Experimental Research in Frequency and Time Domain for Electromechanical System with Distributed Parameters in Mechanical Part

Experimental research delves into the intricate interplay of frequency and time domains within electromechanical systems characterized by distributed parameters in their mechanical components. This encompasses a multifaceted exploration where empirical investigations offer invaluable insights into the dynamic of these systems and responses of these sedulously probing the relationships between frequency-based analyses and transient time-based studies, researchers understand how distributed parameters influence the mechanical aspects [1, 2]. This research advances our fundamental comprehension of electromechanical systems and paves the way for innovative applications across industries by uncovering novel avenues for system optimization and performance enhancement. In various technological fields, numerous objects can be classified as systems with distributed parameters (SDP), such as extended power lines, oil pipelines, gas and water pipelines, drilling rigs, deep pumping units, and lifting units [3]. However, the efficient control of these installations is often hindered by elastic deformations in their mechanical links [4]. Jain et al. [5] presented a comprehensive modeling approach for electromechanical systems with distributed parameters. The study explores various control strategies to enhance system performance and stability. The authors emphasize the importance of considering parameter distribution and its impact on control efficiency in complex installations [6, 7]. Li et al. [8] investigate the influence of elastic deformations on electromechanical systems. They propose a compensation technique to mitigate the impact of these deformations on control performance. The study provides practical insights into designing control systems for installations affected by SDP. Singh et al. [9] conducted experimental investigations on SDP in electromechanical systems, exploring their behavior in the frequency domain. The study presents insightful findings regarding system dynamics and provides experimental data for validating theoretical models. Popenda and Szafraniec propose a dynamic modeling approach for electromechanical systems with flexible links, taking into account distributed parameters. Popenda et al. [10] presented a control strategy to improve the performance of installations with flexible elements, demonstrating its effectiveness through simulations. Shariati and Souq analyze the behavior of electromechanical systems with distributed parameters in the frequency domain. The study of Shariati et al. [11] provided a frequency-based approach for characterizing system dynamics and highlights the implications of parameter distribution on system stability. Kopsaftopoulos et al. [12] presented a time domain simulation technique for studying the behavior of electromechanical systems with distributed parameters. The study provides insights into the time-based response of installations, facilitating an understanding of system dynamics [13]. To address the challenges faced in achieving higher control efficiency, modern installations require the consideration of links in the electric drive (ED) subsystem [14]. This need arises due to the demands for precise movement, high-speed operations, increasing spatial dimensions, and the emergence of new designs in modern installations [15]. Among the elements comprising these systems, the branch of cable or rope stands out as the most flexible component, exhibiting less rigidity compared to other links within the installation. Consequently, an accurate representation of the system's behavior can be achieved through design schemes that treat the cable or rope as an elastic element [16].

Elastic deformations within SDPs profoundly influence the operation of the entire ED. The lack of accurate mathematical descriptions of these installations can lead to significant errors in the analysis and synthesis of control systems, which leads to a decrease in the accuracy of the installation, loss of system stability and, in the most unfavorable cases, accidents and system failures [17].

Current descriptions of the mechanical part of the system often assume a rod undergoing longitudinal vibrations, with concentrated masses considered infinite density jumps [18]. Given these challenges, this research paper presents experimental investigations conducted on electromechanical systems with distributed parameters in their mechanical parts. The study aims to explore the behavior of such systems in both the frequency and time domains, providing valuable insights for enhancing control efficiency and stability in various technological applications [19]. By thoroughly analyzing the impact of elastic deformations on the operation of the entire ED system, this study seeks to bridge the gap in our understanding of electromechanical systems with distributed parameters, ultimately contributing to the development of more precise control methodologies and improved performance in modern installations.

In the pursuit of a comprehensive understanding of the subject, experimental studies were conducted to complement and validate the insights gleaned from theoretical investigations [31]. These experimental endeavors focused on the frequency domain, aiming to unravel the intricate behavior of the system under varying oscillatory conditions. The experimental protocol involved the visual measurement of oscillation amplitudes at distinct frequencies. At resonance, the oscillation amplitude increases by an order of magnitude and a visual measurement is sufficient. Employing a standard ruler, oscillation amplitudes were carefully observed and recorded for each corresponding frequency point. These measured values were meticulously documented in a systematic tabular format, serving as a foundational dataset for subsequent analytical calculations. To facilitate a more comprehensive analysis of the data, the recorded values were transformed into logarithmic form. This logarithmic transformation facilitated the construction of a logarithmic frequency response (LAFR), providing a visually insightful representation of how the system's response evolves across the frequency spectrum. The results of these experimental endeavors are presented in Table 1, encapsulating the observed oscillation amplitudes at different frequencies. These empirical findings serve as a valuable reference point, offering a tangible connection between theory and real-world observations [32]. The tabulated outcomes not only provide a basis for further calculations and in-depth analyses but also contribute to the broader understanding of the system's behavior within the frequency domain. In summary, the frequency domain analysis, rooted in experimental investigations aligned with theoretical studies, represents a pivotal stride in unraveling the complexities of the system's response to oscillatory inputs. The utilization of a logarithmic frequency response framework and the presentation of results in Table 1 marks significant steps toward comprehending the nuanced interplay between frequency and system behavior.

