Effects of Negative Sequence Voltage Subharmonics on Cage Induction Motors

Abstract

In some power systems, voltage waveforms contain, apart from harmonics, interharmonics and subharmonics that are components of frequency less than or not an integer multiple of the fundamental frequency. Voltage subharmonics and interharmonics may be of both a positive and negative sequence, independently of their frequency. Previous papers on induction motors under subharmonics have been generally limited to the components of a positive sequence. This study deals with the effect of negative sequence subharmonics on the work of induction motors. Investigations were performed using the 2D finite element method and an experimental method. Differences between the impact of positive and negative sequence subharmonics are discussed. It was found that negative sequence voltage subharmonics can result in significant current subharmonics, torque pulsations and vibration. Further, because of possible resonance, motors that are comparatively resistant to positive sequence subharmonics might be especially sensitive to negative sequence subharmonics of the same frequency and vice versa.

1. Introduction

Voltage waveforms in public and industrial power networks commonly contain various undesirable components—usually voltage harmonics. Apart from harmonics, they could also contain subharmonics and interharmonics, which are components of frequency that less than or not an integer multiple of the fundamental frequency. Subharmonics (subsynchronous interharmonics) are often considered as a special case of interharmonics. However, in some studies (for instance [1,2,3,4]), they are regarded separately as they exert a specific impact on electrical equipment [5].

Voltage subharmonics (VShs) are generated by renewable sources of energy such as wind power stations [5,6,7,8] and photovoltaic plants [9,10,11], by double frequency systems such as high voltage DC links and inverters [3,12,13,14,15,16], and also by cycloconverters [3,15], arc furnaces [3,15,17] and pulsating loads [3,15,16,18,19]. Voltage subharmonics can also occur in the voltage produced by inverters [4,20,21]. Notably, cyclic voltage fluctuations are considered as a superposition of the fundamental, subharmonic and interharmonic components [5,22,23,24]. It is also worth mentioning that in some systems, VShs reach levels of about 1% of the fundamental component, for example in [8,13,25].

VShs exert an extraordinarily negative effect on various types of electrical equipment. They disrupt the work of light sources, power and measurement transformers, power electronics and control equipment as well as rotating machinery [1,2,3,4,20,21,22,23,24,26,27,28,29,30,31,32,33,34,35]. In induction motors, VShs cause a local saturation of magnetic circuits [30], speed fluctuations [4,23,24,28,29,30,31], increases in power losses and overheating [3,20,26,27,28,31,33,34,35], and torque pulsations [21,23,27,28,29,31], which may result in vibration of an extremely high level [2]. To make the matter worse, the frequency of torque pulsations may correspond to the natural frequency of the first elastic-mode of the rotating mass [21,31], leading to resonance phenomena. Elastic-mode resonance [21,36] causes, among others, a magnification of torque pulsations, possibly even by a factor exceeding 100 [21]. Consequently, apparently inconsiderable torque pulsations may, after their amplification, result in destruction of a power train [21]. Ref. [21] reported fluctuations of the output voltage of an inverter, causing small torque pulsations (1.5% peak-to-peak). Under resonance the torque pulsations were magnified by the factor of 110, resulting in a coupling failure.

Although VShs are regarded as extraordinarily harmful, their permissible levels have not been introduced into power quality standards. The main reason is the lack of sufficient data to determine the permissible levels. For example, the standard EN 50160 Voltage characteristics of electricity supplied by public distribution systems [37] contains the following comment (concerning both subsynchronous and supersynchronous interharmonics): “The level of interharmonics is increasing due to development of the application of frequency converters and similar control equipment. Levels are under consideration, pending more experience”. Similarly, in an informative annex of the standard IEEE-519: IEEE Recommended practices and requirements for harmonic control in electric power systems [38], the limits are provided together with the following comment: “It is important to recognize that the suggested voltage interharmonic limits are based on lamp flicker assessed using the measurement technique (…). These voltage interharmonic limits correlate with a short-term flicker severity P_st value equal to 1.0 (…). The recommended limits (…) are not based on the effects of interharmonics on other equipment and systems such as generator, mechanical systems, motors, transformers, signaling and communication systems, and filters. Due consideration should be given to these effects…”.

Summarizing the above considerations, the introduction of permissible levels of VShs into the power quality standards requires in-depth investigations of the effect of VShs on various types of electrical equipment. Among others, harmful phenomena occurring in induction motors should be fully explained.

