Dynamic Interaction and Beating Phenomena in Electromechanical Rotating Systems: Condition Monitoring and Fault Diagnosis of Industrial Pumps and Variable Frequency Drive-Driven Induction Motors

When two or more oscillatory components with closely spaced frequencies coexist in a structure or rotating machine, a phenomenon known as beating arises [1]. The superposition of these components leads to periodic amplitude variations in the resultant signal, producing a characteristic amplitude modulation pattern [2]. This behavior is widely observed in multi-shaft rotating machinery, lightweight structures, and coupled dynamic systems, and its accurate identification is essential for modal analysis, condition monitoring, and resonance mitigation [3], [4]. Mathematically, beating represents a classical form of amplitude modulation in which a low-frequency envelope modulates a higher-frequency carrier wave [2]. It occurs when mechanical systems are subjected to simultaneous excitations with small frequency differences and has been reported in a wide range of applications, including structural seismic response, vehicle dynamics, and nanoscale and photonic systems [1], [2].

Analytical investigations of beating generally fall into two categories: linear approaches based on small detuning assumptions within classical vibration theory [1], [2], and nonlinear analyses addressing modal interactions and energy exchange between coupled modes [3]. In industrial applications, vibration behavior serves as a key indicator of rotating machinery health. The presence of beating may reflect dynamic imbalance, hydraulic interaction, or unfavorable frequency proximity between mechanical components, making its accurate analysis critical for the early detection of bearing faults, coupling misalignments, and other mechanical abnormalities [5], [6]. Using simplified two-degree-of-freedom models and combined time–frequency analysis, the mechanisms of near-frequency interaction can be effectively characterized [3]. Moreover, beating theory contributes to improved frequency alignment, vibration control strategies, and fault diagnosis in complex mechanical systems. Ongoing research is focused on nonlinear modeling of high-degree-of-freedom systems and emerging applications in biomechanics, microelectromechanical systems, and nanotechnology, further expanding the scope of vibration science [7].

Recent developments in machine condition monitoring have significantly enhanced the effectiveness of vibration-based techniques for fault diagnosis and predictive maintenance of rotating machinery. Vibration analysis remains one of the most reliable tools for assessing the health condition of pumps, compressors, motors, and rolling element bearings, enabling the early detection of mechanical and electrical faults before catastrophic failures occur [8]. Standardized procedures for vibration measurement and severity assessment have improved the consistency and reliability of machine condition evaluation in industrial applications [9]. Detailed investigations of bearing failure mechanisms and their associated vibration signatures have provided valuable diagnostic indicators for identifying defects related to lubrication degradation, fatigue damage, misalignment, and electrical erosion [10]. Furthermore, continuous online monitoring systems incorporating vibration, temperature, and process measurements have demonstrated considerable potential for detecting abnormal operating conditions and supporting maintenance decision-making processes [11]. Previous studies have shown that electrical phenomena associated with inverter-fed drives and PWM systems can contribute to bearing degradation, increased vibration levels, and premature machine failures [12], [13]. In addition, industrial standards and engineering guidelines have established recommendations for improving motor reliability, operational performance, and machinery protection under varying operating conditions [14]. The fundamental principles of vibration measurement, signal interpretation, and dynamic response analysis remain essential for understanding fault-related vibration behavior and implementing effective condition monitoring strategies [15]. Collectively, these developments highlight the critical role of vibration analysis, bearing condition assessment, and continuous monitoring in improving equipment reliability, reducing maintenance costs, and enhancing the safety of industrial rotating machinery [8], [9], [10], [11], [12], [13], [14], [15]. Advanced vibration diagnostic techniques have further improved the capability of detecting and characterizing faults in rotating machinery. Among the available methods, frequency spectrum analysis, envelope detection, and amplitude modulation analysis have demonstrated superior performance in identifying incipient defects before they evolve into severe mechanical failures. Recent studies have shown that envelope spectrum analysis can effectively extract fault-related information from vibration signals and accurately identify localized defects in rolling element bearings under both steady-state and transient operating conditions [16], [17], [18]. Enhanced envelope processing techniques and generalized envelope spectrum methods have further improved the detection of weak fault signatures that are often masked by background noise and complex machine dynamics [19]. Amplitude modulation analysis has also gained significant attention as an effective tool for identifying speed-dependent vibration components and improving fault detection accuracy in industrial rotating equipment [20]. Moreover, advanced signal processing techniques integrating envelope analysis with frequency-domain methods have demonstrated improved performance in diagnosing bearing degradation, rotor defects, and abnormal machine behavior [21]. Recent developments in signal decomposition and feature extraction methods have further enhanced the capability of vibration monitoring systems to detect faults in complex industrial environments characterized by varying loads and operating conditions [22]. These advancements confirm that vibration analysis, FFT spectrum evaluation, and envelope-based diagnostic techniques remain essential tools for predictive maintenance and early fault diagnosis in modern industrial machinery [16], [17], [18], [19], [20], [21], [22].

