Evaluation on Filter Performance of Variational Mode Decomposition and Its Application in Separating Closely Spaced Modes

-e equivalent filter characteristics of variational mode decomposition (VMD) are fully evaluated when applied to the fractional Gaussian noise (fGn) and the application in separating closely spaced modes of vibration system is performed in this paper. VMD is a newly proposed signal decomposition technique, which nonrecursively decomposes a signal into a given number of subsignals (modes), and each mode is mostly compact around a center pulsation. -e filter performance of VMD is largely dependent on the constraint parameter and the initialization of center frequencies. In order to extract the desired modes, criteria for the determination of decomposition parameters are established. -e initial center frequencies could be simply determined by prior estimatedmodal frequencies of the analyzed signal, while the constraint parameter is optimized utilizing a genetic algorithm (GA). A two-degree-of-freedom parametric system is considered to evaluate the capability of VMD in the separation of closely spaced modes. Compared with the noise-assisted versions of empirical mode decomposition (EMD) and wavelet packet transform (WPT), the parameter-optimized VMD can successfully separate the closely spaced modes while recovering the most modal information simultaneously. When introduced to the ground vibration test (GVT) of a horizontal tail, the proposed method successfully extracted the first five oscillation modes and identified the modal parameters accurately.


Introduction
Empirical mode decomposition (EMD) introduced by Huang et al. [1] is widely used to adaptively decompose a signal into separate spectral bands with different oscillation modes. EMD is a data-driven decomposition technique with a self-adaptive expansion basis and can interpret the physical meanings of data generated by nonlinear and nonstationary processes [2]. e subsignals obtained from EMD called intrinsic mode functions (IMFs) are ideally a collection of complete, adaptive, and orthogonal representation for the analyzed signal. e instantaneous modal information of each IMF thus can be derived through the Hilbert transform. Many EMD-based modal parameter identification methods have been proposed and had great impact on a variety of engineering applications [3][4][5][6].
Despite the advantages, EMD is limited by lacking thorough mathematical understandings and some other obvious shortcomings such as sensitivity to noise and sampling. Some noise-assisted versions of EMD such as Ensemble EMD (EEMD) [7], Complementary EEMD (CEEMD) [8], and Complete EEMD with Adaptive Noise (CEEMDAN) [9] have been proposed to refine the decomposition performance. Nevertheless, those attempts to overcome the limitations of EMD do not fundamentally change the essence of EMD as a dyadic filter bank [10]. If the center frequencies of two oscillation modes are close enough, EMD will be incapable of generating desirable IMFs. Two or more oscillation modes sharing the same spectral band in the frequency domain of a single IMF is the so-called mode mixing phenomenon [1], which is inevitable when multiple modes are closely spaced with each other.
Efforts including adaptive filter [11], wavelet packet transform (WPT) [3], and frequency shifting [12] alleviate the mode mixing phenomenon in an ad hoc manner at the cost of the completeness and self-adaptability of EMD.
Variational mode decomposition (VMD) [13] is a newly developed signal decomposition technique well founded on a sound theoretical background and much more robust to sampling and noise. Contrary to EMD, VMD is an entirely nonrecursive algorism where modes are extracted concurrently. In particular, VMD searches for a given number of modes and their respective center frequencies adaptively. Each mode is band-limited around the center frequency which, to some extent, narrows the bandwidth of the extracted modes and eliminates the mode mixing phenomenon. Experiments have shown the superior performance of VMD with regard to tone detection, tone separation, and noise robustness compared to EMD [13].
In practical applications, however, the number of modes is required to be predefined. And the selection of the parameter that controls the tightness of the band limits is partially based on experience. Inappropriate combination of critical parameters will have predictable impact on the nature of detected modes. e suggestion proposed by the author in [13] is checking the spectral overlap or orthogonality between modes, or looking at the residuals.
Recent developments of the VMD method mainly concentrate on the field of fault diagnosis. With the assistance of the particle swarm optimization [14][15][16], multikernel support vector machine [17], the k-nearest neighbour algorithm [18], and other novel algorithms [19][20][21], promising results have been obtained. In a similar manner with the EMD denoising, the VMD has also been introduced to signal denoising based on the detrended fluctuation analysis [22,23]. Moreover, the applicability of the VMD in identifying the electromechanical oscillatory modes has also been demonstrated in [24]. e identification process is based on the time-frequency analysis of nonlinear signals which arise after a large disturbance. e oscillatory power signal is converted into monocomponents through VMD and then the instantaneous modal characteristics are obtained via Hilbert transform.
Without loss of completeness, it is worthwhile to point out that, in addition to the aforementioned methods, other advanced decomposition techniques exist in the literature. Alternatively, the digital Taylor-Fourier transform (DTFT) could also be used to identify low-frequency electromechanical modes in power systems [25]. Instead of solving a variational problem in VMD, the DTFT decomposes a signal into monocomponents by spectral analysis using a filter bank. Some recent works related to chirp mode decomposition (CMD) are also concentrated on the signal decomposition. But the CMD and its improved versions mainly focus on the early fault detection and multifeature extraction of vibration signal with a fast fluctuating instantaneous frequency [26,27]. Likewise, some time-frequency analysis methods [28][29][30] could also filter the target signal into desired monocomponents but with their own limitations.
Considering the modal parameter identification of a vibration system, however, the aforementioned successful applications may not be suitable. In addition, few researches on the selection of decomposition parameters that are applicable to the separation of modes and parameter identification have been found in the literature. Based on that concern, this paper emphasizes the accurate modal parameter identification taking advantage of the filter characteristics of the VMD algorithm. e outline of this paper is as follows. Section 1 is the introduction. Section 2 introduces the primary mathematical principles of VMD and evaluates the filter performance when applied to fractional Gaussian noise (fGn). e predictable impact of the decomposition parameters on the nature of detected modes is also thoroughly investigated. Criteria for the selection of proper decomposition parameters that are applicable to the separation of modes and parameter identification of vibration system are established in Section 3. Section 4 extends the results to the ground vibration test (GVT) of a horizontal tail. Conclusions are listed in the last section.

