Instantaneous Frequency Identification Research Articles

A combination of improved spline-kernelled chirplet transform and multi-synchrosqueezing technique for analyzing nonstationary signals and structural instantaneous frequency identification

To improve the accuracy of structural instantaneous frequency (IF) identified by the spline-kernelled chirplet transform, a combination of an improved spline-kernelled chirplet transform and a multi-synchrosqueezing technique is proposed for analyzing nonstationary signals and structural IF identification in this paper. Firstly, an improved spline-kernelled chirplet transform (ISCT) is put forward by introducing a three parameter Gaussian window function for an existing spline-kernelled chirplet transform. These three parameters are successfully obtained through an optimization algorithm and the energy concentration measure (CM). Then, combined with a multi-synchrosqueezing technique, a so-called improved multi-synchrosqueezing spline-kernelled chirplet transform (IMSSSCT) approach is further presented to improve the time–frequency energy concentration. Finally, the accuracy and practicability of the proposed IMSSSCT method are validated by analytical signals, a single-degree-of-freedom Duffing system, a three-layer shear frame, as well as a cantilever beam experiment. The research results indicate that the IMSSSCT technique can effectively and accurately identify the IFs of nonlinear and time-varying structures.

Instantaneous frequency identification for nonstationary signals of time-varying structures using enhanced synchroextracting wavelet transform and dynamic optimization

Civil engineering structures under ambient excitations are essentially time-varying and nonlinear structural systems and the resultant response signals exhibit non-stationarity. To reveal the time-varying characteristics of non-stationary signals, time frequency analysis methods with high resolutions are required. Although synchroextracting short time Fourier transform (SESTFT) is able to generate more energy concentrated time frequency representations and allow signal reconstruction, its major disadvantage is the window function is fixed. To address this issue, an enhanced synchroextracting wavelet transform (ESEWT) method is proposed to refine frequency bands by combing synchroextracting and continuous wavelet transform. After that, dynamic optimization (DO) is used to extract instantaneous frequency (IF) curves within the refined frequency bands. Two numerical examples and an experimental study case are investigated to illustrate the effectiveness and accuracy of the proposed method. The results demonstrate that the proposed ESEWT is capable of extracting frequency bands more accurately and its combination with DO identifies IFs of non-stationary signals better than current time frequency analysis methods such as SESTFT.

A new perspective on the dynamic forced 2-DOF system with the non-perturbative approach

An elegant and straightforward method based on the equivalent linearized method is offered for the analysis of nonlinear vibrations of forced 2-DOF dynamical systems, in search of a simple good approximate solution with the non-perturbative approach. In the present approach, the 2-DOF system is transformed into a decoupled linear oscillator with instantaneous frequency identification. In this approach, we avoid the restriction on the periodic force coefficient, which was considered to be well below unity in various perturbation methods. This liberation of the amplitude of the periodic force led to the emergence of new patterns of the formation of periodic solution waves. It is found that the existence of the periodic force has an inhibitory effect on the scattering factors unless their frequency approaches the natural frequency of the dynamic system. The validation of the scheme is shown by comparing the numerical findings with the existing results. The supremacy of the proposed scheme is demonstrated by making numerical comparisons for each pattern of the obtained solution. The analytical solutions show high consistency proving the accuracy of the non-perturbative approach. This motivates us to extend the scheme for solving free and forced nonlinear vibrations of more coupled structures. It is noteworthy as a new technique with significant usefulness.

An Enhanced Adaptive Chirp Mode Decomposition for Instantaneous Frequency Identification of Time-Varying Structures

An Enhanced Adaptive Chirp Mode Decomposition for Instantaneous Frequency Identification of Time-Varying Structures

Wave mode analysis of a turbine guide vane-integrated rotating detonation combustor based on instantaneous frequency identification