The beating phenomenon is one of the fundamental oscillatory behaviors encountered in the vibration analysis of rotating machinery. It arises from the superposition of two vibration components with very close frequencies [1]. From a signal analysis perspective, this phenomenon represents a particular case of amplitude modulation, in which the signal amplitude varies periodically over time [2]. As illustrated in Figure 1, beating occurs in vibration signals when two harmonic components with nearly identical frequencies combine, producing a characteristic oscillatory envelope in the resulting signal.

When two harmonic motions with closely spaced frequencies are superimposed, the resulting motion exhibits a beating pattern [1]. Consider:

where, $\delta$ is a small frequency difference. The superposition of these motions can be expressed as Eq. (1) and rewritten in the following form:

This equation is illustrated graphically in Figure 1. It can be observed that the resulting motion, $x(t)$, represents a cosine wave with a frequency of $\frac{(\omega+\delta)}{2}$, which is approximately equal to $\omega$, and with a time-varying amplitude of $2 x \cos \frac{(\omega t)}{2}$. Whenever the amplitude reaches its maximum, a beat occurs. The frequency $\delta$, at which the amplitude periodically varies between 0 and 2 X , is referred to as the beat frequency. This interaction produces oscillations with a varying amplitude, commonly described as amplitude modulation. Moreover, if the frequencies vary with time, frequency modulation can also be considered [2]. If two sinusoidal waves with frequencies $f 1$ and $f 2$ are combined, their mathematical representation can be written as follows:

Using the trigonometric identity, this can be transformed into:

The first part represents amplitude modulation with a frequency $\frac{(f 1-f 2)}{2}$, and the second part represents the main oscillation with the average frequency $\frac{(f 1+f 2)}{2}$. This relationship shows that the oscillation amplitude varies periodically, which is the essence of the beating phenomenon. Therefore, amplitude modulation occurs when two frequencies combine algebraically. In a system with linear behavior, frequencies do not directly sum; thus, for amplitude modulation to occur, some form of irregularity or disturbance must be present in the system. Amplitude modulation can manifest in different forms, one of which is the beating phenomenon, which arises when the amplitudes of two closely spaced frequencies interact constructively and destructively [1]. As described in the Vibration Analysis Handbook, for example, if two frequencies are 29.6 Hz and 25.6 Hz (Figure 2), their amplitudes add when they become in phase, resulting in an increased overall amplitude. As they move out of phase, the amplitudes subtract until a 180-degree phase difference is reached. This continuous process of in-phase and out-of-phase interaction generates a signal with a time varying amplitude [2].

When two frequencies produce a beating effect (Figure 3), a cause-and-effect relationship exists. In such cases, although two frequency components appear in the spectrum, only a single underlying fault is present. The 29.6 Hz component acts as the carrier frequency and reflects the influence of the fault. Amplitude modulation can also occur when two frequencies are not exact multiples of each other. In Figure 3, the signal contains two frequencies along with a second harmonic. These two frequencies are nearly, but not exactly, equal; therefore, the time-domain signal may initially appear steady.

A full beat cycle is shown in Figure 4, where the amplitude modulation is clearly observable. The difference between the first and second frequencies is 0.2 Hz; therefore, one complete beat lasts 5 seconds. As the two frequencies move closer together, the time required to complete a single beat increase. Consequently, it is crucial to determine whether the target frequency is an actual harmonic or merely close to it. In vibration analysis, amplitude modulation is a phenomenon in which the amplitude of a carrier vibration signal (such as the shaft rotational frequency) is altered by another signal (such as periodic impacts caused by bearing defects or mechanical looseness). This interaction produces regular fluctuations in the signal amplitude, which can be observed as an envelope. In the frequency domain, amplitude modulation is characterized by the appearance of sidebands around the carrier frequency. These features are typically indicative of nonlinear or periodic faults in rotating machinery. Figure 5 illustrates a simulation of how amplitude modulation is generated, where the purple waveform represents the modulated amplitude resulting from the red and blue frequency components.

