3. Modal transitory post-processing — option “USURE”#
The characterization of transient measurements is the aim of signal processing. It teaches us that a signal is entirely determined by the data of all its statistical moments. In practice, it is out of the question to calculate all the statistical moments; in post-processing, we limit ourselves to quantities calculated classically in signal processing (simple mean, standard deviation and value RMS). They are characteristic of the signals that one wishes to analyze and compare. Similar signals must necessarily have these first statistical moments similar to each other (the opposite being false). The statistical quantities selected here are well suited to the analysis, comparison or classification of vibration signals under random excitation with shock nonlinearities.
We will therefore examine the averaged quantities and their calculation, distinguishing between the different quantities mentioned in the previous chapter:
travel,
shock forces,
the determination of contact and contact time.
Other composite information may be calculated from the preceding ones, in particular the wear power.
3.1. Statistical block processing#
In order to analyze the stationarity of the signals and the statistical processing carried out on the signals, the temporal signals are divided by blocks. Thus, the post-processing duration defined between the initial instant (INST_INIT) and the final instant (INST_FIN) is divided into a number of time blocks (NB_BLOC) of identical duration. The calculation of the statistical values: mean, standard deviation,… is carried out for each of the blocks, a general value for the signal for all the blocks is also calculated.
In the case of a calculation of the response of a structure to a random loading, this block calculation technique makes it possible to ensure that the transitory calculation phase is over and that the announced value is indeed stationary over an observation time associated with the duration of the calculation.
3.2. Statistical treatments applied to shock movements#
Let us consider the time signal \(\mathrm{Depl}\text{\_}x(t)\), which is archived at a certain frequency \({F}_{\mathrm{acquis}}\) on \(N\) points. The starting data is therefore a vector \(\mathrm{Depl}\text{\_}x(i)\) with \(N\) components.
In this case, the average displacement is defined by:
\(\overline{\mathrm{Depl}\text{\_}x}=\frac{\sum _{1}^{N}\mathrm{Depl}\text{\_}x(i)}{N}\)
This mean value characterizes the central value around which the movement signal evolves. For movements, it will therefore make it possible to determine whether a centered configuration is observed (movements with zero mean), or an eccentric configuration (non-zero mean).
The displacement variance is by definition:
\(\mathrm{var}(\mathrm{Depl}\text{\_}x)=\frac{\sum _{1}^{N}{(\mathrm{Depl}\text{\_}x(i)-\overline{\mathrm{Depl}\text{\_}x})}^{2}}{N}\)
The displacement standard deviation is then equal to:
\(\sigma (\mathrm{Depl}\text{\_}x)=\sqrt{\mathrm{var}(\mathrm{Depl}\text{\_}x)}\)
The standard deviation of a signal characterizes its dispersion around its mean value. A low standard deviation will rather concern a signal with low amplitude variations, a high standard deviation will concern stronger variations.
For a centered variable, i.e. with zero mean, the standard deviation is equal to the mean RMS of the signal (Root Mean Square).
For any variable we define the mean RMS of the signal by:
\(\mathrm{RMS}(\mathrm{Depl}\text{\_}x)=\sqrt{\frac{\sum _{1}^{N}\mathrm{Depl}\text{\_}x{(i)}^{2}}{N}}\)
The absolute minimum and maximum of the signal are also interesting and very simple information to obtain, which determines the extent of the signal.
Figure 3.2-a: Example of a movement signal and visualization statistical quantities
A polar representation of all signals \(\mathrm{Depl}\text{\_}x\) and \(\mathrm{Depl}\text{\_}y\) is also interesting to analyze in the case of an obstacle of circular or similar geometry. Let’s agree to call \(R\) the radial displacement and \(\theta\) the angular displacement, equivalent of \(\mathrm{Depl}\text{\_}x\) and \(\mathrm{Depl}\text{\_}y\) in polar terms.
By definition we have:
\(R(i)=\sqrt{\mathrm{Depl}\text{\_}x{(i)}^{2}+\mathrm{Depl}\text{\_}y{(i)}^{2}}\)
\(\theta (i)=\mathrm{Arctg}(\frac{\mathrm{Depl}\text{\_}y(i)}{\mathrm{Depl}\text{\_}x(i)})\)
Among other things, this representation makes it possible to distinguish between:
orbital movements with permanent contact (mean radial displacement in the order of play and standard deviation of the radial displacement low),
pure impact movements (standard deviation of the large radial displacement, variation of the angular displacement small),
other configurations: orbital movement with impacts…
Note:
In the local coordinate system chosen for the shock obstacles, the quantities called here \(\mathrm{Depl}\text{\_}x\) and \(\mathrm{Depl}\text{\_}y\) here are*:math:mathrm{DYloc}`*and*:math:mathrm{DZloc}`*, the axis*:math:`mathrm{Xloc}`* having been chosen by convention perpendicular to the plane of the obstacle. *
In summary, the post-processing option “ USURE “ of the operator POST_DYNA_MODA_T will determine for local movements \(\mathrm{DYloc},\mathrm{DYloc},\mathrm{DZloc}\) as well as for their polar decomposition \(R\) and \(\theta\) the statistical quantities by blocks with the principle set out above:
average value,
value RMS,
standard deviation,
minimum value,
maximum value.
