Converter-based drive systems with permanent magnet synchronous motors (PMSM) are widely used in many industrial sectors. Consequently, issues related to the monitoring and diagnosis of electrical, magnetic, and mechanical faults are becoming increasingly important due to the growing use of these drives in various devices, including those critical to safety. Modern drive systems consist not only of the motor and magnets but also of a range of specialised devices, such as power supply systems, frequency converters with pulse width modulation (PWM), measurement and control systems, and complex mechanical systems that transmit mechanical torque (shafts, gears, couplings) to drive the given actuator. Each of these components is susceptible to failures that can disrupt normal drive system operation and require appropriate actions, particularly from the control system, to detect anomalies in the drive system [1,2,3]. Modern diagnostic methods enable the detection of both electrical faults (including various types of short circuit, interphase faults, winding connection issues, etc.) and mechanical faults (such as the most common: bearing damage, eccentricity, misalignment of the drive system, unbalance of rotating elements, shaft deflection, etc.), as well as demagnetisation in permanent magnet motors [4]. Lack of proper diagnostics can lead to significant losses, including a substantial reduction in drive system efficiency, decreased positioning accuracy in servodrives and CNC machines, increased energy consumption, and unplanned downtime. According to the ISO 17359 standard [5], the following signals are used for the proper evaluation of motor condition: current, voltage, input and output power, temperature, rotational speed, vibrations, torque, noise, and acoustic techniques. In advanced motor diagnostics, the most commonly used methods include [1,2,3,6]: motor current and voltage analysis, temperature measurement [7] vibration analysis, and noise analysis. The degradation of the motor and the torque transmission system, manifested by various anomalies or asymmetries, leads to unstable operation. This is typically observed through increased levels of vibration, noise, and temperature [2,4,7]. The electric drive is not fully symmetric [8,9]. To differentiate between asymmetry caused by drive system degradation and asymmetry resulting from power supply issues or inherent design characteristics, it is crucial to correctly identify, record and classify the relevant signals for the proper operation of the drive system. Assessing characteristics, particularly those related to current, helps increase detection sensitivity, thereby eliminating design or power supply asymmetries. Analysis and estimation of these changes serve as criteria for effective operation and accurate detection. As highlighted in review studies [1,2,3,4,10] anomalies can occur in various parts of the drive system. In this article, the focus is on detecting the unbalance of a mechanical component in the electric drive system of a two-mass servomechanism with a PMSM motor, which is connected to the load via a long, flexible shaft. In the analysed system, no additional external sensors for current, acceleration, or acoustics were used; instead, sensors that are part of the drive system's basic equipment (such as speed, position, and current measurement) were utilised. The unbalance was modelled by adding additional masses to a rotating disc that was rigidly attached to the motor shaft. The test mass was varied during the experiments. A review of the literature indicates that there are many methods to detect unbalances in drive systems, most of which pertain to single-mass systems using external sensors [2–4,7,10-12]. These include diagnostic signals such as mechanical vibrations (acceleration) [2] noise (acoustic waves) [2,4] current (stator current) [3,11,12] and temperature [7]. These signals allow for the assessment of the current condition of the electric drive system, particularly in the evaluation of the state of bearings, alignment, balancing, and the condition of the foundation. Thanks to modern measuring instruments (dedicated sensors, measurement cards, and microprocessors), it is possible to measure mechanical vibrations (which can have various causes, whether electrical, magnetic, or mechanical) that indicate the condition of the drive system. In industrial environments, fully symmetrical drive systems are rare or non-existent. Even if such a system was initially balanced at a particular workstation, over time, degradation of a component of the system (whether electrical, magnetic, or mechanical) may occur [13]. Therefore, this work focusses on detecting unbalance in the drive system of a two-mass setup with a PMSM motor, using, among other things, the reference current signal from the speed controller. The two-mass model was investigated because it allows for analysing the impact of torsional vibrations on the drive system. One source of these vibrations is a sudden change in the motor rotational speed, which causes twisting in the flexible connection [14]. Vibrations in two-mass systems lead to energy loss, reduced efficiency, and dynamic stresses [15]. These vibrations are also problematic in precise position control. Therefore, diagnosing the unbalance in the system, which significantly exacerbates this issue, becomes crucial. When a mechanical fault occurs in an electric drive, particularly for a PMSM motor operating in a field-orientated control structure, changes in electrical, magnetic and mechanical parameters lead to disturbances in current, voltage, and speed signals, affecting the proper functioning of the frequency control structure [3,10,16,17,18,19,20]. An uncontrolled increase in the level of damage leads to unstable drive operation [17]. Therefore, it is crucial to detect faults as quickly as possible. Depending on the type and power of the drive, this could range from a few to several seconds. As demonstrated in [3,4,16] internal signals from the drive control structure, particularly current, can be utilised effectively for this purpose. Hence, the aim of this work is to develop a method for monitoring and diagnosing unbalance faults in a two-mass precision servodrive system, using signals from the internal field-orientated control structure embedded within the control system. Many existing solutions, such as those described in [2,3,4,10,12,13,21,22,23,24,25,26,27,28,29] use additional sensors to monitor the condition of machines or equipment. Adding these elements to existing systems significantly increases costs during both installation and operation. This requires regular calibration, maintenance, and, in the case of failure, replacement, which introduces another potential source of problems within the system. It often requires modifications to existing infrastructure and integration