DE102019114930B3 - Method and arrangement for monitoring systems - Google Patents

Abstract

Method for monitoring a system (1) with a sensor (2) for generating vibration signals, a frequency spectrum being calculated from the vibration signals with the aid of a Fourier transformation, and from this frequency spectrum a modified frequency spectrum is calculated which is for frequencies of a defined cut-off frequency has a lower frequency resolution than the frequency spectrum of the Fourier transform.

Description

The invention relates to a method for monitoring systems based on vibrations that occur. Technical systems usually include machines with rotating components, such as shafts, rotors, rolling elements or gear wheels. Defects in these components lead to characteristic frequency excitations. One example is a rolling element in a rolling bearing that repeatedly rolls over a damaged bearing point. Another example is the imbalance in a generator.

The characteristic frequency excitations can often be derived from basic frequencies, such as the speed of a shaft or the electrical mains frequency. If the construction of the system or the machine is known, then a list can be drawn up with frequencies which belong to certain potential defects.

A large number of sensors come into consideration for monitoring vibrations, such as displacement sensors, speed sensors, acceleration sensors, pressure sensors or sound sensors.

The WO 2009/021900 A1 discloses, for example, a method for monitoring a wind turbine. An accelerometer for monitoring the vibrations is attached to the impeller hub. To monitor the characteristic frequencies of the individual impeller blades, the signal detected by the sensor is subjected to a Fourier transformation (FFT).

WO 02/084 234 A1 discloses a method and a device for analyzing and synthesizing signal frequencies.

The DE 2 219 085 A discloses a frequency analyzer for examining complex signals which are composed of harmonic frequency components.

Based on the design of a system or machine (e.g. the number and size of the rolling elements, the number of vanes of a pump wheel, the number of teeth of the gear wheels ...) and the respective speed of the components, one can draw conclusions about possible damage frequencies. If the monitoring system detects unusual vibration amplitudes at certain frequencies, then these frequencies can be compared with a list of possible damage frequencies in order to identify possible causes or to diagnose potential damage as early as possible.

However, so-called harmonics are often excited in such cases, i.e. Multiples of the actual damage frequency. The reason for this is to be found in the fact that structures of the monitored system match the frequencies of the harmonics, which leads to resonance effects. Harmonics can therefore dominate the measurement signal.

For this reason, surveillance systems often record relatively high frequencies compared to the base frequencies of the monitored system. For example, hydropower plants often have a low speed of less than 10 Hz. However, due to the design and type of the components and taking into account possible harmonics, frequencies up to several thousand hearts are important if diagnoses are to be derived from measured frequencies.

The computer system that processes this data must sample the vibration signals at a much higher rate. For example, for theoretical reasons, a sampling rate of at least 2000 Hz is required to detect an oscillation at 1000 Hz. In practice, an even higher sampling rate is required to avoid disturbing sampling effects (anti-aliasing).

Therefore, many monitoring systems work for large technical systems with sampling rates greater than 10 kHz. This corresponds to a data accumulation of more than 10,000 measured values per second. To determine the amplitudes of frequencies, a Fourier transformation (FT) is carried out on this database, which converts the time signals into a frequency spectrum. A so-called FFT (Fast Fourier Transform) is usually used for this.

The lowest frequencies of interest define the length of the time window that forms the basis for an FT. If the lowest frequency is f = 10Hz and you want to prove this frequency, then it is advisable to choose the time window at least a multiple of 1 / f. Eg a second. At a Second measurement duration T, the FT achieves a frequency accuracy of 1 / T, ie 1 Hz. This corresponds to an accuracy of only 10% for the diagnosis of the frequency at 10 Hz. Increasing T would increase accuracy, but at the same time increase the amount of data. For example: If the measurement window is increased to 10s, the accuracy of the frequency detection increases by a factor 10 , but also 10 times more data has to be processed to perform and evaluate an FT: in our example 10 * 10000 values per second.

It is clear from what has been said that the amount of data generated by the measurement and preparation steps described determines the effort for the subsequent diagnostic steps, which is reflected in the high demands placed on the performance of the computer systems used. Since in modern monitoring systems the diagnostic steps are often not carried out on site, but instead take place in the cloud, there are additional high demands on the bandwidth of the data transmission devices, which are increasingly working wirelessly.

The inventor has set himself the task of specifying a method for monitoring systems in which the frequency spectra to be evaluated have a significantly smaller data volume than in the known methods without impairing the diagnostic quality of the method.

