Hybrid Empirical and Variational Mode Decomposition of Vibratory Signals

Abstract

Signal analysis is a fundamental field in engineering and data science, focused on the study of signal representation, transformation, and manipulation. The accurate estimation of harmonic vibration components and their associated parameters in vibrating mechanical systems presents significant challenges in the presence of very similar frequencies and mode mixing. In this context, a hybrid strategy to estimate harmonic vibration modes in weakly damped, multi-degree-of-freedom vibrating mechanical systems by combining Empirical Mode Decomposition and Variational Mode Decomposition is described. In this way, this hybrid approach leverages the detection of mode mixing based on the analysis of intrinsic mode functions through Empirical Mode Decomposition to determine the number of components to be estimated and thus provide greater information for Variational Mode Decomposition. The computational time and dependency on a predefined number of modes are significantly reduced by providing crucial information about the approximate number of vibratory components, enabling a more precise estimation with Variational Mode Decomposition. This hybrid strategy is employed to compute unknown natural frequencies of vibrating systems using output measurement signals. The algorithm for this hybrid strategy is presented, along with a comparison to conventional techniques such as Empirical Mode Decomposition, Variational Mode Decomposition, and the Fast Fourier Transform. Through several case studies involving multi-degree-of-freedom vibrating systems, the superior and satisfactory performance of the hybrid method is demonstrated. Additionally, the advantages of the hybrid approach in terms of computational efficiency and accuracy in signal decomposition are highlighted.

1. Introduction

Signal analysis is a broadly explored field of study that has provided fundamental support for significant advancements across various areas since its introduction. This interdisciplinary approach combines principles of mathematics, statistics, and engineering to interpret and manipulate both digital and analog signals for the purpose of estimating relevant information. This process plays a crucial role in a variety of fields, ranging from medicine to engineering and control systems, facilitating the development and optimization of advanced systems in each area [1,2].

From the fundamentals established by the Fourier Transform, which enabled the decomposition of signals into frequency components, signal analysis has undergone significant evolution [1,3]. The Fourier Transform, introduced as a mathematical tool for understanding signal frequency content, has found widespread applications in telecommunications, audio processing, and signal compression [4,5,6]. However, it assumes that signals are stationary and linear, limiting its utility for nonlinear and nonstationary signals, which are frequently encountered in real-world scenarios [7,8].

To address these limitations, new methodologies have emerged, including the Empirical Mode Decomposition (EMD) and the Hilbert–Huang Transform (HHT), which were designed for analyzing nonlinear and nonstationary data [7,9]. The EMD decomposes a signal into intrinsic mode functions (IMFs) adaptively, which can then be analyzed using the Hilbert Transform to extract instantaneous frequencies and amplitudes [10]. This approach has been applied effectively in power quality assessment, where EMD and the Hilbert Transform are used to detect and classify disturbances such as voltage sags and spikes, demonstrating superior accuracy compared to S-transform methodologies [11]. This methodology has proven effective in applications ranging from geophysics, where it provided new insights into water wave dynamics [10,12], to engineering, where it has been applied for fault diagnosis and vibration analysis [13,14,15,16].

The robustness of the HHT has been further validated through applications in biomedical engineering, where it has been used to analyze physiological signals such as brain wave activity and heart rate variability [17,18]. These successes underscore the versatility of EMD and the HHT in handling dynamic and noisy environments, offering a granular view of signal behavior that traditional methods cannot achieve.

However, traditional techniques like the Fourier Transform are often limited in their ability to analyze nonlinear and nonstationary signals, which are common in real-world scenarios [7,8]. Similar to how iterative conformal mapping has been used to simplify complex fluid domains [19], advanced methodologies such as Empirical Mode Decomposition and Variational Mode Decomposition (VMD) have emerged to address these challenges [20]. These approaches are particularly useful in capturing dynamic behaviors and transitions, as seen in the study of instabilities in nonlinear systems like Stokes waves [21].

Further advancements, such as Differential Empirical Mode Decomposition (DEMD), extend the capabilities of EMD by addressing issues like mode mixing and aliasing. DEMD has shown particular promise in applications involving vibration signal analysis, providing superior accuracy in frequency component extraction [22]. Advanced approaches like Empirical Multi-Synchroextracting Decomposition (EMSD) have further enhanced mode decomposition capacity, offering superior precision for dynamic and nonstationary scenarios [23].

Variational Mode Decomposition employs a spectral-domain approach to decompose signals into intrinsic modes. Instead of operating directly in the time domain, VMD optimizes a model based on the spectral distribution of the signals through a cost function [24,25,26]. VMD has found success in various fields, including power systems, where it has been employed to detect high-voltage discharges and enhance power quality [27,28]. Its ability to separate complex frequency components has also proven useful in structural health monitoring and mechanical fault detection [29,30]. Despite its strengths, VMD relies heavily on predefined parameters, which can limit its adaptability in scenarios involving nonstationary or highly nonlinear signals [20].

To overcome the limitations of EMD and VMD individually, this manuscript presents a hybrid approach that leverages the strengths of both Empirical Mode Decomposition and Variational Mode Decomposition. This hybrid EMD–VMD strategy leverages EMD’s ability to detect mode mixing and uses this information to guide the parameter configuration for VMD, improving both the accuracy and efficiency of signal decomposition. Similar hybrid methodologies, such as VMD combined with Hilbert–Huang Transform (VMD–HHT), have demonstrated effectiveness in extracting multidimensional dynamic response characteristics in offshore engineering [31]. This approach is particularly effective for nonlinear and nonstationary signals, where traditional methods often fall short [20,25,32].

Supported by studies addressing cases of complex nonlinear systems [17,33,34,35,36], the hybrid EMD–VMD approach is presented as a practical option with satisfactory results for advanced signal analysis. By integrating the adaptive capabilities of EMD with the spectral precision of VMD, this methodology effectively addresses challenges associated with signals exhibiting nonlinear and nonstationary characteristics, such as time-varying frequencies and amplitudes, quadratic trends, and Gaussian noise [7,37].

In addition to improving accuracy and efficiency in the decomposition of complex signals, the hybrid method demonstrates remarkable flexibility in interdisciplinary applications. Fields such as biomedical signal processing, vibration analysis, and structural monitoring benefit from this strategy, not only in terms of robustness against noise but also in its ability to identify and separate frequency components in dynamic and noisy scenarios [10,27,38]. As a result, this approach stands out as a good option with satisfactory results for complex real-world applications.

Four case studies are presented to illustrate the functionality and application of this hybrid strategy. The first case study serves as an explanatory example of the method presented. Here, a signal composed of six vibratory components with well-defined frequencies is analyzed, demonstrating the step-by-step application of the EMD–VMD hybrid methodology and its effectiveness in detecting mode mixing.

The second case study examines a two-degrees-of-freedom mechanical vibrating system, where a detailed comparison is made between the mathematical development of the vibratory components and their decomposition using the HHT, VMD, and the hybrid EMD–VMD method. This allows for an evaluation of how the hybrid approach improves the estimation of natural frequencies and provides a more accurate understanding of the system’s vibratory response.

The third case study addresses a complex signal composed of six vibratory components with very close frequencies. This scenario is particularly challenging for conventional EMD due to mode mixing. The hybrid EMD–VMD strategy proves its usefulness by detecting mode mixing and providing crucial information about the number of vibratory components, thereby improving the accuracy and efficiency of estimation using VMD.

The fourth case study explores the application of the hybrid EMD–VMD method to a highly complex signal designed to emulate real-world nonlinear systems. This composite signal includes four vibratory components with time-dependent frequencies and amplitudes, a quadratic trend, and additive Gaussian noise. These characteristics introduce significant challenges for traditional analysis methods such as the Fast Fourier Transform (FFT). This case demonstrates the robustness of the hybrid approach in accurately decomposing signals with dynamic, non-periodic behaviors, highlighting its potential for addressing noise and nonlinearity more effectively than conventional techniques.

