4. Definition of the interspectral excitatory power matrix

Seismic excitement is by nature, as we said, random. Thus it can be known not by its temporal expression but in frequency form by a power spectral density also called interspectrum.

When there are several supports, they can be excited by identical or different excitations, the latter case being that of multi-supports.

For \(m\) supports, we define the interspectral power density matrix of order \(m\), or by misnomer the interspectrum of order \(m\), which is a matrix (\(m\times m\)) of complex functions dependent on frequency.

The diagonal terms represent the « auto- » spectral power densities -or autospectra- at the excitation points, the extra-diagonal terms correspond to the interspectral densities between the excitations at two distinct support points (each row or column of the matrix in fact represents a support point in physical mesh or a mode in modal calculation). By definition of these terms, it can be deduced that the manipulated interspectral power density matrices are Hermitian. (See [bib2] or Reference documentation associated with the POST_DYNA_ALEA [R7.10.01] command)

We present below the various Code_Aster commands that make it possible to obtain an interspectral power density matrix.

4.1. Reading on a file

The most basic way to define an interspectral power density matrix is to give, « by hand », the values at the various frequency steps.

The operator LIRE_INTE_SPEC [U4.36.01] is then used.

LIRE_INTE_SPEC reads « interspectrum excitation » in a file. The format of the file in which the interspectral matrix is recorded is simple: the function of each term of the interspectral matrix is described in succession; for each function, one line per frequency is given by indicating the frequency, the real and imaginary parts of the complex number; or the frequency, the module and the phase of the complex number (keyword FORMAT).

Example of an interspectrum excitation file (for a matrix reduced to one term) :

DIM = 1 FONCTION_C I = 1 J = 1 NB_POIN = 4 VALEUR = 2.9999 0.0. 0. 3. 1. 0. 13. 1. 0. 13.0001 0.0. 0. FINSF FIN

4.2. Obtaining an interspectrum from time functions

The interspectral power density matrix can be deduced from time functions. We then use the CALC_INTE_SPEC [U4.36.03] operator in*Code_Aster* [bib3].

From a list of \(N\) time functions, this operator makes it possible to calculate the \(N\mathrm{\times }N\) power interspectrum that corresponds to them.

For each term of the interspectral matrix (\(N\times N\)) the following approach is used [bib3].

To calculate the interspectrum of two signals, the Wiener-Khintchine relationship [bib7] is used, which makes it possible to establish a formula for calculating the power spectral density by the Fourier transform of finite samples of signals \(x(t)\) and \(y(t)\).

Then it comes:

\({S}_{\text{xy}}(f)\text{=}\underset{T\to \mathrm{\infty }}{\text{lim}}\frac{1}{T}\mathrm{E}\mathrm{[}{\mathrm{X}}_{\mathrm{k}}(f,T)\text{.}{\mathrm{Y}}_{\mathrm{k}}\mathrm{\times }(f,T)\mathrm{]}\)

Where \(\{\begin{array}{c}{X}_{k}(f,T)\text{=}\text{TF}[{x}_{k}](f)=\underset{0}{\overset{T}{\int }}{x}_{k}(t){e}^{-\mathrm{i2}\pi f}\text{dt}\\ {Y}_{k}(f,T)\text{=}\text{TF}[{y}_{k}](f)=\underset{0}{\overset{T}{\int }}{y}_{k}(t){e}^{-\mathrm{i2}\pi f}\text{dt}\end{array}\)

are the discrete Fourier transforms of \(«x»\) and \(«y»\).

When interested in signals resulting from measurements, most of the time only signals known discretely are available, in the same way a transitory calculation result is a discrete signal.

An approximation of the interspectrum of the discrete signals \(x[n]\) and \(y[n]\) defined on \(L\) points spaced by \(\Delta t\), divided into \(p\) blocks of \(q\) points is obtained by the relationship:

\(\begin{array}{}{\stackrel{ˆ}{S}}_{\text{xy}}\left[k\right]=\frac{1}{\mathrm{pq}\Delta t}\sum _{i=1}^{p}{X}^{(i)}[k]{Y}^{(i)}\ast [k]\\ {X}^{(i)}[k]=\Delta t\sum _{n=0}^{q}{x}^{(i)}[n]{e}^{\text{-}\mathrm{2i}\pi \text{kn}/q}\\ {Y}^{(i)}[k]=\Delta t\sum _{n=0}^{q}{y}^{(i)}[n]{e}^{\text{-}\mathrm{2i}\pi \text{kn}/q}\end{array}\)

The various blocks may or may not overlap. The values \(p\) and \(q\) are at the user’s choice.

This method is that of the periodogram of WELCH [bib8].

The calculation is done on a window that moves over the domain of definition of the functions. In the command, the user specifies the length of the analysis window, the offset between two successive calculation windows, and the number of points per window.

4.3. Excitations predefined or reconstructed from existing complex functions

One may wish to define an interspectral power density matrix in various ways:

• by white noise: the values are constant

• according to the analytical formula of KANAI - TAJIMI useful in seismic calculation (filtered white noise),

• or by taking over existing complex functions.

