Detection of Rotor Unbalance in Multi-Mass Systems Using FFT and Bispectrum

Unbalance is a phenomenon inherently linked to the operation of rotating components, especially those found in electric machine rotors. The mechanical vibrations produced by an unbalanced rotor propagate through the machine’s housing, leading to reduced durability and decreased operational reliability. Minimising the adverse effects of this issue on the performance of the entire drive system requires early detection of the initial stages of damage. Modern diagnostic methods enable the identification of a wide range of faults that may occur in electrical machines, including primarily mechanical faults (such as rolling bearing wear, misalignment, eccentricity, or unbalance of rotating parts), electrical faults (including, among others, inter-turn short circuits in stator windings and rotor cage damage), as well as magnetic faults related, for example, to demagnetisation processes in permanent-magnet motors (Chen et al., 2019; Krichen et al., 2017; Orlowska-Kowalska et al., 2022; Wang et al., 2014). Mechanical damage to a permanent magnet (e.g., chipping) may cause rotor unbalance and irregularities in the airgap magnetic field. This results in increased pulsations of electromagnetic torque, vibration levels, and noise; therefore, diagnostic methods combining the analysis of mechanical vibration and electrical signals are increasingly used (Calderon-Uribe et al., 2023; Rahman and Uddin, 2017), enabling more effective detection of rotor unbalance at an early stage of its development.

The purpose of diagnostic systems is to detect damage as quickly as possible, allowing the user to plan the necessary repair in advance and avoid costly downtime. Currently, both ready-made industrial diagnostic systems and solutions built in-house using widely available measurement cards and sensors are available, enabling flexible adaptation to the needs of a facility. The latter approach also makes it possible to integrate additional functions, such as artificial intelligence algorithms for automatic classification of operating states and fault prediction (Boudiaf et al., 2018; Ewert et al., 2020; Wolkiewicz and Skowron, 2017). Diagnostic systems are most often based on the analysis of signals, such as current, mechanical vibrations, or noise, and their effectiveness depends not only on the quality of the measurement hardware but also on the applied data processing and analysis methods (Nandi et al., 2005; Niu et al., 2023). Modern approaches also include deep machine learning techniques and data fusion from multiple sensors, which significantly improve the quality of fault detection and classification in rotating machines (e.g., vibration analysis and machine learning techniques in rotating machinery diagnostics) (Tiboni et al., 2022; Tran et al., 2022).

Each diagnostic signal contains symptoms of various faults. Diagnostic systems may detect single faults or identify multiple types of failure. In Ebrahimi et al. (2009), based on the fast fourier transform (FFT) analysis of mechanical vibrations, three distinct faults were identified: broken rotor cage bars, rotor eccentricity, and a damaged rolling bearing. Vibration analysis also makes it possible to detect rotor unbalance. By using mean, maximum, and root mean square (RMS) values, along with additional statistical indicators, such as skewness, kurtosis, and the crest factor, the study (Ribeiro Junior et al., 2020b) demonstrated the possibility of identifying rotor unbalance, rotor cage damage, and rolling bearing failures. Additionally, methods based on the wavelet transform allow precise localisation of signal changes associated with the development of faults, improving early failure detection (Ruqiang et al., 2014). Time–frequency analysis makes it possible to distinguish simultaneous defects in different rotor components, such as eccentricity and unbalance (Jia et al., 2022). Modern approaches also include the use of statistical indicators and signal entropy, which increases the accuracy of fault classification in electrical machines (Zhang and Wang, 2019).

The stator current can be successfully used to detect electrical faults (Diversi et al., 2025; Thomson and Fenger, 2001). In Liang et al. (2018), the application of stator current and mechanical vibration analysis for detecting short-circuit faults in PMSM stator windings is shown. The study demonstrates the ability to detect rotor unbalance in an induction motor supplied by a frequency converter, using FFT and bispectrum analysis of stator current and vibration acceleration signals, as demonstrated in Ewert (2020); Ewert et al. (2022). In Yatsugi et al. (2023), a method is presented to detect and classify various types of induction motor faults based on the characteristics of the stator current signal. Meanwhile, in Thang et al. (2020), the stator current signal under frequency converter supply conditions enables identification of rotor and stator defects. The above research results confirm that the stator current signal contains significant diagnostic information in the case of both mechanical and electrical faults. Moreover, the application of higher-order analyses enables more effective differentiation of rotor unbalance from other defects occurring under frequency converter supply conditions (Thang et al., 2020; Thomson and Fenger, 2001).