Owing to the pressure gain combustion characteristics, the gas turbine can enhance the system thermodynamic cycle efficiency by integrating a rotating detonation combustor (RDC). In this study, an experimental model of a hydrogen/air RDC with a variant installation of turbine guide vanes (TGVs) was developed. A mathematical model was proposed to identify the rotating detonation wave (RDW) propagation direction. The accuracy of the calculated RDW instantaneous velocity was improved, and the range of velocity fluctuations narrowed from 200 to 1700 m s−1 to 800–1600 m s−1. The impact of TGV installation on the rotating detonation characteristics was investigated. Both the RDW frequency and static pressure increased by installing the TGV at the combustor outlet. The propagation direction of the initial detonation wave induced the RDW to propagate along the same direction. TGV installation reduced the discrepancy in the RDW velocity between two opposite directions as well as the propagation direction changing frequency, while the inverse TGV installation achieved better optimization. The high-frequency pressure oscillations attenuated at the downstream of the TGV, and the amplitude varied depending on the TGV installation options. The static pressure decreased by 14%–16% in both TGV installation options when the combustion flow travelled through the TGV.

Structural instantaneous frequency identification based on synchrosqueezing fractional Fourier transform

The fractional Fourier transform can transform the signal or function into any intermediate domain between time and frequency domains called fractional Fourier domain, in which the non-stationary signals can be processed. To be a time-frequency analysis tool, the fractional Fourier transform may face the same ridge line blurred problem for noisy signals as other time-frequency analysis methods. To improve the instantaneous frequency identification accuracy of fractional Fourier transform, this paper proposes the synchrosqueezing fractional Fourier transform by using the mechanism that can rearrange the energy in the time-frequency plane and make the frequencies more concentrated on the real frequency of the signal. The synchrosqueezing fractional Fourier transform and its corresponding inverse transformation are theoretically derived. Then the corresponding instantaneous frequency identification based on synchrosqueezing fractional Fourier transform is formulated. The accuracy and effectiveness of proposed instantaneous frequency method is verified by a numerical example of three-degree-of-freedom damped time-varying structural system and experiments of moving vehicle-beam system with simply supported and continuously supported beam tested in the laboratory. Compared with the fractional Fourier transform without synchrosqueezing transform, synchrosqueezing wavelet transform and Fourier synchrosqueezing transform, it is demonstrated that the synchrosqueezing fractional Fourier transform can effectively improve the time-frequency distribution divergence and enhance the accuracy of the instantaneous frequency identification. The method has a certain anti-noise performance. Due to quick calculation by means of traditional fast Fourier transform and superior orthogonal transform feature, the proposed synchrosqueezing fractional Fourier transform can be used as an effective time-frequency signal processing tool for multi-component non-stationary signals.

An improved local maximum synchrosqueezing transform with adaptive window width for instantaneous frequency identification of time-varying structures

Civil engineering structures usually exhibit time-varying characteristics under operational conditions due to changes in environment and loads. The identification of instantaneous frequencies (IFs) from measured responses of time-varying structures is a challenge when using the local maximum synchrosqueezing transform (LMSST) due to the difficulty in selecting the appropriate window width. In this study, an improved LMSST with adaptive window width (ALMSST) is proposed to circumvent the limitations of LMSST. The window width of ALMSST can be adaptively determined by combining the autoregressive power spectrum-based variational modal decomposition (AR-VMD) and a window width optimization algorithm. The AR-VMD is used to decompose the multi-component signal into mono-component signals. The Rényi entropy is adopted as an evaluation index in the window width optimization algorithm for selecting the optimal window width for each mono-component signal. Therefore, ALMSST can provide a highly concentrated time–frequency (TF) representation for all mono-component signals. Two simulated signals demonstrate that the ALMSST improves the accuracy of identified IFs compared with LMSST. Numerical simulation of a three-story shear building model with time-varying stiffness shows that ALMSST can accurately identify the IFs of time-varying structures under heavy noise. A cable test with linear and sinusoidal varying tension forces and a vehicle-bridge interaction (VBI) model system with different vehicle weights are investigated to verify the applicability of ALMSST to track the IFs of time-varying structures. Numerical simulation and experimental results illustrate that ALMSST performs well in identifying the IFs of time-varying structures.