The task is solved by a method for monitoring plants with the features of the independent claim. Advantageous embodiments result from the dependent claims dependent thereon.

The inventor was guided by the observation that the frequency excitations from rotating components have only a limited frequency stability. For example, with a synchronous motor, the speed depends on the mains frequency, which has a fluctuation range of 0.2%. Since the FT averages within the time window, speed fluctuations within this time window mean that especially high frequencies can no longer be precisely observed. This is because the relative fluctuation range of the excitation frequency leads to the same relative fluctuation range of the excited frequencies. In the example of the synchronous motor, an excited frequency of 500 Hz fluctuates by ± 1 Hz (500 Hz * 0.2%), while an excited frequency of 2000 Hz fluctuates by ± 4 Hz. With an FT, however, the absolute frequency resolution df = 1 / T is constant over the entire frequency range. With a measurement window of 1s, for example, the absolute frequency resolution is df the FT is therefore 1 Hz. This means that this frequency resolution of 1 Hz fits the fluctuation range of the excited frequency of 500 Hz quite well, while this frequency resolution is higher than required for an excited frequency of 2000 Hz, since this excited frequency fluctuates by ± 4 Hz. With the excited frequency of 2000 Hz, a frequency resolution of 4 Hz would be sufficient. This means that the FT produces 4 times as many frequency lines in this frequency range as would be required for further diagnosis. The inventor therefore proposes that after the FT the resulting frequency spectrum is modified in such a way that the number of data points in the frequency spectrum is reduced for higher frequencies. This results in a significantly reduced computing effort for all subsequent diagnostic steps. In addition, the bandwidth required for any subsequent data transmission is reduced.

• 1 Diagramme zur Fourier-Transformation und der erfindungsgemäßen Abwandlung derselben;
• 2 Diagramme mit einen konventionellen Frequenzspektrum nach Fourier-Transformation und das daraus berechnete erfindungsgemäße modifizierte Spektrum;
• 3 Ablaufdiagramm des erfindungsgemäßen Verfahrens;
• 4 Anordnung zur Ausführung des erfindungsgemäßen Verfahrens und Datenträger.

The solution according to the invention is explained below with reference to figures. The following is shown in detail:

• 1 Diagrams of the Fourier transform and the modification thereof according to the invention;
• 2nd Diagrams with a conventional frequency spectrum after Fourier transformation and the modified spectrum according to the invention calculated therefrom;
• 3rd Flow chart of the method according to the invention;
• 4th Arrangement for executing the method and data carrier according to the invention.

• df_j ^mod = df₀, wenn f_j ^mod < f_g ist, und für f_j ^mod ≥ f_g ist df_j ^mod eine monoton wachsende Funktion, d.h. es ist df_j+1 ^mod ≥ df_j ^mod, wobei alle df_j ^mod größer oder gleich dfo sind. Es ist klar, dass sich dabei die Frequenzlinien f_j ^mod des modifizierten Spektrums über das Frequenzband der FT erstrecken, wobei es unerheblich ist, wenn ganz am oberen Ende, d.h. nahe bei der maximalen Frequenz fmax der FT ein kleines Frequenzband der FT im modifizierten Spektrum nicht abgedeckt wird, d.h wenn f_max ^mod < f_max ist.

With the FT, the discrete temporal course of a measured variable is transferred to the frequency domain. The length of the time window under consideration is given by To. The constant time interval between the individual measuring points is determined by the sampling rate R, the number of measuring points recorded in the time window To being given by N = T ₀ * R. After the transfer in the frequency space results in a frequency spectrum, which is composed of N number of pairs (f _i, a _i) (i = 1 ... N), where a _i f, the amplitude at frequency _i indicates. The frequencies f _i are equidistant, where f _{i + 1} = f _i + df ₀ and dfo = 1 / T ₀ . 1 shows this in the top diagram. The modification proposed by the inventor consists in that a modified frequency spectrum is calculated from the frequency spectrum of the FT just described, which at Frequencies above a cut-off frequency f _g contains fewer data points than the frequency spectrum of the FT. The distance between two successive modified frequency points f _i ^mod and f _{j + 1} ^mod is given by df _j ^mod . According to the proposal according to the invention:

• df _j ^mod = df ₀ if f _j ^mod <f _g , and for f _j ^mod ≥ f _g , df _j ^{mod is} a monotonically increasing function, ie it is df _{j + 1} ^mod ≥ df _j ^mod , with all df _j ^{mod are} greater than or equal to dfo. It is clear that the frequency lines f _j ^{mod of} the modified spectrum extend over the frequency band of the FT, it being irrelevant if at the very top, ie close to the maximum frequency fmax of the FT, a small frequency band of the FT in the modified spectrum is not covered, ie if f _max ^mod <f _max .