In drive system diagnostics, the selection of an appropriate diagnostic signal is crucial, as is the method used to process it. The literature presents a wide spectrum of signal processing techniques used in the diagnostics of electrical machines. Advanced time–frequency analysis methods are gaining increasing interest (Ewert et al., 2024), as are approaches based on multiscale transforms (Tran et al., 2022), which enable effective extraction of fault symptoms from non-stationary and non-linear signals. In addition, solutions are described that integrate classical signal analysis methods with intelligent classification algorithms, significantly increasing the reliability of identifying various types of defects (Jia et al., 2022; Łuczak, 2024; Orhan et al., 2025). A review of current diagnostic techniques used in permanent magnet synchronous motors, including time–frequency methods, magnetic flux monitoring, and advanced signal analyses, is presented in Sergakis et al. (2025), while a detailed literature analysis in the field of permanent magnet motor diagnostics is provided in Dogan and Tetik, (2021); Orlowska-Kowalska et al. (2022).

The modelling of unbalance can be realised in several ways. A test mass can be mounted, for example, directly on the rotor, on an additional disc fitted to the motor shaft, or on the motor fan. Moreover, the issue of unbalance concerns various types of electrical machines, which makes its early detection crucial for the reliable and safe operation of drive systems. The literature indicates that an appropriate method to represent unbalance allows for a reliable assessment of the effectiveness of diagnostic methods and the comparison of experimental results with theoretical models. For example, in Ewert, (2020), a method for detecting rotor unbalance in a drive system is presented based on the analysis of mechanical vibration signals and electrical quantities, taking into account frequency converter supply. In Faiz et al. (2017), static and dynamic eccentricity diagnostic methods in electrical machines were analysed, based on modelling the phenomenon and examining characteristic symptoms in diagnostic signals. Particular attention is given to multi-mass systems with long shafts, where unbalance can lead to complex dynamic phenomena and an increase in torsional vibration amplitudes (Ewert et al., 2024; Sergakis et al., 2025).

This work therefore concentrates on the detection of unbalance in a two-mass, long-shaft system incorporating two brushless AC servomotors. Special attention is given to multi-mass systems with long shafts, where unbalance can lead to complex dynamic phenomena, including an increase in torsional vibration amplitudes and couplings between individual masses. In such structures, unbalance of a single component can affect the dynamic response of the entire system, which significantly complicates its unambiguous identification.

To detect rotor unbalance, FFT and bispectrum analyses were applied to the mechanical vibration acceleration signal and to the speed controller’s reference current, enabling the evaluation of both linear and non-linear relationships between the measured quantities. Unbalance was introduced by attaching additional test masses to a rotating disc rigidly coupled to the motor shaft—an approach commonly used in experimental diagnostics of rotating machinery, as it allows for controlled excitation of dynamic disturbances.

Unbalance occurs when a rotating element with an asymmetrical distribution of mass in relation to its rotational centre is set in motion. The result of unbalance are additional centrifugal forces and moments, which adversely affect the bearings. In fact, they contribute to increased mechanical vibrations and noise (AbdulBary et al., 2021).

According to Figure 1, the introduction of an additional test mass m « M mounted at an angle α at a distance of r→ \overrightarrow r from the axis of rotation O of a cylinder with mass M and moment of inertia J causes the shift of the centre of gravity of the rotating element from point O to point O_m by the vector. rm→ \overrightarrow {{r_m}} This vector is referred to as the eccentricity of the centre of gravity (Yamamoto et al., 2016). In this case, unequal balance (unbalance) is defined as the product of the testing mass m and the vector r→ \overrightarrow r on which it is installed (1) (Yamamoto et al., 2016): (1) N→=mr→ \overrightarrow N = m\overrightarrow r

Figure 1.

Graphical illustration of rotor unbalance, (a) generation of centrifugal force, (b) generation of a moment of force.

