2. Theoretical reminder#
This structural modification technique is based on the substructuring method. The first substructure corresponds to the initial structure and the second substructure corresponds to the modification made.
The initial structure is modelled on the basis of its own modes identified experimentally. Except in very specific cases, the measurement points are not located at the level of the interface between the initial structure and the modification to be made. Before coupling these two substructures, it is therefore necessary to go through an intermediate step which consists in condensing the information measured at the interfaces.
The various models that are used during this calculation are: the « measurement » model, the « support » model and the « modification » model. We will describe these various models below.
To simplify the presentation, we use the following notations:
the index \(\mathit{mes}\) relates to measurement points,
index \(\text{int}\) relates to interface nodes,
the exponent « + » indicates the generalized inverse of a matrix,
\(L\): location matrix (observability matrix),
\(y\): displacement field,
\(\Phi\): matrix consisting of the expansion base vectors defined on the support model,
\({\eta }_{\text{sup}}\): generalized coordinates associated with the \(\Phi\) expansion base
\(\psi\): modal matrix identified (proper deformations) on the initial structure,
\({\eta }_{\mathrm{test}}\): generalized coordinates associated with \(\psi\)
2.1. The « measure » model#
We call the « measurement » model the modal model of the initial structure obtained from the information measured. On this model, a mesh consisting of measurement points is defined.
A measured field of movement can be projected on a basis composed of the identified natural modes, i.e.:
\({y}_{\mathrm{mes}}=\psi {\eta }_{\mathrm{test}}\) (1)
We are limited to the case of structures with low damping (for which damping can be modelled mode by mode), for which the modal model of the initial structure is written:
\(\left\{-{\omega }^{2}[\mathrm{Id}]+j\omega [{\beta }_{\mathrm{test}}]+[{\Omega }_{\mathrm{test}}^{2}]\right\}{\eta }_{\mathrm{test}}={f}_{{\eta }_{\mathrm{test}}}\) (2)
Where:
\([\mathrm{Id}]\) |
: |
identity matrix |
\([{\Omega }_{\mathrm{test}}^{2}]\) |
: |
spectral matrix (diagonal matrix whose diagonal terms are the natural pulsations raised to the square) |
\([{\beta }_{\mathrm{test}}]\) |
: |
generalized amortization matrix |
\({f}_{{\eta }_{\mathrm{test}}}\) |
: |
modal loading |
\(\omega\) |
: |
pulse of excitement |
2.2. The « support » model#
We call the « support » model the simplified numerical model of the initial structure.
This model should include the interfaces between the initial structure and the modification. It is used for the expansion and condensation of information measured at the interfaces.
The expansion consists in finding a field defined on this support model, whose restriction on the measurement points is as close as possible to the field measured experimentally. This field is obtained by minimizing the distance between the measured information and the estimated information on the support model.
The estimated displacement field \({\tilde{y}}_{\mathrm{mes}}\) at the measurement points is given by the following relationship:
\({\tilde{y}}_{\mathrm{mes}}={L}_{\mathrm{mes}}\Phi {\eta }_{\text{sup}}\) (3)
The estimated field \({\tilde{y}}_{\mathrm{mes}}\) is of the same type as the measured field \({y}_{\mathrm{mes}}\).
The generalized coordinates \({\eta }_{\text{sup}}\), unknown, are obtained by minimizing the distance between the measured field and the estimated field. Using the least squares technique, we obtain:
\({\eta }_{\text{sup}}={({L}_{\mathrm{mes}}\Phi )}^{\text{+}}{y}_{\mathrm{mes}}\) (4)
It is thus possible to estimate the field \(\tilde{y}\) defined at the nodes of the support model, consistent with the field measured experimentally.
\(\tilde{y}=\Phi {\eta }_{\text{sup}}=\Phi {({L}_{\mathrm{mes}}\Phi )}^{\text{+}}{y}_{\mathrm{mes}}\) (5)
The restriction of this field to interface degrees of freedom is:
\({\tilde{y}}_{\text{int}}={L}_{\text{int}}\tilde{y}={L}_{\text{int}}\Phi {({L}_{\mathrm{mes}}\Phi )}^{\text{+}}{y}_{\mathrm{mes}}={T}_{\mathrm{It}}{y}_{\mathrm{mes}}\) (6)
\({T}_{\mathrm{it}}\) can be defined as the transition matrix between the measurement points and the interface between the initial structure and the modification.