The operator DEFI_INTE_SPEC [U4.36.02] is then used.

4.3.1. Existing complex functions

Under the keyword factor PAR_FONCTION, simply give the name of the function for each pair of indices NUME_ORDRE_I, NUME_ORDRE_J, corresponding to the upper triangular matrix (due to its hermiticity).

4.3.2. White noise

White noise is characterized by a constant value over the entire domain of definition in question. Under the keyword factor CONSTANT, we give this value (VALE_R or VALE_C) on the frequency band [FREQ_MIN, FREQ_MAX] for each pair of indices INDI_I, INDI_J, corresponding to the upper triangular matrix (due to its hermiticity). To perfectly define the function, we specify the interpolation and the extensions.

4.3.3. White noise filtered by KANAI - TAJIMI [bib9]

For a structure supported on the ground, it is common to take the Kanaï-Tajimi power spectral density as an excitation. This spectral density represents the filtering of white noise by the ground, considered as a system with one degree of freedom. The parameters of the formula allow you to adjust the center frequency and the width of the band of the spectrum.

Spectrum \(G(\omega )\) is expressed by the following relationship:

\(\begin{array}{}G(\omega )\text{=}\frac{{\omega }_{g}^{4}\text{+}4{\xi }_{g}^{2}{\omega }_{g}^{2}{\omega }^{2}}{{({\omega }_{g}^{2}-{\omega }^{2})}^{2}\text{+}4{\xi }_{g}^{2}{\omega }_{g}^{2}{\omega }^{2}}{G}_{0}\\ \begin{array}{cc}{\omega }_{g}\text{=}2\pi f& \text{pulsation propre}\\ {\xi }_{g}& \mathrm{amortissement}\mathrm{total}\\ {G}_{0}& \mathrm{niveau}\mathrm{du}\mathrm{bruit}\mathrm{blanc}\mathrm{avant}\mathrm{filtrage}\end{array}\end{array}\)

The user must specify the filter’s natural frequency \({f}_{g}\), the modal damping \({\xi }_{g}\), and the white noise level \({G}_{0}\) (= VALE_R) before filtering; as with any function: interpolation, external profiles, and domain of definition (frequency band).

By default, a current ground is well represented by the values \({f}_{g}=2.5\mathrm{Hz}\) and \({\xi }_{g}=0.6\).

Example of use for white noise filtered by KANAI_TAJIMI :

Interex = DEFI_INTE_SPEC (

DIMENSION: 1 KANAI_TAJIMI: ( NUME_ORDRE_I: 1 index of the term of the density matrix NUME_ORDRE_J: 1 interspectral power FREQ_MOY: 2.5 natural frequency AMOR: 0.6 modal damping VALE_R: 1 white noise level INTERPOL: “LIN” linear interpolation PROL_GAUCHE: “CONSTANT” extension PROL_DROIT: “CONSTANT” FREQ_MIN: 0. domain of definition FREQ_MAX: 200. PAS: 1. ));

4.4. Other types of arousal

The calculations in the preceding paragraphs were carried out under the hypothesis of an arousal in imposed movement on a degree of freedom. With some modifications, it is possible to use the same approach for an excitation in effort [§4.4.1] or by fluid sources [§ 4.4.2], this being expressed in a finite element [§4.4.3] or on a function of the shape of the structure [§4.4.4].

In the rest of this paragraph, we assume the random excitation known and supplied by the user in the form of a DSP, power spectral density.

4.4.1. Case of excitement in imposed forces

Under the keyword EXCIT we have GRANDEUR = EFFO.

When the excitation at the supports is of the imposed force type, the general equation of motion is:

\(M\ddot{X}(t)+C\dot{X}(t)+\text{KX}(t)=\sum _{j=1}^{m}{F}_{j}\)

The response of the structure is then calculated on a base of dynamic modes \(\Phi =\left\{{\phi }_{i,i=\mathrm{1,}n}\right\}\), these modes being calculated by assuming the free excitatory supports. In this case, there is no distinction between absolute, relative and differential movement and no static modes are used.

The modal participation factor is defined as: \({P}_{\text{ij}}=\frac{{}^{T}\text{}{\phi }_{i}{F}_{j}}{{\mu }_{i}}\)

Transient, harmonic, and random responses have the same expressions as the relative movement responses of multi-press arousal in the general case [§3]. (This corresponds to the absence of static modes). Excitatory force is represented in each degree of liberty-support by its DSP in the form of a term equivalent to \({S}_{\ddot{E}\ddot{E}}(\omega )\).

4.4.2. Excitation by fluid sources

Fluid sources appear, for example, in the study of a pipe network. They correspond to active components or to secondary pipe connections. They are generally sources of pressure or sources of flow. These different source types are presented below according to their mathematical formatting and what Code_Aster does in each configuration.

These fluid sources are not directly seismic excitations but can be induced by an earthquake. The resolution of the mechanical problem uses the same methods, due to their random nature, which justifies their presentation here.