In vibration analysis, amplitude modulation and the beating phenomenon are two distinct concepts whose examination in both the time and frequency domains is essential for diagnosing faults in rotating machinery. In beating, two signals with very close frequencies combine, producing an oscillation with an average frequency and an envelope with a low frequency equal to the difference between the two frequencies. In this case, the amplitude periodically fluctuates between zero and its maximum value. However, in amplitude modulation, a high-frequency carrier signal is governed by a lower-frequency modulation signal, such that the carrier amplitude varies in accordance with the modulating signal. Unlike beating, the carrier amplitude in amplitude modulation does not necessarily reach zero. In the frequency domain, beating appears as two spectral lines very close to each other, typically located near $1 \omega$ and $2 \omega$, and their separation defines the beat frequency. In contrast, amplitude modulation produces a central line at the carrier frequency along with two sidebands symmetrically located around it. The presence of these symmetric sidebands is a key signature of modulation and plays a crucial role in diagnosing bearing and gear defects. Figure 6 compares both the amplitude and frequency characteristics of the two phenomena.

Pump P-2002 is a double-suction centrifugal pump driven by an electric motor and is classified as a critical asset in the Masjed-Soleiman Petrochemical Complex. The technical specifications of the pump are presented in Figure 7, while Figure 8 shows the actual view of Pump P-2002 together with its three-dimensional model generated from the plant design software, providing a comprehensive representation of the equipment and its configuration within the process unit. The primary function of the pump is to receive the amine solution from Tower T-2002 and transfer it to Tower T-2001 for carbon dioxide absorption from the process gas stream. Due to its critical role in the process, the pump is equipped with a continuous online monitoring system. In addition to real-time monitoring of operating parameters, periodic offline vibration measurements are routinely performed using a portable vibration analyzer to assess the mechanical condition of the equipment and support predictive maintenance activities.

Condition monitoring consists of a set of activities aimed at evaluating changes in a machine over time based on factors such as vibration, noise, performance, lubrication, and temperature. In predictive maintenance, condition monitoring plays a vital role in preventing severe damage to machinery and ensuring equipment reliability. For critical equipment, data acquisition is performed through both online and offline monitoring systems. Figure 8 illustrates the actual and schematic views of Pump P-2002, providing a clear understanding of the monitored equipment and its configuration within the process unit. Figure 9 presents the online analytical software of the machinery monitoring system (MMS), which enables continuous monitoring and real-time assessment of machine operating conditions. Figure 10 shows the waterfall plot of the pump generated by the MMS. This diagnostic tool provides a comprehensive representation of vibration amplitude variations over time and frequency, allowing the identification of increasing vibration trends and indicating the severity and progression of developing faults within the pump. Such visualization facilitates the detection of abnormal operating conditions and supports predictive maintenance decision-making.

Figure 11 shows the online trend analysis obtained from the MMS. Routine condition monitoring of Pump P-2002, including vibration, ultrasonic, and temperature measurements, was carried out daily on both the drive-end (DE) and non-drive-end (NDE) sides of the pump housing. During data acquisition and mechanical condition assessment, a significant increase in vibration levels was observed and recorded in the horizontal, vertical, and axial directions on both sides of the pump through online and offline measurements. Following the detection of this abnormal vibration behavior, the equipment was further investigated using both the Central Control Room (CCR) monitoring system and a portable vibration analyzer. These analyses were conducted to evaluate trend variations and identify changes in vibration frequency components, thereby determining the root cause and severity of the developing mechanical fault.

When two signals with very close frequencies (202.875 Hz and 202.785 Hz) exist within a system, their superposition produces a combined signal whose amplitude varies over time. These periodic amplitude fluctuations are known as beating [1]. In the following section, a real vibration analysis case from rotating machinery is presented, in which the two frequency components below were identified for the purpose of examining the beating phenomenon:

The frequency difference is as follows:

The beat period is as follows:

Figure 12 illustrates the interaction of the two mentioned frequencies and the resulting amplitude and frequency modulation, as identified through vibration analysis using the Leonova Diamond device.

The time-domain signal exhibits fast oscillations with variable amplitude, manifesting as beats [2]. In the frequency spectrum, this behavior is represented by two closely spaced peaks. These pulsations are also observable in the circular plot shown in Figure 13.

Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, and Figure 19 present the vibration analysis results for the P-2002 unit at the non-drive end axial measurement position, obtained at various temporal intervals. Vibration severity was monitored using root-mean-square and peak-to-peak velocity metrics, which displayed significant surges during specific monitoring periods. Furthermore, indicators such as kurtosis and crest factor were utilized to evaluate impulsive behavior and potential nonlinearities within the vibration signals [7]; elevated values of these metrics serve as precursors to localized mechanical degradation and increased activity at specific frequencies, with sporadic elevations indicating the onset of bearing degradation and the emergence of the beating phenomenon. To ensure data integrity, the sensor output voltage s wereverified, where stable readings confirmed the reliable operation of the instrumentation [4]. Furthermore, high-definition envelope analysis was utilized, which is particularly effective for detecting transient impacts and incipient bearing faults. Elevated high-definition envelope levels were observed at the bearing characteristic frequencies, namely, the ball pass frequency outer race, ball pass frequency inner race, ball spin frequency, and fundamental train frequency, thereby providing clear evidence of initial mechanical failure [5].

To further validate the vibration analysis results, the beating phenomenon was simulated and examined using MATLAB. The simulations demonstrated that the waveform generated by the frequency interaction in the mathematical model closely matches the output obtained from the vibration analyzer. In particular, the amplitude behavior and beat period observed in the real data were consistent with the results produced by the MATLAB model, thereby confirming the accuracy of the frequency analysis and the correct identification of the beating phenomenon. This agreement highlights the reliability of the analytical methods employed for condition monitoring of rotating machinery and verifies the precision of the developed model. The graphical results obtained from the MATLAB analysis are presented in Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, and Figure 25.

The second case study examines the occurrence of the beating phenomenon in an electromotor used to drive a natural-gas compressor equipped with a variable frequency drive(VFD). The equipment under investigation is the two-stage centrifugal compressor C-1202, powered by a medium-voltage electromotor. This compressor receives natural gas at a suction pressure of 34.5 bar and compresses it to a discharge pressure of 43.96 bar. The compressed gas first enters a receiving vessel, then flows into a separation drum, passes through the Mix Point section, and is ultimately routed toward the Primary Reformer. The electromotor is controlled by a variable frequency drive, which increases the operating speed in discrete steps at a ramp rate of 5 Hz/min until reaching the nominal frequency of 50 Hz. A step up Gearbox is installed between the motor and the compressor, providing a speed increase up to 12,200 rpm. A schematic representation of this machinery within the distributed control system is shown in Figure 26.

According to the comprehensive data acquisition program of the vibration condition monitoring system (MMS), periodic measurements of Compressor 1202 are routinely performed. Examination of the recorded data indicated that all operating parameters of the equipment remained within acceptable limits and were consistent with previous measurement cycles. Another round of data collection was conducted on the following morning, and the results showed no significant deviation from the previous readings, suggesting normal operating conditions. Figure 27 presents the G-envelope (GE) spectrum of the motor bearing at the non-drive-end (NDE) side, which did not indicate any abnormal behavior during routine inspections. However, only a few hours after this measurement, the compressor experienced an unexpected shutdown due to a high-temperature trip at the motor rear bearing housing. The evolution of this event is illustrated in Figure 28 and Figure 29, which show the increasing temperature trends of the rear and front motor bearings, respectively. Figure 30 presents the corresponding increase in vibration levels measured at the motor bearing on the non-drive-end side.

To further investigate the root cause of the failure, advanced bearing condition analyses were performed. Figure 31 compares the vibration responses of a healthy bearing (green) and a damaged bearing (blue) at measurement point H2-MDE (horizontal direction at the motor drive end). The comparison includes the time-domain vibration velocity signal (mm/s), displacement signal ($\mu$m), and GE response. As shown in Figure 31, the damaged bearing exhibits increased vibration amplitudes, higher displacement levels, and more pronounced envelope responses compared with the healthy bearing condition. These changes indicate the presence of defect-related impulsive excitations and progressive bearing degradation, confirming the effectiveness of vibration and envelope analysis techniques for fault detection and bearing condition assessment. Further confirmation of the bearing defect was obtained through measurements at point H1-MNDE (horizontal direction at the motor non-drive end).

Figure 32 compares the responses of the healthy bearing (green) and the damaged bearing (red- blue) using GE, vibration velocity, and acceleration analyses. The damaged bearing condition exhibits increased envelope amplitudes, elevated vibration velocity levels, and significantly higher acceleration peaks compared with the healthy bearing. The appearance of pronounced defect-related frequency components in the GE spectrum, together with the increase in impact-related acceleration energy, indicates the development of localized bearing damage. These findings are consistent with the progression of bearing deterioration and further validate the effectiveness of vibration and envelope-based diagnostic techniques for early fault detection.