3.3. Statistics for shock forces#
As for the movements, we suppose to have a discrete signal on \(N\) points: \(\mathrm{Fx}\text{\_}\mathrm{choc}(i)\). The signal obtained should be composed of time ranges where the shock force is zero (no contact) and others where the shock force is significant (effective contact), which is the case during numerical calculations. In fact, for experimental signals, because of the dynamics of the measurement system, a noise level can be observed outside the shock period (cf. [Figure 3.3-a]). It is therefore only necessary to carry out statistical processing when the signal leaves the noise level. This requires the introduction of a detection threshold (SEUIL_FORCE **) which, although superfluous in the digital domain, was reproduced in the post-processing of*Code_Aster*.
Figure 3.3-a: Example of an experimental shock force signal
Given the value \(S\mathrm{max}\), determining the maximum noise level in question, we will then calculate:
The**number of moments in shock*:
\(\mathrm{Nchoc}=\mathrm{card}\left\{i/\mid \mathrm{Fx}\text{\_}\mathrm{choc}(i)\mid >S\mathrm{max}\right\}\)
The**average shock force**over the**total time*:
\(\overline{\mathrm{Fx}\text{\_}\mathrm{choc}}=\frac{1}{N}\cdot (\sum _{i/\mid \mathrm{Fx}\text{\_}\mathrm{choc}(i)\mid >S\mathrm{max}}^{N}\mid \mathrm{Fx}\text{\_}\mathrm{choc}(i)\mid )\)
The**average shock force**reduced to**shock time* is equal to:
\(\mathrm{Fx}\text{\_}\mathrm{choc}=\overline{\mathrm{Fx}\text{\_}\mathrm{choc}}\cdot \frac{N}{\mathrm{Nchoc}}\)
The**average RMSde shock force**over the**total time* is calculated as follows:
\(\mathrm{RMS}(\mathrm{Fx}\text{\_}\mathrm{choc})={(\frac{1}{N}\sum _{i/\mid \mathrm{Fx}\text{\_}\mathrm{choc}(i)\mid >S\mathrm{max}}^{N}{\mathrm{Fx}\text{\_}\mathrm{choc}(i)}^{2})}^{1/2}\)
The**average RMS *** reduced to shock time is equal to:
\(\mathrm{RMS}(\mathrm{Fx}\text{\_}\mathrm{choc})=\mathrm{RMS}(\mathrm{Fx}\text{\_}\mathrm{choc})\cdot \frac{N}{\mathrm{Nchoc}}\)
As for movement signals, we can also focus on the absolute maximum or minimum of the force signal, thus determining its extent. For the normal force, the minimum is always zero, while the tangential force is alternating.
In summary, the post-processing option “USURE” of the POST_DYNA_MODA_T operator will determine for normal and tangential shock forces the statistical quantities by blocks with the principle stated above:
average value calculated on the shock time or the total time,
value RMS calculated on the shock time or the total time,
maximum signal value.
3.4. Statistics for shock times#
The shock time percentage is defined by:
\(\text{\%}\mathrm{Tchoc}=\mathrm{Nchoc}/N\)
Looking at the information available on an experimental system, the shock force signal is the most appropriate for accurately determining the occurrence of a contact. As mentioned above, we feel the need to introduce a maximum noise level, and to count the shock phases when the signal exceeds this threshold (SEUIL_FORCE).
In the figure below, we can distinguish a concept of elementary shock determined as a successive passage above and then below the threshold, and a more general concept of global shock, bringing together several elementary shocks separated by short moments of return below the threshold.
Figure 3.4-a
A characteristic rest time \(\mathrm{Tr}\) (DUREE_REPOS) is therefore introduced; the end of a global shock time occurs if the signal remains at rest for a time at least longer than \(\mathrm{Tr}\). This concept of characteristic rest time \(\mathrm{Tr}\) is of course quite vague and will have to be determined by the user in view of the transitory results. It is nevertheless essential because it alone makes it possible to bring together a very close impact train constituting in fact a single contact phase.