VMD Analysis on fGn
2.1. Fractional Gaussian Noise. Fractional Gaussian noise (fGn) is defined as the increment process of fractional Brownian motion [31]. It is a zero-mean Gaussian stationary process whose autocorrelation sequence where k is a variable that denotes the time lag at which the autocorrelation is measured. As is well known, the special case H � 0.5 reduces to white noise, whereas other values induce nonzero correlations, either negative if 0 < H < 0.5 or positive if 0.5 <H < 1. e description of fGn is mainly determined by the value of Hurst exponent, H, while the variance σ 2 is merely a scale parameter.
Taking the discrete Fourier transform of equation (1), the power spectrum density of fGn is obtained: with |f| ≤ 0.5. If H ≠ 0.5, then S H (f) ∼ Cσ 2 |f| 1− 2H when f approaches zero. It, therefore, follows that fGn is a convenient model for power-law spectrum at low frequencies [32]. If 0 <H < 0.5, the spectrum is high-pass having S H (0) � 0. On the other hand, if 0.5 <H < 1, a "1/f"-type spectral divergence is delineated with S H (0) � ∞.

Brief Review of VMD.
e goal of VMD is to decompose a real-valued input signal f into a given number of modes u k and each is assumed to be mostly compact around a center pulsation ω k . e definition of mode u k in VMD is an amplitude-modulated-frequency-modulated (AM-FM) signal, written as 2 Shock and Vibration which is more mathematically rigorous than the so-called IMF. e search for u k and ω k results in a constrained variational problem as follows: where u k : � u 1 , . . . , u k and ω k : � ω 1 , . . . , ω k are shorthand notations for the set of all modes and their center frequencies, respectively. In order to render the problem unconstrained, both a quadratic penalty and Lagrangian multipliers λ are used. e augmented Lagrangian L is introduced as where α denotes the constraint parameter of the data fidelity that controls the tightness of the band limits. e saddle point of the augmented Lagrangian L is the solution to the original minimization problem of equation (4). Detailed implementation can be found in [13]. Particularly, the modes are updated by simple Wiener filtering directly in Fourier domain which makes the algorism more robust to sampling and noise.