Deep learning-based adaptive mode decomposition and instantaneous frequency estimation for vibration signal

Under time-varying environments and loads, vibration monitoring data of structures exhibit nonstationarity with multiple frequency components, and may have obvious nonlinearity under some extreme loads. The time–frequency analysis method is ideal for analyzing these nonlinear and nonstationary signals. However, the vibration signals of large-scale structures typically contain a variety of frequency components, making it challenging for effective mode decomposition without the interference of false mono-components by existing adaptive mode decomposition methods. False mono-components will severely decrease the accuracy of instantaneous frequency. To address the mode mixing and false mono-components in mode decomposition, a data-driven approach based on deep learning for signal adaptive mode decomposition and instantaneous frequency estimation is proposed in this paper. By transforming the time–frequency analysis problem into an optimization task of a deep neural network, built-in redundant adaptive filters with sparse regularization are designed, then the accurate intrinsic mode functions and time-varying frequencies can be estimated. The proposed method is verified by the numerical nonstationary signal and vibration signal of an experimental long cable model. The results demonstrate the superior capability of the proposed method in decomposing a signal into mono-components adaptively without false mono-components and achieving high accuracy of corresponding instantaneous frequencies. From the examples, an accuracy of 98.52% is achieved for a noise-free signal with spectral overlap. The proposed method is more robust to noise, and the instantaneous frequency identification error is 4.90%, which is much lower than EEMD and VMD under a signal-to-noise ratio of 5 dB.

Investigation of time-varying natural frequencies of high-rise buildings under harsh excitations using a high-resolution combined scheme

High-rise buildings subjected to harsh excitations like typhoons or earthquakes usually present nonlinear features and exhibit time-varying behavior of structural modal properties. Under such circumstances, variation in structural natural frequencies can serve as a critical indicator for structural condition assessment. However, limited by noises contained in measurement records and time-frequency resolution of signal analysis, most instantaneous frequency identification methods are difficult to provide accurate estimation of time-varying natural frequencies of high-rise buildings. To address this issue, a combined scheme based on the variational mode decomposition method and adaptive superlets transform (VMD-ASLT) for time-varying natural frequency identification is presented in this paper. Through numerical simulation studies and field measurements of dynamic responses of a 508 m high skyscraper, the noise robustness and accuracy of the combined scheme for time-varying structural natural frequency estimation are validated. Moreover, the combined scheme is further utilized to investigate the time-varying features of natural frequencies of the skyscraper under typhoon and earthquake excitations. This paper aims to provide an effective tool for time-varying structural natural frequency estimation and further reveal the variation of the dynamical behavior of high-rise buildings under harsh excitations.

Dynamic Analysis of a Rotating Cardan Shaft under the Influence of a Breathing Crack and Dense Fluid Force

To extract any information from a complex data vibration, the time-frequency technique represents a suitable tool and a better indicator of the overall asset health of a rotating machine. Fluid influences on a cracked driveshaft and dynamic analysis of a rotating Cardan shaft are discussed in this paper. The study of the forced vibration signals of the rotating Cardan shaft focused uniquely on the impact of the hydrodynamic forces near the cracked driveshaft and its transmission of power to a driven shaft. By merging each subsystem consisting of the breathing crack mechanism in a rotating shaft, the perturbation function between misaligned shafts and transient hydrodynamic forces results in a strongly nonlinear system of partial differential equations, which becomes complicated when analyzing the dynamic response of the Cardan system. In view of the complexity of the Cardan system, the instantaneous frequency identification based on synchrosqueezing wavelet transform filters is numerically used to provide a reliable indication of the rotating system equipment’s health. A conclusive dynamic analysis through the wavelet synchrosqueezing demonstrated that the Cardan joint and the hydrodynamic resistance forces have a high impact on the system’s vibration characteristics and significantly influence the transmission motion with a continuous increase of the driven shaft vibration amplitudes.