The concept of elementary shock time being defined, statistical treatment on shock time will consist in determining the following information:
number of elementary shocks: \(\mathrm{Nb}\text{\_}\mathrm{choc}\text{\_}\mathrm{elem}\)
number of global shocks: \(\mathrm{Nb}\text{\_}\mathrm{choc}\text{\_}\mathrm{glob}\)
number of elementary shocks per global shock: \(\frac{\mathrm{Nb}\text{\_}\mathrm{choc}\text{\_}\mathrm{elem}}{\mathrm{Nb}\text{\_}\mathrm{choc}\text{\_}\mathrm{glob}}\)
average elementary shock time:
\({\stackrel{ˉ}{T}}_{\mathrm{choc}\text{\_}\mathrm{elem}}=\frac{\mathrm{Nchoc}\cdot \Deltat }{\mathrm{Nb}\text{\_}\mathrm{choc}\text{\_}\mathrm{elem}}\)
average global shock time
\({\stackrel{ˉ}{T}}_{\mathrm{choc}\text{\_}\mathrm{glob}}=\frac{\mathrm{Nchoc}\cdot \Deltat }{\mathrm{Nb}\text{\_}\mathrm{choc}\text{\_}\mathrm{glob}}\)
maximum global shock time the longest global shock time observed on the analyzed block.
In summary, the post-processing option “ USURE “ of the POST_DYNA_MODA_T operator ** will determine the statistical quantities by blocks for shock times with the principle set out above:
mean value of the global shock time, `
maximum value of the global shock time,
mean value of the elementary shock time,
the number of global shocks per second,
the average number of elementary shocks per global shock.
3.5. Wearing power#
The quantity generally calculated in vibrations with shock and friction is the wear power defined by ARCHARD [bib1], which reflects the average power developed by the frictional forces during movement. These forces are the driving force behind frictional wear. The wear power in the case of discrete signals is calculated as follows:
\(\overline{{P}_{\mathrm{usure}}}=\frac{\sum _{i/\mid \mathrm{Fn}\mid >S\mathrm{max}}^{N}\mid \mathrm{Fn}(i)\cdot \mathrm{Vt}(i)\mid }{N}\)
This power can for example be correlated to wear or removal of material through a wear coefficient \({K}_{T}\) by a relationship of the type: \(V(T)={K}_{T}\ast {P}_{\mathrm{usure}}\ast T\) where \(V(T)\) is the volume removed during the period \(T\).
Other more sophisticated wear laws can be used in another post-processing operator: POST_USURE described in [R7.01.10].
3.6. Data structure table POST_DYNA associated with option “USURE”#
A table-like structure for the option USURE ** of the POST_DYNA_MODA_T operator combines the results described above.
This table contains the names of the statistical result sub-tables associated with the various quantities analyzed: displacements, shock forces, shock counts and wear power.
There are 10 access variables in this table:
for the displacement variables: DEPL_X, DEPL_Y,, DEPL_Z,,, DEPL_RADIAL, DEPL_ANGULAIRE, which correspond respectively to the movements in X, Y and Z local and their cylindrical decomposition in the plane of the obstacle.
for the shock force variables: FORCE_NORMALE, FORCE_TANG_1, FORCE_TANG_2, which correspond respectively to the normal forces, tangential to the obstacle, the first being in the plane of the obstacle, the second orthogonal to the plane of the obstacle.
for shock counting variables: STAT_CHOC.
for the wear power variable: PUIS_USURE.
The sub-tables associated with the 10 quantities above contain a certain number of access variables for each shock link:
for displacement variables: MOYEN, ECART_TYPE, RMS,,,, MAXI, MINI, which correspond respectively to the mean, standard deviation, RMS or effective value, maximum and minimum value, of the corresponding displacement variable.
for shock force variables: MOYEN_T_TOTAL, MOYEN_T_CHOC, RMS_T_TOTAL,, RMS_T_CHOC, MAXI, which correspond respectively to the mean values over the total time, mean over the shock time, value RMS or effective mean over the total time, value RMS or effective over the total time, value or effective over the shock time, respectively, which correspond to the maximum value of the corresponding force variable.
for shock counting variables: NB_CHOC_S, NB_REBON_CHOC, T_, T_ CHOC_MOYEN, T_, T_, T_ CHOC_MAXI, T_, T_ REBON_MOYEN, %_T_ CHOC, which correspond respectively to the values of the number of shocks per second, the number of rebounds per shock, the number of rebounds per shock, the average of the number of rebounds per shock, of the average of the shock time, of the average of the shock, of the average of the shock time, of the average of the shock time, of the average of the shock time, of the average of the shock time, of the average of the shock time, of the average of the shock time, of the average of the shock time, the average of the shock time, the average of CHOC_MINI shock.
for the wear power variable: PUIS_USURE which corresponds to the wear power calculated according to ARCHARD.