4. Modal calculation by classical dynamic substructuring#
4.1. Introduction#
For modal problems, the system under study is not subject to any external forces. The dissipation in the solid is not taken into account. The method of modal calculation by substructuration, programmed in Code_Aster, which allows the global problem to be replaced by a simplified problem, proceeds in four stages. First of all, natural modes and static deformations are calculated on each of the substructures composing the system. Then, the global problem is projected onto these fields, and the couplings between the substructures are taken into account, at the level of their interfaces. The reduced problem obtained can then be solved in a conventional manner. Finally, all that remains is to derive/to establish the overall solution by reconstitution.
4.2. Dynamic equations verified by the substructures separately#
We are going to consider a \(S\) structure composed of \({N}_{S}\) substructures denoted by \({S}^{k}\). We assume that each substructure is modelled in finite elements. We have seen that in a dynamic substructuring calculation, the vibratory behavior of the substructures results from the external forces applied to them, and from the bonding forces exerted on them by the other substructures. So, at the level of substructure \({S}^{k}\), and in the case of modal calculation we can write:
: label: eq-4
{M} ^ {k} {ddot {q}}} ^ {k}} + {K} ^ {k} {q} ^ {k} = {f} _ {L} _ {L}} {L} ^ {k}
The field of bond forces is written in complex notation:
\({f}_{L}^{k}(t)=\left\{{f}_{L}^{k}\right\}{e}^{j\omegat }\)
The travel field is written as:
\({q}^{k}(t)=\left\{{q}^{k}\right\}{e}^{j\omegat }\)
The speed and acceleration fields are written as:
\({\dot{q}}^{k}(t)=j\omega \left\{{q}^{k}\right\}{e}^{j\omegat }\)
\({\ddot{q}}^{k}(t)=-{\omega }^{2}\left\{{q}^{k}\right\}{e}^{j\omegat }\)
Finally, substructure \({S}^{k}\) verifies the following equation:
: label: eq-4
({K} ^ {k} - {omega} ^ {2} {2} {M} ^ {k}){{q} ^ {k}} ={{f} _ {L} _ {L} ^ {k}}
The modal synthesis method consists in looking for the unknown field of displacement, resulting from finite element modeling, on an appropriate space, of reduced dimension (Ritz transformation). We saw that for each substructure, this space is composed of dynamic eigenmodes and static deformations:
: label: eq-4
{q} ^ {k} =left [begin {array} {array} {cc} {varphi} _ {i} ^ {k} & {psi} _ {j} ^ {k}end {array}right]right]right]leftleft{begin {array} {array} {cc} {varphi} {c} ^ {array} {k}end {array}right]left{begin {array} {right]left{begin {array} {array} {right]left{begin {array} {array} {right]left{begin {array} {array} {right]left{begin {array} {array} {right]left{begin {array}}right} = {Phi} ^ {k} {eta} ^ {k}
\({\phi }^{k}\) are the modal vectors associated with the dynamic eigenmodes of \({S}^{k}\),
\({\psi }^{k}\) are the modal vectors associated with the static deformations of \({S}^{k}\),
\({\eta }_{i}^{k}\) is the vector of the generalized coordinates associated with the eigenmodes of \({S}^{k}\),
\({\eta }_{j}^{k}\) is the vector of the generalized coordinates associated with the static deformations of \({S}^{k}\),
\({\eta }^{k}\) is the generalized coordinate vector for \({S}^{k}\).
The equation [eq] is projected based on \({S}^{k}\) taking into account [eq]. This allows us to write:
: label: eq-4
({stackrel {} {K}}} ^ {k}} ^ {k}} - {omega} ^ {k} - {omega} ^ {k}){{eta} ^ {k}}} ^ {k}} ={{stackrel {} {f}}} _ {L} ^ {k}}){{eta} ^ {k}} ={{k}} ={{stackrel {} {f}}}
Assuming that dynamic eigenmodes and static deformations are organized as shown by the formula [eq] and considering that the eigenvectors associated with dynamic modes are normalized with respect to the unit modal mass, the generalized mass and stiffness matrices take the following form:
\({\stackrel{ˉ}{M}}^{k}=\left[\begin{array}{cc}\mathrm{Id}& {\varphi }^{{k}^{T}}{M}^{k}{\psi }^{k}\\ {\psi }^{{k}^{T}}{M}^{k}{\varphi }^{k}& {\psi }^{{k}^{T}}{M}^{k}{\psi }^{k}\end{array}\right]{\stackrel{ˉ}{K}}^{k}=\left[\begin{array}{cc}{\lambda }^{k}& {\varphi }^{{k}^{T}}{K}^{k}{\psi }^{k}\\ {\psi }^{{k}^{T}}{K}^{k}{\varphi }^{k}& {\psi }^{{k}^{T}}{K}^{k}{\psi }^{k}\end{array}\right]\)
where:
\(\mathrm{Id}\) is the Identity matrix,
\({\lambda }^{k}\) is the diagonal matrix of the squares of the natural pulsations of the base.
