1. Principle of calculation#

1.1. Determination of a modal base of the system under flow and projection of the excitation#

The calculation of the dynamic response of a system to a turbulent excitation induced by a fluid flow is carried out by respecting the following steps:

first, the modal base of the non-flow system is calculated using the operator CALC_MODES [U4.52.02],

the characteristics of the configuration studied are then defined, to take into account the fluid-structure coupling phenomenon, using the operator DEFI_FLUI_STRU [U4.25.01]. This operator makes it possible, for example, to fill in the speed profiles associated with the fluid excitation zones, for configurations such as “bundle of tubes under cross flow”. It produces a concept of type [type_flui_stru] intended to be used by operators implemented downstream in the command file,

the modal characteristics of the system under flow are then calculated using the operator CALC_FLUI_STRU [U4.66.02]. At the output, we have a modal base for each flow speed,

the definition of turbulent excitation is then done by a call to the operator DEFI_SPEC_TURB [U4.44.31]. The available models are as follows:

  • « correlation length » spectra, specific to configurations of the « bundle of tubes under transverse flow » type, for application to vibrations of GV tubes. The corresponding factor keywords are SPEC_LONG_COR_1, SPEC_LONG_COR_2,, SPEC_LONG_COR_3, and SPEC_LONG_COR_4. These spectra are predefined; however, the user can adjust the settings. This part is developed in paragraph [§2.2],

  • distributed turbulent excitation model. The corresponding keyword factor is SPEC_FONC_FORME. The excitation spectrum is defined by its decomposition over a family of shape functions by providing, on the one hand, an interspectral matrix, and, on the other hand, a list of shape functions associated with this matrix. The associated [interspectrum] and [function] concepts must be generated in advance. In the case of the « control cluster » component, the user can also use a predefined turbulence spectrum, identified on model GRAPPE1. This part is developed in paragraph [§2.3],

  • localized turbulent excitation model. The corresponding keyword factor is SPEC_EXCI_POINT. It is used in the case of an excitation spectrum associated with one or more specific forces and moments. The definition of arousal is then done by providing:

  • an interspectral matrix of excitations (the associated [interspectrum] concept must be generated beforehand),

  • the list of the nodes for applying these excitations,

  • the nature of the excitation applied in each of these nodes (force or moment),

  • the directions of application of the excitations thus defined.

This part is developed in paragraph [§2.4].

The projection of the turbulent excitation spectrum previously defined, on the modal basis of the structure under flow, is then carried out using the operator PROJ_SPEC_BASE [U4.63.14].

1.2. Calculation of the response to turbulent excitation: frequency resolution#

1.2.1. Introduction#

The calculation of the frequency response of the structure or of the fluid-structure coupled system is carried out in three steps:

  1. calculation of modal excitation interspectra,

  2. calculation of modal response interspectra,

  3. recombination on the physical basis.

First, for each mode, the transfer function of the mechanical system is introduced (structure alone or fluid-structure coupled system). Each of the three steps above is then detailed.

1.2.2. Calculation of modal excitation interspectra#

The modal excitation interspectra \({S}_{\text{QiQj}}(f,U)\) are determined by projecting the turbulent excitation spectrum on the modal basis of the mechanical system (structure alone or fluid-structure coupled system). This projection step is detailed in paragraph [§2] for the various models applicable to wire structures.

1.2.3. Calculation of modal response interspectra#

The modal displacement interspectra \({S}_{\text{qiqj}}(f,U)\) are then deduced from the modal excitation interspectra \({S}_{\text{QiQj}}(f,U)\) using the following relationship:

\({S}_{{q}_{i}{q}_{j}}(f,U)={H}_{i}^{\ast }(f,U){S}_{\text{QiQj}}(f,U){H}_{j}(f,U)\) eq 1.2.3-1

where \({H}_{i}^{\text{*}}(f,U)\) designates the conjugate complex of the transfer function \({H}_{i}(f,U)\) of the mechanical system in question. Given a frequency \(f\) and a flow speed \(U\), the transfer function \({H}_{i}(f,U)\) of the mechanical system for mode \(i\) is defined by:

\({H}_{i}(f,U)=\frac{1}{{M}_{i}{\omega }_{i}^{2}(-{(\frac{f}{{f}_{i}})}^{2}+\mathrm{2j}{\xi }_{i}(\frac{f}{{f}_{i}})+1)}\) eq 1.2.3-2

where \({M}_{i}\) designates the modal mass of mode \(i,{\omega }_{i}\) and \({f}_{i}\) respectively denote, at speed \(U\), the pulsation and the natural frequency of mode \(i,{\xi }_{i}\) designates, at speed \(U\), the reduced damping of mode \(i\), and \(J\) designates the complex number such as \({J}^{2}\text{= -}1\).

The calculation of the modal displacement interspectra from the modal excitation interspectra and transfer functions is performed using the operator DYNA_SPEC_MODAL [U4.53.23].

