Reference solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

This test is developed in detail in reference [:ref:`bib1 <bib1>`].

The fundamental dynamic equation, in relative motion of the mass-spring system with respect to the ground, is written as: :math:`\ddot{x}+\frac{k(x)}{m}x=\gamma (t)`.

For a displacement of the form :math:`x=\mathrm{a.sin}(\omega t)` and :math:`\ddot{x}=-a{\omega }^{2}\mathrm{sin}(\omega t)`, we obtain from the equation of motion the shape of the accelerogram:

:math:`\gamma (t)=a\mathrm{sin}(\omega t)\left[-{\omega }^{2}+\frac{{k}_{0}}{m}(1-\frac{\mid a\mathrm{sin}(\omega t)\mid }{{x}_{0}})\right]`

The fundamental frequency :math:`{f}_{0}` of the undamped oscillator is :math:`{f}_{0}=\frac{1}{2\pi }\sqrt{\frac{{k}_{0}}{m}}`.


Benchmark results
----------------------

Fundamental frequency :math:`{f}_{0}` of the undamped oscillator.

Displacements relating to the moments 2, 6, 10, 10, 14 and 18 seconds.


Uncertainty about the solution
---------------------------

None if we calculate the Duhamel integral analytically [:ref:`bib2 <bib2>`].


Bibliographical references
---------------------------


1. P. LALUQUE, P. LABBE, S. PETETIN and A. TIXIER: Seismic response of a reactor building PWR1300 taking into account the separation between the foundation and the ground. Note SEPTEN TA83 .06 (May 1984).


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