Filter Performance of VMD on fGn.
Similar to some previous studies [10,32,33], extensive simulations are carried out on fGn processes with H being set to three typical values 0.2, 0.5, and 0.8, and the variance σ 2 being fixed to 1 for all samples. e data length is taken to be N � 1024 and, for each value of H, 1000 independent sample paths of fGn are generated via the Wood and Chan algorithm [34]. Given the data sets, the VMD is realized for all sample paths resulting in a collection of modes. Spectral analysis is carried out for each mode of each realization. e Fourier spectra of each mode u k are averaged and arranged mode by mode in Figure 1. e number of modes K is set to 5 with uniformly distributed initial center frequencies. at is, ω 1 k � 0, 0.1, 0.2, 0.3, and 0.4 Hz { }. For comparison purposes, the averaged Fourier spectra of IMFs for EMD realizations are also displayed in Figure 2. As evidenced in Figures 1 and 2, the collection of the extracted modes u k by VMD tends to self-organize in a filter bank structure which is similar to what is classically observed in wavelet packet decomposition rather than the wavelet-like structure exhibited in EMD [32]. Practically, all the extracted modes u k are equivalent band-pass filters, which are different from EMD where the filter associated with the 1st mode is essentially high-pass. e spectral band of the analyzed sample paths of fGn is divided into five discrete subfrequency regions almost evenly. at is to say, the bandwidth of each filter is practically identical to each other evolving with the uniformly distributed predominant center frequencies. Moreover, the increase of constraint parameter α narrows the pass-band of each filter and thus minimizes the overlap between the separated modes simultaneously.
In a detailed investigation, the averaged center frequencies and the energy distribution of the extracted five modes are calculated and plotted against the modal order in Figures 3  and 4, respectively. Regardless of the value of α, the energy distribution shown in Figure 4 successfully characterizes the power spectrum density of fGn. As mentioned in the previous subsection, in the case of H � 0.2, the spectrum is high-pass and, therefore, follows an increasing trend of energy proportion versus the modal order; while in the case of H � 0.8, the "1/f"-type spectral divergence is captured; for the special case of H � 0.5, the power spectrum density is a constant which is also true for the energy distribution. e impact of the constraint parameter α on the center frequencies ω k may not be convincing as depicted in Figure 3, yet it still drops some results. e center frequencies ω k skewed either to the right or left characterizing the spectrum of fGn for a small value of α are forced to converge to the uniformly initialized center frequencies ω 1 k with the increase of α.

Impact of Initialization and Constraint Parameter α on
Filter Performance. In order to further verify the impact of constraint parameter α on the convergence, the normalized center frequencies are initialized randomly as where N is the data length which is 1024 in all simulations of this paper. Statistically, the base-2 logarithm of the initial center frequencies is uniformly distributed, as shown in Figure 5, whereas the regular values are concentrated in the low-frequency band.
Distinguished from the results obtained where the center frequencies are initialized uniformly, the averaged Fourier spectra, as illustrated in Figure 6, tend to organize in disorder with merely no statistical law to find, in the case of a larger value of α. Whereas the filter bank structures observed in the case of α � 1e3 coincide with those generated in the previous realizations as the dashed lines depicted in Figure 6 overlap with the blue ones except for the case of H � 0.8. Scatter diagrams of the initial center frequencies ω 1 k versus the converged center frequencies ω n k as shown in Figure 7, where each mode is denoted by a corresponding color, demonstrate a growing linear relationship between each other with the increase of α. Linear stripe patterns that are distinct from each other are visible in Figure 7(a), indicating that the adaptability is successfully recovered regardless of the initialization in the case of a small value of α. Moreover, the histograms plotted in Figure 8 also suggest the same result since the empirical probability density distribution of the converged center frequencies ω n k gradually evolves to that of the ω 1 k , with an increasing value of α. While only the Shock and Vibration 3 convergence analysis in the case H � 0.2 is demonstrated here, the same results could be observed regardless of the Hurst exponent.
In summary, the filter performance of VMD does depend on the initialization as well as the value of constraint parameter α, suggesting that the algorithm does not  necessarily converge to a global minimum. From a different point of view, the selection of α and initial center frequencies ω 1 k could be refined to extract a collection of desired modes.

