Principle of the approach ======================= Position of the problem under consideration and general principle -------------------------------------------------- We are in the case of a **multi-supported structure**, that is to say that the structure has :math:`m` degree of freedom of support, each being subject to its own excitation (not necessarily equal everywhere). It is assumed that the structure is represented by a finite element model with :math:`n` degrees of freedom. We're looking for the answer in a finite (and low) number of :math:`l` degrees of freedom. It is assumed that the **excitation magnitude is of the imposed movement** type and results in a family of :math:`{\mathrm{g}}_{\mathrm{j}}(\mathrm{t})` accelerograms for each of the degrees of liberty-supports :math:`j`, :math:`j=\mathrm{1,}m`. The absolute movement of the structure is **decomposed** classically into **training movement and relative movement.** The calculation of the response in power interspectra is carried out by **modal recombination**. Following this modal calculation, a random dynamic response calculation is divided into three parts: * definition of the exciter power interspectrum, * calculation of the power response interspectrum. These first two parts are the subject of order DYNA_ALEA_MODAL [:external:ref:`U4.53.22 `]. The reproduction of the response power interspectrum on a physical basis is carried out with the command REST_SPEC_PHYS [:external:ref:`U4.63.22 `]. * calculation of statistical parameters from the power interspectrum result. This last step is processed by the POST_DYNA_ALEA [:external:ref:`R7.10.01 `] [:external:ref:`U4.84.04 `] command. Breakdown of the movement -------------------------- The following decompositions and projections are detailed in the reference documentation relating to the resolution by transitory calculation of a seismic calculation [:ref:`R4.05.01 `]. We have only retained the main points here. Let :math:`{X}_{a}` be the absolute displacement vector (of dimension :math:`n`) of all the degrees of freedom of the structure. The total response called **absolute** :math:`{X}_{a}` of the structure is expressed as the sum of a **relative** :math:`{X}_{r}` contribution and the **training** contribution :math:`{X}_{e}` due to anchor movements (subject to accelerations represented by an accelerogram :math:`{g}_{j}(t)` in each of the degrees of liberty-supports :math:`j`, :math:`j\mathrm{=}\mathrm{1,}m`). .. image:: images/Object_6.svg :width: 153 :height: 21 .. _RefImage_Object_6.svg: Let :math:`M,K` and :math:`C` be the mass, stiffness, and damping matrices of the problem, limited to unsupported degrees of freedom. The equation of motion is then written in the coordinate system linked to relative motion: :math:`\begin{array}{c}{{\begin{array}{}\text{M}\ddot{X}\end{array}}_{r}(t)+\text{C {}\dot{X}}_{r}(t)+{\text{K X}}_{r}(t)\text{= -}\text{M}\ddot{{X}_{e}}(t)+{F}_{\text{ext}}\end{array}` :math:`{F}_{\text{ext}}`: vector of external forces In general, the external forces are zero when calculating seismic responses. Breakdown on a modal basis -------------------------------- The response calculation in power interspectra is carried out by **modal recombination** and uses, in imposed motion, a modal base that includes both dynamic modes and static modes. Let :math:`\Phi =\left\{{\phi }_{i,i=\mathrm{1,}n}\right\}` be the :math:`(n,n)` matrix of dynamic modes calculated for the associated conservative system, keeping the m supports locked. Let :math:`\Psi =\left\{{\psi }_{j,j=\mathrm{1,}m}\right\}` be the :math:`(n,m)` matrix of static modes. The :math:`{\Psi }_{j}` mode corresponds to the deformation of the structure under a unitary displacement imposed at the degree of liberty-support :math:`j`, the other degrees of liberty-supports being blocked. The imposed displacement of anchors :math:`{X}_{s}(t)` is linked to :math:`{X}_{e}(t)` by the relationship: :math:`{X}_{e}(t)=\Psi {X}_{s}(t)`. The components of the acceleration of anchor points :math:`{\ddot{X}}_{s}(t)` are the accelerograms :math:`{g}_{j}(t)`, :math:`j=\mathrm{1,}m`. So we can write :math:`\ddot{{X}_{e}}(t)={\Psi \ddot{X}}_{s}(t)=\sum _{j\text{=1}}^{m}{\psi }_{j}{g}_{j}(t)\text{}`. We change the variable :math:`{X}_{r}(t)=\Phi \mathrm{.