Application of a Modal Parameter Identification Method Based on Variational Mode Decomposition in Flight Flutter Testing

Signals of flight flutter testing exhibit non-stationary characteristics, closely spaced modes, and low signal-to-noise ratio, presenting challenges in data processing. In recent years, the variational mode decomposition (VMD) method has emerged as a promising approach to mitigate mode mixing and exhibit robust noise resistance. Therefore, a novel time-frequency domain modal parameter identification method based on VMD is proposed to process impulse response signals in flight flutter testing. The modal frequency and damping ratio are determined through a three-step process: VMD analysis, Hilbert transform, and least square fitting. The efficacy of the proposed method in identifying closely spaced modes and resisting noise is validated through a numerical example. Furthermore, this method is applied to analyze two types of pulse excitation signals in actual flight flutter testing: one induced by the pilot’s shaking stick and the other induced by small rocket excitation. The obtained modal parameters are compared with those from ground vibration tests and specialized software, respectively, to showcase the effectiveness and superiority of the proposed method.


Introduction
Flight flutter testing is mandated for all new aircraft or those undergoing significant structural modifications, establishing itself as a globally acknowledged high-risk subject in the realm of flight testing.The primary objective of this testing is to acquire structural modal parameters (frequency and damping) for extrapolating the aircraft flutter boundaries.Effective excitation to structural modes and precise modal parameter identification are crucial for accurately obtaining these modal parameters and ensuring the safety of flight testing [1].
During flight flutter testing, commonly used excitation methods include turbulent excitation [2], frequency-swept excitation [3], and pulse excitation [4].These methods can be applied individually or in combination depending on specific testing conditions.The resulting response signals often demonstrate non-stationary behavior, presenting multiple modes with closely spaced frequencies and vulnerability to noise interference.This complexity poses challenges for the accurate estimation of modal parameters.
For turbulent and frequency-swept excitations, various modal parameter identification methods are available, while pulse excitation data has received comparatively less attention.Currently, specialized software such as "Prin80" is commonly utilized to analyze pulse excitation data, particularly in the IOP Publishing doi:10.1088/1742-6596/2762/1/012050 2 context of national aircraft certification.This software employs a frequency-domain approach based on polynomial fitting [5].However, it's worth noting that the underlying Fourier transform method is influenced by frequency resolution when processing short-time pulse data, and tends to exhibit suboptimal performance in identifying closely spaced modes.
As modal parameter identification techniques evolve, various time-frequency domain methods have been developed.Among them, the Hilbert-Huang Transform (HHT) [6,7] stands out, comprising empirical mode decomposition (EMD) and Hilbert transform.However, the core EMD technique suffers from mode mixing issues due to its recursive nature.While ensemble empirical mode decomposition (EEMD) [8] can mitigate this problem to some extent, it significantly increases computational complexity.
Variational mode decomposition (VMD) [9], introduced in 2014, represents an adaptive and nonrecursive signal decomposition method.It transforms the mode decomposition challenge into a variational solution problem.Through an iterative search for the optimal solution to the variational model, the original signal is decomposed into a discrete number of components, each closely aligned with corresponding center frequencies.This process dynamically determines the center frequency and bandwidth of each component, enabling effective adaptive separation.Extensive research has validated the VMD method's solid theoretical foundation, showcasing superior anti-noise performance and anti-mode-mixing capabilities.In recent years, its successful application has extended across various domains, including machinery, electronics, biology, and energy, with notable achievements, particularly in the realm of mechanical fault diagnosis [10,11,12].
In view of the above advantages of the VMD method and the needs of actual flight flutter testing, a time-frequency domain method for modal parameter identification based on VMD is proposed in this paper by combining Hilbert transform and least square fitting, which is called 'VMD-HT method'.The practical application in dealing with flutter signals by pulse excitation shows that the VMD-HT method has good identification accuracy and important engineering application value.
Given the noted advantages of the VMD method and the specific requirements of practical flight flutter testing, this paper proposes a novel time-frequency domain approach for modal parameter identification, termed the 'VMD-HT method.'This method combines the variational mode decomposition (VMD) with Hilbert transform and least square fitting.The application of the VMD-HT method to flutter signals induced by pulse excitations demonstrates its commendable accuracy in modal parameter identification, highlighting its significant engineering application value.

Variational mode decomposition
In the VMD algorithm, the intrinsic mode function (IMF) is redefined as an amplitude-modulatedfrequency-modulated signal k u , written as: where ( ) k A t and ( ) k t  are the instantaneous amplitude and phase of the kth IMF respectively at time t.
Then the instantaneous frequency can be obtained as . Assuming that the multi-component signal ( ) f t can be decomposed into K IMF components, each with a central frequency ( ) k t  and limited bandwidth, the constrained variational model can be established as: 3 Where ( ) t  is unit impulse, and j 1   .
The optimal solution for the constrained variational problem stated above in equation ( 2) can be obtained by converting it into a non-constrained problem, and the generalized Lagrange function is given as: Where  is a quadratic penalty term, and  is Lagrangian multiplier.To solve this variational problem, the multiplier alternation algorithm is used to constantly update each mode and its centre frequency until all the components in the frequency domain can be determined.
In the expression above, 1 ˆ( ) It is important to note that the value of K and  need to be predefined when using VMD method.In this paper, K is initially determined by the spectral peak number of the signal, while  is found by research to have little impact on the decomposition results when it is several thousand.Therefore, no additional discussion on is provided here.