Overall, these case studies validate the ability of the hybrid EMD–VMD method to overcome the challenges associated with complex signals, such as nonlinearity and nonstationarity. By integrating the strengths of EMD and VMD, this strategy offers results highlighted in the advanced analysis of nonlinear and nonstationary signals, as well as opening up possibilities for the development of future applications in fields such as engineering, biomedical sciences, and structural monitoring. Its accuracy, efficiency, and robustness against noise confirm its relevance as an essential tool for solving real-world problems in dynamic and noisy systems. Future research will aim to address identified limitations, such as conflicts with extremely close frequencies and high noise levels, as well as develop computational efficiency confirmation indices to further validate the method. This document contributes to the field by presenting an hybrid methodology and demonstrating its effectiveness compared to conventional techniques, laying the groundwork for solutions in the analysis of complex signals.

2. Evolution of the Hilbert–Huang Transform

Signal analysis has seen significant advancements since the introduction of the Fourier Transform, which enabled the decomposition of functions into sums of sines and cosines, facilitating signal analysis in terms of frequency components. This method laid the groundwork for subsequent developments, such as the Fast Fourier Transform, which revolutionized signal processing and found applications in a wide range of disciplines, including electrical engineering, economics, and medicine [1,3,6,8].

However, the Fourier Transform has limitations when applied to nonlinear and nonstationary signals due to its assumption of stationarity [6]. These limitations led to the development of adaptive methods to address the complexity of nonlinear systems.

One such advancement is Empirical Mode Decomposition, which decomposes signals into intrinsic mode functions that adapt to the local characteristics of the signal. The IMF is defined as a function that meets two conditions, where the number of extrema and the number of zero-crossings differ by no more than one, and the mean of the upper and lower envelopes is zero at any point. The IMF is essential in signal decomposition, enabling adaptive analysis of nonstationary and nonlinear signals [7].

This method offers a comprehensive representation of signal components across time and frequency domains. The EMD process includes identifying local extrema, constructing upper and lower envelopes, and iteratively subtracting their mean to yield the IMF. This ensures an accurate decomposition of harmonic vibratory components while maintaining the fundamental characteristics of the original signal.

Further refinements to EMD were made to improve its mathematical framework while maintaining its adaptability [9]. Additionally, EMD has proven effective in separating oscillation modes into stationary sub-signals, making it valuable for analyzing complex, nonlinear, and nonstationary data [12]. EMD has also been applied in various fields, including image compression, where it has been compared to wavelet packet decompositions [24].

Despite its strengths, EMD faces challenges, particularly the mode mixing problem, which can hinder the clear separation of signal components. To address this, an enhanced methodology called Ensemble Empirical Mode Decomposition (EEMD) was developed, specifically designed to overcome mode mixing [25]. EEMD enhances signal feature extraction and has proven effective in decomposing signals into local periodic oscillations [13].

The Hilbert Transform also plays a crucial role in signal analysis by converting real signals into their analytic form, allowing for the better estimation of instantaneous envelope and phase [25]. Its applications span various fields, including theoretical physics, engineering, and modulated signal analysis [28,29].

The Hilbert–Huang Transform integrates EMD with the Hilbert Transform to provide a unified approach to signal analysis [7,10]. This method enables the adaptive decomposition of signals into intrinsic components, which can then be analyzed in detail using the Hilbert Transform. The HHT has proven to be particularly effective in analyzing nonlinear and nonstationary signals, surpassing traditional methods such as the Fourier Transform in these contexts [39,40].

The Hilbert–Huang Transform has been widely used in various applications, such as fault diagnosis, vibration analysis, and structural health monitoring, where it enables early fault detection and improves machinery performance [30,41,42]. Additionally, it has been applied in diverse areas, including the analysis of animal vocalizations and the study of macroeconomic indicators [43,44].

Further advancements in signal decomposition have been achieved with the introduction of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN), which improves the separation of signal components by addressing mode mixing through the use of adaptive noise [45]. This method has been applied in various fields, such as biomedical signal analysis and vibration monitoring [46,47,48,49].

In more recent developments, Variational Mode Decomposition (VMD) has been introduced as an advanced technique for adaptive signal decomposition, addressing mode mixing through a variational approach [20]. This technique represents a significant evolution in signal analysis methods as illustrated in the Figure 1, offering a robust alternative to EMD for analyzing nonlinear and nonstationary signals.

Since its introduction, the HHT has been applied in diverse fields, including environmental and geophysical sciences, neurology, and technological innovation. In environmental science, it has facilitated the analysis of PM10 particle concentration fluctuations in bulk loading ports [50] and has played a crucial role in identifying areas prone to landslides in Taiwan [51]. In the context of natural disaster prevention, its use in analyzing infrasound signals for early warning systems for tsunamis highlights the potential of the HHT to contribute to safety and preparedness [52].

In neurology, the HHT has enabled advances in understanding cerebral blood flow regulation [53] and the analysis of electroencephalographic signals, opening up new possibilities in the study of brain activity and the development of advanced brain-computer interfaces [54,55]. In technological innovation, the HHT has spurred advancements in structural health monitoring and fault diagnosis in machinery, proving exceptional for detecting structural damage [56] and diagnosing faults in induction machine bearings [16,57].

In engineering and materials science, the HHT has been applied to innovative solutions such as gas detection using EFPI fiber signals and stochastic analysis of vibratory signals from induction motors for fault detection [58,59]. The integration of the HHT with artificial intelligence techniques for structural health monitoring and the improvement in feature selection for fault diagnosis in ball bearings underscores its synergy with advanced statistical methods [50,60].

In geophysics and seismology, the HHT has enhanced our understanding of complex natural phenomena, from magnetic properties linked to density irregularity structures to the study of blasting vibration effects in tunnels [61,62]. Its application in estimating lightning current parameters and calculating seismic intensity parameters for designing reinforced concrete structures showcases its contribution to safety and effectiveness in analyzing natural phenomena and their effects on infrastructure [63,64].

Since its introduction, the HHT has undergone numerous studies aimed at enhancing its precision and effectiveness. For example, the Cole–Hopf transformation has been pivotal in linking linear and nonlinear equations in Banach spaces, providing a basis for understanding the relationship between the evolution operator and its infinitesimal generator within a nonlinear semigroup framework [65]. The trend toward big data analysis has become apparent, with the HHT being applied more extensively thanks to its integration with big data technologies, improving the detection of complex patterns and prediction in various fields [66,67].

The algorithms underlying EMD and its variants, such as EEMD and CEEMDAN, have been refined to improve their efficiency and precision, including the development of adaptive approaches for sifting and parameter selection. Comprehensive reviews of the evolution, challenges, and solutions surrounding the HHT highlight its versatile applications in fields ranging from mechanical engineering to biomedicine [68,69]. These studies emphasize how the HHT has evolved to address complex signal analysis problems by adapting to the inherent characteristics of the analyzed signals.

The HHT continues to evolve, addressing challenges and finding applications in a wide range of fields. Its development reflects a continuous effort by the scientific community to understand and analyze the inherent complexity of dynamic systems. The global COVID-19 pandemic accelerated the adoption of digital health technologies and telemedicine, where the HHT found important applications in the analysis of biomedical signals for remote patient monitoring and treatment personalization. The combination of the HHT with emerging deep learning models emerged as a promising area of research, offering significant improvements in complex data classification and analysis and facilitating the development of advanced diagnostic systems [2,70].

Looking forward, the HHT is expected to play a crucial role in predictive analysis, leveraging its ability to decompose and analyze complex signals to predict future events and trends more accurately. Its ongoing development signifies its potential to expand its applications to complex systems in environmental sciences, quantum physics, and beyond. This historical overview underscores the ongoing role of the HHT as a dynamic and adaptable tool in data analysis, prepared to meet emerging challenges in science and technology. The HHT and its related developments continue to evolve, driven by the need for analytical tools capable of addressing the increasing complexity of data in the modern era.