Induction motors under VShs have been the subject of numerous studies [2,3,4,20,21,23,24,25,26,27,28,29,30,31,33,34,35]. However, the previous works were generally devoted to two cases of subharmonic occurrence. The first one concerns supplying a voltage containing a single subharmonic component of a positive sequence [3,24,31,32,33,34]. The other deals with sinusoidal voltage fluctuations in frequency that are less than the frequency of the fundamental voltage component [2,23,24,28,29,30]. It should be noted that such a modulation results in the presence of positive sequence subharmonics and interharmonics (based on [16]). Some results of research on the impact of negative sequence VShs on an induction motor were provided in [3,24,26,35]. Among them [24], summarized results of the investigations on speed fluctuations and flicker attenuation, and [3,26,35] dealt with excessive heating and thermal aging of the insulation system.

In summary, the results of the investigations on torque pulsations and vibration have not been presented in the previous works on negative sequence VShs. Additionally, the previous studies generally did not deal with the effect of the moment of load inertia and the frequency of VShs on the current subharmonics. The main contribution of this paper is the assessment of torque pulsations, vibration and currents of the induction motor under negative sequence VShs. The assessment should be useful for determining the permissible subharmonic levels and introducing them into the power quality standards.

2. Sequence of Voltage Subharmonics

The sequence of voltage harmonics is strictly connected with their order (frequency); for instance, the fifth voltage harmonic is of a negative sequence, while the seventh harmonic is of a positive sequence. In contrast, the sequence of VShs (voltage subharmonics) does not depend on their frequency but on their origin. In practice, at any frequency, VShs can be both of a positive and negative sequence [16].

A significant source of VShs are double conversion systems [15,16]. They contain an AC/DC rectifier and a DC/AC inverter, connected through a DC link, which is in practice, a capacitor or a reactor, whose task is to suppress ripples. If the capacitor or reactor were of infinite value, the double conversion system would generate only voltage harmonics of frequencies f_h [15,16]:

$$f_h=(p_{1}n\mp 1)f_1$$

(1)

where p₁ is the pulse number of the rectifier, f₁ is the line frequency and n is an integer.

In fact, as the capacitor or reactor are of the finite value, ripples occur in the DC link [15]. For a current source inverter their frequency can be expressed as [15]:

$$f_r=n{p}_2{f}_o$$

(2)

where p₂ is the pulse number of the inverter and f_o is the inverter output frequency.

Further, for voltage source inverters, frequencies f_r generally depend on the PWM modulation strategy [14,15].

The ripples in the DC link cause a modulation (with frequency f_r) of both the fundamental and higher current harmonics supplying double conversion systems [15,16]. At the same time, if the sine of frequency f is modulated with frequency f_m, two additional components of frequencies f + f_m and f − f_m occur in the waveform (based on [5,22,23]). Consequently, the input current of a double conversion system contains subharmonics and interharmonics of frequencies (f_sh, f_ih) equal to (based on [15,16]):

$$f_{sh/ih}=f_1\mp f_r,$$

(3a)

$$f_{sh/ih}=f_h\mp f_r{,}^{ }$$

(3b)

If the frequency described by (3a) and (3b) is of a positive value, the sequence of subharmonics and interharmonics is the same as that of the current harmonic being modulated (based on [16]). Further, if (3a) or (3b) indicate negative frequency, the sequence of a subharmonic/interharmonic changes from positive to negative or vice versa (based on [16]). For example, the interharmonic of frequency f₁ + f_r is always of a positive sequence (for the full current balance), whereas the component of frequency f₁ − f_r could be of either of a positive or negative sequence, depending on the sign of the above expression.

Other sources of subharmonic and interharmonic contamination are asynchronous and synchronous motors driving loads of pulsating anti-torque, for example reciprocating compressors [18,19]. The motor input current contains, among others, components of frequencies (based on [16,18]):

$$f_{ih/sh}=f_1\mp f_a$$

(4)

where f_a is the frequency of anti-torque pulsations.