Separation of Modes Based on VMD
3.1. Parameter-Optimized VMD. Inspired by the conclusion of the previous section, criteria for the optimization of α and the initialization of center frequencies ω 1 k are proposed in this section, in order to extract a collection of desired modes of vibration system. For an actual dynamic structure containing closely spaced modes, the identification of modal frequency is much more efficient and accurate than the damping ratio, since each significant peak in frequency domain could be roughly taken as one natural mode. Hence, the predominant center frequencies of the equivalent band-pass filters in VMD realization are to be initialized by the prior identified modal frequencies utilizing the simple peak-picking method and there follows the number of modes K. However, the desired value of α must be able to separate the coupled spectrum of closely spaced modes and retain the most modal information simultaneously. e strategy suggested here is optimizing the value of α through genetic algorithm (GA). e constructed fitness function is described as where r(u k , f) denotes the correlation coefficient between a mode u k , f and the analyzed signal f; ε denotes the penalty factor that is taken as the number of duplicate modes by checking the matrix of correlation coefficients for the extracted modes u k . Duplicate modes are detected when the correlation coefficient r(u i , u j ) between two separated modes is extremely large. e optimal value of α opt satisfies α opt � argmin α (fitness). e instantaneous modal information of each monocomponent subsignal u k thus can be derived through the Hilbert transform. Primary steps of the parameter-optimized VMD realization are as follows: Step 1. Preestimate the modal frequencies ω i , i � 1, 2, . . . , M of the analyzed signal f using simple peak-picking method.
Step 2. Initialize the center frequencies ω 1 k � ω i and there follows the number of modes K � M.
Step 3. Search for the optimal value α opt � argmin α (fitness) through GA.   Step 4. Decompose the signal f into a desired number of modes u k with the parameter-optimized VMD.
Step 5. Derive the analytic signal u k,A of each mode by Hilbert transform and obtain u k,A (t) � A k (t)e jϕ (t) .
Supposing the signal f is the impulse response of a small damping vibration system, we have ω(t) � dϕ(t)/dt � ω k ����� 1 − ζ 2 k and ln A k (t) � −ζ k ω k t + c. us, the modal frequency ω k and damping ratio ζ k are identified by least squares method.

Numerical Experiment.
A 2-dof parametric system as shown in Figure 9 is designed to evaluate the performance of the proposed method on the separation of modes. e dynamics of the system are described by equation (8): where the δ(t) is the impulsive load. e two natural frequencies derived are By changing the value of μ, the ratio c of the two natural frequencies is altered. Given the designed values of damping ratio (ζ 1 , ζ 2 ), the damping matrix C is reconstructed as where M denotes the mass matrix and K denotes the stiffness matrix.
In the present numerical experiment, the displacement response u 2 is recorded as the output signal with a sampling frequency F s � 1024 Hz. e structural parameters are chosen with m � 1 kg, k � 8000 N/m, c � 0.8, ζ 1 � 0.01, and ζ 2 � 0.015. Besides, an additional zero-mean Gaussian white noise is added to the output response simulating the measurement noise.
e signal-to-noise ratio (SNR) is set to 20 dB. e EEMD and CEEMDAN are realized on the contaminated response signal. Eliminating the first two IMFs containing high-frequency noise and the last several trends, the primary modes are plotted in Figures 10 and 11, respectively. e results are unsatisfying because both realizations not only fail to separate the two oscillation modes but also generate additional pseudocomponents. Contrary to the noise-assisted versions of EMD realizations, the wavelet  packet transform (WPT) identifies the situation as a single AM-modulated signal as evidenced in Figures 12(b) and 12(c). e best tree is shown in Figure 12(a) corresponding to the decomposition result down to level 5 using the db4 wavelet. In this case, the best tree reduces to the wavelet tree, and the WPT merely plays the role of signal denoising. Different from the failed attempts to decompose the signal into separate monocomponents, the proposed parameter-optimized VMD successfully captures the two oscillation modes. As illustrated in Figure 13(b), although there is a slight mixture of modal contribution considering the relative amplitude, the decomposed results obtained by the optimized VMD outperform the implementation of EEMD, CEEMDAN, and WPT. e desired two oscillation modes are extracted concurrently while preserving the dynamics of the signal. Both resonances and antiresonances remain at the same frequency lines, which is not the case of the obtained using WPT in Figure 12; even this appears to be the smoother and better reconstructed spectra. e Poly-MAX identification results in Figure 14, which will be discussed later, further support the argument since the antiresonance is present, apparently at the same frequency line as identified by the two extracted modes in Figure 13. Compared with the theoretical time history of the desired modes in Figure 13(a), the error is mainly concentrated on the left boundary due to the simple mirror extension of the signal as suggested in [13]. e boundary issues are also classically observed in EMD which could be only partially alleviated by more mathematically attempts. Furthermore, the obtained Hilbert spectrum as shown in Figure 13 e damping ratios are ζ 1 � 1.01% and ζ 2 � 1.47% with the percentage error of 1.42% and 2.31%. For comparison purpose, traditional modal parameter identification is conducted utilizing the PolyMAX method in frequency domain with the aid of stabilization diagram [35]. e results are illustrated in Figure 14. e identified modal frequencies are f 1 � 14.29 Hz and f 2 � 17.83 Hz with the percentage error of 0.38% and 0.18%. e damping ratios, however, are ζ 1 � 2.00% and ζ 2 � 2.97% with the considerable percentage error of 100.42% and 97.81%.
Compared with some other adaptive signal decomposition methods and the modal parameter identification using PolyMAX in frequency domain, the proposed parameter-optimized VMD is superior in separating closely spaced modes and retaining the most modal information simultaneously. While only the case c � 0.8 at the noise level of SNR � 20 dB is displayed in this section, more complete numerical experiments within the range 0.5 ≤c ≤ 0.9 under the noise condition of SNR � 20 dB and 10 dB are carried out. e identified modal parameters are listed in Table 1. It appears that the identification of modal frequency is much more accurate than the damping ratio. Despite that, the maximum percentage error of damping identification is still within 5% which is acceptable in engineering practice. e complete investigation further demonstrates the effectiveness of the presented method based on VMD in separating closely spaced modes of vibration system.

