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


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

The problem is handled by a one-degree-of-freedom model. The column is considered to be an undamped and non-heavy slender beam of stiffness :math:`k=3E{I}_{Z}/{l}^{\mathrm{3 }}=\mathrm{3,942}{.10}^{\mathrm{7 }}N/m`. The superstructure located at the top of the column is modelled by a point mass :math:`m=\mathrm{43,8}{10}^{\mathrm{3 }}\mathrm{kg}`.

The two load cases lead to the calculation of the response of a system to a degree of freedom subjected to acceleration :math:`\gamma (t)` of any form:

:math:`\ddot{{x}_{r}}+{\omega }^{2}{x}_{r}=-\gamma (t)` with :math:`\omega =\sqrt{\frac{k}{m}}=\sqrt{\frac{3E{I}_{z}}{m{l}^{3}}}` the natural frequency of the system and :math:`{x}_{r}` the relative displacement of point :math:`B` with respect to point :math:`A`. The solution is obtained by integrating the Duhamel integral [:ref:`bib3 <bib3>`]:

:math:`{x}_{r}(t)=-\frac{m}{\omega }\underset{0}{\overset{t}{\int }}\gamma (t)\mathrm{sin}\omega (t-\tau )d\tau`


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

Displacement relative to point :math:`B`.

For a triangular imposed acceleration, we can calculate the Duhamel integral analytically [:ref:`bib3 <bib3>`]:

:math:`t<{t}_{0}`: :math:`{x}_{r}=-\frac{{P}_{0}}{{\omega }^{2}{t}_{0}}(t-\frac{\mathrm{sin}\omega t}{\omega })`

:math:`{t}_{0}<t<2{t}_{0}`: :math:`{x}_{r}=-\frac{{P}_{0}}{{\omega }^{2}{t}_{0}}(2{t}_{0}-t-\frac{2\mathrm{sin}\omega (t-{t}_{0})}{\omega }-\frac{\mathrm{sin}\omega t}{\omega })`

:math:`{t}_{0}<t<2{t}_{0}`: :math:`{x}_{r}=-\frac{{P}_{0}}{{\omega }^{3}{t}_{0}}(2\mathrm{sin}\omega (t-{t}_{0})-\mathrm{sin}\omega (t-2{t}_{0})-\mathrm{sin}\omega t)`


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

None if we calculate the Duhamel integral analytically [:ref:`bib3 <bib3>`]. In order of the precision of the numerical integration method used to calculate the Duhamel integral ([:ref:`bib1 <bib1>`], [:ref:`bib2 <bib2>`]): Simpson method with 40 points per period.


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


1. R.W. Clough and J. Penzien: Dynamics of structures New York, Mac Graw-Hill, 1975, p.102-105


2. Guide VPCS AFNOR Technical- 1990


3. J.S. Przemieniecki: Theory of matrix structural analysis New York, MacGraw-Hill, 1968, p.351-357