For frequency f_a less than the line frequency f₁, the generated subharmonics and interharmonics are only of a positive sequence, otherwise of both negative and positive (based on [12,16]). Notably, the frequency of voltage fluctuations due to electromechanical loads is often less than the network frequency f₁ [39]; consequently, the above components are expected to be of a positive sequence. It is worth adding that in the case of an inverter-driven motor, fluctuations of the anti-torque cause ripples in the DC link of frequency f_r = f_a, and consequently, the supply current contains current subharmonics and interharmonics of the frequencies described by (3a) and (3b) (based on [12]).

Apart from of AC motors, subharmonics and interharmonics originate from the work of other fluctuating loads. Some of them can be roughly modeled with time-varying resistance (based on [16]):

$$R\left(t\right)=R_0\left(1-rsin\left(2\pi f_{z}t\right)\right)$$

(5)

where f_z is the load varying frequency and r is the coefficient less than unity.

The frequency of the generated subharmonics and interharmonics can be expressed as (based on [16]):

$$f_{sh/ih}=f_1\mp n{f}_z$$

(6)

If the term f₁ ± nf_z is greater than zero, the subharmonics and interharmonics components are of a positive sequence, otherwise of a negative sequence (based on [16]). Notably, some loads containing power electronic devices may fluctuate with a frequency f_z significantly greater than the line frequency, even exceeding 150 Hz [39].

In summary, various loads in a power system may generate subharmonics of both a positive and negative sequence.

3. Methodology

For the investigations, two methods were employed: the finite element method (FEM) and an experimental method. FEM calculations on currents and torque pulsations were performed for two cage induction motor types: SLgm 315 ML2B (200 kW) and TSg100L-4B (3 kW). These were denoted as motor 1 and motor 2, respectively. The chosen parameters of the motors under investigations are presented in

Table 1. Chosen parameters of the investigated motors.

Motor	Type	Rated Power (kW)	Rated Speed (rpm)	Rated Voltage (V)	Rated Current (A)
motor 1	SLgm 315 ML2B	200	2982	400 Δ	346
motor 2	TSg100L-4B	3	1420	380 Δ	6.9
motor 3	3SIE100L4B	3	1465	400 Y	6.3

. For the two-dimensional FEM analysis, the ANSYS Electronics Desktop environment (ANSYS Electromagnetics Suite 18.0.0—for motor 1) and the ANSYS Electronics Desktop environment 2022R1 (for motor 2) were used. The applied tau-type meshes consisted of 8082 and 22,094 triangle elements for motor 1 and motor 2, respectively. The parameters of the motor models were identified on the basis of the construction data and the results of measurements [31,33]. It is worth mentioning that one of the co-authors is an employee of the manufacturer of motor 1 and took part in its design process. Field computations were carried out with a transient-type solver. More details concerning the applied models are presented in the authors’ previous works [31,33,34].

The experimental setup consisted of a multi-machine system for generation of subharmonics, power quality analyzers and the two investigated motor types (3SIE100L4B, 3 kW, see Table 1), which were equipped with a system of vibration measurement. One of the motors was uncoupled and embedded on a rigid, steel frame and the other was coupled with an unloaded DC generator. In this study, they are referred to as motor 3 Case A and motor 3 Case B, respectively.

The tested motors were supplied with a multi-machine system composed of two synchronous generators coupled via a transformer (based on [40]). One of the alternators produced the fundamental voltage harmonic, and the other injected the subharmonic component. The parameters of the generators are specified in [2].

For the vibration measurement and the analysis, the Bruel&Kjear (B&K) system was used, which consists of a standalone four-channel data acquisition module (B&K type 3676-B-040), an accelerometer calibrator (B&K type 4294), a computer with the installed BK Connect software and a three-axis accelerometer (B&K type 4529-B) mounted on an additional steel stand [2]. It is worth adding that the accelerometer was calibrated before each experiment and that the vibration measurements were performed in accordance with the chief provisions of the standard [41]. Further, the voltage and current frequency components were measured with a computer-based power quality analyzer and a power quality estimator-analyzer [42] designed at Gdynia Maritime University for commercial purposes and certified by the Polish Register of Shipping. A simplified diagram of the measurement system is presented in Figure 1.

The next graph (Figure 2) presents a comparison of the measured and computed current subharmonics for motor 2 versus the frequency of VShs. The experimental tests and the FEM computations were performed for VShs with a value of 1%, and the uncoupled motor and the value of the fundamental voltage component corresponding to the nominal flux. The comparison proves that the applied numerical model is of sufficient accuracy for the paper’s purpose. An additional model validation is provided in [33,34] for motor 2 and positive sequence VShs (motor 1 was not experimentally tested because of its high rated power).