As shown in Figure 1a, setting the element with the additional test mass m into rotational motion at angular velocity ω generates a centrifugal force Fc→ \overrightarrow {{F_c}} (2) (Yamamoto et al., 2016): (2) Fc→=mr→ω2 \overrightarrow {{F_c}} = m\vec r{\omega ^2}

The cause of the increase in vibration levels is the presence of an unbalanced centrifugal force exerted on the machine housing. Since the unbalance depends on the angular velocity ω an increase in the amplitude at the rotational frequency can be observed in the mechanical vibration spectrum (3) (Korkua et al., 2010): (3) funb=fr=ω2π=n60 {f_{{\rm{unb}}}} = {f_r} = {\omega \over {2\pi }} = {n \over {60}} where f_unb is the fault frequency that indicates the occurrence of rotor unbalance, f_r is the rotational frequency of the rotor, and n its rotational speed.

The second approach considered (Figure 1b) involves analysing the effect of an additional test mass m on the generation of torque caused by gravitational interaction (4) (Ewert et al., 2024; Kim, 2009). In this case, centrifugal force is not included in the control system. The torque acting on the test mass m can be expressed by the following formula: (4) Tunb=Jdωdt=mgrcosφ {T_{{\rm{unb}}}} = J{{d\omega } \over {dt}} = mgr\;{{\cos}}\left( \varphi \right) where φ is the angle of rotation.

If we assume that the rotating element is a cylinder with mass M and radius R, then the angular acceleration resulting from the unbalance caused by the test mass m is a cosine function of the angle and is directly proportional to the magnitude of the unbalance (4). During rotation, there is an additional component of angular acceleration that is directly related to the test mass and the instantaneous angular velocity ω (5): (5) dωdt=mgr0,5MR2cosφ=mgr0,5MR2cosωt {{d\omega } \over {dt}} = {{mgr} \over {0,5M{R^2}}}{{\cos}}\left( \varphi \right) = {{mgr} \over {0,5M{R^2}}}{{\cos}}\left( {\omega t} \right)

Due to the small value of the mass m relative to the mass M of the rotating element, this acceleration will have a small amplitude. The disturbance generated by the unbalance should be compensated for by the speed-control system, which should be visible as an additional component of the active current at angular velocity ω in the control signal. Therefore, both in the mechanical vibration acceleration signal and in the control signal, the additional component described by Eq. (3) should be sought.

The bispectrum is a higher-order transform which, unlike the power spectrum, does not lose information about the phase shift of the analysed signals (Hachemi Benbouzid, 2000). Its advantage lies in its ability to improve the signal-to-noise ratio, which can be particularly important when analysing symptoms with relatively small amplitudes (Sun et al., 2019). The bispectrum of two variables (f₁ and f₂) is expressed as the product of the Fourier transforms (6) X(f) of a single variable (Argenti et al., 2011; Matsuoka and Ulrych, 1984): (6) B3f1,f2=Xf1Xf2X*f1+f2 {B_3}\left( {{f_1},{f_2}} \right) = X\left( {{f_1}} \right)X\left( {{f_2}} \right){X^*}\left( {{f_1} + {f_2}} \right) where:

• X*() denotes the complex conjugate, 0≤f1≤f2≤fs2,f1+f2≤fs2. 0 \le {f_1} \le {f_2} \le {{{f_s}} \over 2},{f_1} + {f_2} \le {{{f_s}} \over 2}.

• f_s – sampling frequency.

From a computational point of view, a characteristic feature of the bispectrum is the symmetry of its arguments, which means that changing their order does not affect the calculated value. (Matsuoka and Ulrych, 1984). The most important symmetries of the bispectrum are (7) (Argenti et al., 2011; Matsuoka and Ulrych, 1984; Sun et al., 2019): (7) B3f1,f2=B3f2,f1=B3−f2,−f1==B3f1,−f1−f2=B3−f1−f2,f2==B3−f1−f2,f1=B3f2,−f1−f2 \matrix{{{B_3}\left( {{f_1},{f_2}} \right) = {B_3}\left( {{f_2},{f_1}} \right) = {B_3}\left( { - {f_2}, - {f_1}} \right) = } \hfill \cr { = {B_3}\left( {{f_1}, - {f_1} - {f_2}} \right) = {B_3}\left( { - {f_1} - {f_2},{f_2}} \right) = } \hfill \cr { = {B_3}\left( { - {f_1} - {f_2},{f_1}} \right) = {B_3}\left( {{f_2}, - {f_1} - {f_2}} \right)} \hfill \cr }

Due to the above relationship, the calculations can be limited to a single region, called the principal domain (Region I) (Schmidt, 2020), shown in Figure 2.

