fig. 30. As observed in Fig. 29, the shapes of the PSD are corrected. The coherence between point 1 and point 2 is improved compared with that in Fig. 28. However, we can observe that the improvement of the CSD is not obvious with the application of the FIR filter, the shape distortion of CSD can still be observed, and this issue is worth further study in the future. We can see the minor kurtoses reduction again in Fig. 30, the averaged kurtoses of both points are slightly less that ...
Citations
... on-line control on kurtoses by adjusting these parameters. In the stationary non-Gaussian random test area, the study on the subject is quite complete (Smallwood, 2005;Cui et al., 2020). ...
The methods of generating stationary random signals, both Gaussian and non-Gaussian, are quite complete, but the researches on the non-stationary signals are insufficient. Especially, the current methods seldom provide mathematical bases about the kurtoses of the produced signals such that the generations of non-stationary non-Gaussian signals with the desired kurtoses are inefficient, which also decrease the flexibility of the real-time control in shaker table tests. In the article, the amplitude modulation method is employed to realize the signal synthesis. The carrier waves of the method are investigated considering the bursts overlapping situations. At first, the explicit equations between the kurtoses of the synthesized signals and the three crucial parameters (the offset, the distance between a pair of adjacent bursts and the parameter of the Beta-distributed random variables) are deduced for the carrier waves with both overlapped bursts and non-overlapped busts. Meanwhile, to solve the power spectral density variation led by the amplitude modulation method, an explicit expression of a rescaling parameter is also proposed. Furthermore, the impacts of the three parameters are investigated; the focus of the investigation is on how the kurtoses of the synthesized signals are changed by the parameters. Based on the results of the investigation, a test procedure is put forward to apply the proposed equations in a shaker table test. The control process of the test demonstrates that the real-time kurtoses control can be achieved efficiently with the help of the newly proposed equations.
... An analytical relation between kurtosis, amplitude, and phase at specific frequencies was presented later to make this method applicable in a closed-loop control [7]. From the perspective of time-varying PSD and PDF, a non-stationary non-Gaussian stochastic process simulation method based on the zero-memory nonlinear translation relationship between non-Gaussian and Gaussian stochastic processes is proposed by Cui et al. [8]. Fei et al. [9] presented a method to synthesize non-Gaussian random vibration that is characterized by running RMS (root mean square). ...
... where G m ðf n Þ is the damage equivalent PSD after m iterations (7) Calculate the ERS in frequency domain using the updated PSD and Equation (12) and compare with the ERS envelope calculated in time domain (8) Update the predetermined test time and PSD level to match both the FDS and ERS envelope of the field data ...
... To determine the effects of damping ratio and fatigue exponent on the calculation of response spectra and PSD synthesis, the pseudovelocity FDS of field data were calculated with different values of Q (10, 25, 50) and b (4,8,12), as shown in Figure 7. From Figure 7, we can see that the FDS decreases and the dispersion increases, as the value of b increases. The value of Q has little effects on FDS (compared with the effect of b). ...
The response spectra are widely used in the damage assessment of non-Gaussian random vibration environments and the derivation of damage equivalent accelerated test spectrum. The effectiveness of the latter is strongly affected by modal parameter uncertainties, multiple field data processing, and the nonsmooth shape of the derived power spectral density (PSD). Optimization of accelerated test spectrum derivation based on dynamic parameter selection and iterative update of spectrum envelope is presented in this paper. The extreme response spectrum (ERS) envelope of the field data is firstly taken as the limiting spectrum, and the corresponding relationship between damping coefficient, fatigue exponent, and damage equivalent PSD under different test times is constructed to achieve the dynamic selection of uncertain parameters in the response spectrum model. Then, an iterative update model based on the weighted sum of fatigue damage spectrum (FDS) error is presented to reduce the error introduced by the nonsmooth shape of the derived PSD. The case study shows that undertest can be effectively avoided by the dynamic selection of model parameters. The weighted error is reduced from 80.1% to 7.5% after 7 iterations. Particularly, the error is close to 0 within the peak and valley frequency band.
... e time-domain procedure for the super-Gaussian test used in the submission was presented by Cui et al. [16]. e formulation of this procedure starts with expressing the system transfer function using state-space matrices. ...
Random vibration environmental testing employs the specified statistical properties of the real world vibration to reproduce the desired excitations on the shaker table for fatigue test purposes. Smooth and safe operation is the essential requirement for a long-duration test. Traditionally, the windowing and overlap-add (WOA) method is applied to the acceleration signals of the shaker table, and previous studies have indicated that this operation reduces the kurtoses of the processed signals. To protect the test equipment from abrupt changes in the input voltage, the WOA method is proposed to operate on the input voltage signals in a frame-by-frame form for super-Gaussian environmental testing. To figure out the impacts of the proposed operation on the response kurtoses of a shaker table, we express the system transfer function in the time domain, and the WOA method is analysed considering the transfer function of a dynamic system. Based on the analysis, a further study is made to explain the mechanism of the kurtosis decrease due to the WOA method. Through the study, we find that the kurtosis reduction conclusion is not applicable to all types of super-Gaussian signals, and the kurtoses can be invariable and even increased by allocating the positions of the high-excursion peaks of super-Gaussian signals when the WOA method is applied. A window function is recommended for zero-memory nonlinear (ZMNL) transformation to move the positions of the high-excursion peaks of a super-Gaussian signal, providing a novel way of adjusting kurtosis when WOA method is applied. The proposed WOA method and window function are first verified in a single-input-single-output (SISO) numerical simulation to test their effectiveness under different reference kurtoses. Then, they are evaluated in a two-input-two-output shaker table test. The test results demonstrate that the proposed window function can prevent the kurtosis decrease with the application of the WOA method.