4. Results

4.1. Preliminary Remarks

All the numerical computations and experimental tests were carried out for VShs (voltage subharmonics) values that were 1% of the fundamental voltage component and the rated load torque. The results of the FEM analyses chosen to be presented in this study correspond to the work with a load of the negligible moment of inertia (NMI). Justification for this case is included in the authors’ previous works [31,32]. Further, the current and torque components are presented in relation to the rated current (I_rat) or the rated torque (T_rat).

4.2. Effect of Negative Sequence Subharmonics on Currents

Noxious phenomena occurring in induction motors under VShs, such as vibration, torsional vibration, an increase in power losses, overheating and speed fluctuations [2,3,4,20,21,23,24,26,27,28,29,30,31,33,34,35] are generally interconnected with current subharmonics and interharmonics flowing through windings.

Below, Figure 3 and Figure 4 present a comparison of current spectra for motor 1 supplied with the voltage containing VShs of frequency f_sh = 43 Hz and different sequences. For the injection of a positive sequence voltage subharmonic (Figure 3), the current contains, among others, a subharmonic with a value equal to 12.6% of the rated current, accompanied by the current interharmonic with a value that is 10.6% of I_rat and frequency 57 Hz. The presence of current interharmonic results from the speed fluctuations and resonance phenomena [24,31,33]. Further, for the injection of a negative sequence voltage subharmonic (Figure 4), the current subharmonic is equal 8.3% of I_rat and the 57-Hz current interharmonic, which is merely 0.9% of I_rat. It is worth adding that the presence of the third harmonic in the phase stator winding current is typical for delta-connected motors.

The difference between the effects of a positive and negative sequence VShs results mostly from the frequency of torque pulsations (f_p). For positive sequence VShs, the magnetic fields induced by the fundamental and subharmonic components rotate in the same direction, and the frequency of the torque pulsations is (based on [24]):

$$f_p=f_1-f_{sh}$$

(7)

For negative sequence VShs they rotate in the opposite direction and the frequency of torque pulsations is equal:

$$f_p=f_1+f_{sh}$$

(8)

In practice, for positive sequence VShs, the frequency f_p is within the range 0–f₁, while for negative sequence subharmonics, it is within the range f₁–2f₁. At the same time, for the frequency of torque pulsations f_p close to the natural frequency of the rigid-body mode (f_Nr-b) [18], resonance may occur [2,18,31,33]). The frequency f_Nr-b is in the single-tens-of-Hz range for small motors, while for powerful ones, it is a few Hz (based on the authors’ experience [2,18,31,32,33]). Under the rigid-body resonance, the speed fluctuations boost current subharmonics and interharmonics, which results in a further magnification of torque pulsations and speed fluctuations. Consequently, current subharmonics and interharmonics may reach extremely high values [2,33]. Such resonance phenomena were reported for both asynchronous and synchronous motors under positive sequence VShs [2,18,23,31,33], whereas for negative sequence VShs, they are not expected. Another difference between positive and negative sequence VShs concerns the slip. For positive sequence VShs, it is less than zero, and for negative sequence VShs, it is always greater than two (for perfect idling).

In Figure 5, the current subharmonics versus the frequency of negative sequence VShs are presented. For motor 1, the current subharmonics are as high as 32.9% of I_rat for a frequency of f_sh = 10 Hz, whereas for greater frequencies f_sh they are much lower. The shape of this characteristic can be explained by the fact that low-frequency current subharmonics are suppressed mostly by windings resistance, which takes a comparatively low value for medium- and high-power induction motors (based on [31]). Further, for motor 2, the current subharmonics are significantly lower and change approximately linearly from 11.5% of I_rat for f_sh = 10 Hz to 7.0% of I_rat for f_sh = 45 Hz. For comparison, for positive sequence VShs and the work with the high moment of load inertia, the current subharmonic is about 30% higher for f_sh = 10 Hz and much lower for f_sh = 45 Hz.

It should be added that the authors also carried out numerical investigations for various moments of load inertia as well for the work with a constant rotational speed, which corresponds to the work with a load of the high moment of inertia [31]. The results of the calculations show that for negative sequence VShs, the moment of load inertia has a negligible impact on current subharmonics and interharmonics. On the contrary, their values significantly depend on the moment of load inertia for positive sequence VShs [31,33].