with monitoring software. Furthermore, data from different sensors can be difficult to interpret, often requiring the use of advanced analysis algorithms and personnel with specialised qualifications. The use of non-invasive measurement methods in the drive system, as proposed in the article, reduces costs because utilising existing control signals does not incur additional expenses related to the purchase, installation, and maintenance of new sensors. The non-invasive approach does not require intervention in existing mechanical or electrical systems, thereby eliminating the risk of introducing new sources of failure. Signals used for control are typically already well validated and monitored, which enhances the system's reliability. Since these signals are already part of the control system, using them for fault detection is much simpler and less risky. Fewer components in the system mean fewer potential points of failure, which increases the overall reliability of the system, not just the drive system. For example, in [20] an online diagnostic method for minor Interturn Short Circuits (ISCF) in low-speed Permanent Magnet Synchronous Motors (PMSM) used a technique based on the decomposition of the zero-sequence voltage vector, which is measured within the internal system. Many studies measure voltage, current, or speed signals, such as in [12] where current and speed measurements, along with frequency analysis, were used to diagnose mechanical faults in synchronous machines, with additional sensors used. In [30] a model-based scheme was proposed to detect and identify faults in a gear drive with a steady state PMSM motor. A state-space model of the system was proposed and the Recursive Least Squares (RLS) algorithm was used for parameter estimation. Fault detection was performed using a thresholding method on the residual current spectrum to assess the frequency characteristic of a particular fault. Both invasive and non-invasive methods are applied in this context. For example, in [11,31] the stator current, after Park's transformation, was analysed for rotor unbalance using the discrete wavelet transform. This work focused on the latter method. From a diagnostic perspective, both invasive and non-invasive signals, as well as the methods for processing them, are of equal or similar importance. Thus, the effectiveness of the diagnostic process depends not only on the diagnostic signals analysed [7-9,12,14-15,30] but also on the processing methods applied. For example, in [18], although analysis of Root Mean Square (RMS) vibration signals detects individual faults and helps to determine the technical condition of the machine, unfortunately it cannot specify the cause of its malfunction. In [19] the Fast Fourier Transform (FFT) algorithm was used to detect broken rotor cage bars, eccentricity, and damaged bearings in an induction machine based on mechanical vibration analysis. In [32] the potential for using average, maximum and RMS signals, as well as cross-correlation coefficients, kurtosis, and peak values, for detecting rotor unbalance was presented, with an MLP neural network employed to differentiate these faults. In [33] phase currents and a vibration sensor were used to detect stator faults in a PMSM motor, where wavelet analysis of vibrations was able to identify up to 15% of turn-to-turn shorts in a single phase. Mechanical vibrations are a versatile and widely used diagnostic signal, as demonstrated in [34] where they were also used to detect electromagnetic faults. An example of applying vibroacoustic to identify misalignment in a drive system consisting of a motor, a cylindrical gear and a worm gear is presented in [35]. To detect unbalance in a PMSM drive system, mechanical vibration signals can be analysed using FFT, bispectrum, full spectrum, and orbit shape analysis, as described in [22]. In [21] mechanical vibrations were analysed with FFT to detect PMSM unbalance, with additional modelling of demagnetisation and dynamic eccentricity. In [23], FFT and bispectral analysis of mechanical vibrations and stator current were used to detect rotor unbalance in an induction motor powered by a frequency converter. FFT analysis has limitations because it does not account for changes in process over time [24,25,26,36,37] which is why the Short-Time Fourier Transform (STFT) is used to detect faults in dynamic states, enabling the identification of both the type of fault and its timing. In [36] STFT was used to detect stator faults in a PMSM machine by analysing the stator current, its envelope, and the spatial vector module of the stator current, highlighting STFT's advantages over traditional FFT analysis. In [24] the possibility of detecting the rotor unbalance in an induction motor was demonstrated using STFT analysis of the stator current vector, with results compared to an approach based on FFT analysis of the stator current. In [25] the application of STFT to detect mechanical faults, such as blade-to-stator friction in turbochargers, was demonstrated by analysing vibration signals during machine start-up and braking, showcasing the method's superiority over traditional FFT analysis. In [26] STFT was used to analyse noise from both an undamaged motor and motors with unbalance, misalignment, and bearing damage, allowing for precise determination of the timing of the fault. In [37] the use of variable window STFT was proposed to detect winding faults in PMSM by analysing stator current signals and automating the detection process using a convolutional neural network. It should be noted that the presented literature review does not cover two-mass systems nor does it analyse the reference current signal from the speed controller. In the existing literature, the unbalance is modelled in various types of electric machines and drives. Specifically, an this phenomenon has been identified in induction motors [23,24,26] PMSMs [11,21,22,31,36] tidal turbines [13], commutator motors in impact drills and mixers [27] and washing machines [28,29]. The literature analysis indicates that the issue of unbalance affects all rotating machines. Consequently, this work focusses on detecting unbalance in a two-mass system with a long shaft consisting of two AC servodrives (PMSMs). The non-invasive standard equipment in an automated control system, the reference current signal in the q axis of the speed controller, was analysed using STFT. The study investigated the impact of varying the test mass installed in single-mass and two-mass systems on the level of unbalance in the drive system, specifically examining how the moving average window affects this issue. The influence of different neural network architectures was also analysed, and the optimal solution for the given example was proposed, with experimental validation of this choice in relation to the single-mass system.