The middle diagram of the 1 shows this fact by the solid line. The dashed line shows the constant course of df with the conventional FT. How exactly the monotonous increase in df runs above the cut-off frequency is initially not important for achieving data reduction. However, it is clear that an excessive reduction can lead to a loss of quality. We will deal with this further below. In the middle diagram of 1 the increase is linear, although other monotonically increasing curves are also suitable. To achieve a data reduction must f _g apparently less than the maximum frequency f _max = (N-1) * df ₀ . It is clear that the following applies to the modified frequency lines f _j ^mod : f _j ^mod f fmax, although, as mentioned, it can happen that f _max ^mod <f _max .

The size 1 / df as a function of frequency can be used to visualize the resulting reduction in the amount of data f because the area under the 1 / df curve is a measure of the number of data points. In the lower diagram the 1 This fact is shown, the solid line representing the course of 1 / df according to the modification proposed by the inventor, and the dashed line representing the constant course of the conventional FT. The area under the solid line is significantly smaller than the area under the dashed line. The modification proposed by the inventor thus leads to a reduction in the amount of data which corresponds to the area between the two lines. In the example shown, the reduction is approximately 1/3, ie Nmod is approximately 2/3 of N. The index j of the modified frequency values then naturally runs from 1 to N ^mod .

From what has been said it is clear that for frequencies up to the cut-off frequency f _g the modified frequency values match the frequency values of the FT: f _j ^mod = f _i (for j = i). The same applies in this frequency range for the associated amplitudes: a _j ^mod = a _i (for j = i). Above the cutoff frequency f _g the modified frequency values are usually not exactly on one frequency point of the FT.

There are now several options, such as the amplitudes at the modified frequency points above f _g can be calculated from the amplitudes of the FT. One possibility is to use interpolation. All known types of interpolation can be used. Further possibilities result from the definition of intervals I _j ^mod which ^cover the entire frequency range above f _g cover, with exactly one modified frequency value f _j ^mod in each interval I _j ^mod : $${\text{l}}_{\text{j}}{}^{\text{mod}}=\left[{\text{G}}_{\text{j}}{}^{\text{mod}},{\text{G}}_{\text{j}+\text{1}}{}^{\text{mod}}\right],{\text{where g}}_{\text{j}}{}^{\text{mod}}<{\text{f}}_{\text{j}}{}^{\text{mod}}<{\text{G}}_{\text{j}+\text{1}}{}^{\text{mod}}$$

To do this, the interval limits must of course lie between the associated successive modified frequency values and be selected so that there are no gaps between successive intervals, which results in the above formula from the fact that the upper limit of the jth interval is equal to the lower limit of the j + 1st interval. Even with these restrictions, there are still endless possibilities to define such intervals. However, it is advantageous if the frequency lines of the modified spectrum are arranged approximately in the middle in these intervals. This can be achieved if the interval limits are formed from the mean of two successive modified frequency values. Both arithmetic and geometric mean values can be used: $${\text{G}}_{\text{j}+\text{1}}{}^{\text{mod}}=\left({\text{f}}_{\text{j}}{}^{\text{mod}}+{\text{f}}_{\text{j}+\text{1}}{}^{\text{mod}}\right)/\mathrm{2nd}\text{or}{\text{. G}}_{\text{j}+\text{1}}{}^{\text{mod}}=\text{root}\left({\text{f}}_{\text{j}}{}^{\text{mod}}*{\text{f}}_{\text{j}+\text{1}}{}^{\text{mod}}\right)$$

Thus, now, for example, which are formed to f _j ^mod associated amplitude a _j ^mod from all amplitudes a _i of the FT whose associated frequency values f _i in the interval I _j ^mod fall. If several f _i fall in an interval I _j ^mod , then a _j ^{mod can be formed} from the mean of the associated a _i . It is also possible that a _j ^{mod is} simply formed from the sum of these a _i . Then the curve of the frequency spectrum changes, but since only temporal changes in the spectra are relevant for diagnostic purposes, this results not a disadvantage, while a summation requires less computing capacity than an averaging. For the same reason, many other metrics can be used to calculate a _j ^mod from the a _i falling in the interval.

If a f _i falls exactly on an interval limit, then the rule can simply be established that this frequency is always assigned to the previous or always the following interval.