Through detailed investigations conducted on the motor non-drive end side (MNDE) bearing of the induction motor to identify the root causes of failure, it was established that the observed damage pattern manifested as localized spots and pitting along the rolling path and rolling elements correlates directly with the passage of electrical currents through the bearing [10], [11]. In this scenario, transient local temperature rises at the points of electrical discharge entry, combined with rolling elements passing over these regions, cause localized plastic deformation and micro-pitting on both the inner and outer bearing rings as well as several rolling elements, a damage mechanism commonly termed electrical pitting or electrical erosion [10]. The flow of electrical currents through the bearing and shaft path in motors supplied with sinusoidal waveforms is a well-documented phenomenon. The concentration of these currents within the bearings primarily originates from bearing construction characteristics, material properties, and inherent asymmetries in the current distribution [14]. The low-frequency nature of shaft and bearing currents under such operating conditions induces stray currents flowing through conductive paths, including the shaft, motor housing, and bearings. Under these circumstances, implementing insulating measures or employing insulated or hybrid bearings has proven effective in mitigating shaft and bearing low-frequency currents [10-11]. One of the most critical contributors to this class of electrical bearing failures is the inadequate performance and quality of input filters in variable frequency drive systems powering the motor. The use of variable frequency drives combined with pulse width modulation output waveforms leads to elevated transient voltage s and high-frequency currents and ultimately trigge rs the bearing currents electrical discharge machining effect. Consequently, bearing failure remains one of the most prevalent issues in motors operated via variable frequency drives [12-13].

Figure 33 shows the bearing with insulating coating. Insulated bearings are widely employed in induction motors to prevent circulating currents from flowing through the motor bearings, which can cause premature bearing damage. Variable frequency drives, commonly used to control induction motors, are known to increase the risk of bearing failure due to induced electrical currents. This study examines the mechanisms by which circulating currents propagate through motor bearings under variable frequency drive operation and explores effective mitigation strategies to prevent associated bearing damage. Specifically, the use of insulated bearings on the non-drive end side serves as an effective barrier, disrupting the current path that typically extends from the shaft through the inner race, rolling elements, and outer race to the motor housing. The path interruption provided by insulated bearings significantly reduces the risk of electrical discharge and pitting within the bearing components. Bearing failure rates under these conditions can vary considerably depending on operational and environmental factors; however, empirical studies indicate that a substantial fraction of failures occur within 3 to 12 months following system startup. Given that many modern alternating current motors utilize sealed bearings to exclude contaminants, electrical degradation has emerged as the predominant cause of bearing failure in induction motors driven by variable frequency drives ( Figure 34) [10].

The presence of the beating phenomenon in the time-domain signal is evident in vibration analyses due to the proximity of two frequency components [12], [13], specifically Two times running speed (2X) and Two times Line Frequency (2LF), separated by 37.5 cycles per minute. Beating occurs when an exciter oscillates at a frequency close to the natural frequency of a composite system, or alternatively, when two or more oscillators operate at near-yet-distinct frequencies. In such scenarios, a frequency wave phenomenon, akin to resonance, emerges. As depicted in Figure 35 and Figure 36, one signal acts as a resultant of another, where the frequencies of these two signals differ by a small margin. Consequently, at certain times, one signal leads, while at others, it lags, causing constructive and destructive interference between them.

As observed in Figure 36, an increase in the electromotor’s speed leads to an augmented frequency separation between the 2LF and 2X components [12], [13], reaching approximately 37.5 cycles per minute. This wider frequency gap attenuates the intensity of the beating phenomenon, consequently resulting in a significant reduction of the overall equipment vibration levels. Given that the presence of the 2LF component in this context, referring to the variable frequency drive(VFD) system’s 2LF within the vibration spectrum can indicate electrical anomalies in the system, its analysis is of paramount importance. To analyze this phenomenon, the individual frequencies are converted from cycles per minute to Hertz as follows:

The resulting beat frequency ($f_{\text {beat}})$, which represents the rate of the amplitude modulation, is calculated in Eq. (13):

The corresponding beat period $T_{\text {beat}}$ representing the time interval between successive vibration peaks, is determined by:

The resulting time-dependent vibration velocity, $x(t)$, which characterizes the superposition of these two signals, can be modeled using the sum-to-product trigonometric identity. Based on the average frequency ($\approx$93.749 Hz) and the modulation frequency ($\approx$0.312 Hz), the equation is expressed as:

where, $A$ represents the amplitude of the individual frequency components. This mathematical model confirms that the vibration velocity oscillates at the average frequency of the two components, while its magnitude is modulated by a slow-moving envelope defined by the beat frequency.