For the purposes of this publication, a simplified unbalance model was adopted, which focuses on taking into account the influence of the weight of the unbalance mass on the creation of the moment of force coming from the force of gravity [38]. This approach does not take into account the centrifugal force because the control system on which the method was developed does not have any information about the current value of this force. The idea is presented in Figure 1, which shows the action of the gravitational force mg on an unbalanced mass m, located at a distance r from the axis of rotation of a cylinder with mass M, moment of inertia J and radius R. It was assumed that this cylinder rotates with constant angular velocity ω relative to the stationary x-y coordinate system. The acceleration due to gravity is set in the vertical direction, parallel to the y axis. The value of the torque acting on the mass m can be expressed by the formula (1)

Most often it happens $m\ll M$. During the operation of the system, the unbalance moment $T_u$ affects the rotational motion of the cylinder in accordance with Newton's second law of dynamics (2)

Assuming that the rotating mass has the shape of a cylinder with mass M and radius R (3)

Hence, the angular acceleration resulting from unbalance is a cosine function of the angle and directly proportional to the amount of unbalance. This means that during operation there is an additional component of angular acceleration, closely related to the unbalanced mass and the instantaneous speed ω (4)

In absolute terms, remembering that in general $m\ll M$, it is expected that this acceleration will have a small amplitude, but the speed control system will compensate for the resulting disturbance, which should result in the appearance of an additional component with angular speed ω in the control signal. Therefore, it is reasonable to analyse the frequency of the control signal, which should be expected to contain an additional component of rotational frequency, which has its source in the unbalance.

The model of the mechanical part of the single-mass system was implemented in Matlab - Simulink environment to verify the concept for the single-mass object, which were presented in subsection 7.1.