Alternatively, such intervals can also be formed for the frequency spectrum of the FT: $${\text{l}}_{\text{i}}=\left[{\text{f}}_{\text{i}}-{\text{df}}_0/\mathrm{2,}{\text{f}}_{\text{i}}+{\text{df}}_0/\mathrm{2nd}\right]$$

The modified amplitudes a _j ^mod are then calculated from the amplitudes a _{i of} the FT, whose associated intervals I _i overlap with the respective interval I _j ^mod . If an interval I _i completely falls within an interval I _j ^mod , then the associated amplitude a _i is used 100% to calculate the amplitude a _j ^mod . In the other case, ie if an interval I _i falls partly in the interval I _j ^mod and partly in the interval I _{j + 1} ^mod , then the associated amplitude a _i is weighted accordingly both for calculating the amplitude a _j ^mod and used to calculate the amplitude a _{j + 1} ^mod , ie a _i is distributed over the associated intervals. It is advantageous if the sum of the weights is 100%. In general, the weighting factor for an amplitude a _{i is given} by the percentage overlap of the interval I _i with the interval I _j ^mod .

Instead of the amplitudes a _i and a _j ^mod , the squares of the amplitudes can be used analogously to what has been said above. One then speaks of the so-called power spectrum. When using power spectra, the simple summation mentioned above is particularly advantageous for calculating the a _j ^mod , since in a power spectrum the sum of the frequency lines corresponds to the variance of the time signal. By using the sum formation, this then continues to apply to the modified spectra, so that the frequency lines of the modified spectrum can also be interpreted physically as a contribution of the respective frequency to the overall variance. It is also possible that the modified spectrum is first calculated as a power spectrum and then the modified spectrum with amplitudes is created by root pulling.

2nd shows in the upper diagram a frequency spectrum, which was calculated with a conventional FT from the measured time signal. The lower diagram shows the modified frequency spectrum calculated from it. A cut-off frequency of 100 Hz was used. The number of data points in the upper diagram is approx. 130,000, whereas the modified spectrum in the lower diagram only comprises 1000 data points, ie only approx. 0.8%, which represents a very considerable saving. The spectrum of the FT is a power spectrum, and the modified spectrum was created from it using summation metrics. It can be seen that the modified spectrum deviates in shape from the frequency spectrum according to FT with increasing frequency: The y values of the modified spectrum are above the cutoff frequency increasingly above the y values of the spectrum according to FT.

It should also be mentioned that the method according to the invention can be used just as well to increase the frequency resolution at low frequencies without increasing the amount of data. This can be achieved in that the saving of data points at the frequencies above the cut-off frequency is used to increase the frequency resolution at the frequencies below the cut-off frequency by the length of the time window To being increased accordingly. Both advantages can also be combined by not using the full saving of data points at higher frequencies to increase the frequency resolution at lower frequencies.

3rd shows a flow chart of the method according to the invention. Since the individual process steps each use certain parameters, these parameters are first discussed below. These include the time window To, the sampling rate R, the cutoff frequency f _g and the course of the at least sectionally monotonous increase in df ^mod above the cutoff frequency f _g . These parameters or sizes are determined in each case before the respective method step in which they are used. This can be done, for example, in one or more initialization steps. The latter two quantities are required for the calculation of the modified frequency spectrum. The size of the savings in data points also results from them. The lower the cutoff frequency f _g and the steeper the increase in df ^mod , the greater the saving of data points. There are several ways in which this determination can be made.

In a system to be monitored, in which the fluctuation range B of the excitation frequency is known, the following procedure has proven to be particularly advantageous. The cut-off frequency is set to f _g = df ₀ / B, where dfo = 1 / T ₀ . The rise of df ^mod above the cutoff frequency f _g is given by df _j ^mod = B * f _j ^mod , where f _{j + 1} ^mod = f _j ^mod + df _j ^mod . It is therefore a linear increase with the slope B. This ensures that above the cutoff frequency the modified spectrum has the relative frequency resolution df _j ^mod / f _j ^mod = B, ie the relative frequency resolution corresponds to the fluctuation range of the excitation frequency. This also ensures that the greatest possible saving of data points results without there being a loss of quality in the subsequent diagnostic steps.

