On the decomposition of the multichannel system response in "pseudo-specific" forms of oscillations
The “object of research” plastered with dozens of sensors during full-scale dynamic tests (for example, when studying vehicle vibroactivity) easily provides us with a large amount of data obtained, but what to do with them is often not very clear. The same is true for simulation modeling of dynamic processes of systems with a large number of degrees of freedom.
It may not be entirely clear to those who do not regularly encounter a problem, but, as a rule, it is almost meaningless to watch the corresponding animation of the process stochastic in time and space. Where it breaks or why it shakes so much is usually not visible. What was invented except animation, I will tell you below, and I will recommend this.
By the elementary procedure, one can obtain the spatial “forms” of oscillations, and it is precisely those that actually manifest themselves in these loading conditions and the intensity of their manifestation (dispersion; if desired, the processes themselves).
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Fig.1 The decomposition of the multichannel response in pseudo-forms. "String in a viscous medium" (see figure 2)
In addition to animation, well-known "pairwise" studies (mutual spectra, etc.), which, of course, do not give a general picture.
Such methods are quite obvious: by “cutting out” very narrow bands of the spectrum corresponding to the resonant peaks in the spectrograms, spectrogram (again, “pairwise”) obtain almost-nearly harmonic (highly correlated) vibrations.
In addition to the obvious technical complexity, the lack of such methods is that it is usually necessary to “successfully” select a “reference” channel, which is not always possible. As a result, very large errors occur in the mutual phases of processes. In addition, the selected spatial form of oscillations and without such errors due to the properties of the linear system may be “complex” (non-phase), that its value for subsequent subjective analysis usually (not always) significantly reduces.
A 4-channel study of torsional oscillations (carrier system of an average bus), distinguished into one process is actually known using the spatial form of oscillations, determined by the researcher’s example - the readings of two sensors located diagonally at the corners of the body were added (taken from “+”) they were deducted (taken from "-") the testimony of the other two, located axisymmetrically.
Even with a small number of sensors, this approach may not be entirely good. For example, the frame of a truck is usually much more malleable in the sub-engine (front) part, respectively, the torsion shape, perhaps, should have been assumed in a more complex form (probably, with some “boosting” coefficients for the front sensors - and at the same time the issue of justifying the adopted models). With a larger number of observation channels, this is all very complicated and ill-founded.
Explicit selection (in separate processes) of several forms of oscillations (modes), pre-determined from a different experiment or simulation, was once suggested by me (probably not only by me) and was immediately called by my chief then “the modal decomposition method”. It was supposed to analyze the results of field tests in this way, and to determine the vectors of own forms using bench tests (or finite element modeling).
Actually, they did about it, but at the same time it turned out that.
It is clear that “bench experimenters”, as opposed to “polygon ones”, have a true perturbing process available for simultaneous observation. This gave them a great advantage - it was possible to determine the true transfer functions and use the available hardware and software to experimentally determine their own shapes and frequencies of oscillation of the structure (based on the curvfitting, with “cutting out” the “eye” of the frequency ranges and indicating the system how many to find; also art rather than science, but still).
But. They obviously avoided exploring the type of car assembly, preferring individual frames, carrying bodies, etc. - objects with very low damping. We are talking about modal bench testing - the definition of natural frequencies and forms. Not about simulation. In principle, plausible eigenmodes of oscillations on slightly more strongly damped objects can be obtained experimentally, but it is noticeably more complicated.
Very valuable information turned out to be about these “preferences”. If it is very, very short - only the conclusions, which we managed to gradually come to: it is hardly worth striving to expand the response precisely into the true own forms of the structure, despite the “prettiness” of the idea.
If it is “on the fingers”, then the reason, apparently, with a high probability, is this — depending on the external disturbance; the “own modes of oscillations” (more precisely, stationary “modal” processes corresponding to their own modes of oscillation) will most likely be one way or the other. otherwise they are correlated with each other, actually forming some other spatial forms of oscillations that are to some extent different from their own, and this manifests itself, naturally, in a strong dependence on the damped structure.
