3. The dynamic random response

3.1. Reminder on power spectral densities [bib2]

3.1.1. Definitions

Let it be a probabilistic signal defined by its probability density \({p}_{x}({x}_{1},{t}_{1},\dots ,{x}_{n},{t}_{n})\). This probability density makes it possible to calculate the moment functions of the signal.

First order moment or signal expectancy:

\({\mu }_{X}(t)=E\left[X(t)\right]=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}X{p}_{x}(x,t)\text{dx}\)

Moments of order 2 or intercorrelation of two signals:

\({\rho }_{\text{XY}}({t}_{1},{t}_{2})=E\left[X({t}_{1})\overline{Y({t}_{2})}\right]=\underset{-\infty }{\overset{+\infty }{\int }}x\stackrel{ˉ}{y}p(x,{t}_{1};y,{t}_{2})\text{dxdy}\)

When the signal is stationary, the cross-correlation only depends on \(\tau ={t}_{2}-{t}_{1}\).

We write it \({R}_{\text{XY}}(t)=E\left[X(t)\overline{Y(t-\tau )}\right]\)

Power spectral density and interspect

We define \({S}_{\text{XY}}(\omega )\) the power interspectrum or interspectral power density between two stationary probabilistic signals by the Fourier transform of the intercorrelation function, which is written as:

\({S}_{\text{XY}}(\omega )=\frac{1}{2\pi }\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{R}_{\text{XY}}(\tau ){e}^{\text{-}i\omega \tau }d\tau\)

The inverse formula is written as: \({R}_{\text{XY}}(t)=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{S}_{\text{XY}}(\omega ){e}^{i\omega \tau }d\omega\)

\({S}_{\text{XY}}(\omega )\) is generally complex and verifies the symmetry relationship: \({S}_{\text{YX}}(\omega )=\overline{{S}_{\text{XY}}(\omega )}\).

When \(X=Y\), \({S}_{\text{XX}}(\omega )\) is called power autospectrum or density power spectral (DSP). This function has the property of being real and always positive.

3.1.2. Relationships between DSP and other signal characteristics

Note:

Most of the time, the signal is defined for a limited time, its Fourier transform does not exist, we then define a Fourier transform estimated over a period of length \(T\) by:

\({\stackrel{ˆ}{X}}_{T}(\omega )=\frac{1}{2\pi }\underset{-T/2}{\overset{T/2}{\int }}X(t){e}^{\text{-}i\omega \tau }d\tau\).

We then have the following relationships with this estimated Fourier transform:

\(\begin{array}{}{S}_{\text{XY}}(\omega )=\underset{T\to +\infty }{\text{lim}}\frac{2\pi }{T}E\left[{\stackrel{ˆ}{X}}_{T}(w)\overline{{\stackrel{ˆ}{Y}}_{T}(\omega )}\right]\\ {S}_{\text{XX}}(\omega )=\underset{T\to +\infty }{\text{lim}}\frac{2\pi }{T}E\left[{\stackrel{ˆ}{X}}_{T}(w)\overline{{\stackrel{ˆ}{X}}_{T}(\omega )}\right]\end{array}\)

Link between power autospectrum and signal strength:

The power of a signal is equal to its variance. For a centered signal, the variance is: \({\sigma }_{X}^{2}={R}_{\text{XX}}(0)\).

So we have: \({\sigma }_{X}^{2}={R}_{\text{XX}}(0)=\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{S}_{\text{XX}}(\omega )d\omega\).

3.2. The equations of motion

The total response of the structure is determined by the relationship: \({\ddot{X}}_{a}(\omega )=H(\omega )\ddot{E}(\omega )\),

where \(\ddot{E}(\omega )\) is the vector of \(m\) lines consisting of the excitations represented by the Fourier transforms of the accelerograms \({g}_{j}(t)\) with the \(m\) degrees of liberty-supports,

\(H(\omega )\) is the transfer matrix defined by \(H(\omega )={\omega }^{2}\text{ph}(\omega )\Phi +\Psi\)

where \(p\) is the participation factor matrix,

\(H(\omega )\) the vector of modal transfer functions \({h}_{i}(\omega )\)

\(\Phi\) base of dynamic modes

\(\Psi\) base of static modes

it has \(n\) rows (= number of free degrees of freedom of the structure) and \(m\) columns.

3.2.1. « Interspectral-excitation » matrix

NOTE:

This term « interspectral-excitation matrix » is abusive: it means « interspectral excitation power density matrix » .

It is assumed that seismic excitation can be considered as a stationary signal - taking into account the relationships between representative times - and centered. This makes it possible to use a certain number of results from the probabilistic analysis. We are then interested in the stationary response of the system to a stationary excitation.