The modeling of the pipe network is supposed to be carried out using a vibro-acoustic beam from Code_Aster.

The response to fluid sources is calculated within the framework of the response to imposed forces (cf. [§4.4.1]), in this framework we are interested in magnitude responses such as « displacement » (GRANDEUR = DEPL_R under the keyword REPONSE).

The sources of pressure and force, for reasons of modeling fluid sources, are represented by dipoles [bib5], so it is necessary to give two application points.

Volume flow source: GRANDEUR **= SOUR_DEBI_VOLU under the keyword** EXCIT **

A volume flow rate is expressed in \({m}^{3}/s\), its power spectral density in \({({m}^{3}/s)}^{2}/\mathrm{Hz}\).

A flow-volume source is considered, in the \(P-\phi\) formulation of fluid pipe elements, to be an effort imposed on the \(\phi\) degree of freedom of the source application node [R4.02.02].

The user provides the DSP of flow volume \({S}_{\text{vv}}(\omega )\), the DSP \({S}_{\text{vv}}^{\text{'}}(\omega )\) applied in effort on the degree of freedom \(\phi\) is: \({S}_{\text{vv}}^{\text{'}}(\omega )\text{=}{(\rho \omega )}^{2}{S}_{\text{vv}}(\omega )\)

where \(\rho\) is the density of the fluid.

Mass flow source: GRANDEUR **= SOUR_DEBI_MASS under the keyword** EXCIT **

A mass flow rate is expressed in \(\mathit{kg}\mathrm{/}s\), its power spectral density in \({(\mathrm{kg}/s)}^{2}/\mathrm{Hz}\). Mass flow is the product of volume flow by the density of the fluid.

The user provides the DSP of mass flow \({S}_{\text{mm}}(\omega )\), the DSP \({S\text{'}}_{\text{mm}}(\omega )\) applied in effort on the degree of freedom \(\phi\) is: \({S\text{'}}_{\text{mm}}\omega ={\omega }^{2}{S}_{\text{mm}}\omega\)

Pressure source: GRANDEUR **= SOUR_PRESS under the keyword** EXCIT **

A pressure source is applied in*Aster* in a dipole \({P}_{1}{P}_{2}\).

For a pressure source whose DSP is \({S}_{\text{PP}}(\omega )\), expressed in \({\mathrm{Pa}}^{2}/\mathrm{Hz}\), Aster constructs an interspectral power density matrix \({S\text{'}}_{\text{pp}}(\omega )\) which is applied as a force imposed on the degree of freedom \(\phi\) of the points \({P}_{1}\) and \({P}_{2}\).

\({S\text{'}}_{\text{PP}}(\omega )={S}_{\text{PP}}(\omega )(\begin{array}{cc}{(\frac{S}{\text{dx}})}^{2}& -{(\frac{S}{\text{dx}})}^{2}\\ -{(\frac{S}{\text{dx}})}^{2}& {(\frac{S}{\text{dx}})}^{2}\end{array})\)

where \(S\) is the fluid section, \(\text{dx}\) the distance between the two points \({P}_{1}\) and \({P}_{2}\).

Source of force : GRANDEUR **= SOUR_FORCE under the keyword** EXCIT **

The force simply corresponds to the product of the pressure by the fluid section of the tube: \(F=\mathrm{PS}\). It is therefore also applied to a dipole \({P}_{1}{P}_{2}\).

For a force source whose DSP is \({S}_{\text{FF}}(\omega )\), expressed in \({N}^{2}/\mathrm{Hz}\), Aster applies the force imposed on the degree of freedom \(f\) of the points \({P}_{1}\) and \({P}_{2}\), (distant from \(\mathrm{dx}\)), the interspectral power density matrix \({S\text{'}}_{\text{FF}}(\omega )\) as follows:

\({S\text{'}}_{\text{FF}}(\omega )={S}_{\text{FF}}(\omega )(\begin{array}{cc}{(\frac{1}{\text{dx}})}^{2}& -{(\frac{1}{\text{dx}})}^{2}\\ -{(\frac{1}{\text{dx}})}^{2}& {(\frac{1}{\text{dx}})}^{2}\end{array})\)

4.4.3. Excitation distributed over a form function

If the spectral power density of the excitation \(E(\omega )\) corresponds to an effort imposed on a function of the form \({f}_{i}\), \(E(\omega )\) gives the frequency dependence of the level of the excitation.

The spatial weighting of the effort is represented in Code_Aster by a field at the nodes that does not depend on the frequency: keyword CHAM_NO under the keyword factor EXCIT. This field at the nodes is an « assembled vector ». From the theoretical point of view, the calculation formalism is the same as before (imposed force excitation [§4.4.1]), for a force vector in the second member equal to \({f}_{i}\).

4.5. Apps

These different types of excitement are included in the validation tests, and are presented by examples in the report [bib6]. In particular fluid-type excitations are in the test: pipe subjected to random fluid excitations [V2.02.105] (SDLL105). The excitations on form functions are tested in the test case: beam subjected to distributed random excitation [V2.02.106] (SDLL106).