As a brief explanation “by contradiction” - two different “undamped” harmonics are uncorrelated, of course, always in any way. (Explanation. The aforementioned effect of “mixing own forms” is observed in the absence of “essential nonlinearities”, and beyond the cases of “complex forms” with “disproportionate” damping in linear systems.)
Based on the above, it is proposed to present the response of the system as the sum of several correlation-independent processes such as some forms of vibration (bending, torsional, etc.). Such an analysis is very simple, as a rule, it gives very valuable results. The resulting forms may be similar to the system's own oscillation forms, but they may not, but they always reflect the behavior of the system under the conditions of real external influence. and they reflect as simply and informatively as possible, and their sequence, ranked by their dispersions, is an approximating series.
To obtain the decomposition of interest to us is considered (1),
or, the same, compactly, (2),
where x is the experimentally obtained n-channel physical response of a system with a duration of N samples, q is an m-channel real “modal” process with mutually uncorrelated channels, H is a matrix of columns describing the sought-for common-mode (“real”) oscillations.
Generally speaking, m <n is expected later (the number of forms is less than the number of channels of the physical response), but initially we assume m = n, then if H and q have rank n, (2) is satisfied in any case without a residual, we obviously arrive at
or in more detail
That is, the search for the desired such “pseudospecific” vibration forms and the corresponding uncorrelated (at zero lag) processes q reduces to the spectral decomposition (by eigenvectors and numbers) of the symmetric matrix Rxx [0] (composed of the values of auto- and cross-correlations of the processes x at lag Δτ = 0) Its own orthonormal vectors give us the desired forms, and the real eigenvalues - the variances of the corresponding processes (besides the significant ones, there will most likely be ~ 0 variances, therefore in practice usually m <n). If desired, one can also construct the processes themselves in the form of time realizations, find their spectra, etc.
To clarify the above, a model of a linear mechanical system with five degrees of freedom, approximately imitating the behavior of a string in a viscous medium, is considered (Fig. 2),
Fig.2 Test system.
oscillating under the influence of a stochastic disturbance f 1 [i] of the pink noise type (Fig. 3), sampling interval Δt = 0.01 s, realization length 25000 samples. (Hereinafter, the SPM estimates are obtained by the Welch method, the segment length is 1024, the Hamming window and 60% segment overlap.)
Fig. 3 Spectrum of external disturbance.
A fragment of the temporal realization of the disturbance is shown in Fig.4.
Fig.4. Fragment of the time realization of the incoming disturbance
In continuous time, the model is a diffuser system (6),
where M = 1 kg, k = 10kg / s, T = 2000H, l = 1m.
The corresponding (of course, what) matrices are numerically equal to the following:
One of the frequency response systems is depicted in Fig.5.
Fig.5. Frequency response test system
(Explanation - there is nothing wrong with the number of resonant peaks, as an explanation of this frequency response of the system with conditionally reduced damping (k = 1kg / s) is shown in Figure 5)
Fig.6 Explanation of the five resonances.
For simulation simulation in discrete time, a finite difference model was used (KR models work well for small sampling intervals, as was shown ). The resulting response spectrum is shown in Fig.7.
Fig.7 Response spectrum (channel x 1 )
Further using the correlating method, the values of auto- and cross-correlation of the response channels were found.
The results of the application of the method under discussion: the variances of pseudo-modal processes Rqq and the pseudospecific forms H were found:
Quantitatively significant (according to the values of their variances in Rqq), apparently, are the first three processes in the decomposition, animation of the entire multichannel process and the corresponding first three modes of oscillations in 32 samples (from 6714 to 6746 counting, i.e. from 67.13 to 67.63 from the current time) is shown in Figure 1 at the beginning of the article. As you can see, these pseudo-forms have a certain similarity with the eigenmodes of vibration of the string.
Accumulated experience indicates the following features of this method.
It may not be entirely clear to those who do not regularly encounter a problem, but, as a rule, it is almost meaningless to watch the corresponding animation of the process stochastic in time and space. Where it breaks or why it shakes so much is usually not visible. What was invented except animation, I will tell you below, and I will recommend this.
By the elementary procedure, one can obtain the spatial “forms” of oscillations, and it is precisely those that actually manifest themselves in these loading conditions and the intensity of their manifestation (dispersion; if desired, the processes themselves).