VMD-HT Method
The decoupling dynamic equation of discrete multi-degree-of-freedom system is expressed as: For continuous systems, the steady-state solution of the equation under the zero initial value condition is obtained using the Duhamel integral to derive the displacement response of the kth modal coordinates.By finding the second-order derivative for the displacement, the acceleration response at any point in physical coordinates can be further derived as: where It can be observed in equation ( 7) that the acceleration signal is composed of single-frequency amplitude-modulation harmonics, which can be decomposed by VMD to obtain different IMFs.The single component in equation ( 7) can be written as: Performing the Hilbert transform of k u to get k U , then the analytic signal is established, i.e.
( ) Therefore, Instantaneous amplitude ( ) k A t and instantaneous phase ( ) k t  are introduced.Then, select the applicable data of ( ) to intercept (interception method is shown in numerical example), and instantaneous amplitude logarithm and instantaneous phase are obtained: Utilizing equation (10), the least square fitting of the interception data is carried out, and the slopes of fitting curve for instantaneous amplitude logarithm and the instantaneous phase are obtained respectively: (11) Finally, the natural frequency and damping ratio of each mode can be calculated by:

Numerical example
To evaluate the ability of the VMD-HT method to identify closely spaced modes and resist noise, an impulse response signal resembling equation ( 7) is synthesized.The original composite signal signal f , expressed in equation ( 13), comprises modes with frequencies of 8Hz, 8.3Hz, and 16Hz, respectively, along with additional Gaussian noise with a signal-to-noise ratio (SNR) of 10dB.The signal has a duration of 5s and is sampled at a rate of 256 Hz.After obtaining the instantaneous amplitude of each IMF through Hilbert transform, we selectively intercept the data corresponding to the exponential attenuation part of the IMF for subsequent fitting, ensuring greater accuracy in the fitting result.Figure 2 demonstrates the selection of the instantaneous amplitude data of the falling edge based on the shape of IMF1.Subsequently, through least square fitting of the instantaneous amplitude logarithm and instantaneous phase of the intercepted data, the fitting curves in Figure 3 are generated, and the slope of the lines is calculated using equations (10,11).The effective fitting ensures the accuracy of modal parameter determination.Finally, the frequency and damping ratio of the three modes are obtained according to the equations Firstly, we employ the VMD-HT method to identify the modal parameters of the aforementioned signals.Here, K=3 and α=2000 are chosen.The time history and frequency spectrum of the original s ignal and Intrinsic Mode Functions (IMFs) obtained by VMD are displayed in Figure 1.Notably, due t o the limited frequency resolution of 0.2Hz for the 5s signal, distinguishing the two closely spaced modes, 8.0 Hz and 8.3 Hz, in the spectrum of the original signal proves challenging.However, VMD can still successfully extract them as IMF1 and IMF2.Additionally, the aforementioned signal is also analyzed using the polynomial fitting method in Prin80 software, a common tool for pulse data in flight flutter testing, as illustrated in Figure 4.The red curves represent the fitting result of the frequency response function for different modes.Due to the influence of poor frequency resolution, the peak near 8Hz in the spectrum is easily processed into a single mode, highlighting a limitation of the frequency domain method.However, in this case, we treat the peak as two modes to assess its ability to recognize closely spaced modes.Furthermore, the disparities between the results obtained from the two methods and the theoretical values are compared in table 1.In general, accurately identifying damping is more challenging than frequency.As observed, the identification accuracy of the two methods for mode 2 and mode 3 is essentially the same.However, for mode 1, the VMD-HT method demonstrates higher accuracy in identifying its damping ratio, indicating that the VMD-HT method holds more promising advantages in processing multimode impulse signals with noise.

Application in flight flutter testing
Pulse excitation is a prevalent technique in flight flutter testing.The subsequent examples showcase the practical application of the VMD-HT method, specifically in the context of Pilot's shaking stick excitation for low-frequency modes and small rocket excitation for high-frequency modes.