In systems equipped with variable frequency drives, the switching nature and generation of common-mode voltages create conditions conducive to unwanted electrical currents flowing through the shaft and bearing paths [13]. If adequate protective measures are absent, these currents discharge through the bearings. This phenomenon instigates damage such as pitting, micro-welding, and surface indentations on the bearing’s raceways and rolling elements, ultimately diminishing the bearing’s service life and increasing the failure rate of rotating equipment. To prevent such degradation, the implementation of insulated bearings on the non-drive end of the electromotor, or shaft isolation, is recommended as an effective countermeasure. Following the implementation of corrective actions, which included addressing the identified root causes and installing an insulated bearing on the non-drive end, the equipment’s performance was reassessed. The resultant data, presented in the vibration plots of Figures \ref{fig37} and \ref{fig38}, indicate that the equipment has remained in stable operational service to date without any precursors of failure.

The beating phenomenon, as a fundamental manifestation of wave interference in vibrational systems, plays a crucial role in the condition monitoring of rotating machinery. The results of this study indicate that the occurrence of beating in an engineering context signifies energy exchange between two closely spaced vibrational modes or dynamic interaction between two unbalanced sources within the system [4], [8].

Based on the conducted mathematical analyses, the beat frequency is directly related to the absolute difference between the frequencies of the two involved components ($f b=|f 1-f 2|$), and the effective signal amplitude varies as a periodic function of time. From a signal analysis perspective, the beating phenomenon is essentially a form of amplitude modulation, wherein the carrier wave constitutes the average frequency, and the ‘modulating wave’ represents the difference frequency [2].

In the examination of numerical and experimental examples (such as frequencies of 29.6 Hz and 25.6 Hz), it was observed that the resulting frequency spectrum (computed via the fast Fourier transform) exhibits not a single peak, but rather two adjacent and distinct peaks. This characteristic is considered a distinctive ‘frequency signature’ for the beating phenomenon [3]. This research emphasizes that in the analysis of industrial vibrations, accurately differentiating between the system’s true harmonics and ‘nearby but distinct’ frequencies (beating frequencies) is vital to avoid diagnostic errors [4].

Ultimately, the correct identification of these patterns in the condition monitoring of equipment such as pumps and turbines provides an efficient tool for detecting closely spaced modes, mechanical anomalies (like rotor imbalance or misalignment), and combined vibrations [8]. Future horizons in this domain, by leveraging advanced signal processing algorithms, will enable more precise monitoring of amplitude variations even in the presence of environmental noise [7], [8].

The beating phenomenon, which arises from the interaction of closely spaced vibration frequencies, plays a fundamental role in understanding the dynamic behavior of rotating machinery. This study demonstrates that the combined use of time-domain analysis, frequency-domain techniques, and analytical modeling provides an effective framework for identifying closely spaced excitation sources and interpreting complex vibration responses. Such an integrated approach is essential for diagnosing transient vibration events, detecting misalignment, and identifying electromechanical abnormalities in industrial systems.

The developed analytical model, based on the superposition of adjacent frequency components, shows strong agreement between theoretical predictions and experimental observations. Key parameters such as beat frequency and amplitude modulation characteristics are accurately captured by the proposed formulation, confirming its capability to represent complex vibrational behavior. Furthermore, the combined application of high-resolution FFT analysis and envelope-based evaluation provides a robust tool for distinguishing between interacting excitation sources in rotating machinery.

The case study on variable-frequency-drive-driven equipment clearly illustrates the practical implications of the proposed methodology. The interaction between 2X and 2LF components was identified as the main source of beating and abnormal vibration amplification, highlighting the influence of coupled electromechanical effects in industrial systems. The implemented corrective actions, including the elimination of current leakage paths and the installation of an insulated bearing at the non-drive end, successfully mitigated the beating phenomenon and restored stable operating conditions.

Overall, the results confirm that the spectral and temporal characteristics of beating provide valuable diagnostic information for early fault detection and condition monitoring of rotating machinery. Future work may focus on extending the proposed framework to variable load conditions, multi-source excitation scenarios, and nonlinear system behavior to further improve predictive capability in industrial vibration analysis.