Fractional Fourier transform: Time-frequency representation and structural instantaneous frequency identification

The fractional Fourier transform involving a rotation angle as an additional variable is capable to transform a signal to any fractional Fourier domain in between time and frequency domain. With insight into the important role of rotation angle and time–frequency projection of fractional Fourier transform, the time–frequency representation of fractional Fourier transform is in this paper theoretically derived in terms of the window. It is demonstrated that the fractional Fourier transform in time–frequency domain is an algorithm of classical Fourier transform combined the chirp window that is adaptive dilation and translation. The window parameter properties are investigated in details and compared with those of wavelet window. To realize the structural instantaneous frequency identification, the analytical ridge point and corresponding ridge line based on the time–frequency representation of fractional Fourier transform is defined and presented. The fractional Fourier transform based instantaneous frequency identification algorithm is established accordingly by extracting ridge line. The effectiveness of proposed method is verified by a numerical multi-component non-stationary signal, a three-story frame on the shaking table excited by non-stationary inputs with different signal-to-noise ratios and a cable experiment with the time-varying tensioned forces. The identified instantaneous frequency results are compared with those obtained from short-time Fourier transform and wavelet transform. Both numerical and experimental verifications have demonstrated that the proposed fractional Fourier transform based method is capable of identification of instantaneous frequencies involving non-stationary signals with a sound robustness to noise. The fractional Fourier transform can be an alternative tool to process the non-stationary signals in time–frequency domain.

Instantaneous Frequency Estimation Method for the Vibration Signal of Rotating Machinery Based on STFTSC Algorithm

In this paper, an instantaneous frequency identification method known as STFTSC is developed by combining short-time Fourier transformation and the seam carving (SC) algorithm (widely used in image processing). In this method, the STFT is applied to analyze the time-frequency energy distribution of a vibration signal under variable-speed conditions. Subsequently, the energy gradient is calculated through the Sobel operator based on the time-frequency energy distribution. The energy gradient distribution contains multiple ridges, which are corresponding instantaneous frequencies of different orders. Finally, the targeted ridge extraction is transformed into an optimization problem, and the dynamic programming algorithm (DP) is used to search the targeted ridge with the minimum energy gradient to estimate the instantaneous frequency. The effectiveness of the proposed method is validated by a simulation experiment, moreover, a rotating machinery fault simulation test bench is employed to validate the method, which is then compared with polynomial chirplet transformation (PCT) and analyzed during the test process. The results show that the STFTSC instantaneous frequency estimation algorithm has a higher extraction accuracy and better application value in engineering problems than the PCT.

Structural Instantaneous Frequency Identification Based on the Fractional Fourier Transform

In order to identify the instantaneous frequency of time-varying signals, this paper derives the theoretical relationship between the frequency in a signal and rotational angle α in a fractional Fourier transform. It is demonstrated that the fractional Fourier transform is actually an algorithm of ordinary Fourier transform combined with telescopic translation window. A general expression of signal instantaneous frequency in the fractional Fourier domain is thereafter formulated so that structural instantaneous frequency can be extracted accordingly. The feasibility and reliability of proposed method is verified by a simulated nonlinear frequency modulation signal and a numerical example of a three-degree-of-freedom damped time-varying structure system. The results show that the proposed method is in good agreement with the theoretical values and the method has a certain degree of anti-noise capability. Subsequently, the proposed method is capable of identifying the instantaneous frequency of time-varying structures.

An improved time-frequency analysis method for structural instantaneous frequency identification based on generalized S-transform and synchroextracting transform

In this paper, an improved time-frequency analysis (TFA) method based on generalized S-transform (GST) and synchroextracting transform (SET), named improved synchroextracting generalized S-transform (ISEGST) is proposed to improve the accuracy of structural instantaneous frequency (IF) identification. Namely improved generalized S-transform (IGST), a new transform is first put forwarded by optimizing the window function of GST. The parameters of window function in IGST are obtained by the energy concentration measure (CM) and the optimization methodology. Applying IGST in association with SET, the ISEGST is then derived. Compared with the TFA methods in the literature, the performance of ISEGST in numerical signals has been significantly improved, especially in the presence of noise. In simulation, a nonlinear Duffing system and a two-layer shear frame model are used respectively to verify the effectiveness of the proposed method. Moreover, a time-varying cable test is conducted. Acceleration responses of the structure under linear and sinusoidal varying tension force are collected respectively. Then, the ISEGST is used for IF identification. Numerical simulation and experimental results show that the ISEGST can effectively identify IF of structures with non-stationary response signals, and it has high accuracy and good stability.

