General presentation ===================== Reminders ------- The dynamic fluid forces exerted on a moving structure can be classified into two categories: *the forces**independent** of the movement of the structure, at least in the range of small displacements; these are mainly random forces generated by turbulence or the two-phase nature of the flow, *the fluid forces**dependent**on the movement of the structure, called "**fluid-elastic forces**", responsible for fluid-structure coupling. In this document, we are interested in the four **fluid-elastic** force models integrated into the CALC_FLUI_STRU operator. The computer aspects related to the integration of these models have been the subject of specification notes [:ref:`bib. 1 `], [:ref:`bib. 2 `]. Modeling ------------ The dependence of fluid-elastic forces on the movement of the structure results, for low amplitudes, in **a transfer matrix between the fluid-elastic force and the** **displacement vector**. The projection of the equation of motion of the coupled fluid-structure system on the modal basis of the structure alone is written, in the **Laplace domain**: :math:`\{[{M}_{\mathit{ii}}]{s}^{2}+[{C}_{\mathit{ii}}]s+[{K}_{\mathit{ii}}]-[{B}_{\mathit{ij}}(U,s)]\}(q)=({Q}_{t})` **eq. 1.2-** 1 +-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |where|:math:`[{M}_{\mathit{ii}}],[{C}_{\mathit{ii}}]\mathit{et}[{K}_{\mathit{ii}}]` refer respectively to the diagonal matrices of structural mass, damping and stiffness in air; | +-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |:math:`(q)` refers to the vector of generalized movements in air; | +-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ | |:math:`({Q}_{t})` is the vector of generalized random excitations (independent forces); | +-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ |and |:math:`[{B}_{\mathit{ij}}(U,s)]` represents the fluid-elastic force transfer matrix, projected on the modal basis of the structure alone. This matrix depends in particular on | + + + | | | + + .. image:: images/Object_6.svg + | | :width: 17 | + + :height: 15 + | | | + + + | | | + + + | | | + +, the characteristic speed of the flow, as well as on the frequency of the movement through the Laplace variable + | | | + + + | | .. image:: images/Object_7.svg | + + :width: 17 + | | :height: 15 | + + + | | | + + + | | | + + + | | | + + + | |. | +-----+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+ In principle, :math:`[{B}_{\mathit{ij}}(U,s)]` is any matrix whose extra-diagonal terms, if they are not zero, introduce a coupling between modes. On the other hand, the terms of :math:`[{B}_{\mathit{ij}}(U,s)]` evolve in a non-linear manner with the complex frequency :math:`s`. Each fluid-elastic force model is associated with a specific transfer matrix. In all cases, the formulation of the modal problem under flow can be characterized by the relationship [:external:ref:`éq. 1.2-1 <éq. 1.2-1>`]. For the various types of configurations that can be simulated using the operator CALC_FLUI_STRU, the representations of fluid-elastic force transfer matrices are explained in the rest of this document.