3. Hybrid Methodology EMD–VMD

The hybrid methodology integrates Empirical Mode Decomposition and Variational Mode Decomposition to enhance the estimation of components in signals with multiple harmonic vibrations, particularly those characterized by closely spaced, nonlinear, and nonstationary frequencies. This approach leverages the adaptability of Empirical Mode Decomposition in extracting signal features and the precision of Variational Mode Decomposition in separating components, effectively addressing mode mixing.

3.1. Step 1: Estimation of IMFs Using EMD

Empirical Mode Decomposition decomposes the signal into a set of intrinsic mode functions by identifying the inherent oscillatory modes present in the signal with multiple harmonic vibratory components. Each IMF is a fundamental oscillatory component within the signal, allowing it to be broken down into functions that represent oscillatory characteristics at different scales or frequencies.

As shown in Figure 2, the signal exhibits multiple harmonic vibratory components, each representing distinct oscillatory characteristics that can be analyzed through the Empirical Mode Decomposition process. This is essential for analyzing nonstationary and nonlinear signals, in which frequencies and amplitudes vary over time. The process of obtaining the IMFs involves several key steps including the identification of local extrema, the construction of upper and lower envelopes, the mean envelope calculation, the IMF estimation, and the iteration with residue calculation.

Identification of Local Extrema: The first step in EMD is to identify all local maxima and minima of the signal $x\left(t\right)$, as depicted in Figure 3. These extrema are mathematically defined by the following sets:

$$T_{\max }=\{t\in \mathbb{R}|x\left(t\right)>x(t-\Delta t)\mathrm{and}x\left(t\right)>x(t+\Delta t)\},$$

$$T_{\min }=\{t\in \mathbb{R}|x\left(t\right)<x(t-\Delta t)\mathrm{and}x\left(t\right)<x(t+\Delta t)\}.$$

Here, $T_{\max }$ represents the times at which $x\left(t\right)$ reaches a local maximum, and $T_{\min }$ represents the times at which $x\left(t\right)$ reaches a local minimum.

The Figure 3 corresponding values of the signal $x\left(t\right)$ at these times are used to form the sets of ordered pairs:

$$X_{\max }=\left\{(t,x\left(t\right))\right|t\in X_{\max }\},X_{\min }=\left\{(t,x\left(t\right))\right|t\in X_{\min }\}.$$

These pairs, $X_{\max }$ and $X_{\min }$, serve as the nodes for constructing the upper and lower envelopes of the signal.

Construction of Upper and Lower Envelopes: Once the local extrema are identified, the next step is to construct the upper $e_{\max }\left(t\right)$ and lower $e_{\min }\left(t\right)$ envelopes of the signal using cubic spline interpolation. For the sets of maxima $X_{\max }=\left\{(t_i,x_{\max }\left(t_i\right))\right\}$ and minima $X_{\min }=\left\{(t_i,x_{\min }\left(t_i\right))\right\}$, the cubic spline interpolation ensures that each segment between successive extrema is smoothly connected:

$$S_i\left(t\right)=a_i+b_i(t-t_i)+c_i{(t-t_i)}^2+d_i{(t-t_i)}^3$$

The coefficients $a_i,b_i,c_i,$ and $d_i$ are determined by the following conditions:

-

The spline passes through all data points:

$$S_i\left(t_i\right)=x_{\max }\left(t_i\right),S_i\left(t_{i+1}\right)=x_{\max }\left(t_{i+1}\right)$$

-

Continuity of the first and second derivatives at each interior point:

$${\dot{S}}_i\left(t_{i+1}\right)={\dot{S}}_{i+1}\left(t_{i+1}\right),{\ddot{S}}_i\left(t_{i+1}\right)={\ddot{S}}_{i+1}\left(t_{i+1}\right)$$

-

Natural spline boundary conditions (second derivative is zero at endpoints):

$$\ddot{S}\left(t_1\right)=0,\ddot{S}\left(t_n\right)=0$$

Solving this system yields the spline functions $S_i\left(t\right)$ for each interval, which together form the upper envelope $e_{\max }\left(t\right)$. The lower envelope $e_{\min }\left(t\right)$ is constructed similarly using the minima.

Figure 4 illustrates the identified upper $e_{\max }\left(t\right)$ and lower $e_{\min }\left(t\right)$ envelopes of the signal, constructed from local maxima and minima.

Mean Envelope Calculation: The mean of the upper and lower envelopes is computed to estimate the local mean of the signal:

$$m\left(t\right)={\displaystyle \frac{e_{\max }\left(t\right)+e_{\min }\left(t\right)}{2}}$$

Figure 5 presents the use of envelopes to capture the oscillation range in the signal, and these are used to calculate the mean envelope $m\left(t\right)$.

IMF Estimation: The mean $m\left(t\right)$ is subtracted from the original signal to obtain the first IMF candidate:

$$h_1\left(t\right)=x\left(t\right)-m\left(t\right)$$

This result $h_1\left(t\right)$ is iteratively refined by recalculating maxima and minima, adjusting the signal until it meets the criteria for an IMF in Figure 6.

This candidate is iteratively refined by repeating steps 1–3 on $h_1\left(t\right)$ until it meets the criteria of an IMF: the number of extrema and zero crossings must differ at most by one, and the mean of the envelopes must be zero at every point in Figure 7.

Iteration and Residue Calculation: Once an IMF $c_1\left(t\right)$ is extracted, it is subtracted from the signal to obtain the residue:

$$r_1\left(t\right)=x\left(t\right)-c_1\left(t\right)$$

This residue serves as the new signal for further decomposition. This process is repeated iteratively to extract multiple IMFs until the residue becomes a monotonic function or satisfies a stopping criterion. The EMD process results in a set of IMFs that represent the oscillatory modes within the signal in Figure 8.

3.2. Step 2: Detection of Mode Mixing and Calculation of Gain k

A known limitation of EMD is mode mixing, where different frequency components are combined within a single IMF. To detect mode mixing, we analyze each IMF by comparing the number of maxima $N_{\max }\left(c_i\right)$, minima $N_{\min }\left(c_i\right)$, and zero crossings $N_{\mathrm{cz}}\left(c_i\right)$:

$$\Delta N_{\mathrm{ext}\text{-}\mathrm{cz}}=\left|N_{\max }\left(c_i\right)+N_{\min }\left(c_i\right)-2N_{\mathrm{cz}}\left(c_i\right)\right|$$

When this equation exceeds a predefined threshold, it indicates mode mixing in the IMF. The gain k is then calculated to adjust the number of modes for VMD:

$$k=k_{\mathrm{EMD}}+\Delta N_{\mathrm{ext}\text{-}\mathrm{cz}}$$

where

-

$k_{\mathrm{EMD}}$ is the initial number of IMFs extracted by EMD;

-

$\Delta N_{\mathrm{ext}\text{-}\mathrm{cz}}$ represents the detected degree of mode mixing.

This equation is crucial for informing the VMD process, ensuring an appropriate number of modes for accurate signal decomposition.

3.3. Step 3: Application of VMD Using Calculated Gain K

If mode mixing is detected, the methodology transitions to VMD for refined decomposition. VMD separates the signal into modes by solving a variational problem that minimizes the bandwidth of each mode:

• Initialization: The number of modes K is estimated using EMD preprocessing, detecting the number of intrinsic mode functions and the mixing present in these signals.
• Variational Problem Formulation: VMD seeks to decompose the signal $x\left(t\right)$ into K modes $u_k\left(t\right)$ by minimizing their bandwidth in the frequency domain [20]:

  $$\underset{\left\{u_k\left(t\right)\right\},\left\{{\omega }_k\right\}}{min}\sum _{k=1}^K\left\{{∥{\partial }_t\left[\left(\delta \left(t\right)+j{\displaystyle \frac{1}{\pi t}}\right)\ast u_k\left(t\right)\right]e^{-j{\omega }_{k}t}∥}_2^2+\alpha {∥u_k\left(t\right)∥}_2^2\right\}$$

  where

  -

  $\alpha$ is a regularization parameter;

  -

  $\delta \left(t\right)$ is the Dirac delta function;

  -

  $u_k\left(t\right)$ and ${\omega }_k$ are the modes and their central frequencies.