Autoregressive spectrum-guided variational mode decomposition for time-varying modal identification under nonstationary conditions

Modal identification is an important research subject in structural health monitoring. It is a mature subject for modal identification of time-invariant and stationary conditions. Because time-varying properties are valuable for damage detection and time-varying system identification, and because the environmental loads do not always satisfy the assumption of stationarity, it is necessary to develop accurate identification methods for time-varying structures, structures under nonstationary excitations, or the coupling of the two conditions. The variational mode decomposition (VMD) method, which adaptively decomposes the signal into narrow-band intrinsic modes, shows the potential for time-varying and nonstationary conditions. However, parameter selection greatly affects the accuracy of mode decomposition. In this study, a new modal identification method based on an autoregressive spectrum-guided variational mode decomposition method is proposed for the application of time-varying and nonstationary conditions. First, the determination of the initial center frequencies for VMD is proposed using an autoregressive spectrum-guided method to greatly improve the accuracy. Second, a method to select the optimal balancing factor is developed to improve the decomposition accuracy by controlling the proper bandwidth. Subsequently, an instantaneous frequency identification method is presented by detecting the ridge of the time–frequency distribution. Finally, studies of numerical and actual structures demonstrate the capacity of the proposed method for practical applications of time-varying cases and nonstationary conditions.

Time-varying characteristics analysis of vehicle-bridge interaction system based on modified S-transform reassignment technique

The health monitoring of bridge under moving vehicles demands a high-efficient time-varying characteristic analysis method to extract the frequency fluctuations caused by vehicle-bridge dynamic interaction effect. Characteristics identification of structures becomes a challenging task especially when time-varying vibration modes are involved. For this purpose, an instantaneous frequency identification technique based on modified S-transform reassignment is presented in this paper. Firstly, the general approximation between structural instantaneous frequency and S-transform is deduced. The Gaussian window is employed in S-transform and a novel first-order frequency function with two parameters is added into the Gaussian window function to improve the time–frequency resolution. The added function makes the Gaussian window scalable-sliding and the optimal values of the parameters are selected based on time–frequency concentration criterion. Considering the Rényi entropy as the measurement of criterion, the information performance extracted from S-transform spectrogram is evaluated. Then the time–frequency reassignment technique is applied to improve the readability of S-transform spectrogram to assure the accuracy of the extracted instantaneous characteristics, which enhances the time–frequency concentration efficiently. The proposed method is verified by numerical simulation, the results obtained by the spectrograms derived from short-time Fourier transform, wavelet transform, modified S-transform and reassigned modified S-transform are compared. Afterwards, the time-varying frequency components related to the vehicle-bridge interaction are extracted from the response signal. Numerical simulations involving different parameter analysis and laboratory experiments are conducted to validate the flexibility of the technique in vehicle-bridge system. The effect of vehicle weight, speed and road surface roughness on the identified results are discussed. Both numerical and experimental results have demonstrated that a higher time–frequency concentration in the spectrogram can be observed of the proposed modified S-transform reassignment method than other time–frequency methods, and subsequently a higher accuracy of instantaneous frequency identification is achieved which provides a good tool for time-varying characteristic analysis in bridge health monitoring.