In summary, for negative sequence VShs, current subharmonics practically do not depend on the moment of load inertia, and in some cases, are significantly less than for positive sequence VShs. The differences result mostly from other frequencies of torque pulsations and the absence of the rigid-body mode resonance for negative sequence VShs.

4.3. Torque Pulsations under Negative Sequence Subharmonics

Torque pulsations induce vibrations [43,44], which may reach an extraordinary level under VShs [2]. To make the matter worse, under the elastic-mode resonance, the amplitude of torque pulsations could be significantly magnified, which may lead to a mechanical failure of a power train [21,31].

Below, in Figure 6, the pulsating torque component (ΔT_p) caused by VShs is presented. In practice, the characteristics shown in Figure 6 correspond to the torque component of the frequency described with Equation (8). For motor 1 and the frequency f_sh = 10 Hz, the component ΔT_p is 41% of T_rat. With the increasing frequency f_sh, the pulsating torque component decreases nonlinearly to about 10% of T_rat for f_sh = 45 Hz. Further, for motor 2, the component ΔT_p (Figure 6) is much less affected by f_sh than in the case of motor 1 and takes significantly lower values (between 8.6% and 15.7% of T_rat).

As mentioned in the previous subsection, the resonance of the rigid-body mode was not observed under negative sequence VShs. At the same time, resonance phenomena reported under positive sequence VShs [23,31,33] may significantly increase the value of the torque component ΔT_p. For example, for motor 2 under the resonance, the pulsating torque component was about 60% of T_rat.

It is also worth adding that the FEM analyses on torque pulsations were carried out for a constant rotational speed. The differences between the torque pulsations for the constant rotational speed and a load of the negligible moment of inertia are inconsiderable, similar to the case of currents (see the previous subsection).

In summary, the torque pulsations caused by negative sequence VShs may take significant values, up to about 40% of T_rat in the case of the investigated motors. The main difference between the effect of negative and positive sequence VShs is the lack of resonance phenomena (of the rigid-body mode) for negative sequence VShs.

4.4. Vibration under Negative Sequence Subharmonics

The results of the empirical investigations on vibration are presented below. As the highest vibration under VShs occurs for an idling motor (for example [2,32]), the experimental research concerned no-load. The appropriate tests were performed for motor 3, Case A (an uncoupled motor) and Case B (a motor coupled with a DC generator)—see Section 3. All the presented measurement points correspond to the averaged value vibration velocity obtained in three independent tests. The averaging was performed directly in the BK Connect software. Further, the assessment of vibration severity was carried out in accordance with the standard [45] because the threshold values of evaluation zones [41,45] are not univocally specified in its current version [41].

The measured broad-band vibration velocity [41,45] is shown in Figure 7 and Figure 8 for Case A and Case B, respectively. For Case A, the maximal velocity is 4.46 mm/s for frequency f_sh = 16 Hz and the vertical direction. For Case B (Figure 8), the maximal vibration velocity is 4.23 mm/s for frequency f_sh = 20 Hz and the longitudinal direction. Notably, for electrical machines under power quality disturbances, the most severe vibration is often observed in the transverse direction [2,43,44], but it also may occur in the longitudinal direction [32,43].

For both Case A and Case B, the measured vibration velocity generally falls into the evaluation Zone C [41,45], which corresponds to the vibration velocity within the range 1.8–4.5 mm/s for small electric motors [45]. It should be stressed that vibration within Zone C is considered unacceptable for long-term continuous operation [41,45]. Additionally, for Case A (Figure 7), the maximal vibration velocity (4.46 mm/s) is close to the threshold value of the evaluation Zone D (4.5 mm/s) [45]. According to the standards [41,45], the vibration within Zone D is “normally considered to be of sufficient severity to cause damage to the machine”.

Although the measured vibration velocity should be considered excessive, it was significantly less than for the same motor under positive sequence VShs [2]. For a positive sequence and Case A, the vibration velocity was as high as 7.38 mm/s and for Case B, it was 6.95 mm/s. The details concerning the investigations under positive sequence VShs are presented in the authors’ previous study [2]. The difference between the effect of positive and negative sequence VShs can be explained by two reasons. The first is the lack of the rigid-body resonance for negative sequence subharmonics, which can significantly amplify torque pulsations (see the previous subsections). The other reason could be a different response of the mechanical structure to torque pulsations of various frequencies [43,44]. As mentioned above, for positive and negative sequence VShs, torque pulsations are of different frequencies.