  This equation ensures that each mode has minimal bandwidth, which reduces mode mixing.
• Iterative Decomposition: VMD iteratively updates the modes and their central frequencies. The equation adjusts the modes iteratively to ensure that they remain within specific frequency bands, and the modes are updated as

  $${\widehat{u}}_k^{n+1}\left(\omega \right)={\displaystyle \frac{\widehat{x}\left(\omega \right)-{\sum }_{i\ne k}{\widehat{u}}_i^n\left(\omega \right)}{1+2\alpha {(\omega -{\omega }_k^n)}^2}}$$

  and the equation dynamically adjusts the central frequencies as follows:

  $${\omega }_k^{n+1}={\displaystyle \frac{{\int }_0^{\infty }\omega {\left|{\widehat{u}}_k^{n+1}\left(\omega \right)\right|}^{2}d\omega }{{\int }_0^{\infty }{\left|{\widehat{u}}_k^{n+1}\left(\omega \right)\right|}^{2}d\omega }}$$
• Convergence and Extraction: The iterative process continues until convergence is achieved. The modes $u_k\left(t\right)$ then provide a refined decomposition of the original signal, addressing mode mixing more effectively than EMD alone.

3.4. Step 4: Post-Processing with Hilbert Transform

After the application of VMD, the Hilbert Transform is applied to each mode to extract the instantaneous amplitude $A\left(t\right)$ and frequency $\omega \left(t\right)$. The Hilbert Transform $\mathcal{H}\left\{u_k\left(t\right)\right\}$ of each mode $u_k\left(t\right)$ is given by

$$\mathcal{H}\left\{u_k\left(t\right)\right\}={\displaystyle \frac{1}{\pi }}\mathrm{P}.\mathrm{V}.{\int }_{-\infty }^{\infty }{\displaystyle \frac{u_k\left(\tau \right)}{t-\tau }}d\tau$$

where P.V. denotes the Cauchy principal value.

The analytical signal is then formed as

$$z_k\left(t\right)=u_k\left(t\right)+j\mathcal{H}\left\{u_k\left(t\right)\right\}$$

The instantaneous amplitude $A\left(t\right)$ and frequency $\omega \left(t\right)$ are derived as

$$A\left(t\right)=|z_k\left(t\right)|=\sqrt{u_k^2\left(t\right)+{\mathcal{H}}^2\left\{u_k\left(t\right)\right\}}$$

$$\omega \left(t\right)={\displaystyle \frac{d}{dt}}arg\left(z_k\left(t\right)\right)$$

3.5. Advantages of the Hybrid Approach

By integrating EMD and VMD with adaptive gain calculation, this hybrid methodology enhances the precision and efficiency of signal decomposition:

• Enhanced Precision: The method identifies and corrects mode mixing, ensuring accurate separation of vibratory components.
• Adaptive Decomposition: It dynamically adjusts the number of modes, tailoring the decomposition to the signal’s characteristics.
• Computational Efficiency: By utilizing VMD selectively based on EMD results, it reduces unnecessary computation, improving efficiency.

This comprehensive and technical approach provides a robust solution for the estimation of vibratory components in complex signals.

3.6. Flowchart for Hybrid Methodology

To further illustrate the hybrid EMD–VMD methodology, Figure 9 provides a visual representation of the process. This flowchart outlines the key steps involved in the methodology, from the initial signal input to the final reconstruction of the signal using the combined IMFs and modes.

The flowchart in Figure 9 provides a step-by-step visual guide to the hybrid methodology, illustrating how the EMD and VMD techniques are integrated to enhance signal decomposition. This visual representation helps in understanding the sequential flow and decision points within the methodology, making it easier to grasp the overall process and its components.

The hybrid EMD–VMD methodology offers a robust and efficient approach for the decomposition of complex signals. By combining the strengths of EMD and VMD, this method addresses the limitations of traditional techniques, providing a more accurate and computationally efficient solution for estimating vibratory components. The detailed algorithm and flowchart further illustrate the implementation and benefits of this approach, highlighting its potential applications in various fields of signal analysis.

4. Cases Studies

To support the theoretical framework and validate the hybrid strategy for signal analysis, four comprehensive case studies are presented. These case studies are used to demonstrate the practical applicability and effectiveness of combining Empirical Mode Decomposition and Variational Mode Decomposition to enhance the estimation of harmonic vibratory components in complex signals.

The first case study involves a signal composed of six harmonic vibratory components with closely spaced frequencies. Such a signal poses significant challenges for the conventional EMD technique, as mode mixing can lead to inaccurate decompositions. Here, the hybrid method depicted its strength by utilizing EMD to detect mode mixing, which in turn informs the VMD process about the number of harmonic vibratory components. This strategic approach ensures more precise and efficient estimation, overcoming the limitations of both standalone EMD and VMD techniques.

The second case study focuses on a two-degrees-of-freedom mechanical system. In this scenario, a detailed comparison is conducted between the analytical estimation of harmonic vibratory components and their decomposition using the Hilbert–Huang Transform, the variational method, and the hybrid method of EMD and VMD. This analysis aims to highlight the advantages of the hybrid method in accurately capturing and decomposing the inherent oscillatory behavior of the system, providing deeper insights into its dynamic characteristics.

The third case study examines a six-degrees-of-freedom mechanical system, where the objective is to extract the natural frequencies using different methods, including EMD, VMD, and the hybrid approach. These estimated frequencies are then compared with the analytically calculated natural frequencies. This comparison highlights the hybrid method’s ability to closely match the analytically derived frequencies, showcasing its precision in estimating harmonic vibratory components in complex mechanical systems. The hybrid method proves to be more accurate and computationally efficient than standalone EMD or VMD in this scenario, reinforcing its superiority in complex signal analysis.

These case studies not only showcase the robustness of the hybrid method of EMD and VMD in handling nonlinearity and nonstationarity but also underscore its superiority in estimating and analyzing harmonic vibratory components compared to traditional Fourier Transform techniques. By applying the mean squared error as a performance index, the case studies further highlight the accuracy and efficiency of the hybrid method. The findings reinforce the utility of the HHT and the hybrid approach in various scientific and practical applications, particularly in interpreting complex signals with significant changes in amplitude and frequency.

4.1. Case 1: Estimation of Vibratory Components Using EMD, VMD, and the Hybrid EMD–VMD Method

For the first case, a set of vibratory components with close frequencies was selected to present a challenge for the technique and highlight the advantages over conventional methods. A signal composed of six vibratory components with different amplitudes and frequencies was defined. The parameters of the signal are presented in

Table 1. Amplitudes $F_i$ and frequencies ${\Omega }_i$ of the vibratory components.

i	${\mathit{F}}_{\mathit{i}}\left[\mathit{u}\right]$	${\mathbf{\Omega }}_{\mathit{i}}$ [rad/s]	${\mathit{f}}_{\mathit{i}}$ [Hz]
1	6	157.08	25
2	2	94.25	15
3	5	62.83	10
4	1.5	31.42	5
5	4	18.85	3
6	2	6.28	1

:

The signal was generated by summing these vibratory components, as shown in:

$$\mathcal{F}\left(t\right)=\sum _{i=1}^6{F}_{i}cos\left({\Omega }_{i}t\right)$$

The generated signal consists of the sum of six vibratory components, each defined by its respective amplitude and frequency. The sampling interval is defined as a 10 s window consisting of 1000 samples, ensuring adequate temporal resolution for the representation of the signal’s frequency components. Figure 10 illustrates the generated signal.

The generated signal was decomposed using three different decomposition methods: Empirical Mode Decomposition, Variational Mode Decomposition, and a hybrid method combining both techniques. First, the EMD algorithm was applied to decompose the signal into intrinsic mode functions. This technique allows the decomposition of the signal into intrinsic vibratory components without assuming any a priori basis.