The procedure described in the previous section can be modified by choosing a higher cutoff frequency and / or a less steeply increasing curve of df ^mod above the cutoff frequency. In other words, if the course of df ^{mod is} below the course of df ^mod described in the previous section. There is no loss of quality with this definition either, although the saving of data points is less great. For example, it could be known that particularly important resonance frequencies are located in the range of the frequency f = dfo / B, which one would definitely like to examine with the highest possible frequency resolution. Then one would advantageously choose a cut-off frequency above these resonance frequencies. Similar considerations can lead to the need to evaluate particularly important resonance frequencies above the cutoff frequency with increased accuracy. Then the rise of df ^mod in this area would be made less pronounced, or completely suspended, so that df ^mod forms a plateau there. Or one could even switch back to smaller df ^mod in this area, ie that df ^{mod changes} from a monotonously increasing course to a local minimum and then possibly continues to grow monotonously again. In general, therefore, df ^{mod is} monotonically increasing at least in sections above the cutoff frequency, all df _j ^{mod being} greater than or equal to dfo.

In a system in which the fluctuation range B of the excitation frequency is not known or can only be determined with considerable effort, the parameters in question can also be determined using a heuristic method. For this purpose, a small amount of artificial damage is briefly introduced into the system, and measurement data are recorded, which are then Fourier transformed. The artificial damage is reflected in the frequency spectrum, as a comparison with data from the undamaged system shows. Starting from a high cut-off frequency and an associated flat linear increase of df ^mod (the extrapolation of the rise line should go through the zero point) and the respective calculation of the modified spectrum, the cut-off frequency is successively reduced until the artificial damage no longer or only very much is difficult to detect in the modified spectrum. A slightly higher cut-off frequency is then selected so that damage can still be reliably detected. Artificial damage can be generated, for example, by wrapping a gear tooth of the system with adhesive tape. If necessary, the parameters determined in this way can be modified in accordance with the previous section.

The position of the frequency lines f _j ^{mod of} the modified spectrum is given by specifying the parameters mentioned. In embodiments of the method according to the invention which use intervals, they are calculated together with the frequency lines.

It is particularly advantageous if, in the case of a linear increase in df _j ^mod, the interval limits g _j ^{mod are formed} with the geometric mean, since the following relationships then apply: $${\text{f}}_{\text{j + 1}}{}^{\text{mod}}/{\text{f}}_{\text{j}}{}^{\text{mod}}=\text{Constant};{\text{G}}_{\text{j}+\text{1}}{}^{\text{mod}}/{\text{G}}_{\text{j}}{}^{\text{mod}}=\text{Constant};{\text{f}}_{\text{j}}{}^{\text{mod}}/{\text{G}}_{\text{j}}{}^{\text{mod}}=\text{Constant}$$

In the process step, which with V1 with the aid of a sensor, which is arranged at a suitable point in the system, vibration signals are generated and transmitted to a signal processing unit. The vibration signals from the signal processing unit with the sampling rate R detected. For the procedural step V1 is therefore the parameter R required.

The signal processing unit comprises a computing unit which is suitable for carrying out the FT described above on the basis of the data recorded from the transmitted vibration signals and for calculating the modified frequency spectrum from the resulting frequency spectrum. For this purpose, there is a computer program in the computing unit with program steps for performing the arithmetic operations mentioned.

The FT is carried out on the time window To in the process step which is carried out with V2 is designated. For the procedural step V2 the To parameter is therefore required.

The modified frequency spectrum is calculated in the process step, which is carried out with V3 is designated. The parameters come here f _g , the position of the frequency lines f _j ^mod and, if applicable, the interval limits for use.

With with V4 The method step referred to is at least one diagnostic step which uses the modified frequency spectrum as input. As already mentioned, the diagnostic steps can either be carried out on site or remotely.

In practice, the process steps V1 , V2 and V3 can be implemented using different asynchronous computing processes, for example. The data transfer from V1 on V2 , such as V2 on V3 can be done using queues, for example. As mentioned above, each process has its own default values.

4th shows an arrangement for performing the method according to the invention and a data carrier on which the program code for performing the method according to the invention is stored. The disk is with 5 designated. The disk 5 is a computer readable medium. The arrangement comprises the system to be monitored, which is designated by 1. The facility includes 1 a sensor for generating vibration signals, which with 2nd is designated. The arrangement further comprises a signal processing unit, which with 3rd is designated. The sensor 2nd is with the signal processing unit 3rd connected. The connection can also be made wirelessly. The signal processing unit 3rd comprises a computing unit, which is designated by 4. A computer program with program steps which cause the arrangement to carry out the method according to the invention is stored in the computing unit.