')
| Original multichannel process  | Decomposition   |
Fig.1 The decomposition of the multichannel response in pseudo-forms. "String in a viscous medium" (see figure 2)
In addition to animation, well-known "pairwise" studies (mutual spectra, etc.), which, of course, do not give a general picture.
Such methods are quite obvious: by “cutting out” very narrow bands of the spectrum corresponding to the resonant peaks in the spectrograms, spectrogram (again, “pairwise”) obtain almost-nearly harmonic (highly correlated) vibrations.
In addition to the obvious technical complexity, the lack of such methods is that it is usually necessary to “successfully” select a “reference” channel, which is not always possible. As a result, very large errors occur in the mutual phases of processes. In addition, the selected spatial form of oscillations and without such errors due to the properties of the linear system may be “complex” (non-phase), that its value for subsequent subjective analysis usually (not always) significantly reduces.
A 4-channel study of torsional oscillations (carrier system of an average bus), distinguished into one process is actually known using the spatial form of oscillations, determined by the researcher’s example - the readings of two sensors located diagonally at the corners of the body were added (taken from “+”) they were deducted (taken from "-") the testimony of the other two, located axisymmetrically.
Even with a small number of sensors, this approach may not be entirely good. For example, the frame of a truck is usually much more malleable in the sub-engine (front) part, respectively, the torsion shape, perhaps, should have been assumed in a more complex form (probably, with some “boosting” coefficients for the front sensors - and at the same time the issue of justifying the adopted models). With a larger number of observation channels, this is all very complicated and ill-founded.
Explicit selection (in separate processes) of several forms of oscillations (modes), pre-determined from a different experiment or simulation, was once suggested by me (probably not only by me) and was immediately called by my chief then “the modal decomposition method”. It was supposed to analyze the results of field tests in this way, and to determine the vectors of own forms using bench tests (or finite element modeling).
Actually, they did about it, but at the same time it turned out that.
It is clear that “bench experimenters”, as opposed to “polygon ones”, have a true perturbing process available for simultaneous observation. This gave them a great advantage - it was possible to determine the true transfer functions and use the available hardware and software to experimentally determine their own shapes and frequencies of oscillation of the structure (based on the curvfitting, with “cutting out” the “eye” of the frequency ranges and indicating the system how many to find; also art rather than science, but still).
But. They obviously avoided exploring the type of car assembly, preferring individual frames, carrying bodies, etc. - objects with very low damping. We are talking about modal bench testing - the definition of natural frequencies and forms. Not about simulation. In principle, plausible eigenmodes of oscillations on slightly more strongly damped objects can be obtained experimentally, but it is noticeably more complicated.
Very valuable information turned out to be about these “preferences”. If it is very, very short - only the conclusions, which we managed to gradually come to: it is hardly worth striving to expand the response precisely into the true own forms of the structure, despite the “prettiness” of the idea.
If it is “on the fingers”, then the reason, apparently, with a high probability, is this — depending on the external disturbance; the “own modes of oscillations” (more precisely, stationary “modal” processes corresponding to their own modes of oscillation) will most likely be one way or the other. otherwise they are correlated with each other, actually forming some other spatial forms of oscillations that are to some extent different from their own, and this manifests itself, naturally, in a strong dependence on the damped structure.
As a brief explanation “by contradiction” - two different “undamped” harmonics are uncorrelated, of course, always in any way. (Explanation. The aforementioned effect of “mixing own forms” is observed in the absence of “essential nonlinearities”, and beyond the cases of “complex forms” with “disproportionate” damping in linear systems.)
Based on the above, it is proposed to present the response of the system as the sum of several correlation-independent processes such as some forms of vibration (bending, torsional, etc.). Such an analysis is very simple, as a rule, it gives very valuable results. The resulting forms may be similar to the system's own oscillation forms, but they may not, but they always reflect the behavior of the system under the conditions of real external influence. and they reflect as simply and informatively as possible, and their sequence, ranked by their dispersions, is an approximating series.
To obtain the decomposition of interest to us is considered (1),
or, the same, compactly, (2),
| H q = x (2) |
where x is the experimentally obtained n-channel physical response of a system with a duration of N samples, q is an m-channel real “modal” process with mutually uncorrelated channels, H is a matrix of columns describing the sought-for common-mode (“real”) oscillations.