Subsequently, the VMD method was applied to perform mode decomposition. VMD addresses the mode mixing problem by decomposing the signal into a specific number of modal components, providing a more robust solution in the presence of noise and closely spaced frequency components.

The hybrid method combines EMD and VMD to improve the accuracy of vibratory component estimation. First, EMD is applied to identify potential mode mixing by comparing the number of maxima and minima with the number of zero crossings in the signal. Then, VMD, guided by the information obtained from EMD, is used to determine the appropriate number of components to estimate.

The results of the decompositions are presented in Figure 11. Each of the six estimated vibratory components is compared with the original components of the signal. The graphs show comparisons between the original vibratory components and those obtained by each decomposition method. The estimated frequencies ${\widehat{\Omega }}_i$ and amplitudes ${\widehat{F}}_i$ of the components are evaluated in Figure 12 and Figure 13, respectively, demonstrating that the hybrid method offers greater accuracy in the estimation of vibratory components, especially in signals with closely spaced frequencies and nonlinear components.

Additionally, the instantaneous amplitudes ${\widehat{F}}_i$ and frequencies ${\widehat{\Omega }}_i$ of the estimated components were analyzed. The following figures present the estimated amplitudes and frequencies for each of the six vibratory components.

To effectively illustrate the outcomes attained through the various decomposition and analysis methods introduced in this study, a comprehensive table is constructed, as shown in

Table 2. Comparative analysis of frequency and amplitude estimates derived from multiple signal decomposition methods.

	Reference	EMD	VMD	Hybrid EMD–VMD
Frequency [rad/s]
${\Omega }_1$	157.08	163.73	157.14	157.08
${\Omega }_2$	94.25	62.71	94.37	94.25
${\Omega }_3$	62.83	31.54	62.71	62.83
${\Omega }_4$	31.42	17.53	18.65	31.42
${\Omega }_5$	18.85	6.91	7.29	18.91
${\Omega }_6$	6.28	2.32	6.41	6.41
Amplitude [u]
$F_1$	6.00	6.25	6.00	6.00
$F_2$	2.00	5.32	2.12	2.00
$F_3$	5.00	3.50	5.01	5.00
$F_4$	1.50	2.40	3.32	1.50
$F_5$	4.00	2.00	2.10	4.01
$F_6$	2.00	0.20	2.01	2.01

. This table consolidates the results, displaying the average values derived from each analytical strategy, allowing for a clear comparative overview.

The techniques covered include EMD, VMD, and the hybrid EMD–VMD method. Each of these methods offers a unique approach to signal analysis, tailored to extract specific features such as frequency and amplitude from the data. The inclusion of average values in the table facilitates an intuitive understanding of the general performance and effectiveness of each method across different metrics.

This structured presentation not only aids in assessing the relative strengths and limitations of each technique but also provides a solid foundation for further discussion and analysis. By comparing the aggregated results, readers can discern patterns and the efficacy levels of each strategy, contributing to more informed decisions in future applications of these methods in similar contexts.

To strengthen the results and the performance of the method, the mean squared error (MSE) was used as a measure of accuracy. The MSE quantifies the average of the squares of the errors—that is, the average squared difference between the estimated values and the reference values. The formula for calculating MSE is given by

$$\mathrm{MSE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{({\mathrm{Reference}}_i-{\mathrm{Method}}_i)}^2$$

(1)

where ${\mathrm{Reference}}_i$ is the reference value, ${\mathrm{Method}}_i$ is the estimated value from each method, and n is the number of observations.

The MSE values, comparing the estimates from each method (EMD, VMD, and Hybrid EMD–VMD) to the reference values, are summarized in

Table 3. Mean squared error of frequency and amplitude estimates.

	EMD	VMD	Hybrid EMD–VMD
Frequency [rad/s]
${\Omega }_1$	1.1025	0.0001	0.0000
${\Omega }_2$	25.6004	0.0009	0.0000
${\Omega }_3$	24.9604	0.0004	0.0000
${\Omega }_4$	4.9281	4.0969	0.0000
${\Omega }_5$	3.61	3.4249	0.0001
${\Omega }_6$	0.3969	0.0004	0.0004
Amplitude [u]
$F_1$	0.0625	0.0000	0.0000
$F_2$	11.1024	0.0144	0.0000
$F_3$	2.25	0.0001	0.0000
$F_4$	0.81	3.3124	0.0000
$F_5$	4.00	3.61	0.0001
$F_6$	3.24	0.0001	0.0001

. These results offer insights into the relative accuracy and effectiveness of the decomposition techniques applied in this study.

The MSE values indicate the degree of deviation of the estimated frequencies and amplitudes from the reference values. Lower MSE values represent higher accuracy of the decomposition method. The hybrid method consistently shows the lowest MSE values across all components, demonstrating its superior performance in accurately extracting the oscillatory components compared to the EMD and VMD methods.

4.2. Case 2: Vibratory Component Analysis in a Two Degrees of Freedom System

The focus of this investigation is the accurate decomposition of vibratory modes using various methodologies, which are then compared against the analytical response of the system. The vibratory system is described by

$$M\ddot{X}+KX=0$$

with

$$\begin{array}{cc}\hfill M& =\left[\begin{array}{cc}{m}_1& 0\\ 0& m_2\end{array}\right],K=\left[\begin{array}{cc}{k}_1+k_2& -k_2\\ -k_2& k_2+k_3\end{array}\right]\hfill \\ \hfill X& =\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\hfill \end{array}$$

where M represents the inertia matrix, K is the stiffness matrix, and X denotes the position vector.

The responses for position, velocity, and acceleration are given by

$$\begin{array}{cc}\hfill x_1\left(t\right)& =X_1^{\left(1\right)}cos({\omega }_{n1}t-{\phi }_1)+X_1^{\left(2\right)}cos({\omega }_{n2}t-{\phi }_2)\hfill \\ \hfill x_2\left(t\right)& =r_1{X}_1^{\left(1\right)}cos({\omega }_{n1}t-{\phi }_1)+r_2{X}_1^{\left(2\right)}cos({\omega }_{n2}t-{\phi }_2)\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\dot{x}}_1\left(t\right)& =-X_1^{\left(1\right)}{\omega }_{n1}sin({\omega }_{n1}t-{\phi }_1)-X_1^{\left(2\right)}{\omega }_{n2}sin({\omega }_{n2}t-{\phi }_2)\hfill \\ \hfill {\dot{x}}_2\left(t\right)& =-r_1{X}_1^{\left(1\right)}{\omega }_{n1}sin({\omega }_{n1}t-{\phi }_1)-r_2{X}_1^{\left(2\right)}{\omega }_{n2}sin({\omega }_{n2}t-{\phi }_2)\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\ddot{x}}_1\left(t\right)& =-X_1^{\left(1\right)}{\omega }_{n1}^{2}cos({\omega }_{n1}t-{\phi }_1)-X_1^{\left(2\right)}{\omega }_{n2}^{2}cos({\omega }_{n2}t-{\phi }_2)\hfill \\ \hfill {\ddot{x}}_2\left(t\right)& =-r_1{X}_1^{\left(1\right)}{\omega }_{n1}^{2}cos({\omega }_{n1}t-{\phi }_1)-r_2{X}_1^{\left(2\right)}{\omega }_{n2}^{2}cos({\omega }_{n2}t-{\phi }_2)\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill X_1^{\left(1\right)}& =\sqrt{{\left({\displaystyle \frac{x_{20}-r_2{x}_{10}}{r_1-r_2}}\right)}^2+{\left({\displaystyle \frac{v_{20}-r_2{v}_{10}}{(r_1-r_2){\omega }_{n1}}}\right)}^2}\hfill \\ \hfill X_1^{\left(2\right)}& =\sqrt{{\left({\displaystyle \frac{r_1{x}_{10}-x_{20}}{r_1-r_2}}\right)}^2+{\left({\displaystyle \frac{r_1{v}_{10}-v_{20}}{(r_1-r_2){\omega }_{n2}}}\right)}^2}\hfill \end{array}$$

$$\begin{array}{cc}\hfill tan{\phi }_1& ={\displaystyle \frac{v_{20}-r_2{v}_{10}}{{\omega }_{n1}(x_{20}-r_2{x}_{10})}}\hfill \\ \hfill tan{\phi }_2& ={\displaystyle \frac{r_1{v}_{10}-v_{20}}{{\omega }_{n2}(r_1{x}_{10}-x_{20})}}\hfill \end{array}$$