In particular, we deduce from [éq 1.2.3-2] the relationship linking the modal shift autospectra to the modal excitation autospectra:

\({S}_{\text{qiqi}}(f,U)={\mid {H}_{i}(f,U)\mid }^{2}{S}_{\text{QiQi}}(f,U)\) eq 1.2.3-3

where \({\mid {H}_{i}(f,U)\mid }^{2}\) is the square of the module of \({H}_{i}(f,U)\)

1.2.4. Recombination on a physical basis#

Given a flow speed \(U\), the physical displacement interspectrum \({S}_{{u}_{1}{u}_{2}}({x}_{1},{x}_{2},f)\) at the abscissa points \({x}_{1}\) and \({x}_{2}\), at the frequency \(f\), is obtained by modal recombination. This operation is written as:

\({S}_{{u}_{1}{u}_{2}}({x}_{1},{x}_{2},f)=\sum _{i=1}^{N}\sum _{j=1}^{N}{\phi }_{i}({x}_{1}){\phi }_{j}({x}_{2}){S}_{{q}_{i}{q}_{j}}(f,U)\) eq 1.2.4-1

Where \(N\) designates the number of modes in the base; \({\phi }_{i}({x}_{k})\) is the component at the discretization point \({x}_{k}\) of the deformation of the \(i\) th mode along the spatial direction in question.

Recombination on a physical basis is carried out using the operator REST_SPEC_PHYS [U4.63.22]. The space direction in question is specified at the time of calling this operator.

1.2.5. Statistical elements#

Modal variance \({\sigma }_{i}^{2}(U)\), associated with speed \(U\), is expressed as follows:

\({\sigma }_{i}^{2}(U)=2\underset{0}{\overset{\infty }{\int }}{S}_{{q}_{i}{q}_{j}}(f,U)\text{df}\) eq 1.2.5-1

At flow speed \(U\), the response value RMS \({\sigma }_{\text{RMS}}(x)\) at a point \(x\) of the structure is given by:

\({\sigma }_{\text{RMS}}(x)=\sqrt{\sum _{i=1}^{N}{\phi }_{i}^{2}(x){\sigma }_{i}^{2}(U)}\) eq 1.2.5-2

Where \(N\) designates the number of modes in the base and \({\phi }_{i}(x)\) is the component at point \(x\) of the deformation of the \(i\) th mode in the direction of space considered.

This operation is performed by the operator POST_DYNA_ALEA [U4.84.04].

1.3. Calculation of the response to turbulent excitation: temporal resolution#

The temporal resolution takes place according to the sequence of the following operations:

1.3.1. Interspectral density factorization#

The operator GENE_FONC_ALEA [U4.36.05] realizes the factorization of the interspectral density of modal excitations \({S}_{\text{QiQj}}(f,U)\), before applying the Monte Carlo method.

1.3.2. Generation of random modal excitations#

The operator GENE_FONC_ALEA [U4.36.05] generates random modal excitations \({Q}_{i}(t)\) by performing draws using the Monte Carlo method. The RECU_FONCTION [U4.32.03] operator allows you to retrieve each of the \({Q}_{i}(t)\) evolutions.

1.3.3. Modifying a modal base and projecting#

The operator MODI_BASE_MODALE [U4.66.21] modifies the modal base of the structure by substituting for the initial characteristics those obtained for a given flow speed.

The operator PROJ_MATR_BASE [U4.63.12] allows the projection of the mass and stiffness matrices assembled on the new modal base previously defined.

1.3.4. Definition of obstacles#

The definition of the geometry of the obstacles is carried out, if necessary, using the operator DEFI_OBSTACLE [U4.44.21].

1.3.5. Dynamic resolution#

The dynamic transient calculation for mode \(i(1\le i\le N)\) is performed using a numerical integration diagram with the DYNA_TRAN_MODAL [U4.53.21] operator.

\({M}_{\text{ii}}{\ddot{q}}_{i}(t)+{C}_{\text{ii}}{\dot{q}}_{i}(t)+{K}_{\text{ii}}{q}_{i}(t)={Q}_{i}(t)\) eq 1.3.5-1

Where \({M}_{\text{ii}},{C}_{\text{ii}}\text{et}{K}_{\text{ii}}\) respectively designate the generalized mass, damping, and stiffness associated with the \(i\) th mode; \({q}_{i}(t)\text{et}{Q}_{i}(t)\) respectively designate the generalized displacement and excitation associated with the \(i\) th mode.

1.3.6. Ritz screening#

The reproduction on a physical basis is carried out using a Ritz projection:

\(U(x,t)=\sum _{i=1}^{N}{u}_{i}(x){q}_{i}(t)\) eq 1.3.6-1

\(U(x,t)\) refers to the assembled vector of physical displacements; \({u}_{i}(x)\) is the assembled vector defining the \(i\) th modal form and \({q}_{i}(t)\) the generalized displacement following the \(i\) th mode.

This last operation is performed using the REST_GENE_PHYS [U4.63.31] operator.