Dynamic problem
==================

This problem is that of calculating the evolution of a cable structure.




Iterative time integration algorithm
--------------------------------------------

The discretized form of [eq] and [eq] complete, which must be satisfied at every node and at every moment is:

 :math:`{F}_{\text{int}}(t)-{F}_{\text{iner}}(t)={F}_{\text{ext}}(t)` [:ref:`11 <11>`]

The time integration algorithm is of the Newmark [bib] and [bib] type. Suppose that the state of the cable (fields :math:`u,\dot{u}\mathrm{et}\ddot{u}` at the nodes) is known at time :math:`t` and that we have just calculated an approximate value of these fields at the nth iteration of time :math:`t+\Delta t`.


* If these values satisfy [eq], except for one tolerance, they are taken as field values at time :math:`t+\Delta t`.


* Otherwise, we are looking for displacement correction :math:`\Delta {u}^{n+1}`, to which, according to the Newmark algorithm, the speed and acceleration corrections correspond to:


*

:math:`\Delta {\dot{u}}^{n+1}=\frac{\gamma }{\beta \Delta t}\Delta {u}^{n+1}`

and

:math:`\Delta {\ddot{u}}^{n+1}=\frac{1}{\beta \Delta {t}^{2}}\Delta {u}^{n+1}`

such as:

:math:`\left[{K}_{M}^{n}+{K}_{G}^{n}+\frac{1}{\beta \Delta {t}^{2}}M\right]\Delta {u}^{n+1}={F}_{\text{ext}}(t+\Delta t)-{F}_{\text{int}}^{n}(t+\Delta t)+{F}_{\text{iner}}^{n}(t+\Delta t)`

In cable movement analysis, the Newmark algorithm may be unstable. This is why we use the algorithm called HHT, defined in [:ref:`bib5, bib7 <bib5, bib7>`], in which the two Newmark parameters are functions of a third parameter :math:`\alpha`:

:math:`\begin{array}{c}\gamma =\frac{1}{2}-\alpha \\ \beta =\frac{{(1-\alpha )}^{2}}{4}\\ \alpha \le 0\end{array}`

For :math:`\alpha =0`, the algorithm is that of Newmark, known as the "trapezium rule". But for :math:`\alpha` being slightly negative, in practice: :math:`\alpha \ge -0.3`, there appears to be numerical amortization that stabilizes the calculation.

Determining the initial acceleration and initializing the fields at the start of a new time step are shown in [bib].




Comparison of short circuit calculations and tests
-----------------------------------------------------

To validate this cable modeling, we compared dynamic calculations by*Code_Aster* to short circuit test results [:ref:`bib8 <bib8>`]. These were carried out at the EDF Electrical Engineering Laboratory on an experimental structure representative of station configurations [Figure]. Three cables stretched between two gantries distant from :math:`\mathrm{102 }m` are shorted, in the foreground, by a shunt arranged on insulating columns.

At the other gantry, they are powered by a three-phase current of :math:`35\mathrm{kA}` during :math:`\mathrm{250 }\mathrm{ms}`. The evolution has been recorded:


* from the tension of the cables to their anchoring on the gantries, using dynamometers;


* the movement of the mid-points of the ranges, identified by signaling spheres, using fast cameras. We see the glass cage of one of these cameras mounted on a gantry, to the left of [Figure].


.. image:: images/100000000000086F00000ADB5EEEB0DEEA7E928E.png
    :width: 5.5055in
    :height: 7.0819in


.. _RefImage_100000000000086F00000ADB5EEEB0DEEA7E928E.png:

Figure 3: Overview of the short circuit test facility

The [Figure] gives the comparison for an anchoring tension and for the displacement of the middle of a cable.


.. image:: images/100000000000017B00000262BC7426500EFF2A8A.png
    :width: 3.9425in
    :height: 6.3382in


.. _RefImage_100000000000017B00000262BC7426500EFF2A8A.png:

Figure 4: Comparisons of *Code_Aster* calculations and short circuit tests