5. Harmonic response by classical dynamic substructuring#
5.1. Introduction#
For harmonic problems, the system under study is subject to any force that is spatially random, but sinusoidal in time. The shape of the load, the frequency of excitation and the modal properties each play an essential role. It is also necessary to take into account the dissipation in the solid, which can be translated by the introduction of a damping matrix. The method of harmonic 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 classically. Finally, all that remains is to derive/to establish the overall solution by reconstitution.
5.2. 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 \({S}^{k}\), we can write:
: label: eq-5
{M} ^ {k} {ddot {q}}} ^ {k}} ^ {k} + {C} ^ {k}} ^ {k} + {K} ^ {k} {q} {q} ^ {k} {q} ^ {k} = {f} ^ {k} = {f} {q} {q} {q} {q} {q} {q} {q} {q} {q} {q} ^ {k} = {f} ^ {k} = {f} ^ {k} = {f} ^ {k} = {f} ^ {k} = {f} ^ {k} = {f} {k}
In a harmonic problem, a dynamic loading is imposed, spatially random, but sinusoidal with a pulse \(\omega\) in time. We are then interested in the stabilized response of the system, without taking into account the transitory part.
The field of external forces is written as:
\({f}_{\mathrm{ext}}^{k}(t)=\left\{{f}_{\mathrm{ext}}^{k}\right\}{e}^{j\omega t}\)
The field of bonding forces is written as:
\({f}_{L}^{k}(t)=\left\{{f}_{L}^{k}\right\}{e}^{j\omega t}\)
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 of motion:
: label: eq-5
({K} ^ {k} +jomega {C} ^ {c} ^ {k} - {omega} ^ {k})left{{q} ^ {k}right}right} =left{{f}} =left{{f}} _left{{f}} =left{{f}}} =left{{f}} =left{{f}} =left{{f}} =left{{f}} =left{{f}} =left{{f}} =left{{f}} =left{{f}} =left{{f}}} =left{{f}}} =left{{f}}}}
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-5
{q} ^ {k} =left [begin {array} {cc} {cc} {varphi} ^ {k} & {psi} ^ {k}end {array}right]left{begin {array}{array} {c} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {array} {c} {eta} _ {eta} _ {eta}{eta} _ {eta} _ {eta} _ {j} ^ {k}end {array}right}rightarray} {array} {array} Phi} ^ {k} {eta} ^ {k}
\({\varphi }^{k}\) 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-5
({stackrel {} {K}}} ^ {k}} +jomega {stackrel {} {C}} ^ {k} - {omega} ^ {2} {stackrel {} {M} {M}} {M}} {M}}}} ^ {k})left{{stackrel {} {M}}} ^ {k})left{{stackrel {} {M}}} ^ {k})left{{stackrel {} {m}}} ^ {k})left{{stackrel {} {f})left{{stackrel {} {m}}} {M}}} ^ {k})left{} _ {mathrm {ext}} ^ {k}right} +left{{stackrel {} {f}}} _ {L} ^ {k}right}
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:
\({\stackrel{ˉ}{M}}^{k}=\left[\begin{array}{cc}\mathrm{Id}& {\varphi }^{{k}^{T}}{M}^{k}{\psi }^{k}\\ {\psi }^{{k}^{T}}{M}^{k}{\varphi }^{k}& {\psi }^{{k}^{T}}{M}^{k}{\psi }^{k}\end{array}\right]{\stackrel{ˉ}{K}}^{k}=\left[\begin{array}{cc}{\lambda }^{k}& {\varphi }^{{k}^{T}}{K}^{k}{\psi }^{k}\\ {\psi }^{{k}^{T}}{K}^{k}{\varphi }^{k}& {\psi }^{{k}^{T}}{K}^{k}{\psi }^{k}\end{array}\right]\)
In the case of the Craig-Bampton method, it is shown that the normal modes and the constrained modes are orthogonal with respect to the stiffness matrix whose extra-diagonal terms are, therefore, zero. However, this property is not used in the algorithm programmed in*Code_Aster*.