The amplitude ratios are

$$\begin{array}{cc}\hfill r_1& ={\displaystyle \frac{-m_1{\omega }_{n1}^2+k_1+k_2}{k_2}}\hfill \\ \hfill r_2& ={\displaystyle \frac{-m_1{\omega }_{n2}^2+k_1+k_2}{k_2}}\hfill \end{array}$$

with

$$\begin{array}{cc}\hfill a_0& =k_1{k}_2+k_1{k}_3+k_2{k}_3\hfill \\ \hfill a_2& =k_1{m}_2+k_2{m}_1+k_2{m}_2+k_3{m}_1\hfill \\ \hfill a_4& =m_1{m}_2\hfill \end{array}$$

The natural frequencies are determined by

$$\begin{array}{ccc}\hfill {\omega }_{n1}& =& \sqrt{\left|{\displaystyle \frac{-a_2+\sqrt{a_2^2-4a_0{a}_4}}{2a_4}}\right|}\hfill \\ \hfill {\omega }_{n2}& =& \sqrt{\left|{\displaystyle \frac{-a_2-\sqrt{a_2^2-4a_0{a}_4}}{2a_4}}\right|}\hfill \end{array}$$

The two-degrees-of-freedom vibratory mechanical system is characterized by the following parameter values in

Table 4. Parameter values of the two-degrees-of-freedom system.

Parameter	Value
$m_1$	$0.5\mathrm{kg}$
$m_2$	$0.5\mathrm{kg}$
$k_1$	$500\mathrm{N}/\mathrm{m}$
$k_2$	$500\mathrm{N}/\mathrm{m}$
$k_3$	$500\mathrm{N}/\mathrm{m}$

.

The amplitude ratios are given by

$$\begin{array}{ccc}\hfill r_1& =& {\displaystyle \frac{-m_1{\omega }_{n1}^2+k_1+k_2}{k_2}}=1\hfill \\ \hfill r_2& =& {\displaystyle \frac{-m_1{\omega }_{n2}^2+k_1+k_2}{k_2}}=1\hfill \end{array}$$

The natural frequencies can be obtained using the following formulas:

$$\begin{array}{ccc}\hfill {\omega }_{n1}& =& \sqrt{\left|{\displaystyle \frac{-a_2+\sqrt{a_2^2-4a_0{a}_4}}{2a_4}}\right|}=50\mathrm{rad}/\mathrm{s}\hfill \\ \hfill {\omega }_{n2}& =& \sqrt{\left|{\displaystyle \frac{-a_2-\sqrt{a_2^2-4a_0{a}_4}}{2a_4}}\right|}=86.60\mathrm{rad}/\mathrm{s}\hfill \end{array}$$

The position responses $x_1\left(t\right)$ and $x_2\left(t\right)$, derived from the system equations, are shown in Figure 14, highlighting their time variations based on system dynamics.

Using the Fourier Transform, the frequency and amplitude components of $x_1\left(t\right)$ and $x_2\left(t\right)$ are quantified. Figure 15 shows the Fourier spectra for both signals.

The signals $x_1\left(t\right)$ and $x_2\left(t\right)$ are decomposed into their vibratory components ${\widehat{x}}_{11}$, ${\widehat{x}}_{12}$, ${\widehat{x}}_{21}$, and ${\widehat{x}}_{22}$ using documented methods, as shown in Figure 16, which highlights the intrinsic modes extracted.

The frequency parameters of $x_1\left(t\right)$ and $x_2\left(t\right)$ are shown in Figure 17, providing a detailed understanding of their oscillatory dynamics.

The amplitude parameters of $x_1\left(t\right)$ and $x_2\left(t\right)$ are shown in Figure 18, providing a clearer view of their vibratory behaviors. These amplitudes are estimated from the extracted modes.

Finally,

Table 5. Detailed comparative analysis of frequency and amplitude estimates derived from multiple signal decomposition methods.

	Reference	EMD	VMD	Hybrid EMD–VMD	FFT
Position 1
${\Omega }_{\mathbf{11}}$ [rad/s]	50.00	50.14	44.17	49.95	50.20
${\Omega }_{\mathbf{12}}$ [rad/s]	86.60	86.52	86.71	86.58	86.39
${\mathit{X}}_{\mathbf{1}}^{\left(\mathbf{1}\right)}$ [m]	0.05	0.051	0.036	0.05	0.043
${\mathit{X}}_{\mathbf{1}}^{\left(\mathbf{2}\right)}$ [m]	0.05	0.048	0.052	0.05	0.042
Position 2
${\Omega }_{\mathbf{21}}$ [rad/s]	50.00	50.33	44.05	49.95	50.20
${\Omega }_{\mathbf{22}}$ [rad/s]	86.60	86.83	86.71	86.58	86.39
${\mathit{X}}_{\mathbf{2}}^{\left(\mathbf{1}\right)}$ [m]	0.05	0.051	0.034	0.049	0.043
${\mathit{X}}_{\mathbf{2}}^{\left(\mathbf{2}\right)}$ [m]	0.05	0.048	0.035	0.049	0.042

consolidates the average values derived from the decomposition and analysis methods, offering a clear comparative overview. This structured presentation highlights the effectiveness of each method in estimating position, velocity, and acceleration features, facilitating informed decisions for future applications.

The techniques covered include Empirical Mode Decomposition, Variable Mode Decomposition, Fourier Transform and the hybrid EMD–VMD method. Each of these methods offers a unique approach to signal analysis, tailored to estimating specific features such as frequency and amplitude from the data. The inclusion of average values in the table facilitates an intuitive understanding of the general performance and effectiveness of each method across different metrics such as position, velocity, and acceleration.

This structured presentation not only aids in assessing the relative strengths and limitations of each technique but also provides a solid foundation for further discussion and analysis. By comparing the aggregated results, readers can discern patterns and the efficacy levels of each strategy, contributing to more informed decisions in future applications of these methods in similar contexts.

The following table presents the MSE values for each method (EMD, VMD, EMD–VMD, and FFT) compared to the reference values for the frequencies and amplitudes of both position and velocity signals.

As shown in

Table 6. Mean squared error for frequency and amplitude estimates.

	EMD	VMD	Hybrid EMD–VMD	FFT
Position 1
${\Omega }_{\mathbf{11}}$ [rad/s]	0.0009	0.8546	0.0000	0.0016
${\Omega }_{\mathbf{12}}$ [rad/s]	0.0001	0.0004	0.0000	0.0001
${\mathit{X}}_{\mathbf{1}}^{\left(\mathbf{1}\right)}$ [m]	0.0000	0.0002	0.0000	0.0001
${\mathit{X}}_{\mathbf{1}}^{\left(\mathbf{2}\right)}$ [m]	0.0000	0.0000	0.0000	0.0001
Position 2
${\Omega }_{\mathbf{21}}$ [rad/s]	0.0004	0.8784	0.0000	0.0016
${\Omega }_{\mathbf{22}}$ [rad/s]	0.0016	0.0004	0.0000	0.0001
${\mathit{X}}_{\mathbf{2}}^{\left(\mathbf{1}\right)}$ [m]	0.0000	0.0003	0.0000	0.0001
${\mathit{X}}_{\mathbf{2}}^{\left(\mathbf{2}\right)}$ [m]	0.0000	0.0002	0.0000	0.0001

, the MSE values indicate the accuracy of each method. Lower MSE values correspond to higher accuracy and better performance. The hybrid EMD–VMD method consistently shows lower MSE values across most metrics, demonstrating its effectiveness in accurately capturing the signal characteristics compared to the other methods. This validates the superiority of the hybrid method in handling complex signals with closely spaced frequencies and varying amplitudes.