We note \({S}_{\ddot{E}\ddot{E}}(\omega )\) the matrix of power interspectra corresponding to the excitation. Its data is explained in chapter 4.

For the record, we recall here that it is calculated from Fourier transformations of accelerations. It’s a matrix (\(m\times m\)). The term ij corresponds to the interspectrum between the signals \({\ddot{E}}^{i}\) and \({\ddot{E}}^{j}\), i.e. again between the Fourier transforms of the accelerograms \({g}_{i}\) and \({g}_{j}\).

3.2.2. Random dynamic response

We have seen that the power interspectrum between two probabilistic signals is the Fourier transform of the intercorrelation function of the two signals. It is applied to the total response of the structure:

\({S}_{{{\ddot{X}}_{a}\ddot{X}}_{a}}(\omega )=\frac{1}{2\pi }\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{R}_{{{\ddot{X}}_{a}\ddot{X}}_{a}}(\tau ){e}^{-i\omega \tau }d\tau =\frac{1}{2\pi }\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}E\left[{\ddot{X}}_{a}(t){}^{T}\text{}\overline{{\ddot{X}}_{a}(t-\tau )}\right]{e}^{\text{-}i\omega \tau }d\tau\)

We then work in the time domain to express the cross-correlation function of the total response \({R}_{{\ddot{X}}_{a}{\ddot{X}}_{a}}(t,t\text{'})\).

We note \(h(t)\) the impulse response of the system: \(h(t)={\text{TF}}^{\text{-}1}[H(\omega )]\)

and \(\ddot{e}(t)\) the inverse Fourier transform of the exciting DSP: \(\ddot{e}(t)={\text{TF}}^{\text{-}1}[\ddot{E}(\omega )]\)

By Fourier transform inverse of the relationship: \({\ddot{X}}_{a}(\omega )=H(\omega )\ddot{E}(\omega )\)

We have \({\ddot{\mathrm{X}}}_{\mathrm{a}}(t)\mathrm{=}\mathrm{h}\mathrm{\times }\ddot{\mathrm{e}}(t)\mathrm{=}\underset{R}{\mathrm{\int }}\mathrm{h}(u)\ddot{\mathrm{e}}(t\mathrm{-}u)\text{du}\)

\(\begin{array}{}{R}_{{\ddot{X}}_{a}{\ddot{X}}_{a}}(t,t\text{'})=E\left[{\ddot{X}}_{a}(t){}^{T}\text{}\overline{{\ddot{X}}_{a}(t\text{'})}\right]\\ {R}_{{\ddot{X}}_{a}{\ddot{X}}_{a}}(t,t\text{'})=E\left[\underset{R}{\int }h(u)\ddot{e}(t-u)\text{du}{}^{T}\text{}\overline{\underset{R}{\int }h(v)\ddot{e}(t\text{'}-v)\text{dv}}\right]\\ {R}_{{\ddot{X}}_{a}{\ddot{X}}_{a}}(t,t\text{'})=E\left[\underset{R}{\int }\underset{R}{\int }h(u)\ddot{e}(t-u){}^{T}\text{}\overline{\ddot{e}(t\text{'}-v)}{}^{T}\text{}\overline{h(v)}\text{dv}\text{du}\right]\end{array}\)

In this analysis, we assume the deterministic system, so we can get the impulse response out of the calculation of the mathematical expectation. It comes:

\({R}_{{\ddot{X}}_{a}{\ddot{X}}_{a}}(t,t\text{'})=\underset{R}{\int }\underset{R}{\int }h(u)E[\ddot{e}(t-u){}^{T}\text{}\overline{\ddot{e}(t\text{'}-v)}]{}^{T}\text{}\overline{h(v)}\text{dv}\text{du}\)

Excitation is assumed to be a stationary process, so cross-correlation only depends on the time difference \(\tau \text{=}t\text{-}t\text{'}\):

\({R}_{\ddot{E}\ddot{E}}(t-t\text{'}-u+v)=E[\ddot{e}(t-u){}^{T}\text{}\overline{\ddot{e}(t\text{'}-v)}]={R}_{\ddot{E}\ddot{E}}(\theta )\text{pour}\theta =t-t\text{'}-u+v=\tau -u+v\)

Hence \({R}_{{\ddot{X}}_{a}{\ddot{X}}_{a}}(t,t\text{'})=\underset{R}{\int }\underset{R}{\int }h(u){R}_{\ddot{E}\ddot{E}}(\theta ){}^{T}\text{}\overline{h(v)}\text{dv}\text{du}={R}_{{\ddot{X}}_{a}{\ddot{X}}_{a}}(\tau )\) which justifies the approach in retrospect.