By replacing the measured field \({y}_{\mathrm{mes}}\) with its expression (1), we get:
\({\tilde{y}}_{\text{int}}={T}_{\mathrm{It}}{y}_{\text{int}}={T}_{\mathrm{It}}\Psi {\eta }_{\mathrm{test}}\) (7)
Either:
\({\eta }_{\mathrm{test}}={({T}_{\mathrm{It}}\Psi )}^{\text{+}}{\tilde{y}}_{\text{int}}\) (8)
If we write down: \(B={({T}_{\mathrm{It}}\Psi )}^{\text{+}}\), we get:
\({\eta }_{\mathrm{test}}=B{\tilde{y}}_{\text{int}}\) (9)
By multiplying equation (2) on the left by \({B}^{T}\), and by replacing \({\eta }_{\mathrm{test}}\) with its expression (9), the dynamic stiffness \({\stackrel{ˆ}{Z}}_{\mathrm{test}}\) of the initial structure is given by the following relationship:
\(\left\{-{\omega }^{2}{B}^{T}B+j\omega {B}^{T}[{\beta }_{\mathrm{test}}]B+{B}^{T}[{\Omega }_{\mathrm{test}}^{2}]B\right\}{\tilde{y}}_{\text{int}}={\stackrel{ˆ}{Z}}_{\mathrm{test}}{\tilde{y}}_{\text{int}}={f}_{\text{int}}^{I}\) (10)
With: \({f}_{\text{int}}^{I}\): force applied to the interface
2.3. The « modification » model#
We call the « modification » model the numerical model of the change made to the initial structure.
If we note \({Z}_{x}\), its dynamic stiffness, by partitioning the unknowns into internal degrees of freedom \({y}_{i}\) and external degrees of freedom \({y}_{e}\) (which are composed of interface degrees of freedom), the modification model can be rewritten as follows:
\({Z}_{x}q=\left[\begin{array}{cc}{Z}_{\mathrm{ii}}& {Z}_{\mathrm{ie}}\\ {Z}_{\mathrm{ei}}& {Z}_{\mathrm{ee}}\end{array}\right]\left[\begin{array}{c}{y}_{i}\\ {y}_{e}\end{array}\right]=\left[\begin{array}{c}{f}_{i}^{M}\\ {f}_{e}^{M}+{f}_{\mathrm{eext}}^{M}\end{array}\right]\) (11)
Index \(i\) relates to internal degrees of freedom, index \(e\) relates to external degrees of freedom, and \({f}_{\mathrm{eext}}^{M}\) refers to external force applied to external degrees of freedom.
2.4. Coupling of the two substructures#
The dynamic stiffness of the coupled model is obtained by considering the continuity of the movements at the interfaces:
\({\tilde{y}}_{\text{int}}\mathrm{=}{y}_{e}\) (12)
and the balance of efforts at the interfaces:
\({f}_{\text{int}}^{I}+{f}_{e}^{M}\mathrm{=}0\) (13)
That is: \(\left[\begin{array}{cc}{Z}_{\mathrm{ii}}& {Z}_{\mathrm{ie}}\\ {Z}_{\mathrm{ei}}& {Z}_{\mathrm{ee}}+{\stackrel{ˆ}{Z}}_{\mathrm{test}}\end{array}\right]\left[\begin{array}{c}{y}_{i}\\ {y}_{e}\end{array}\right]=\left[\begin{array}{c}{f}_{i}^{M}\\ {f}_{\mathrm{eext}}^{M}\end{array}\right]\) (14)
The resolution of this system makes it possible to calculate the behavior of the modified structure.
Note: in section 4, the interface compatibility equations are written on a modal basis, i.e. equations (12) and (13) are not verified exactly: they are projected onto the subspace composed of the base of the extended modes of the structure:
\({\Phi }^{T}{\tilde{y}}_{\text{int}}\mathrm{=}{\Phi }^{T}{y}_{e}\) (15)
\({\Phi }^{T}{f}_{\text{int}}^{I}+{\Phi }^{T}{f}_{e}^{M}\mathrm{=}0\) (16)
Since there are generally more degrees of interface freedom than eigenmodes used, relationships (15) and (16) cannot be ensured exactly.
2.5. Retro-projection of the field onto the measurement points#
The field can be evaluated at the points where the measurement was carried out, by retro-projection of the field at the interfaces obtained on the coupled model.
Using relationships (1), (9) and (12), the field at the measurement points is written:
\({y}_{\mathrm{mes}}=\Psi {\eta }_{\mathrm{test}}=\Psi B{\tilde{y}}_{\text{int}}=\Psi B{y}_{e}\) (17)
2.6. Expansion base quality control#
The quality of the results obviously depends on the uncertainty in the measurement, but also on the ability of the expansion base to represent the behavior of the real structure and the transmission of information at the interface level.
The user’s physical sense influences the choice of this expansion base. It is difficult to know at first glance the size of the expansion base allowing to obtain the best estimate of the coupled behavior. The quality of the expansion base can be measured by re-estimating the field at the interface by static expansion on the initial model from the field resulting from the coupled model.
This indicator does not make it possible to directly estimate the quality of the results obtained on the coupled model, but it indicates the relevance of the reconstruction of the field at the interface.