Benchmark solution
=====================

Calculation method used for the reference solution
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The aim is to compare the response of a symmetric system consisting of two mass-spring systems that are identical to the response of a mass-spring system. Both problems, described in detail in reference [:ref:`bib2 <bib2>`], are addressed by the same accelerogram.

First, the natural frequencies :math:`{f}_{i}`, the associated eigenvectors normalized with respect to the modal mass :math:`{\Phi }_{\text{Ni}}` and the static modes :math:`\Psi` of the system (analytical values) are calculated. The generalized response of the multi-supported system is then calculated by analytically solving the Duhamel integral [:ref:`bib1 <bib1>`]. Finally, the relative displacement of the shock nodes is restored on a physical basis, which allows us, after having calculated the field of the training displacements, to calculate the field of the absolute displacements.

We calculate function :math:`\mathit{diff}` defined as being the difference between the absolute movement of the shocking node on a mobile obstacle and that of the shocking node on a fixed obstacle. We check that it is really zero for various moments.

For G modeling, only problem 2 is modeled. There are no reference solutions other than material data (discharge criterion and stiffness).


Benchmark results
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Relative and absolute displacements at shock nodes.


Uncertainty about the solution
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Comparison between two equivalent models.


Bibliographical references
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1. J.S. PRZEMIENIECKI: Theory of matrix structural analysis New York, MacGraw-Hill, 1968, pp. 351-357.


2. Fe. WAECKEL: Use and validation of developments made to calculate the seismic response of multi-supported structures — HP52 /96.002.