High‐resolution time‐frequency representation for instantaneous frequency identification by adaptive Duffing oscillator

Time-frequency analysis of structural vibration responses provides essential information for structural system identification, modal updating, and condition assessment. However, spurious peaks introduced by the strong noise will significantly increase the false positive rate as well as compromise the sparse time-frequency signal representation. This paper proposes a high-resolution time-frequency representation approach for nonstationary signals polluted with strong noise. With a high sensitivity to detect slight frequency shift and an immunity to noise effect, the intermittent chaotic of Duffing oscillator system is introduced to accurately identify the time-varying instantaneous frequencies. Furthermore, an adaptive Duffing oscillator array is adopted to improve the instantaneous frequency identification efficiency. The feasibility and effectiveness of the proposed method are verified with numerical and experimental studies. Numerical studies are conducted on a multicomponent synthetic nonstationary signal as well as on a two-story shear building with time-varying stiffness under seismic loads. In experimental validations, the acceleration response of a laboratory bridge model under moving vehicle load is also analyzed by using the proposed approach to obtain the instantaneous frequency variations induced by the bridge–vehicle interaction. These results are compared with those obtained from a method based on empirical wavelet transform and Hilbert transform (EWT-HT), to highlight the superiority of the proposed approach in obtaining high-resolution time-frequency analysis results for nonstationary signals with strong noise.

A combined method for instantaneous frequency identification in low frequency structures

Civil engineering structures such as high-rise buildings and long-span cable-supported bridges usually exhibit low-frequency characteristics and the resultant response signals may be non-asymptotic (the amplitude change rate is higher than its phase change rate) and even have closely-spaced frequency components. The identification of satisfactory instantaneous frequencies of such response signals is a challenge faced by standard synchrosqueezing wavelet transform. The paper aims to propose a combined method to address this challenge. The proposed method combines an extended analytical mode decomposition (AMD) method, a recursive Hilbert transform and a zoom synchrosqueezing wavelet transform (consisting of frequency-shift operation and partial zoom synchrosqueezing operation). In the method, a multi-component signal, which may consist of closely spaced frequency components, is firstly decomposed into several mono-component signals by the extended AMD, and the extracted mono-components are then demodulated into asymptotic signals by recursively using the Hilbert transform. After that, a frequency-shift operation is employed to improve time resolution and a partial zoom synchrosqueezing operation is then applied to improve the frequency resolution in the narrow low frequency range of interest. Two numerical examples, an experiment on an aluminum cantilever beam with abrupt mass reduction and an experiment on a cable with time-varying tension forces are provided to illustrate the effectiveness, accuracy and robustness of the method.

An Instantaneous Frequency Identification Algorithm for Time-Varying Frequency Signals

A high-precision instantaneous frequency (IF) identification algorithm is proposed in this paper. The algorithm is based on the relationship between the signal phase and time, and uses Taylor’s expansion to parameterize the time-varying frequency. It can accurately identify the IF of various frequency-modulated (FM) signals. The simulation results show that high identification precision can be obtained when the frequency of the signal is stable or modulated by linear and nonlinear functions. Compared to the conventional IF identification algorithms Hilbert transform (HT) and Wigner-Ville distribution (WVD), the proposed algorithm performs better in accuracy. Besides, the proposed algorithm is not sensitive to the amplitude-modulation behavior, and it shows better precision than the Direct Quadrature (DQ) algorithm, which is an IF identification algorithm of the amplitude-modulated signal. The calculation amount of the proposed algorithm is not related to the sampling rate. Increasing the sampling rate can make the identification precision higher without changing the calculation speed. It can provide the theoretical basis for the IF identification of a grid-connected inverter and other automatic devices in the power system.

Instantaneous Frequency Identification Using Adaptive Linear Chirplet Transform and Matching Pursuit

An instantaneous frequency identification method of vibration signal based on linear chirplet transform and Wigner-Ville distribution is presented. This method has an obvious advantage in identifying closely spaced and time-varying frequencies. The matching pursuit algorithm is employed to select optimal chirplets, and a modified version of chirplet transform is presented to estimate nonlinear varying frequencies. Because of the high time resolution, the modified chirplet transform is superior to the original method. The proposed method is applied to time-varying systems with both linear and nonlinear varying stiffness and systems with closely spaced modes. A wavelet-based identification method is simulated to compare with the proposed method, with the comparison results showing that the chirplet-based method is effective and accurate in identifying both time-varying and closely spaced frequencies. A bat echolocation signal is used to verify the effectiveness of the modified chirplet transform. The result shows that it will significantly increase the accuracy of nonlinear frequency trajectory identification.