Figure 2.

The principal domain and symmetry regions of the bispectrum.

Laboratory tests were performed on a dual-mass system which consisted of two permanent magnet synchronous motors (PMSM) (Brushless AC Servomotor 95UMB301CASAA Control Techniques Unimotor) with a rated power of 3 kW, three pairs of poles, and a rotational speed of 3000 rpm. The elastic element connecting the two motors was a long shaft of 6 mm diameter. A disc is installed in the tested motor, in which the test masses are installed with a radius of 42 mm. To minimise mechanical vibrations generated during operation, the drive system is fixed on a concrete foundation. Figure 3 shows a photograph of the laboratory setup of the tested drive system with an elastic coupling, which allows for the modelling of rotor unbalance. The mechanical structure of the test stand did not include any adjustment elements; in particular, there was no possibility of aligning its components, and no preliminary balancing procedure was performed. Consequently, even before the addition of the test masses, the system could exhibit, and indeed exhibited, characteristics typical of an unbalanced and misaligned mechanical system.

Figure 3.

Photograph of laboratory stand for unbalance research in two-mass system. PMSM, permanent magnet synchronous motors.

During the experiments, mechanical vibration acceleration is recorded using a National Instruments NI-9234 data acquisition card. Mechanical vibrations were measured with a type 4506 three-axis piezoelectric accelerometer type 4506 from Brüel & Kjaer, mounted on the motor’s mounting bracket. The reference current I_qREF and the motor angular velocity ω_motor were recorded using a NI USB-9215 data acquisition card. All signals were recorded for 10 s with a sampling frequency of 12.8 kS/s. The current I_qREF is generated by a vector control system implemented on an Analog Devices SHARC DSP 26369 signal processor. The motor is powered by a laboratory three-phase inverter.

Rotor unbalance is modelled using four test masses: 7.5 g, 15.9 g, 24 g, and 45 g. The tested drive system operated without load at a constant angular velocity of 60 rad/s. Two cases were investigated. In the first, the influence of the test mass on the operation of the single motor (single-mass system) is assessed, while in the second, the effect of the same unbalance on the operation of the two-mass system with an elastic coupling is examined. All experiments were conducted under steady-state conditions.

The motor tested operated within the control system as shown in Figure 4, which included two feedback loops: an inner current regulation loop and an outer speed regulation loop with an active disturbance rejection control (ADRC) controller (Wicher and Nowopolski, 2017). During laboratory experiments, the parameters of the speed and current control loops were not modified.

Figure 4.

Block diagram of the control system including modules for measuring current and mechanical vibration acceleration.

4.1.

Preliminary assumptions

The use of a triaxial piezoelectric accelerometer enabled simultaneous measurement of mechanical vibration acceleration in three axes: radial (X-axis), tangential (Y-axis), and axial (Z-axis). Preliminary analysis of the measured signals showed that in the radial direction, vibrations were effectively damped by the massive concrete foundation on which the drive system is mounted. In the axial direction, the introduction of a long shaft-shaped mechanical coupling adversely affected the measured signal. These disturbances significantly hindered the detection of unbalance symptoms. Only in the tangential direction (Y-axis) were unbalance symptoms visible in the mechanical vibration acceleration signal for both the single-mass motor system and the two-mass drive system.

The use of a high sampling rate for the recorded signals (12.8 kS/s) had a negative impact on the computation time of the bispectrum transform (from several dozen seconds to several minutes). Based on the conducted tests, the sampling rate of the recorded signals was reduced to 200 S/s without changing the recording duration. The applied resampling process did not affect the resolution of the bispectrum or FFT transforms but instead reduced the computation time to several dozen seconds. Moreover, the analysed frequencies were within the range up to 30 Hz, that is, below the Nyquist frequency, which after resampling was 100 Hz.

The configuration parameters of the bispectrum transform used are listed in Table 1.

