# Hybrid Empirical and Variational Mode Decomposition of Vibratory Signals
## 1 Introduction
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).