Computational Complexity.
e computational complexity of a VMD algorithm depends on the initialization of the center frequency of each mode and the recursive Fast Fourier Transform (FFT). As is well known, the computational complexity of the FFT algorithm is O (N log N), where N is the length of the transform domain signal or the number of points in DFT. For the extraction of the k number of modes, the computational complexity of the VMD realization will be the sum of the required computations for the initialization of center frequencies and k O(N log N) [24]. Besides, the presented optimized version of VMD adds an extra requirement, which increases the computational cost and it is also subjective, in terms of which optimization technique should be deployed, not to mention its specific parametrization. Table 2 shows the comparison of the computational cost of the presented decomposition techniques. e computational complexity of the EMD/EEMD has proven to be equivalent to that of the FFT but with a larger factor [36]. As an improved version of EEMD, the CEEMDAN algorithm does not fundamentally change the core structure of EEMD but improves the procedure of the noise-enhanced realization of EMD [9]. erefore, in terms of computational complexity, the presented versions of EMD along with WPT are identical to each other. It should be noted that the specific computational complexity of VMD has not been found in the literature and will not be given in this paper, which is beyond the scope of this research. Despite that, the complexity of the VMD algorithm exceeds that of the presented decomposition techniques, not to mention the optimized version.
A numerical experiment is performed to verify the actual computational cost of the decomposition techniques including the original VMD algorithm and the optimized version. e experiment is conducted on a laptop computer with 2.6 GHz CPU and 8 GB memory under the MATLAB environment. e simulation is repeated 100 times for each decomposition technique and the average execution time for each implementation is obtained. e test signal is the displacement response of the 2-dof parametric system recorded at the sampling rate of 1024 Hz and the ratio of the two natural frequencies c set to 0.8, which is the same as that demonstrated in the previous section. Moreover, the specific parameter settings are consistent with those demonstrated in the previous section to obtain the same results as illustrated in Figures 10-13.
As shown in Table 2, without considering the decomposition results, the EMD and WPT are both computationally efficient. e execution time of EEMD and CEEMDAN, however, differs greatly with that of EMD, which is because both algorithms require extensive noiseenhanced realizations of EMD to alleviate the mode mixing phenomenon [7][8][9]. When it comes to each realization, the  Table 2 is also the same in terms of the order of magnitude. Relatively, the execution time of the original VMD implementation is increased by one order of magnitude, where the parameter α is arbitrarily taken as α � 2000. While only the case α � 2000 is demonstrated in this section for the sake of readability, in a more complete experiment, the value of α has little impact on the execution time when the number of modes is fixed to K � 2. Compared with the original VMD implementation, the presented optimized version increases the computational cost by two orders of magnitude.
is means that after hundreds of VMD realizations, the optimal parameter could be finally determined, which is inevitable no matter what optimization technique should be deployed, because, basically, the optimization is the process of trial and error. Although the presented optimized VMD consumes much more time than other decomposition techniques, the decomposition results of the optimized VMD outperform the compared techniques in the separation of closely spaced modes while preserving the most modal information at the same time. Since the priority of modal analysis is the accuracy of the identified modal parameters, the additional computational cost of the presented optimized VMD could be tolerant in practical applications, whereas the compared methods lose their validity in the same situation.

