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


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

First, the natural frequencies :math:`{f}_{i}` and the associated eigenvectors :math:`{\phi }_{\text{Ni}}` normalized with respect to the mass matrix are calculated. The generalized response of the mono-excited system is then calculated by analytically solving the Duhamel integral [:ref:`bib1 <bib1>`]. Finally, the displacement relative to point :math:`D` is restored on a physical basis.

**Natural frequency calculation**

The mass and stiffness matrices are as follows:

:math:`M=\left[\begin{array}{ccc}m& 0& 0\\ 0& m& 0\\ 0& 0& m\end{array}\right]`, :math:`K=k\left[\begin{array}{ccc}2& -1& 0\\ -1& 2& -1\\ 0& -1& 1\end{array}\right]`

Natural frequencies :math:`\omega` are the solution of equation :math:`\mathrm{det}[K-{\lambda }^{2}M]=0`, i.e. :math:`{\lambda }^{3}-5{\lambda }^{2}+6\lambda -1=0` where :math:`\lambda =\frac{\omega }{{\omega }_{0}}` and :math:`\omega =\sqrt{\frac{k}{m}}`.

**Calculation of the generalized response of the mono-excited system**

:math:`\gamma (t)={\mathrm{a.t}}^{2}` with :math:`a={2.10}^{5}`.

In the absolute coordinate system, the fundamental equation for the dynamics of the undamped mass-spring system is written: :math:`M\ddot{{X}_{a}}+K{X}_{a}=0`.

Absolute displacement :math:`{X}_{a}` breaks down into a uniform translational training displacement :math:`{X}_{e}` and into a relative displacement :math:`{X}_{r}`: :math:`{X}_{a}={X}_{r}+{X}_{e}`.

The equation of motion in the relative coordinate system is then written: :math:`M\ddot{{X}_{r}}+K{X}_{r}=-M\Psi \ddot{{X}_{s}}=Q`

with :math:`\ddot{{X}_{s}}=\gamma (t)={\mathrm{a.t}}^{2}` and :math:`\Psi =\left[\begin{array}{}1\\ 1\\ 1\end{array}\right]` and therefore :math:`Q={\mathrm{a.t}}^{2}m\left[\begin{array}{}1\\ 1\\ 1\end{array}\right]`.

The equation of motion projected on the basis of dynamic modes normalized with respect to the mass matrix is written as:

:math:`\ddot{{\alpha }_{i}}(t)+{\omega }_{i}^{2}{\alpha }_{i}(t)=\frac{{\Phi }_{i}^{T}\mathrm{.}M\mathrm{.}\Psi }{{\Phi }_{i}^{T}\mathrm{.}M\mathrm{.}{\Phi }_{i}}\gamma (t)=-{p}_{i}(t)\gamma (t)`.

The answer of this linear system, at instant :math:`t`, is given by the Duhamel integral:

:math:`\ddot{{\alpha }_{i}}(t)=\frac{1}{{\omega }_{i}}{\int }_{0}^{t}-{p}_{i}(t)\gamma (t)\mathrm{.}\mathrm{sin}{\omega }_{i}(t-\tau )d\tau =-\frac{{p}_{i}(t)}{{\omega }_{i}}{\int }_{0}^{t}{\mathrm{a.t}}^{2}\mathrm{sin}{\omega }_{i}(t-\tau )d\tau`.

However, according to [:ref:`bib1 <bib1>`], :math:`{\int }_{0}^{t}a\mathrm{.}{t}^{2}\mathrm{sin}{\omega }_{i}(t-\tau )d\tau =\frac{\alpha }{{\omega }_{i}}\left[{t}^{2}+\frac{2}{{\omega }_{i}}(\mathrm{cos}{\omega }_{i}t-1)\right]`.

So :math:`{X}_{r}={\Phi }_{i}\cdot {\alpha }_{i}=-\underset{i}{\Sigma }\frac{a\mathrm{.}{p}_{i}(t)\mathrm{.}{\Phi }_{i}}{{\omega }_{i}^{2}}\left[{t}^{2}+\frac{2}{{\omega }_{i}}(\mathrm{cos}{\omega }_{i}t-1)\right]`.


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

The three natural frequencies of the system and the relative displacement :math:`{x}_{r}` at the point :math:`D` are taken as reference results, for various times between :math:`0` and :math:`\mathrm{0,1}s`.


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

None if we calculate the Duhamel integral analytically [:ref:`bib1 <bib1>`], [:ref:`bib2 <bib2>`].


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


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


2. S.P. TIMOSHENKO, D.H. YOUNG and W. WEAVER: Vibrations problems in engineering 4th edition, New York, Wiley & Sons, 1974, pp. 284-321