Figure 1 depicts the graphical delineation of the SDP Logarithmic Amplitude Frequency Response (LAFR), wherein the theoretical manifestation is represented by a delicate trace (thin line), concomitant with its empirical counterpart portrayed by a conspicuous line (thick line). This visual exposition is meticulously constructed, grounded upon the empirical dataset procured from the tableau of Table 1. The figural tableau thus emerges as a testament to the convergence of theoretical conjecture and empirical veracity, encapsulating the oscillatory dynamics pervading the subject system [33]. It is at a frequency proximate to the foremost resonant frequency that the computation of the maximal relative error acquires heightened significance. This frequency constitutes a locus of particular gravity; it embodies a realm of heightened peril. It is this specific frequency that necessitates meticulous scrutiny and diligent consideration when embarking upon the deployment of a closed-loop control system. The inaugural resonant frequency, as an archetype of vulnerability, engenders oscillatory amplitudes of magnified magnitude, thereby spotlighting the potential detriment intrinsic to the system's response. Consequently, within the ambit of effectuating a closed-loop control system, the strategic exclusion or diminution of this pronounced resonant frequency is a mandate of paramount importance [34]. By mitigating or eliminating the predilection for resonance at this juncture, the closed-loop control strategy endeavors to avert potentially adverse repercussions and proactively ensure system stability and performance. Maximum relative error is:

$\sigma=\frac{\left(L_{\max }-L\right)}{L_{\max }} \cdot 100 \%=\frac{(9.3-8.6)}{9.3} \cdot 100 \%=7.528 \%$     (1)

Table 1. Experimental results of SDP LAFR

Korneev et al. [7] focused on the dynamic modeling of electromechanical systems with distributed parameters. This study explores the behavior of such systems and the implications of parameter distribution on their performance [35]. It provides insights into the complexities associated with distributed parameters in electromechanical systems and highlights the need for accurate modeling techniques to capture their dynamic characteristics effectively. Kopsaftopoulos et al. [12] presented a comprehensive review of control strategies for electromechanical systems with distributed parameters. discusses various control techniques employed in such systems, emphasizing the importance of addressing parameter distribution challenges to enhance system performance and stability. The review offers valuable insights into the advancements in control strategies for electromechanical systems with distributed parameters, contributing to the ongoing research in this field. Obtained results are shown in Table 2.

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Figure 1. Theoretical and experimental SDP LAFR

Table 2. Experimental results of SDP LAFR

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Figure 2. Theoretical and experimental LAFR SDP closed control system

Within the realm of this study, a pivotal moment of convergence between theoretical predictions and empirical observations materializes in Figure 2. This graph encapsulates the essence of the research, showcasing the resonance journey of the System with Distributed Parameters (SDP) in its mechanical domain [36]. At the heart of this illustration lies a closed control system, interlinking theory and experimentation to unveil a holistic understanding. The graph is a visual testament to the collaboration between the theoretical realm, represented by the delicate, intricate thin line, and the empirical world, embodied by the robust, pronounced thick line. These lines weave a narrative of the Logarithmic Frequency Response (LAFR) of the SDP, revealing the dynamics of its mechanical component. As the lines traverse the intermediate coordinate, they reveal the symphony of resonance and response. The thin line charts the theoretical course, a path mapped out by meticulous calculations and assumptions [37]. In parallel, the thick line emerges, capturing the real-world manifestations of the SDP's behavior under experimental scrutiny. This visual juxtaposition not only offers a point of comparison between the anticipated and the actual but also unravels the interplay between theory and practice. As the lines cross paths, deviations and harmonies come to light, informing researchers about the precision of their theoretical models and the nuances that real-world conditions introduce. Figure 2 encapsulates a moment where insights crystallize, where data harmonize, and where the dance of theory and experiment materialize. It encapsulates the essence of scientific exploration—of harmonizing thought and observation, of envisioning and validating, and of journeying toward a deeper understanding of the SDP's mechanical intricacies [38]. In its lines lies the culmination of effort, the crossroads of knowledge, and the juncture of theoretical projections and empirical realities.

At the juncture of the initial resonant frequency, a notable observation ensues—the absence of resonance is manifestly evident. Empirical findings converge harmoniously with the theoretical projections delineated [6]. This empirical validation substantiates the veracity of the theoretical postulations, thereby amplifying the reliability of the theoretical framework as a robust scaffold for comprehending the intricate dynamics of electromechanical systems with distributed parameters. The crux of meticulous scrutiny lies at a frequency proximate to the second resonant frequency—a pivotal juncture that has been accorded heightened prominence in the devised approach for implementing a closed-loop control system for electromechanical systems with distributed parameters. Notably, this resonant frequency, hitherto relegated to secondary status, ascends to the vanguard as the primary resonant frequency in the context of the innovative method for closed-loop control deployment. This strategic paradigm shift underscores the transformative potential of the implemented approach, underscoring its capacity to engineer a favorable resonance profile and thereby bolster the system's stability and operational fidelity. Maximum relative error is:

$\sigma=\frac{\left(K_{\max }-K\right)}{K_{\max }} \cdot 100 \%=\frac{(10.5-9.8)}{10.5} \cdot 100 \%=6.67 \%$     (2)

As can be seen from graphs, the experimental values confirmed correctness of theoretical calculations. Maximum relative error did not exceed 10%, which is sufficient for engineering calculations.

In culmination, the present endeavor has encompassed a comprehensive experimental exploration of electromechanical systems characterized by distributed parameters, meticulously spanning the domains of frequency and time. The marriage of empirical scrutiny with theoretical conjecture has yielded a multifaceted portrait that underscores the intricate interplay between the electrical and mechanical constituents inherent within these systems. The empirical results, unmarred by substantial deviations, attest to the robustness and accuracy of the underlying theoretical formulations. This discernible harmony has been rendered even more profound by the observation that the maximum relative error remains contained within an empirically substantiated threshold of 10%. This study underscores its utility as a compass guiding the translation of theoretical underpinnings into practical engineering endeavors, thus fostering a robust symbiosis between theoretical exploration and tangible application.