The two – mass object is a popular control object in drive automation [39,40,41], since it reflects the characteristics of typical connections found in the mind. The block diagram of the dual-mass system is shown in Figure 2. The description intentionally omits the loss factor inside the connection, as it is generally negligible.

The equations describing the dynamics of an undamped system are described by the relations (5)

Such a system is characterized by two frequencies: resonant ω_R and anti-resonant ω_A, which result from the physical properties of the object (6)

In the described drive system after applying the Clarke and Park transform, the electromagnetic torque T_E is proportional to the current in the q-axis: T_E = k_φ i_q Assuming that the time constant of the current control system (τ_E) is negligible with respect to the resonance frequency of the system, it can be assumed to a good approximation that the current in the q-axis (i_q) is equal to the set current (i_{q ref}), which is the output signal from the speed control system. In addition to the above, disturbance moments such as friction moment, external load, as well as moments resulting from mechanical defects: misalignment, unbalance and runout can act on the two – mass object on both sides, these moments are denoted T_D1 and T_D2 (total disturbance). Taking into account the aforementioned phenomena, the equation of motion of the dual-mass system subjected to interference moments is presented by equation (7)

A block diagram of the control system is provided in Figure 3.

The speed control system is based on the Active Disturbance Rejection Control [42] method, the block diagram is shown in Figure 4. The essential part is the Extended State Observer, which estimates the disturbance$\widehat{f}$affecting the object in real time, which is then decoupled in the REJECTOR block, so that from the controller's point of view the control object is reduced to a first-order integrator. The ADRC method provides the possibility to shape the bandwidth for the regulator and the disturbance decoupling part independently, with the ESO bandwidth generally exceeding the regulator bandwidth by at least 4 times [43,44]. This causes such a control system to respond quickly to the occurrence of a disturbance, while at the same time making it possible to limit the dynamic response to the setpoint, which is desirable for a vibrating system (the two-mass system described). Thus, the ADRC structure makes it possible to respond efficiently (thanks to an observer with a sufficiently wide bandwidth) to even small values of disturbance moments, a property desirable for unbalance detection.

Many proposed methods for diagnosing electrical motor faults, including rotor unbalance or misalignment issues in drive systems, are based on frequency domain analysis using FFT and searching for frequency components characteristic of specific faults [45,46]. The Fourier transform assumes that the analysed information is periodic and stationary. This function is then decomposed into a fixed number of sinusoidal signals with specified frequencies. Unfortunately, in practice, signals recorded from real-world objects often do not meet these key assumptions. This method relies on sinusoidal periodic functions that represent a single frequency, which results in a loss of important temporal information about the occurrence of individual frequency events. Additionally, the amplitude values of the signal in the frequency domain are averaged, which means that short-duration events present in the signal may be invisible in the FFT spectrum. In fault diagnosis, knowing the exact moment of a fault's occurrence is crucial, as it can help determine the cause by analysing other system parameters from the period before the fault occurred. To overcome this limitation, alternative methods are used that perform time-frequency analysis, such as Discrete and Continuous Wavelet Transform (DWT) [47,48], Hilbert-Huang Transform (HHT) [49,50] or Short-Time Fourier Transform (STFT) [51,52]. In this article, the STFT of the reference current signal I_{q ref} from the Field Oriented Control (FOC) system is proposed as a method for extracting symptoms of rotor unbalance and assessing its degree. The STFT method involves sliding a window of a specified length (usually a power of 2) with a certain step along the signal and analysing its frequency content using the FFT. A key feature of this method is its time-frequency resolution: a larger window provides better frequency resolution but poorer time resolution; conversely, a smaller time window implies better time resolution but worse frequency resolution. This trade-off can be managed by carefully selecting the parameters and type of window function. The STFT computes the Fourier transform of the function x(t) with a symmetric window function with real values w(t), which is shifted in time t along the signal and modulated with frequency ω. The continuous form of STFT can be expressed by the following equations (8) [51,52]:

During measurements with a constant sampling frequency, the signal obtained is discrete and therefore STFT can be expressed as (9) [51,52]: where: N represents the number of FFT samples (points), n is the index of the input sample in the time domain, dependent on the length of the window function, x[n] denotes the input signal, w[n] is the window function, m indicates the position of the window function, H signifies the overlap between consecutive windows, and k is the frequency index.