}q(t)`, :math:`q(t)` is the vector of the generalized coordinates. By premultiplying the equation of motion by :math:`{}^{T}\text{}\Phi`, we obtain - in the absence of external forces other than seismic excitation - the equation projected on the basis of dynamic modes: :math:`{}^{T}\text{}\Phi M\Phi \ddot{q}(t)+{}^{T}\text{}\Phi C\Phi \dot{q}(t)+{}^{T}\text{}\Phi K\Phi q(t)=-{}^{T}\text{}\Phi M\Phi {\ddot{X}}_{s}(t)` It is assumed that the damping matrix is a linear combination of mass and stiffness matrices (assumption of constant Rayleigh damping on the structure or Basile hypothesis allowing diagonal damping). The base :math:`\Phi`, which orthogonalizes the matrices :math:`M` and :math:`K`, therefore also orthogonalizes the matrix :math:`C`. Given this hypothesis, the previous equation is broken down into :math:`n` decoupled scalar equations in the form: .. image:: images/Object_22.svg :width: 153 :height: 21 .. _RefImage_Object_22.svg: For :math:`i=\mathrm{1,}n` where we noted: :math:`{\mu }_{i}={}^{T}\text{}{\Phi }_{i}M{\Phi }_{i}` the modal mass :math:`{k}_{i}={}^{T}\text{}{\Phi }_{i}K{\Phi }_{i}` modal rigidity :math:`{\omega }_{i}=\sqrt{\frac{{k}_{i}}{{\mu }_{i}}}` the modal pulse :math:`{\xi }_{i}=\frac{{}^{T}\text{}{\Phi }_{i}\mathrm{.}C\mathrm{.}{\Phi }_{i}}{2{\mu }_{i}{\omega }_{i}}` reduced modal depreciation" :math:`{p}_{\text{ij}}=\frac{{}^{T}\text{}{\Phi }_{i}\mathrm{.}M\mathrm{.}{\Psi }_{j}}{{\mu }_{i}}` the modal participation factor of support :math:`j` over dynamic mode :math:`i`. Solution :math:`{q}_{i}(t)` of this equation corresponds to the response of the dynamic mode :math:`i` to the entire seismic excitation. We can further decompose the problem by introducing the unknown :math:`{d}_{\text{ij}}(t)` solution of the differential equation: :math:`{\ddot{\mathrm{d}}}_{\text{ij}}+2{{\xi }_{i}{\omega }_{i}\dot{\mathrm{d}}}_{\text{ij}}+{\omega }_{i}^{2}{\mathrm{d}}_{\text{ij}}\mathrm{=}{g}_{j}(t)`, this last equation corresponds to the response of the dynamic mode :math:`i` to acceleration .. image:: images/Object_28.svg :width: 153 :height: 21 .. _RefImage_Object_28.svg: . The relative displacement on the physical basis is then expressed: :math:`{X}_{r}(t)=-\sum _{i=1}^{n}\sum _{j=1}^{m}{p}_{\text{ij}}{d}_{\text{ij}}(t){\varphi }_{i}` Information on the position of the fulcrum is contained in the modal participation factor. Harmonic response ------------------ The total response of the structure was therefore decomposed into a relative contribution and a differential contribution due to the movements of the anchors such as: :math:`\begin{array}{}{X}_{a}(t)\text{=}{X}_{r}(t)\text{+}{X}_{e}(t)\\ \text{avec}\\ \{\begin{array}{}{\ddot{X}}_{e}(t)\text{=}{\Psi \ddot{X}}_{s}(t)\text{=}\sum _{j=1}^{m}{\psi }_{\text{j}}{g}_{\text{j}}(t)\\ {X}_{r}(t)\text{= -}\sum _{i\text{=}1}^{n}\sum _{j=1}^{m}{{p}_{\text{ij}}d}_{\text{ij}}(t){\varphi }_{i}\text{}\text{où}{d}_{\text{ij}(t)}\text{est solution de}\text{}{\ddot{d}}_{\text{ij}}\text{+}2{\xi }_{i}{\omega }_{i}{\dot{d}}_{\text{ij}}\text{+}{\omega }_{i}^{2}{d}_{\text{ij}}\text{=}{g}_{j}(t)\end{array}\end{array}` The resolution of this last one differential equation by the Fourier transform method involves modal transfer functions :math:`{h}_{i}(\omega )` such as:. :math:`{h}_{i}(\omega )=\frac{1}{{\omega }_{i}^{2}-{\omega }^{2}+2i{\xi }_{i}{\omega }_{i}\omega }` So we get: :math:`{d}_{\text{ij}}(\omega )={h}_{i}(\omega )\text{.}{g}_{j}(\omega )\text{et}{\ddot{d}}_{\text{ij}}(\omega )\text{=-}{\omega }^{2}{h}_{i}(\omega )\text{.}{g}_{j}(\omega )` The total harmonic response of the structure is deduced from the preceding formulas by modal recombination. :math:`\begin{array}{}{\ddot{X}}_{a}(\omega )={\ddot{X}}_{r}(\omega )+{\ddot{X}}_{e}(\omega )\\ {\ddot{X}}_{a}(\omega )={\omega }^{2}\sum _{i=1}^{n}\sum _{j=1}^{m}{p}_{\text{ij}}{h}_{j}(\omega ){\varphi }_{j}(\omega ){j}_{i}+\sum _{j\text{=1}}^{m}{\Psi }_{j}{g}_{j}(\omega )\end{array}` The following complex matrix :math:`(n,m)`, referred to as the transfer matrix :math:`H(\omega )`, is then made to appear: :math:`H(\omega )\text{=}{\omega }^{2}P\mathrm{.}h(\omega )\Phi \text{+}\Psi` where :math:`P` is the participation factor matrix, :math:`h(\omega )` is the vector of the modal transfer functions :math:`{h}_{i}(\omega )`. The total response of the structure is :math:`{\ddot{X}}_{a}(\omega )=H(\omega )\ddot{E}(\omega )` where :math:`\ddot{E}(\omega )` is the vector of m lines consisting of Fourier transforms of accelerations .. image:: images/Object_40.svg :width: 153 :height: 21 .. _RefImage_Object_40.svg: at the :math:`m` degrees of freedom - supports. We can see that this expression determines the acceleration response. This then requires integrating the answer twice to obtain the displacement, this problem is presented in [:ref:`bib4 `]. One of the additional advantages of the method that we propose here is to get rid of this difficulty.