Table 1.

Bispectrum parameters	Value/method
Bispectrum estimator	Direct FFT (Direct FFT-based averaged segment method)
Segment length	2000
Overlap	50%
Window type	Hamming
Averaging/smoothing strategy	Ensemble averaging
Scaling/normalisation	Raw bispectrum

The most significant differences occurred in the signals measured for the single-mass and two-mass systems. The introduction of an elastic coupling increased the background level of the measured signals and contributed to higher amplitudes of the analysed frequencies. Figure 5 presents three-dimensional bispectra of vibration acceleration measured along the Y-axis for the single-mass system (Figure 5a) and the two-mass system (Figure 5b). The presented bispectra correspond to an electric drive with an installed test mass of 45 g. To facilitate comparison of both cases, identical axis ranges were used in the figures. Due to the range of the analysed symptoms (up to (3f_r, 3f_r) ≈ (29 Hz, 29 Hz)), the bispectra shown in Figures 5 and 6 were limited to 45 Hz.

Figure 5.

Three-dimensional bispectra of mechanical vibration acceleration signal measured along the Y-axis of the drive with an installed test mass of 45 g: (a) single-mass system, (b) two-mass system.

A similar comparison was performed for the I_qREF current. Figure 6a presents the three-dimensional bispectrum of the current measured for the single-mass system with an installed test mass of 45 g. The presented bispectrum is free from disturbances that are visible in the two-mass system (Figure 6b). To facilitate the interpretation of the three-dimensional plots, identical ranges were set for each axis.

Figure 6.

Three-dimensional bispectra of the I_qREF current signal of the drive with an installed test mass of 45 g: (a) single-mass system, (b) two-mass system.

Despite the occurring disturbances, the symptoms indicating the presence of rotor unbalance are visible in the presented figures. To assess the usefulness of the individual symptoms for detecting rotor unbalance, they were subjected to detailed analysis, which is presented in Chapter 5.

Unbalance of rotating components adversely affects the entire drive system. Unfortunately, there are no instruments that allow direct measurement of imbalance during operation, making it necessary to measure the secondary effects of mechanical asymmetry.

This article introduced a method for identifying rotor unbalance in a two-mass system connected by an elastic coupling. The proposed approach combines FFT analysis with bispectral examination of mechanical vibration acceleration and the I_qREF current signal from the control system. It was shown that using a higher-order transform, such as bispectrum analysis, increases the number of symptoms characteristic of unbalance. A greater number of symptoms should positively impact the effectiveness of unbalance detection.

Bispectrum analysis provides a 2–-3 times higher increase in the amplitude of characteristic frequencies compared to an equivalent comparison based on FFT of mechanical vibrations. For the I_qREF current, changes in the amplitudes of the analysed symptoms are at a similar level. Using the I_qREF current to detect drive system imbalance does not require additional external measurement sensors and can be implemented directly in a signal processor.

A disadvantage of bispectrum analysis compared to classical FFT analysis is its computational complexity, which results in longer processing times. This issue can be mitigated by reducing the sampling rate of the analysed signal, as demonstrated in the article.

The proposed diagnostic methods allow detection of rotor unbalance in a single-mass system caused by a small test mass of 7.5 g, which corresponds to approximately 0.22% of the rotor mass. For the two-mass system, using mechanical vibration signals enables detection of unbalance caused by a similar test mass, corresponding to about 0.085% of the rotating elements’ mass. Analysis of the I_qREF current allows detection of unbalance caused by a 24 g test mass, or approximately 0.27% of the rotating elements’ mass in the two-mass system. These values were obtained at an angular velocity of 60 rad/s, corresponding to about 19% of the nominal speed.

The proposed signal processing methods for vibration acceleration and I_qREF current allow effective detection of rotor unbalance in both single- and two-mass systems at an early stage. The detection process does not require high sampling frequencies; for a system rotating at 60 rad/s, a sampling rate of 200 S/s provides sufficient accuracy.

To improve the effectiveness of unbalance detection in a two-mass system with a flexible coupling over a wide range of angular speeds and for small unbalance-inducing masses (approximately 0.085% of the mass of the rotating components), simultaneous analysis of the I_qREF signal and mechanical vibrations is recommended.