Generally speaking, m <n is expected later (the number of forms is less than the number of channels of the physical response), but initially we assume m = n, then if H and q have rank n, (2) is satisfied in any case without a residual, we obviously arrive at
| H · Rqq [0] · H T = Rxx [0] (3) |
or in more detail
That is, the search for the desired such “pseudospecific” vibration forms and the corresponding uncorrelated (at zero lag) processes q reduces to the spectral decomposition (by eigenvectors and numbers) of the symmetric matrix Rxx [0] (composed of the values of auto- and cross-correlations of the processes x at lag Δτ = 0) Its own orthonormal vectors give us the desired forms, and the real eigenvalues - the variances of the corresponding processes (besides the significant ones, there will most likely be ~ 0 variances, therefore in practice usually m <n). If desired, one can also construct the processes themselves in the form of time realizations, find their spectra, etc.
| q = H T x (5) |
To clarify the above, a model of a linear mechanical system with five degrees of freedom, approximately imitating the behavior of a string in a viscous medium, is considered (Fig. 2),
Fig.2 Test system.
oscillating under the influence of a stochastic disturbance f 1 [i] of the pink noise type (Fig. 3), sampling interval Δt = 0.01 s, realization length 25000 samples. (Hereinafter, the SPM estimates are obtained by the Welch method, the segment length is 1024, the Hamming window and 60% segment overlap.)
Fig. 3 Spectrum of external disturbance.
A fragment of the temporal realization of the disturbance is shown in Fig.4.
Fig.4. Fragment of the time realization of the incoming disturbance
In continuous time, the model is a diffuser system (6),
where M = 1 kg, k = 10kg / s, T = 2000H, l = 1m.
The corresponding (of course, what) matrices are numerically equal to the following:
One of the frequency response systems is depicted in Fig.5.
Fig.5. Frequency response test system
(Explanation - there is nothing wrong with the number of resonant peaks, as an explanation of this frequency response of the system with conditionally reduced damping (k = 1kg / s) is shown in Figure 5)
Fig.6 Explanation of the five resonances.
For simulation simulation in discrete time, a finite difference model was used (KR models work well for small sampling intervals, as was shown ). The resulting response spectrum is shown in Fig.7.
Fig.7 Response spectrum (channel x 1 )
Further using the correlating method, the values of auto- and cross-correlation of the response channels were found.
The results of the application of the method under discussion: the variances of pseudo-modal processes Rqq and the pseudospecific forms H were found:
Quantitatively significant (according to the values of their variances in Rqq), apparently, are the first three processes in the decomposition, animation of the entire multichannel process and the corresponding first three modes of oscillations in 32 samples (from 6714 to 6746 counting, i.e. from 67.13 to 67.63 from the current time) is shown in Figure 1 at the beginning of the article. As you can see, these pseudo-forms have a certain similarity with the eigenmodes of vibration of the string.
Accumulated experience indicates the following features of this method.
- The method is sensitive to extraneous noise, incl. high frequency. There is a danger of getting “random” results (with a large random error) and seeming irreproducibility of experimental results. It is necessary to filter out the non-working (quantitatively insignificant) part of the frequency range of a multichannel signal. (At one time, work in this direction was even stopped due to the reports of experimenters about the “absence” of the correlation between channels - the effect of strong noise)
- The best results were obtained when the number of observation channels was significantly greater than the number of expected apparent vibration modes (the number of resonance peaks in the studied range), i.e. with m << n.
- At high Q-factors, the pseudo-forms, apparently, tend to the true proper forms of oscillations of the system. The results obtained on such systems are more stable in the case of noise.
- When a pair of eigenvalues Rxx [0] coincides or even comes together, the pair of corresponding pseudoforms apparently is determined to within their independent linear combinations — such forms cannot be separated between themselves under given experimental conditions.
- Apparently, by comparing the true forms (found by other methods) and the pseudo-forms under consideration, with some probability one can find the points of application of the external perturbation (especially with a small-channel perturbation), if such a task is relevant.
Source: https://habr.com/ru/post/449528/
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