Recent research has discovered that the kurtoses of non-Gaussian stress loadings can significantly impact the fatigue life of in-service structures. To predict potential damage, a fatigue life estimation method based on Gaussian damage and a kurtosis-related corrective coefficient has been developed. However, the transmission of kurtoses from excitations to responses has not been well studied, particularly in multi-input cases. This hinders efforts to accelerate fatigue damage by controlling input kurtoses. Therefore, this paper aims to establish a kurtoses transmission model for a linear structure under multiple uncorrelated stationary non-Gaussian excitations generated by zero-memory non-linear method. Firstly, a single-input kurtosis transmission equation is proposed to establish the relationship between excitation kurtosis and response kurtosis. Using this equation, the response kurtosis is formulated with input kurtosis and the system parameters of a linear structure. Secondly, the response kurtosis of multi-input cases is deduced, and the study shows that the response kurtosis is the weighted sum of the kurtosis induced by each excitation. Finally, numerical validation and experimental studies are conducted to verify the accuracy of both the single-input and multi-input kurtoses transmission models. The validations demonstrate that the proposed models can predict the response kurtoses with satisfactory precision, regardless of the input spectral shape and the number of uncorrelated input forces.
The amplitude modulation method is used to create non-stationary and non-Gaussian signals with desired kurtosis values, which act as non-stationary excitations for shaker-table fatigue tests. However, the non-stationary signals produced by the method have been proved to possess a relatively narrow band of kurtosis values, thus limiting its potential in accelerated life test. To solve the problem, the kurtosis model of a non-stationary signals produced by the amplitude modulation method is investigated. It is found that the feature of the beta distribution imposes restrictions on the kurtosis range. Based on the finding, two types of updates are put forward in this article. The first one is to utilize the gamma-distributed random numbers to scale the amplitudes of the modulating waves. The second update is to employ stationary non-Gaussian signals, instead of the stationary Gaussian signals, to synthesize the non-stationary and non-Gaussian signals. The two alternatives are all verified with numerical and experimental examples. The results demonstrate that both methods can widen the band of kurtosis. But the non-stationary excitations produced by the second method turn out to have different kurtosis transferring rates in a linear time-invariant dynamical system. Especially, the excitations created with sub-Gaussian signals have larger kurtosis transferring rates than the excitations made by the original amplitude modulation method do, making them suitable to be applied in an accelerated fatigue test.
Joint exploitation of multiple features is a recognized way to implement the effective detection of small targets on the sea in high-resolution maritime radars at the dwelling mode. The tri-feature-based and feature-compression-based detectors use the time-consuming convex hull learning and decision. In this article, a fast feature-fusion-based detector using seven salient features is proposed, where the convex hull learning and decision are replaced by simple threshold determination and decision. It is found that the seven features of sea clutter can be modeled well by the Burr type XII and
t
-distributions. From the fitted distribution, each feature is normalized to approximate the standard Gaussian distribution by a nonlinear transformation. In the seven-dimensional normalized feature space, an analytical method is given to calculate the optimal weights of feature fusion by the maximization of the interclass distance of the two-class samples. Owing to the normalization of the seven features, the feature fusion loss is significantly reduced comparing with the direct fusion of seven features. The fast feature-fusion-based detector was evaluated on the open and recognized radar databases and offshore experimental data using an unmanned aerial vehicle with a corner reflector. The results show that the fast detector attains competitive performance with the existing best feature-based detector.
Zero-memory non-linear (ZMNL) methods have long been a type of very useful tools in creating super-Gaussian random excitations for fatigue test. The major drawback of this method is that the magnitude distortion can be seen in both auto-power spectral density (ASD) and cross power-spectral density (CSD) of the excitation if the dynamic range of the target power spectral density (PSD) is large. To study the problem, we introduced Fourier series of a Gaussian signal into a cubic system, which is a common mathematical model shared by most of the ZMNL functions. We found that the cubic system produces a component leading to the distortion of both ASD and CSD. Meanwhile, if the shape of the reference ASD is flat in a frequency band, the distortion will not happen in the band. Based on this findings, a novel method is proposed in this article. It employs a pseudo reference PSD with flat ASDs to generate non-Gaussian signals in order to avoid the PSD distortions. Designed FIR filters are then applied to rescale the ASDs of the generated signals. After, iterative processes are introduced to recover the kurtoses of these filtered signals. The method is applied on a two-input and two-output numerical model, the feasibility and the availability of the method is verified. Finally, two control schemes based on the proposed pseudo reference PSD method are provided for a shaker table test. Test results show that both control schemes can eliminate the mismatch between the target PSD and the measured PSD of a shaker table.