4.3. Case 3: Vibratory Component Analysis in a Six Degrees of Freedom System

The objective of this study is the precise decomposition of vibratory modes using different methodologies, which are then compared with the analytical response of the system. This vibratory system is described by the coupled second-order differential equations:

$$\begin{array}{ccc}\hfill m_1{\ddot{x}}_1+c_1{\dot{x}}_1+\left(k_1+k_2\right)x_1-k_2{x}_2& =& 0\hfill \\ \hfill m_2{\ddot{x}}_2+c_2{\dot{x}}_2+k_2{x}_2-k_2{x}_1+k_3{x}_2-k_3{x}_3& =& 0\hfill \\ \hfill m_3{\ddot{x}}_3+c_3{\dot{x}}_3-k_3{x}_2+\left(k_3+k_4\right)x_3-k_4{x}_4& =& 0\hfill \\ \hfill m_4{\ddot{x}}_4+c_4{\dot{x}}_4-k_4{x}_3+\left(k_4+k_5\right)x_4-k_5{x}_5& =& 0\hfill \\ \hfill m_5{\ddot{x}}_5+c_5{\dot{x}}_5-k_5{x}_4+\left(k_5+k_6\right)x_5-k_6{x}_6& =& 0\hfill \\ \hfill m_6{\ddot{x}}_6+c_6{\dot{x}}_6-k_6{x}_5+\left(k_6+k_7\right)x_6& =& 0\hfill \end{array}$$

The characteristic equation of the system is determined by

$$det(M{s}^2+K)=0$$

The parameter values are presented in

Table 7. Parameter values of the six-degrees-of-freedom system.

Parameter	Value
Masses [kg]
$m_1$	0.1
$m_2$	0.25
$m_3$	0.15
$m_4$	0.2
$m_5$	0.3
$m_6$	0.25
Stiffness Coefficients [N/m]
$k_1$	1000
$k_2$	300
$k_3$	350
$k_4$	450
$k_5$	500
$k_6$	750
$k_7$	400

.

The natural frequencies are derived from the roots of the characteristic equation:

$$\begin{array}{c}115.54i,-115.54i,\\ 93.120i,-93.120i,\\ 82.682i,-82.682i,\\ 60.299i,-60.299i,\\ 38.941i,-38.941i,\\ 20.982i,-20.982i\end{array}$$

Thus, the natural frequencies for this set of parameters are

$$\begin{array}{ccc}\hfill {\omega }_{n1}& =& 20.982[\mathrm{rad}/\mathrm{s}]\hfill \\ \hfill {\omega }_{n2}& =& 38.941[\mathrm{rad}/\mathrm{s}]\hfill \\ \hfill {\omega }_{n3}& =& 60.299[\mathrm{rad}/\mathrm{s}]\hfill \\ \hfill {\omega }_{n4}& =& 82.682[\mathrm{rad}/\mathrm{s}]\hfill \\ \hfill {\omega }_{n5}& =& 93.120[\mathrm{rad}/\mathrm{s}]\hfill \\ \hfill {\omega }_{n6}& =& 115.54[\mathrm{rad}/\mathrm{s}]\hfill \end{array}$$

In this analysis, the Empirical Mode Decomposition, Variational Mode Decomposition, and hybrid EMD–VMD methods were used to estimate the natural frequencies. These methods enable the decomposition of system signals into their natural frequency components, providing insights into the vibrational dynamics of the system. The accuracy of the frequency estimates obtained by each method is evaluated in

Table 8. Comparative analysis of frequency and amplitude estimates derived from multiple signal decomposition methods.

	Reference	EMD	VMD	Hybrid EMD–VMD
Frequency [rad/s]
${\Omega }_{\mathbf{1}}$	20.982	21.01	21.18	21.05
${\Omega }_{\mathbf{2}}$	38.941	35.62	41.82	40.15
${\Omega }_{\mathbf{3}}$	60.299	5.98	58.00	58.80
${\Omega }_{\mathbf{4}}$	82.682	21.00	58.00	83.20
${\Omega }_{\mathbf{5}}$	93.120	92.10	91.15	92.36
${\Omega }_{\mathbf{6}}$	115.54	92.10	111.02	113.82

, showing the performance and ability of each technique to identify the vibrational characteristics of the system.

The accuracy of each method was further evaluated by calculating the percentage error, shown in

Table 9. Detailed comparative analysis of percentage errors derived from multiple signal decomposition methods.

	EMD Error (%)	VMD Error (%)	Hybrid EMD–VMD Error (%)
${\Omega }_{\mathbf{1}}$	0.13%	0.94%	0.32%
${\Omega }_{\mathbf{2}}$	8.53%	7.39%	3.11%
${\Omega }_{\mathbf{3}}$	90.08%	3.81%	2.49%
${\Omega }_{\mathbf{4}}$	74.59%	29.85%	0.63%
${\Omega }_{\mathbf{5}}$	1.09%	2.12%	0.82%
${\Omega }_{\mathbf{6}}$	20.27%	3.92%	1.47%

, and the mean squared error (MSE) of the frequency estimates, presented in

Table 10. Mean squared error of frequency estimates.

	EMD	VMD	Hybrid EMD–VMD
${\Omega }_{\mathbf{1}}$	0.0008	0.0391	0.0046
${\Omega }_{\mathbf{2}}$	11.0803	8.2764	1.4617
${\Omega }_{\mathbf{3}}$	2921.44	5.2769	2.24
${\Omega }_{\mathbf{4}}$	3825.63	613.49	0.2689
${\Omega }_{\mathbf{5}}$	1.0404	3.8750	0.5776
${\Omega }_{\mathbf{6}}$	552.0700	20.4416	2.95

. The results demonstrate the superiority of the hybrid EMD–VMD method for complex signal decomposition, achieving lower average errors and higher accuracy compared to the EMD and VMD methods.

These results underscore the effectiveness of the hybrid method in accurately capturing the vibratory characteristics of the system, demonstrating its utility in frequency estimation for complex systems with closely spaced frequency modes. This serves as a reference for the application of these methods in vibration analysis and the improvement of systems in similar contexts.

4.4. Case 4: Estimation of Variable Amplitudes and Frequencies

Building upon studies that analyze complex nonlinear systems [35,36,38,71], this case study proposes the analysis of a signal composed of four vibratory components with time-varying frequencies and amplitudes, a quadratic trend, and additive Gaussian noise. These components exhibit nonlinear and non-periodic behaviors, which pose a challenge for traditional methods such as the Fast Fourier Transform [7,37]. The decomposition was performed using Variational Mode Decomposition, Empirical Mode Decomposition, and the hybrid EMD–VMD method.

The composite signal $\mathcal{F}\left(t\right)$ is defined as the sum of the following components:

1. Component ${\mathcal{F}}_1$: A cosine function with a phase shift applied after $t=0.5$ s:

$${\mathcal{F}}_1\left(t\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}t<0.5,\hfill \\ 1.5cos(200·\pi t)\hfill & \mathrm{if}t\ge 0.5.\hfill \end{array}\right.$$

This component is absent during the first half of the time interval. The hybrid EMD–VMD method excels in accurately identifying this absence, thanks to its adaptability to changes in the signal, such as nonstationarity and variations in amplitude and frequency.

2. Component ${\mathcal{F}}_2$: A cosine function present in the first half of the signal:

$${\mathcal{F}}_2\left(t\right)=\left\{\begin{array}{cc}2cos(120·\pi t)\hfill & \mathrm{if}t<0.5,\hfill \\ 0\hfill & \mathrm{if}t\ge 0.5.\hfill \end{array}\right.$$

This component, like ${\mathcal{F}}_1\left(t\right)$, is nonstationary. The hybrid EMD–VMD method allows for the identification and representation of this component, aiding in the analysis of signal behavior at every analyzed time instant.