Pilot's shaking stick excitation
The pilot's shaking stick is a commonly employed and easily implemented excitation method that doesn't necessitate additional devices on the aircraft.However, due to the low frequency of the pilot's manual excitation, it is unable to stimulate the high-frequency modes on the aircraft.Consequently, this method is typically utilized as an auxiliary means of excitation.During actual flight tests, for certain low-frequency modes on the wing, the pilot deflects the aileron by shaking the stick.This action applies pulse excitation to the aircraft, eliciting the corresponding impulse response signal.The same approach is taken for exciting the tail modes.Figure 5 illustrates an example of employing the pilot's shaking stick to deflect the ailerons and subsequently acquiring the pulse response signals of wings during the flight flutter testing of civil aircraft.The vibration on the leading edge of the right wing, as depicted in Figure 5, is chosen as the original signal and analyzed using the VMD-HT method.6 shows the IMFs obtained through VMD analysis, using K=2 and  =2000.The spectrum of IMFs aligns well with that of the original signal, as observed in Figure 7, highlighting the beneficial effect of the VMD method in extracting components.
original singnal 0 0.5 Additionally, table 2 presents a comparison between the calculated results of this method and those computed by the Prin80 software.The table also provides the two modal frequencies derived from the ground vibration test, which are used to determine the modal types.Due to increased stiffness caused by aerodynamic forces in the air, the frequency of bending modes of the wing will be higher than those measured on the ground.The damping observed in the ground test is much smaller than that in the flight test, making it not suitable for reference here.The results show that the identification outcomes of the VMD-HT method for low-frequency modes are reasonable and reliable, consistent with the results obtained from the software and the ground vibration test.This demonstrates that the VMD-HT method can be effectively employed for the data processing of pilot's shaking stick excitation.

Small rocket excitation
A small rocket, sometimes referred to as "bonkers" in early literature, serves as an excitation device in flight flutter testing.It operates by igniting gunpowder through electric ignition, creating a pulse force, as illustrated in Figure 8.This device is affixed to the wing, tail, or an external component of the aircraft.Upon activation, sensors on the aircraft measure the resulting pulse acceleration response.In an actual flight test, as illustrated in Figure 9, four sensors are symmetrically positioned at the tip and center of both the left and right wings, respectively.The installation of a small rocket on the left wing induces an upward pulse force during excitation, effectively stimulating the antisymmetric modes of the wing.All four sensors recorded distinct responses with a sampling rate of 256 Hz.Subsequently, their data were individually analyzed using the VMD-HT method and Prin80 software for comprehensive evaluation.In this context, we selected K=5 and  =2000 for the VMD analysis.
Recognizing the pivotal role of the VMD analysis quality in ensuring the accuracy of the VMD-HT method, Figures 10 and 11 present the modal decomposition results obtained from sensor LW2's data, successfully capturing five key modes spanning low to high frequencies.Finally, Figure 12 illustrates a three-dimensional graph simultaneously comparing modal parameters (frequency and damping ratio) derived from different sensors using both methods.
During flight flutter testing, it is essential to observe the variation trend of modal parameters.However, in general, there are discrepancies in the identification of the same mode among different wing sensors.If the differences are excessively significant, engineers may face challenges in selecting the appropriate modal parameters.Based on the results presented in Figure 12, it is evident that, compared to Prin80 software, the frequency and damping ratio obtained using the VMD-HT method show greater similarity across different sensor data, particularly for modes 2-4.Moreover, there is an enhanced consistency in identification results among symmetrically mounted sensors.In contrast, Prin80 software yields widely varying modal parameters for the same mode with sensors in different positions.The study demonstrates that the VMD-HT method, applied to small rocket excitation, improves the stability and robustness of modal parameter identification in flight flutter testing, regardless of sensor positions on the wings.

Conclusions
This study proposes a VMD-HT method for identifying modal parameters in impulse signals during flight flutter testing.Numerical results indicate that the VMD-HT method outperforms the traditional frequency-domain method found in the specialized software Prin80, especially in identifying closely spaced modes.Furthermore, the VMD-HT method is applied to impulse signals induced by the pilot's shaking stick and small rocket excitation during actual flight flutter testing, respectively.Through comparison and analysis, it is shown that the VMD-HT method effectively identifies both lowfrequency and high-frequency structural modes, enhancing the accuracy and robustness of modal parameter identification in engineering applications.This method proves to be a valuable tool for postprocessing and analyzing flight flutter data.
) Where k  , k m and k Φ denote the damping ratio, mass, and modal vector of the kth mode, respectively.( ) k Q t denotes the unit impulse force vector applied at 0 x x  .

Figure 1 .Figure 2 .
Figure 1.Time history (Left) and frequency spectrum (Right) of the original signal and IMFs

Figure 3 .
Figure 3.The Fitting curve for the instantaneous amplitude logarithm and instantaneous phase

Figure 4 .
Figure 4. Diagram of frequency response function by using the Prin80 software

7 Figure 5 .
Figure 5. Diagram of aileron deviation and the corresponding vibration of the wing

Figure
Figure6shows the IMFs obtained through VMD analysis, using K=2 and  =2000.The spectrum

Figure 6 .Figure 7 .
Figure 6.Time history of the original signal and IMFs obtained by VMD

Figure 8 .
Figure 8.The picture of small rocket excitation device in flight flutter testing

Figure 9 .Figure 10 .Figure 11 .
Figure 9. Position diagram of sensors and the small rocket on the wing

Figure 12 .
Figure 12.Comparison of modal parameters obtained by the two methods between different sensors

Table 1 .
Comparison between the calculated value and the theoretical value (SNR=10dB)

Table 2 .
The comparison between the calculated value and the ground test