Ground Vibration Test of a Horizontal Tail.
Ground vibration test (GVT) of a horizontal tail using a hammering method is conducted to analyze the dynamic characteristics of the structure. e test horizontal tail as pictured in Figure 15 consists of a metal beam and composite frames that provide the airfoil shape. e wing structure was manufactured by 3D printing and covered with composite skin. In order to check the vertical bending and torsion modes, the acceleration sensor was fixed at a selected point while the hammer excited the structure at different locations to estimate the frequency response functions (FRFs). e test data obtained in the case of the distribution of acceleration sensor and excitation point as illustrated in Figure 15 is adopted to verify the effectiveness of the proposed method based on VMD. e sampling frequency of the test data is 1024 Hz with a 5-second duration. e Fourier spectrum, or rather FRF, of the response signal is shown in Figure 16(b) where only the first five modes are considered here. A low-pass filter with the passband frequency of 80 Hz is designed to remove the highfrequency response. e selection of the filter band is a manual process. e modal frequencies of the five natural modes are arbitrarily estimated by the peak-picking method N is the data length and NR denotes the number of realizations.

Excitation point
Acceleration sensor Figure 15: Test horizontal tail.  e optimized value of α through GA is 1.31e5 in this case. Nevertheless, some stateof-the-art optimization algorithms such as particle swarm optimization (PSO), simulated annealing (SA), and immune algorithm (IA) could also be utilized to search for the optimal value of α while no significant impact on the outcome has been found by the author. e extracted modes and the corresponding Fourier spectra are depicted in Figure 17 where obviously each component contains a single oscillation mode. Despite the boundary issues, the Hilbert spectrum as shown in Figure 18 further demonstrates that the frequency pattern is successfully captured without overbinning or underbinning. e modal information of the extracted modes is derived by Hilbert transform and detailed in Table 3. With the purpose of comparison, the PolyMAX method in the frequency domain is also employed to identify the modal parameters of the low-pass filtered test data. e results are also listed in Table 3 and the stabilization diagram is plotted in Figure 19.
Compared with the theoretical values calculated via the finite element method (FEM), the first five modal frequencies of a single test identified are quite accurate for both methods where all the percentage errors are within 0.2%. However, the performance of the two methods on the identification of the damping ratio differs greatly. e identification errors of the proposed method are dramatically small with the maximum value within 3%. e Poly-MAX identification in the frequency domain behaves relatively badly for the vibration test. Again, the experimental results demonstrate the effectiveness of the proposed method in the separation of modes and recovering the most modal information at the same time.

Conclusions
In this paper, the equivalent filter characteristics of VMD are studied when applied to the fGn, and the application in the separation of closely spaced modes of vibration system is performed. Extensive numerical simulations have been carried out and demonstrated the differences in filter performance between VMD and EMD. Contrary to EMD, the VMD possesses the filter characteristics similar to WPT with almost identical bandwidth. Moreover, the initialization along with the constraint parameter α does have a great impact on the decomposition performance of VMD. For a small value of α, the self-adaptability of VMD is recovered. In addition, regardless of the initialization, the spectral characteristics of the inspected signal are successfully captured. Meanwhile, the increase of α reinforces the linear relationship between the initial center frequencies and the converged frequencies. In other words, the algorithm is forced to converge to the local minimum in the case of a large value of α. Taking advantage of the filter performance, criteria for the optimization of α and the initialization of center frequencies ω 1 k are established to extract a desired collection of modes. Center frequencies are to be initialized by the prior identified modal frequencies utilizing the simple peak-picking method, whereas the value of α is optimized through GA based on the proposed fitness function. Both numerical simulations and experimental tests have demonstrated the capability of the proposed method based on VMD in the separation of modes while retaining the most modal information at the same time. However, the presented optimized VMD adds an extra requirement which greatly increases the computational cost. Considering the accuracy of the identified closely spaced modes, however, the presented optimized VMD could be applied in engineering practice, whereas the compared methods lose their validity in the same situation.
Although all the analyzed response signals in this paper are obtained under impulse excitation, for random responses, techniques like random decrement technique (RDT) and natural excitation technique (NeXT) could be used to extract the free decaying response and thus complete the applicability of the proposed method.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.