3. Component ${\mathcal{F}}_3$: A chirp signal with a smooth frequency variation from 20 Hz to 15 Hz:

$${\mathcal{F}}_3\left(t\right)=3cos\left(40\pi t-{\displaystyle \frac{\pi }{3}}log(1+e^{30(t-0.45)})\right)$$

This signal exhibits a continuous transition in its frequency, characteristic of nonstationary systems. While conventional methods struggle to capture these variations, the hybrid EMD–VMD method accurately estimates the instantaneous frequency over time, demonstrating its ability to address the complexity of these systems.

4. Component ${\mathcal{F}}_4$: A quadratic trend modulated by an exponential decay and a sinusoidal term with a frequency of 6 Hz:

$${\mathcal{F}}_4\left(t\right)=\left(7t^2{e}^{-t/5}+1\right)sin(2\pi ·6t).$$

This component exhibits a significantly varying amplitude over time, characteristic of nonlinear systems. The hybrid method effectively captures this dynamic behavior, demonstrating its suitability for analyzing signals with complex temporal variations.

5. Gaussian Noise $\mathcal{N}\left(t\right)$: White noise with a level equal to 10% of the total amplitude of the signal:

$$\mathcal{N}\left(t\right)=0.1·\mathcal{F}\left(t\right)·\mathcal{G}(0,1),$$

where $\mathcal{G}(0,1)$ is Gaussian white noise with zero mean and unit variance.

The composite signal is then given by

$$\mathcal{F}\left(t\right)={\mathcal{F}}_1\left(t\right)+{\mathcal{F}}_2\left(t\right)+{\mathcal{F}}_3\left(t\right)+{\mathcal{F}}_4\left(t\right)+\mathcal{N}\left(t\right).$$

The composite signal $\mathcal{F}\left(t\right)$ is shown in Figure 19, highlighting its complexity with time-varying frequencies and amplitudes, as well as additive noise.

The FFT spectrum of the composite signal is shown in Figure 20. While FFT captures the dominant frequencies, it fails to reflect the time-varying characteristics of the signal components.

The comparison between the original components and the reconstructed modes for EMD, VMD, and the hybrid EMD–VMD method is presented in Figure 21. Each subfigure corresponds to a component: ${\mathcal{F}}_1$, ${\mathcal{F}}_2$, ${\mathcal{F}}_3$, and ${\mathcal{F}}_4$. The results depicted the ability of the hybrid method to accurately reconstruct the original signal components.

The reconstruction errors for each IMF, shown in Figure 22, are calculated as:

$$e_{{\mathcal{F}}_i}={\mathcal{F}}_i-{\widehat{\mathcal{F}}}_i,$$

where ${\mathcal{F}}_i$ and ${\widehat{\mathcal{F}}}_i$ denote the original and estimated components, respectively.

The instantaneous frequencies estimated by each method are shown in Figure 23. The hybrid method accurately tracks the time-varying nature of the frequencies, highlighting its robustness in handling nonstationary components. Unlike EMD, which struggles to resolve overlapping frequencies, and VMD, which requires predefined parameters that limit its adaptability, the hybrid method dynamically adapts to the frequency variations present in the signal. This capability is particularly evident in components with smooth transitions, such as ${\widehat{\mathcal{F}}}_3\left(t\right)$, where the hybrid approach demonstrates superior accuracy in reconstructing the instantaneous frequency trajectory.

The instantaneous amplitudes for each component are presented in Figure 24. The hybrid method provides accurate amplitude tracking, further validating its effectiveness. Notably, for components like ${\widehat{\mathcal{F}}}_4\left(t\right)$, where the amplitude varies significantly due to modulation, the hybrid method captures these variations with precision, surpassing the limitations of both EMD and VMD in scenarios involving rapid amplitude fluctuations. This accuracy underscores the method’s ability to reconstruct the original signal components faithfully, even under challenging conditions such as additive Gaussian noise and nonstationarity.

Overall, the hybrid EMD–VMD method proves to be a robust and versatile tool for analyzing complex signals with time-varying frequencies and amplitudes. It not only excels in tracking the reference trajectories of instantaneous frequencies and amplitudes but also minimizes reconstruction errors. These advantages highlight its suitability for applications where traditional methods like FFT, standalone EMD, or VMD fail to provide sufficient resolution or adaptability. By combining the strengths of EMD and VMD, the hybrid method offers an approach for addressing the limitations of individual techniques, making it a valuable contribution to the field of signal processing.

5. Conclusions

A hybrid algorithm for the estimation of uncertain arbitrary frequency harmonic vibration components in mechanical systems is described. This hybrid estimation approach can be implemented for nonstationary vibration system signals. By integrating the Empirical Mode Decomposition and Variational Mode Decomposition algorithms, the hybrid approach exhibits superior performance in both accuracy and computational efficiency compared to traditional methods in specific cases, particularly when analyzing harmonic vibratory signals with nonlinear and nonstationary behavior. While the hybrid estimation strategy shows improvements over EMD and VMD when used independently, and even over the Fast Fourier Transform in certain variable-frequency operational scenarios, it is not suggested that the hybrid method universally outperforms the FFT. Instead, this study highlights its ability to address specific challenges, such as mode mixing and closely spaced frequencies, where traditional methods may fall short.

The results across the case studies, quantified using the mean squared error, consistently approached zero, indicating the method’s ability to provide highly accurate frequency and mode estimations in complex vibratory systems. In the first case study, involving a two-degrees-of-freedom mechanical system, the hybrid estimation algorithm demonstrated exceptional precision in computing natural frequencies and harmonic vibration modes. This precision, while highly effective for these scenarios, depends on the careful calibration of parameters and the inherent characteristics of the analyzed signals, which can limit its generalizability to other signal types without additional adjustments.

The second case study, which involved a signal composed of harmonic vibratory components with closely spaced frequencies, highlighted the hybrid algorithm’s ability to effectively address the challenge of mode mixing. By leveraging EMD for detecting mode mixing and VMD for refining the estimation of individual components, the hybrid approach enhanced both the accuracy and efficiency of the analysis. This case underscores its potential for applications in vibration monitoring and fault detection, where precise mode separation is critical. However, its dependence on accurate initial conditions and computational resources for parameter optimization presents limitations for real-time applications.

The hybrid framework successfully leverages the strengths of EMD in breaking down signals into adaptive mode functions and the robustness of VMD in maintaining decomposition precision in the presence of noise and mode mixing. Nevertheless, this method shares some limitations with EMD and VMD, such as its sensitivity to noise levels and difficulties in handling signals with extremely similar frequencies in possibly perturbed, uncertain nonlinear vibration systems. These limitations point to opportunities for further optimization and enhancement, particularly in developing adaptive parameter selection mechanisms and improving computational efficiency.

Future work should focus on addressing these limitations by exploring automated methods for parameter tuning and extending the hybrid approach to handle more diverse signal characteristics in uncertain multi-degree-of-freedom vibrating systems. Additionally, the development of computational efficiency confirmation indices and comparisons across a broader range of scenarios, including highly noisy environments, will further validate the method’s utility and expand its applicability.

Finally, the hybrid EMD–VMD algorithm establishes itself as a powerful and efficient tool for the detailed analysis of arbitrary frequency harmonic vibration signals, overcoming the challenges of nonlinearity and nonstationarity in specific contexts. While not universally superior to the FFT or standalone EMD and VMD, its ability to integrate the strengths of these methods demonstrates its relevance for tackling complex vibration system signal analysis challenges. This hybrid approach has the potential to significantly advance the understanding and interpretation of harmonic vibratory phenomena, offering robust solutions for a range of engineering applications, including structural health monitoring, machinery diagnostics, and vibration control.