In the case of the Craig-Bampton method, it is shown that the normal modes and the constrained modes are orthogonal to the stiffness matrix whose extra-diagonal terms are, therefore, zero. However, this property is not used in the algorithm programmed in*Code_Aster*.
4.3. Dynamic equations verified by the global structure#
The dynamic equations that the global structure verifies are:
\((\left[\begin{array}{ccccc}{\stackrel{ˉ}{\stackrel{ˉ}{K}}}^{1}& & & & \\ & {}^{\text{.}\text{.}\text{.}}& & & \\ & & {\stackrel{ˉ}{\stackrel{ˉ}{K}}}^{k}& & \\ & & & {}^{\text{.}\text{.}\text{.}}& \\ & & & & {\stackrel{ˉ}{\stackrel{ˉ}{K}}}^{{N}_{s}}\end{array}\right]-{\omega }^{2}\left[\begin{array}{ccccc}{\stackrel{ˉ}{\stackrel{ˉ}{M}}}^{1}& & & & \\ & {}^{\text{.}\text{.}\text{.}}& & & \\ & & {\stackrel{ˉ}{\stackrel{ˉ}{M}}}^{k}& & \\ & & & {}^{\text{.}\text{.}\text{.}}& \\ & & & & {\stackrel{ˉ}{\stackrel{ˉ}{M}}}^{{N}_{s}}\end{array}\right])\left\{\begin{array}{c}{\eta }^{1}\\ {}^{\text{.}\text{.}\text{.}}\\ {\eta }^{k}\\ {}^{\text{.}\text{.}\text{.}}\\ {\eta }^{{N}_{s}}\end{array}\right\}=\left\{\begin{array}{c}{\stackrel{ˉ}{f}}_{L}^{1}\\ {}^{\text{.}\text{.}\text{.}}\\ {\stackrel{ˉ}{f}}_{L}^{k}\\ {}^{\text{.}\text{.}\text{.}}\\ {\stackrel{ˉ}{f}}_{L}^{{N}_{s}}\end{array}\right\}\)
To which, we must add the link equations (according to [eq]):
\(\forall k,l\text{}{L}_{{S}^{k}\cap {S}^{l}}^{k}\left\{{\eta }^{k}\right\}={L}_{{S}^{k}\cap {S}^{l}}^{l}\left\{{\eta }^{l}\right\}\)
This system is solved by double dualization of boundary conditions [R3.03.01]. Its final formulation therefore involves the vector of Lagrange multipliers \(\lambda\) and can be written in the condensed form:
: label: eq-4
begin {array} {c} (stackrel {}} {mathrm {K}}}mathrm {-} {omega} ^ {2}stackrel {} {mathrm {M}}})mathrm {M}}})mathrm {M}}})mathrm {M}}})mathrm {M}}})mathrm {M}}})mathrm {M}}})mathrm {M}}})mathrm {M}}})mathrm {M}}})mathrm {M}}})mathrm {M}}})mathrm {M}}})mathrm {M}}})lambdamathrm {=} 0\mathrm {L}mathrm {L}mathrm {}lambdamathrm {}}mathrm {=} 0end {array}
The problem defined by equation [eq] is symmetric. On the other hand, its dimension is determined by the number of modes taken into consideration (dynamic modes and static deformations). It is therefore necessary to solve a classical modal problem, of reduced size, to which a linear constraint equation is associated. Its resolution is therefore not a problem.
4.4. Implemented in Code_Aster#
4.4.1. Study of substructures separately#
Dynamic eigenmodes are calculated with the operator CALC_MODES [U4.52.02]. The conditions at the link interfaces are applied with the operator AFFE_CHAR_MECA [U4.44.01]. The operator DEFI_INTERF_DYNA [U4.64.01] allows you to define the connection interfaces of the substructure. The operator DEFI_BASE_MODALE [U4.64.02] allows you to calculate the complete projection base of the substructure.
The operator MACR_ELEM_DYNA [U4.65.01] calculates the generalized stiffness, mass, and possibly damping matrices of the substructure, as well as the bond matrices.
4.4.2. Assembly and resolution#
The full structure model is defined by the operator DEFI_MODELE_GENE [U4.65.02]. Its numbering is done by the operator NUME_DDL_GENE [U4.65.03]. The generalized mass, stiffness and possibly damping matrices of the complete structure are assembled according to this numbering with the operator ASSE_MATR_GENE [U4.65.04].
The calculation of the eigenmodes of the complete structure is performed by the operator CALC_MODES [U4.52.02].
4.4.3. Restitution on a physical basis#
The return of results on a physical basis is identical to the case of modal calculation. It involves the operator REST_GENE_PHYS [U4.63.31].
To reduce the duration of graphic processing during visualizations, it is possible to create a coarse mesh using the operator DEFI_SQUELETTE [U4.24.01]. This mesh, ignored during the calculation, is used to support visualizations.