The result of the STFT transformation is often presented in the form of a spectrogram, which is a three-dimensional plot of the spectral density function of the Fourier transform. It is described by the following equation (10):

The object of the research was a two-mass drive system consisting of two permanent magnet synchronous motors with three pairs of poles, a rated power of 3 kW and a rotational speed of 3000 rpm. The motors were connected to each other by a long shaft with a diameter of 6 mm. A disc was mounted on the drive side, to which a test mass was mounted at a radius of 42 mm. The drive system was attached to a concrete foundation, the purpose of which was to reduce the vibrations that occur during the operation. During the research, the set current I_{q ref} and the angular speed of the motor ω_motor were recorded using the NI USB-9162 card with a sampling frequency of 12.8 kS/s. Signals were measured for 10 s. The I_{q ref} current was generated from the vector control system implemented (Figure 5) in the Analog Devices SHARC DSP 26369 signal processor. The motor was powered by a laboratory 3-phase converter. During laboratory testing, unbalance was modelled with additional test masses: 7.5 g,, 15.9 g, 24 g and 45 g mounted at a radius of 42 mm from the centre of rotation. The drive system was operated unloaded, with a constant angular velocity of 60 rad/s. The study checked the effect of the test mass on the operation of the engine itself (single-mass system) and the two-mass system with PMSMs. The purpose of the laboratory tests was to obtain the diagnostic patterns necessary for the process of training neural networks to detect rotor unbalance and determine its level. These patterns were to be obtained from the control system and not from additional external sensors (measurement systems).

The purpose of the simulation studies was to confirm the diagnostic symptoms indicative of rotor unbalance occurring in the reference current I_{q ref}. During these studies, the same masses that were also used during laboratory tests were used to model the unbalance. The effect of the test mass on the I_{q ref} current is shown in Figure 6, from which it can be seen that the additional test mass increases the amplitude of the rotational frequency f_r. Apart from the rotational frequency, there are no additional harmonics in the I_{q ref} current spectrum up to 45 Hz. Moreover, with simulation time, the value of the rotational frequency amplitude does not change. The restriction of the band to 45 Hz is due to the fact that a symptom of rotor unbalance is an increase in rotational frequency amplitude. In the case of the modelled motor, the set speed was about 60 rad/s, which is equivalent to a frequency f_r of about 9.6 Hz. The set spectrogram range allows observing up to a fourth multiple of the rotational frequency.

This article explores the detection of rotor unbalance in a two-mass system with a PMSM using the STFT applied to the reference current signal from the speed controller. The significance of monitoring and diagnosing faults in drive systems has increased with their widespread use in industrial sectors, particularly for safety-critical applications. Modern drive systems, which include motors, power supply systems, frequency converters, and mechanical components, are prone to various failures that can alter the operation. Diagnosing these faults is essential to prevent efficiency losses, reduced accuracy, increased energy consumption, and unplanned downtime. Effective diagnostic methods include analysing current, voltage, temperature, vibrations, and noise. In particular, the STFT method offers advantages over traditional Fourier transform techniques by providing better time-frequency resolution, which is crucial for identifying and diagnosing faults in dynamic conditions. This study uses the reference current signal and investigates how the varying test masses impact unbalance levels in both single-mass and two-mass systems. It also evaluates different neural network architectures to optimise fault detection. The noninvasive approach of using existing control signals (coming directly from the control system), rather than additional sensors, reduces costs and simplifies the diagnostic process while enhancing system reliability.

The use of neural networks makes it possible to automate the process of detecting the rotor unbalance of a two-mass system, as well as determining its level. The simple MLP network structures used (networks with one and two hidden layers) were characterised by a high unbalance detection efficiency of over 95% and an unbalance level determination efficiency of approximately 93%. A prerequisite for such high efficiency is the use of a smoothing filter with at least 15 samples. The approach proposed in this paper for detecting rotor unbalance of a two-mass system can be implemented in a PMSM control system, which will be analysed by the authors in the future.