As a type of dissipation, we only consider viscous damping (this is the only one supported by the substructuring tools in Code_Aster). Two methods can be used to take this amortization into account:
Rayleigh damping applied at the elementary level, which consists in assuming that the elementary damping matrix \({C}_{e}\) associated with each finite element of the model is a linear combination of the elementary mass and stiffness matrices \({K}_{e}\) and \({M}_{e}\):
\({C}_{e}={\alpha }_{e}{K}_{e}+{\beta }_{e}{M}_{e}\)
The damping matrix is then assembled \({C}^{k}\) and then projected on the basis [eq]:
\({\stackrel{ˉ}{C}}^{k}=\left[\begin{array}{cc}{\varphi }^{{k}^{T}}{C}^{k}{\varphi }^{k}& {\varphi }^{{k}^{T}}{C}^{k}{\psi }^{k}\\ {\psi }^{{k}^{T}}{C}^{k}{\varphi }^{k}& {\psi }^{{k}^{T}}{C}^{k}{\psi }^{k}\end{array}\right]\)
the proportional damping applied to the dynamic natural modes of each substructure. The resulting matrix is therefore an incomplete diagonal (it is not possible to associate proportional damping with static deformations):
\({\stackrel{ˉ}{C}}^{k}=\left[\begin{array}{cc}{\xi }^{k}& 0\\ 0& 0\end{array}\right]\)
5.3. Dynamic equations verified by the global structure#
The dynamic equations that the global structure verifies are:
\(\begin{array}{}\left[\begin{array}{ccccc}{\stackrel{ˉ}{M}}^{1}& & & & \\ & \mathrm{...}& & & \\ & & {\stackrel{ˉ}{M}}^{k}& & \\ & & & \mathrm{...}& \\ & & & & {\stackrel{ˉ}{M}}^{{N}_{S}}\end{array}\right]\left\{\begin{array}{c}{\ddot{\eta }}^{1}\\ \mathrm{...}\\ {\ddot{\eta }}^{k}\\ \mathrm{...}\\ {\ddot{\eta }}^{{N}_{S}}\end{array}\right\}+\left[\begin{array}{ccccc}{\stackrel{ˉ}{C}}^{1}& & & & \\ & \mathrm{...}& & & \\ & & {\stackrel{ˉ}{C}}^{k}& & \\ & & & \mathrm{...}& \\ & & & & {\stackrel{ˉ}{C}}^{{N}_{S}}\end{array}\right]\left\{\begin{array}{c}{\dot{\eta }}^{1}\\ \mathrm{...}\\ {\dot{\eta }}^{k}\\ \mathrm{...}\\ {\dot{\eta }}^{{N}_{S}}\end{array}\right\}+\left[\begin{array}{ccccc}{\stackrel{ˉ}{K}}^{1}& & & & \\ & \mathrm{...}& & & \\ & & {\stackrel{ˉ}{K}}^{k}& & \\ & & & \mathrm{...}& \\ & & & & {\stackrel{ˉ}{K}}^{{N}_{S}}\end{array}\right]\left\{\begin{array}{c}{\eta }^{1}\\ \mathrm{...}\\ {\eta }^{k}\\ \mathrm{...}\\ {\eta }^{{N}_{S}}\end{array}\right\}\\ =\left\{\begin{array}{c}{\stackrel{ˉ}{f}}_{\mathrm{ext}}^{1}\\ \mathrm{...}\\ {\stackrel{ˉ}{f}}_{\mathrm{ext}}^{k}\\ \mathrm{...}\\ {\stackrel{ˉ}{f}}_{\mathrm{ext}}^{{N}_{S}}\end{array}\right\}+\left\{\begin{array}{c}{\stackrel{ˉ}{f}}_{L}^{1}\\ \mathrm{...}\\ {\stackrel{ˉ}{f}}_{L}^{k}\\ \mathrm{...}\\ {\stackrel{ˉ}{f}}_{L}^{{N}_{S}}\end{array}\right\}\end{array}\)
or in harmonic:
\(\begin{array}{}(\left[\begin{array}{ccccc}{\stackrel{ˉ}{\stackrel{ˉ}{K}}}^{1}& & & & \\ & {}^{\text{.}\text{.}\text{.}}& & & \\ & & {\stackrel{ˉ}{\stackrel{ˉ}{K}}}^{k}& & \\ & & & {}^{\text{.}\text{.}\text{.}}& \\ & & & & {\stackrel{ˉ}{\stackrel{ˉ}{K}}}^{{N}_{s}}\end{array}\right]+j\omega \left[\begin{array}{ccccc}{\stackrel{ˉ}{\stackrel{ˉ}{C}}}^{1}& & & & \\ & {}^{\text{.}\text{.}\text{.}}& & & \\ & & {\stackrel{ˉ}{\stackrel{ˉ}{C}}}^{k}& & \\ & & & {}^{\text{.}\text{.}\text{.}}& \\ & & & & {\stackrel{ˉ}{\stackrel{ˉ}{C}}}^{{N}_{s}}\end{array}\right]-{\omega }^{2}\left[\begin{array}{ccccc}{\stackrel{ˉ}{\stackrel{ˉ}{M}}}^{1}& & & & \\ & {}^{\text{.}\text{.}\text{.