It is also worth adding that for Case A and positive sequence VShs, the maxima of the current subharmonic and vibration velocity occur for the same frequency corresponding to the rigid-body resonance [2]. In contrast, for negative sequence VShs, the characteristics of vibration velocity (Figure 7) and current subharmonics (Figure 9) differ significantly.

In summary, negative sequence VShs may cause excessive vibration. However, in the case of the investigated motor, the vibration levels are significantly lower than under a positive sequence VShs.

5. Discussion

The previous works [2,3,4,20,21,23,24,26,27,28,29,30,31,33,34,35] show the extraordinarily harmful impact of VShs (voltage subharmonics) on induction motors. These analyses were generally limited to positive sequence VShs; however, in power network subharmonics, both sequences may occur. In this study the effect of negative sequence VShs on induction motors was examined.

The negative sequence VShs may cause a flow of current subharmonics of significant values, high torque pulsations and excessive vibration, similar to the effects of positive sequence VShs. Nevertheless, there are some differences between the effects of positive and negative sequence VShs on induction motors. As discussed in the previous section, the differences generally result from the frequency of torque pulsations f_p. For positive sequence subharmonics, the frequency f_p is between null and the line frequency, whereas for the negative sequence, it is between the line and twice the line frequency. For negative sequence VShs, the frequency f_p is usually considerably greater than the natural frequency f_Nr-b (see Section 3), and the resonance of the rigid-body mode does not occur. Consequently, torque pulsations, the vibration velocity and an increase in windings temperature [33,35] might be remarkably less than for the same motors under positive sequence VShs. Further, for positive sequence VShs, current subharmonics and torque pulsations depend significantly on the moment of load inertia [23,31,33], whereas for negative sequence they are practically independent.

Apart from of the rigid-body mode resonance of the rotating mass, other kinds of resonances might also occur, leading to excessive vibration or torsional vibration. It should be stressed that “excessive motor electromagnetic vibration is very often a result of a resonance condition on the structure of an entire unit or on the motor components, such as a stator core or frame” [43]. Further, as mentioned in Section 1, the elastic-mode resonance of the rotating mass is particularly dangerous for a power train structure [21,36]. Resonance phenomena could magnify the amplitude of torque pulsations, even by a factor exceeding 100 [21], leading to a shaft or coupling destruction [21,31]. For some power trains, the natural frequency of the first elastic mode is below the line frequency (especially for high power motors), and for others, it is above (based on [21,31,36] and the authors’ calculations). The former might be exposed to the resonance under positive sequence VShs, and the latter under negative sequence VShs. As a result, a power train resistant to positive sequence VShs could be particularly susceptible to negative sequence VSs of the same frequency and vice versa.

In summary, for the investigated motors, negative sequence VShs caused significantly lower torque pulsations and vibration than positive sequence VShs. On the other hand, the appearance of negative sequence VShs carries the additional risk of damage to the power train because of possible resonance phenomena.

6. Conclusions

In a power network, both a positive and negative sequence voltage subharmonic (VShs) could be expected. The sequence of VShs is not connected with their frequency, as in the case of harmonics, but with their origin [16]. The results of investigations presented here show that negative sequence VShs may cause a flow of significant current subharmonics, high torque pulsations and vibration at an unacceptable level. Notably, for the investigated motors, the intensity of these undesirable phenomena was usually considerably less than for positive sequence VShs. For negative sequence VShs, the rigid-body mode resonance of the rotating mass is not expected, current subharmonics and torque pulsations are practically independent of the moment of load inertia, contrary to positive sequence VShs [31]. The differences between positive and negative sequence VShs result chiefly from another frequency of torque pulsations. For a positive sequence, it is between null and the line frequency, and for a negative sequence, it is between the line frequency and the double line frequency. Due to another frequency of torque pulsations, negative sequence subharmonics might cause various resonance phenomena in motors that are comparatively resistant to positive sequence VShs. In particular, the occurrence of the elastic-mode resonance of the rotating mass [21,31,36] can result in the destruction of a power train.