6. Description of document versions

Version Aster	Author (s) Organization (s)	Description of changes
6	G. JACQUART * (EDF R&D/ AMV)	Initial text
7.4	L. RATIER, G. JACQUART * (EDF R&D/ AMA, EDF/CNPE by Tricastin)

Consider the following discrete system with three masses:

The stiffness and mass matrices are:

\(M=(\begin{array}{ccc}m& 0& 0\\ 0& m& 0\\ 0& 0& m\end{array})\text{}K=(\begin{array}{ccc}k& -k& 0\\ -k& \mathrm{2k}& -k\\ 0& -k& \mathrm{2k}\end{array})\)

That is: \({\omega }_{0}^{2}=\frac{k}{m}\)

The specific modes and their pulsation are worth:

\(\begin{array}{c}\begin{array}{}{\omega }_{1}^{2}=\mathrm{0,}\text{198}{\omega }_{0}^{2}\text{,}{m}_{1}=\mathrm{1,}\text{841}\text{}m\text{,}{\Phi }_{1}=(\begin{array}{c}1\\ \mathrm{0,}\text{802}\\ \mathrm{0,}\text{445}\end{array})\\ \end{array}\\ \begin{array}{}{\omega }_{2}^{2}=\mathrm{1,}\text{555}{\omega }_{0}^{2}\text{,}{m}_{2}=\mathrm{2,}\text{863}\text{}m\text{,}{\Phi }_{2}=(\begin{array}{c}1\\ -\mathrm{0,}\text{555}\\ -\mathrm{1,}\text{247}\end{array})\\ \end{array}\\ {\omega }_{3}^{2}=\mathrm{3,}\text{247}{\omega }_{0}^{2}\text{,}{m}_{3}=\mathrm{9,}\text{296}\text{}m\text{,}{\Phi }_{3}=(\begin{array}{c}1\\ -\mathrm{2,}\text{247}\\ \mathrm{1,}\text{802}\end{array})\end{array}\)

Let’s compare the responses of the system modeled by a single eigenmode with or without static correction:

We note that static correction makes it possible to correct the low-frequency response, the model with 1 mode plus correction fits perfectly to the exact low-frequency solution. On the other hand, at high frequency (beyond the first mode), this correction leads to an enormous overestimation of the response. The use of static correction should therefore be used with caution and in the context of narrow-band excitation.

Let’s look at what the method of adding a static mode gives.

If a unit force is applied to point 1, the static deformation is equal to:

\({\Psi }_{s}=\frac{1}{k}(\begin{array}{c}3\\ 2\\ 1\end{array})\)

The projected mass and stiffness matrices that are obtained are as follows:

\(\stackrel{ˆ}{M}=(\begin{array}{c}\mathrm{1,}\text{841}m\frac{\mathrm{5,}\text{049}}{{\omega }_{0}^{2}}\\ \frac{\mathrm{5,}\text{049}}{{\omega }_{0}^{2}}\text{14}\text{}\frac{m}{{k}^{2}}\end{array}\text{})\text{}\text{et}\text{}\stackrel{ˆ}{K}=(\begin{array}{c}\mathrm{0,}\text{365}k\text{}1\\ \text{}1\text{}\frac{3}{k}\text{}\end{array})\)

The natural frequencies of this system are:

\({\omega }_{1}^{2}\mathrm{=}\mathrm{0,198}{\omega }_{0}^{2}\) and \({\omega }_{1}^{2}\mathrm{=}\mathrm{1,667}{\omega }_{0}^{2}\)

The response of the system modelled with a clean mode and a static mode is as follows:

We notice that we correct very well at low frequency (static correction effect), we model the dynamics of the system well beyond the first mode taken into account. On the other hand, the effect of the second mode is poorly represented (shift on the frequency).

Consider the following discrete system with three masses:

The stiffness and mass matrices are:

\(M=(\begin{array}{ccc}m& 0& 0\\ 0& m& 0\\ 0& 0& m\end{array})\text{}K=(\begin{array}{ccc}k& -k& 0\\ -k& \mathrm{2k}& -k\\ 0& -k& \mathrm{2k}\end{array})\)

Let’s make this system non-linear by adding an internal force term between \(\mathrm{x1}\) and \(\mathrm{x2}\) cubic:

\(F=k\text{.}{(\mathrm{x1}-\mathrm{x2})}^{3}\)

Let’s try to evaluate the response of this system to a forced excitation with a frequency close to the first natural frequency of the linear system (we chose \(\omega =\mathrm{0,}\text{18}{\omega }_{0}\)), with a large amplitude \({F}_{m}=3\text{.}k\).

In this configuration, the response of the system cannot be evaluated by the response of the linear system (the cubic term is far too important), it is necessary to implement a non-linear calculation with pseudo-forces as shown in [§3.2].

We can see in the report [bib8] the curves of the transient results of this method, taking into account one or 2 natural modes of the initial linear system.

With a single mode of its own, we notice that we are making a relatively large error (sometimes reaching 50%), on the other hand, it is satisfactory to note that the extremes of the vibrations are fairly well predicted. One could have hoped that by exciting below the first natural frequency it would have sufficed to have only one natural mode to model the response of the system, we can see here that this is not the case. As is often observed for non-linear systems, the system also responds with superharmonics of the excitation frequency.

On the other hand, by taking 2 natural modes to model the response of this structure at 3 ddl, we obtain a very satisfactory result (a few% error on the amplitude), for the eye it is difficult to distinguish the difference. This shows that by choosing a sufficiently rich projection base, it is possible, using a pseudoforce method, to model a complex dynamic system with non-linearities very well.