}}& & & \\ & & {\stackrel{ˉ}{\stackrel{ˉ}{M}}}^{k}& & \\ & & & {}^{\text{.}\text{.}\text{.}}& \\ & & & & {\stackrel{ˉ}{\stackrel{ˉ}{M}}}^{{N}_{s}}\end{array}\right])\left\{\begin{array}{c}{\eta }^{1}\\ {}^{\text{.}\text{.}\text{.}}\\ {\eta }^{k}\\ {}^{\text{.}\text{.}\text{.}}\\ {\eta }^{{N}_{s}}\end{array}\right\}\\ =\left\{\begin{array}{c}{\stackrel{ˉ}{f}}_{\mathrm{ext}}^{1}\\ {}^{\text{.}\text{.}\text{.}}\\ {\stackrel{ˉ}{f}}_{\mathrm{ext}}^{k}\\ {}^{\text{.}\text{.}\text{.}}\\ {\stackrel{ˉ}{f}}_{\mathrm{ext}}^{{N}_{s}}\end{array}\right\}+\left\{\begin{array}{c}{\stackrel{ˉ}{f}}_{L}^{1}\\ {}^{\text{.}\text{.}\text{.}}\\ {\stackrel{ˉ}{f}}_{L}^{k}\\ {}^{\text{.}\text{.}\text{.}}\\ {\stackrel{ˉ}{f}}_{L}^{{N}_{s}}\end{array}\right\}\end{array}\)
To which must be added 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-5
begin {array} {} (stackrel {} {K} {K} +j {omega} _ {o}stackrel {} {C} - {omega} _ {o} ^ {2}stackrel {} {2}stackrel {} {2}\ stackrel {2}stackrel {2}stackrel {} {M}) {stackrel {M}) {stackrel {stackrel} {M}) {stackrel {stackrel}}} _ {o} ^ {T}lambda = {{stackrel {stackrel {} {f}} _ {text {ext}}}}}}}} _ {o}\ L {{lambda}}} _ {o} =0end {array} =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 harmonic problem, of reduced size, to which a linear constraint equation is associated. Its resolution is therefore not a problem.
5.4. Implemented in Code_Aster#
5.4.1. Study of substructures separately#
The parameters \({\alpha }_{e}\) and \({\beta }_{e}\) of Rayleigh damping are introduced, where appropriate, by the operator DEFI_MATERIAU [U4.43.01].
The treatments of the substructures are identical to the case of modal calculation.
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. Rayleigh damping is taken into account by completing operand MATR_AMOR. Proportional damping is introduced by operand AMOR_REDUIT.
Harmonic loading is defined, at the substructure level, by the operators AFFE_CHAR_MECA [U4.44.01] (application of force on the mesh), CALC_VECT_ELEM [U4.61.02] (calculation of the associated elementary vectors) and ASSE_VECTEUR [U4.61.23] (assembly of the load vector on the substructure mesh).
5.4.2. Assembly and resolution#
As in the case of modal calculation, the complete 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 loads are projected onto the bases of the substructures to which they are applied, then assembled using the numbering from NUME_DDL_GENE [U4.65.03] by the operator ASSE_VECT_GENE [U4.65.05].
The harmonic response of the complete structure is calculated by the operator DYNA_LINE_HARM [U4.53.11].
5.4.3. 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] and possibly the operator DEFI_SQUELETTE [U4.24.01] (creation of a « skeleton » mesh).
5.5. Conclusion#
The method for calculating harmonic response by modal synthesis available in Code_Aster is based on that of modal substructuring, which is also programmed. It consists in expressing all the equations in a space of reduced dimension, made up of modes of the various substructures, by a Rayleigh-Ritz method. The definition of these fields is the one used for modal substructuring and includes normal modes as well as other static or harmonic modes. The procedure used results in a projection of the matrices and the second member onto the restricted space.