We now include this expression in the expression for the power spectral density of the response:

\({S}_{{\ddot{X}}_{a}{\ddot{X}}_{a}}(\omega )=\frac{1}{2\pi }\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}{R}_{{\ddot{X}}_{a}{\ddot{X}}_{a}}(\tau ){e}^{\text{-}i\omega \tau }d\tau =\frac{1}{2\pi }\underset{\text{-}\infty }{\overset{\text{+}\infty }{\int }}\underset{R}{\int }\underset{R}{\int }h(u){R}_{\ddot{E}\ddot{E}}(\tau -u+v){}^{T}\text{}\overline{h(v)}{e}^{\text{-}i\omega \tau }\text{dv}\text{du}\mathrm{dt}\)

By distributing the mute integration variables we show the respective Fourier transforms of \(h(u),{R}_{\ddot{E}\ddot{E}}(t-u+v),{}^{T}\text{}\overline{h(v)}\), it finally comes:

\({S}_{{\ddot{X}}_{a}{\ddot{X}}_{a}}=H(\omega )\mathrm{.}{S}_{\ddot{E}\ddot{E}}(\omega )\mathrm{.}\overline{{}^{T}\text{}H(\omega )}\)

with \(H(\omega )={\omega }^{2}\text{P.h}(\omega )\Phi +\Psi\)

Taking into account the relationships between the Fourier transforms of displacement, speed and acceleration, we also have:

\(\begin{array}{}{S}_{{\dot{X}}_{a}{\dot{X}}_{a}}=\frac{-1}{{\omega }^{2}}H(\omega ){S}_{\ddot{E}\ddot{E}}(\omega )\overline{{}^{T}\text{}H(\omega )}\\ {S}_{{X}_{a}{X}_{a}}=\frac{1}{{\omega }^{4}}H(w){S}_{\ddot{E}\ddot{E}}(\omega )\overline{{}^{T}\text{}H(\omega )}\end{array}\)

These relationships make it possible to express the response of the structure by the DSP of displacement or speed.

Notes:

• According to the expression given to \(H(\omega )\) , we express the DSP of the total, relative or differential displacement (respectively speed or acceleration):

absolute movement: \(H(\omega )={\omega }^{2}\text{P. h}(\omega )\Phi \text{+}\Psi\)

relative movement: \(H(\omega )={\omega }^{2}\text{P. h}(\omega )\Phi \text{}\)

differential movement (i.e. training) : \(H(\omega )=\Psi\)

• It is customary, when calculating with Code_Aster, to restrict the transfer function matrix to the rows of \(l\) degrees of liberty of observation. This makes it possible to lighten the calculations as soon as \(l\) is small in front \(n\) .

3.3. Application in Code_Aster

The entire spectral approach for seismic calculation is processed in command DYNA_ALEA_MODAL [U4.53.22]. The data is grouped under three key words factors and one simple keyword.

The modal base consists of dynamic modes calculated by the command CALC_MODES [U4.52.02] stored in a mode_meca concept retrieved by the keyword factor BASE_MODALE, on the one hand; static modes calculated by the command MODE_STATIQUE [U4.52.14] stored in a mode_stat type concept retrieved by the simple keyword MODE_STAT, on the other hand. The keyword factor BASE_MODALE also has the arguments that make it possible to determine the frequency band or the modes used for the calculation and the corresponding depreciations.

The data corresponding to the excitation are collected under the keyword factor EXCIT (cf. paragraph [§4]): the type of excitation is specified in it in the sense of the GRANDEUR: excitation while moving or in effort, the node (s) NOEUD and excited components NOM_CMP, the name of the excited intercepts or autospectra INTE_SPEC, complex functions previously read or calculated, respectively by the operators LIRE_INTE_SPEC [U4.36.01] or CALC_INTE_SPEC [U4.36.03] and stored in an interspectrum table with the tabl_intsp concept that apply in each excited degree of freedom.

Under the keyword factor REPONSE are the data related to the choice of discretization.

Command DYNA_ALEA_MODAL provides the response in the form of power spectral density on a modal basis. To obtain the reproduction of DSP on a physical basis, we will use REST_SPEC_PHYS [U4.63.22], which makes it possible to specify the type of magnitude of the response (displacement or effort), at the « observation points » (node-component) of the result. In the presence of a displacement-type response, it will also be specified here whether the response corresponds to absolute, relative or differential displacement.

REST_SPEC_PHYS provides an interspectrum table that contains, according to the user’s request, the interspectral matrix in motion \({S}_{\text{XX}}\),, , or in acceleration \({S}_{\ddot{X}\ddot{X}}\) for an expression in the absolute coordinate system (index \(a\)), the relative coordinate system (index), the relative coordinate system (index \(r\)) or the training matrix (index \(e\)). \({S}_{\dot{X}\dot{X}}\)

Each previous « combination » requires a specific call to the REST_SPEC_PHYS command.