Claims (14)

Method for monitoring a system (1) with a sensor (2) for generating vibration signals, the method comprising steps V1, V2, V3 and V4, and wherein in step V1 with the aid of the sensor (2), vibration signals are generated and sent to a Signal processing unit (3) are transmitted, wherein the signal processing unit (3) detects the vibration signals with a predefined sampling rate R, and wherein in step V2 an FT is carried out on a predefined time window T ₀ by a computing unit (4) on the detected vibration signals To obtain frequency spectrum (f _i , a _i ), and wherein in step V4 at least one diagnostic step is carried out, characterized in that in step V3 a predefined cut-off frequency f _g and a plurality of predefined frequencies f _j ^mod are used, the frequencies being different f _j ^mod extend over the frequency band of the FT and for frequencies below f _g the frequencies f _j ^mod with the frequency line ien f _{i of} the FT agree, and for frequencies ≥ f _g the distance between successive frequencies df _j ^mod = f _{j + 1} ^mod -f _j ^mod increases monotonically at least in sections, with all df _j ^mod ≥ 1 / T ₀ and f _g is below the maximum frequency of the FT, and wherein a modified frequency spectrum (f _j ^mod , a _j ^mod ) is calculated in step V3, the amplitudes a _j ^mod matching the amplitudes a _{i of} the FT for frequencies below f _g , and for Frequencies ≥ f _g, the amplitudes a _j ^mod are calculated from the amplitudes a _{i of} the FT, and wherein in step V4 the modified frequency spectrum is used to carry out the diagnostic step. Procedure according to Claim 1 , wherein in step V3 for frequencies ≥ f _g the amplitudes a _j ^mod are calculated from the amplitudes a _{i of} the FT by interpolation. Procedure according to Claim 1 , wherein before step V3 intervals I _j ^{mod are} calculated which cover the entire frequency range above f _g , with exactly one modified frequency value f _j ^mod being in each interval I _j ^mod : I _j ^mod = [g _j ^mod , g _{j +1} ^mod ], where g _j ^mod <f _j ^mod <g _{j + 1} ^mod , and wherein in step V3 the amplitude a _j ^{mod is} calculated from the amplitudes a _{i of} the FT, whose frequency f _i into the interval I _j ^mod fall. Procedure according to Claim 1 Intervals I _j ^mod and I _{i are} calculated before step V3, each covering the entire frequency range above f _g , with exactly one modified frequency value f _j ^mod being in each interval I _j ^mod : I _j ^mod = [g _j ^mod, g _{j + 1} ^mod], where g _j ^mod <f _j ^mod <g _{j + 1} ^mod, and the intervals I _i by [f _i -df _0/2, f _i + df _0/2] where , and wherein in step V3 the amplitude a _j ^{mod is} calculated from the amplitudes a _{i of} the FT, whose associated intervals I _i overlap with the respective interval I _j ^mod , wherein the amplitudes a _i are taken into account with a weighting factor that corresponds to the percentage overlap of the interval I _i corresponds to the interval I _j ^mod . Procedure according to Claim 4 , the limits of the intervals I ^{mod being formed} by the arithmetic mean of two successive modified frequency values fmod. Procedure according to Claim 4 , the limits of the intervals I ^{mod being formed} by the geometric mean of two successive modified frequency values fmod. Procedure according to one of the Claims 3 to 6 , wherein in step V3 the amplitude a _j ^{mod is formed} by calculating an average value from the corresponding amplitudes a _i . Procedure according to one of the Claims 3 to 6 , wherein in step V3 the amplitude a _j ^{mod is formed} by calculating a sum of the corresponding amplitudes a _i . Method according to one of the preceding claims, wherein instead of the amplitudes a _i and a _j ^mod the squares of these sizes are used. Method according to one of the preceding claims, wherein for frequencies ≥ f _g the distance between successive frequencies df _j ^mod = f _{j + 1} ^mod -f _j ^mod increases linearly. Procedure according to Claim 10 , the cut-off frequency being given by f _g = 1 / (B * To), where B is the range of fluctuation of an excitation frequency of the system (1), and the rise in df ^mod above the cut-off frequency f _g by df _j ^mod = B * f _j ^{mod is} given, where f _{j + 1} ^mod = f _j ^mod + df _j ^mod . Arrangement for monitoring a system (1), comprising a sensor (2) for generating vibration signals, a signal processing unit (3) with a computing unit (4), which are set up to carry out the method steps according to one of the preceding claims. A computer program comprising instructions which cause the arrangement of the preceding claim to carry out the method steps according to one of the Claims 1 to 11 executes. Computer-readable medium (5) on which the computer program Claim 13 is saved.

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